Patent Publication Number: US-2019178332-A1

Title: Damper apparatus

Description:
TECHNICAL FIELD 
     The aspects of preferred embodiments disclosed herein relate to a damper apparatus including an input element to which a torque from an engine is transferred, and an output element. 
     BACKGROUND ART 
     Hitherto, a double path damper for use in conjunction with a torque converter is known as this type of damper apparatus (see, for example, Patent Document 1). In this damper apparatus, a vibration path ranging from an engine and a lock-up clutch (32) to an output hub (37, 39) is divided into two parallel vibration paths B and C. Each of the two vibration paths B and C includes a pair of springs, and a separate intermediate flange (36, 38) disposed between the pair of springs. A turbine (34) of the torque converter is coupled to the intermediate flange (36) of the vibration path B in order to vary the natural frequencies of the two vibration paths. The natural frequency of the intermediate flange (36) of the vibration path B is smaller than the natural frequency of the intermediate flange (38) of the vibration path C. In this damper apparatus, when the lock-up clutch (32) is connected, a vibration from the engine enters the two vibration paths B and C of the damper apparatus. When the engine vibration having a certain frequency reaches the vibration path B including the intermediate flange (36) coupled to the turbine (34), the phase of the vibration in a range from the intermediate flange (36) of the vibration path B to the output hub (37, 39) deviates by 180 degrees from the phase of the input vibration. At this time, the vibration that enters the vibration path C is transferred to the output hub (37, 39) without causing a phase shift (deviation) because the natural frequency of the intermediate flange (38) of the vibration path C is larger than the natural frequency of the intermediate flange (36) of the vibration path B. Thus, the vibration at the output hub (37, 39) can be damped by causing the deviation by 180 degrees between the phase of the vibration transferred from the vibration path B to the output hub (37, 39) and the phase of the vibration transferred from the vibration path C to the output hub (37, 39). 
     RELATED ART DOCUMENTS 
     Patent Documents 
     
         
         Patent Document 1: Published Japanese Translation of PCT Application No. 2012-506006 (JP 2012-506006 A) 
       
    
     SUMMARY 
     In order to improve the vibration damping performance of the double path damper disclosed in Patent Document 1 above, it is necessary to appropriately set the natural frequencies of the vibration paths B and C by adjusting the spring rates of elastic bodies on both sides of each intermediate flange and the weight of each intermediate flange. When an attempt is made to appropriately set the natural frequencies of the vibration paths B and C by adjusting the spring rates of the elastic bodies, however, the stiffness of the entire double path damper fluctuates significantly. When an attempt is made to appropriately set the two natural frequencies by adjusting the weight of the intermediate flange and the weight of the turbine coupled to the intermediate flange, the weights of the flange and the turbine and furthermore the weight of the entire torque converter increase. Thus, in the double path damper described above, it is not easy to appropriately set the natural frequencies of the vibration paths B and C so as to improve the vibration damping performance. Depending on the frequency of the vibration to be damped, the vibration cannot satisfactorily be damped even by the damper apparatus disclosed in Patent Document 1. In this type of damper apparatus, there is a demand to improve the durability of constituent members, and to suppress an increase in the number of components and an increase in the size of the apparatus. 
     It is therefore an aspect of a preferred embodiment disclosed herein to provide a damper apparatus in which an increase in the number of components and an increase in the size can be suppressed and the vibration damping performance can further be improved while the durability of constituent members is improved. 
     A damper apparatus disclosed herein is a damper apparatus including an input element to which a torque from an engine is transferred, and an output element. The damper apparatus includes a first intermediate element, a second intermediate element, a first elastic body configured to transfer the torque between the input element and the first intermediate element, a second elastic body configured to transfer the torque between the first intermediate element and the output element, a third elastic body configured to transfer the torque between the input element and the second intermediate element, a fourth elastic body configured to transfer the torque between the second intermediate element and the output element, and a fifth elastic body configured to transfer the torque between the first intermediate element and the second intermediate element. At least one of the first and second intermediate elements includes a single member on which a first torque transfer portion disposed between the first and second elastic bodies or between the third and fourth elastic bodies and a second torque transfer portion configured to exchange the torque with the fifth elastic body are both formed. 
     In the damper apparatus, two natural frequencies can be set as a whole in a state in which deflections of all of the first to fifth elastic bodies are permitted. The two natural frequencies are appropriately set by adjusting the stiffness of the fifth elastic body. Thus, the vibration damping performance of the damper apparatus can further be improved. At least one of the first and second intermediate elements includes the single member on which the first torque transfer portion disposed between the first and second elastic bodies or between the third and fourth elastic bodies and the second torque transfer portion configured to exchange the torque with the fifth elastic body are both formed. Thus, the increase in the number of components and the increase in the size of the damper apparatus can be suppressed. In the damper apparatus disclosed herein, a force applied to the first torque transfer portion from the first and second elastic bodies or from the third and fourth elastic bodies may be opposite to a force applied to the second torque transfer portion from the fifth elastic body. Thus, when at least one of the first and second intermediate elements includes two members coupled to each other and the first torque transfer portion is formed on one of the two members while the second torque transfer portion is formed on the other, shear forces acting on coupling portions of the two members increase, and the durability of at least one of the first and second intermediate elements may decrease. When the first and second torque transfer portions are provided on the single member described above, the single member can receive the two forces acting in opposite directions. Thus, the durability of at least one of the first and second intermediate elements can further be improved. As a result, in the damper apparatus disclosed herein, the increase in the number of components and the increase in the size can be suppressed while the durability of at least one of the first and second intermediate elements is improved. 
    
    
     
       BRIEF DESCRIPTION OF THE DRAWINGS 
         FIG. 1  is a schematic structural diagram illustrating a starting apparatus including a damper apparatus disclosed herein. 
         FIG. 2  is a sectional view illustrating the starting apparatus of  FIG. 1 . 
         FIG. 3  is a schematic diagram for describing average attachment radii of first to fourth elastic bodies of the damper apparatus disclosed herein. 
         FIG. 4  is a schematic diagram illustrating a main part of the damper apparatus disclosed herein. 
         FIG. 5  is a schematic diagram illustrating a main part of the damper apparatus disclosed herein. 
         FIG. 6  is a schematic diagram illustrating torque transfer paths of the damper apparatus disclosed herein. 
         FIG. 7  is an explanatory diagram exemplifying a relationship between the rotation speed of an engine and theoretical torque fluctuations of output elements of damper apparatuses. 
         FIG. 8  is an explanatory diagram exemplifying a relationship between a stiffness of the first elastic body of the damper apparatus disclosed herein and a natural frequency on a low speed rotation side, a frequency at an anti-resonance point, and an equivalent stiffness of the damper apparatus. 
         FIG. 9  is an explanatory diagram exemplifying a relationship between a stiffness of the second elastic body of the damper apparatus disclosed herein and the natural frequency on the low speed rotation side, the frequency at the anti-resonance point, and the equivalent stiffness of the damper apparatus. 
         FIG. 10  is an explanatory diagram exemplifying a relationship between a stiffness of the third elastic body of the damper apparatus disclosed herein and the natural frequency on the low speed rotation side, the frequency at the anti-resonance point, and the equivalent stiffness of the damper apparatus. 
         FIG. 11  is an explanatory diagram exemplifying a relationship between a stiffness of the fourth elastic body of the damper apparatus disclosed herein and the natural frequency on the low speed rotation side, the frequency at the anti-resonance point, and the equivalent stiffness of the damper apparatus. 
         FIG. 12  is an explanatory diagram exemplifying a relationship between a stiffness of a fifth elastic body of the damper apparatus disclosed herein and the natural frequency on the low speed rotation side, the frequency at the anti-resonance point, and the equivalent stiffness of the damper apparatus. 
         FIG. 13  is an explanatory diagram exemplifying a relationship between a moment of inertia of a first intermediate element of the damper apparatus disclosed herein and the natural frequency on the low speed rotation side, the frequency at the anti-resonance point, and the equivalent stiffness of the damper apparatus. 
         FIG. 14  is a sectional view illustrating a starting apparatus including another damper apparatus disclosed herein. 
         FIG. 15  is a sectional view illustrating a starting apparatus including still another damper apparatus disclosed herein. 
         FIG. 16  is a front-side elevation illustrating a main part of the damper apparatus of  FIG. 15 . 
     
    
    
     BEST MODES 
     Next, modes for carrying out aspects of preferred embodiments disclosed herein are described with reference to the drawings. 
       FIG. 1  is a schematic structural diagram illustrating a starting apparatus  1  including a damper apparatus  10  disclosed herein.  FIG. 2  is a sectional view illustrating the damper apparatus  10 . The starting apparatus  1  illustrated in  FIG. 1  is mounted on a vehicle including an engine (internal combustion engine in this embodiment) EG serving as a motor. In addition to the damper apparatus  10 , the starting apparatus  1  includes a front cover  3  coupled to a crankshaft of the engine EG, a pump impeller (input-side fluid transmission element)  4  fixed to the front cover  3 , a turbine runner (output-side fluid transmission element)  5  rotatable coaxially with the pump impeller  4 , a damper hub  7  serving as a power output member coupled to the damper apparatus  10  and fixed to an input shaft IS of a transmission (power transfer apparatus) TM that is an automatic transmission (AT), a continuously variable transmission (CVT), a dual clutch transmission (DCT), a hybrid transmission, or a speed reducer, a lock-up clutch  8 , and the like. 
     In the following description, an “axial direction” is basically an extending direction of a central axis CA (axis center; see  FIG. 3 ) of the starting apparatus  1  and the damper apparatus  10  unless otherwise specified. A “radial direction” is basically a radial direction of the starting apparatus  1 , the damper apparatus  10 , and rotational elements of the damper apparatus  10  and the like, that is, an extending direction of a straight line extending from the central axis CA of the starting apparatus  1  and the damper apparatus  10  in a direction orthogonal to the central axis CA (direction of a radius) unless otherwise specified. A “circumferential direction” is basically a circumferential direction of the starting apparatus  1 , the damper apparatus  10 , and the rotational elements of the damper apparatus  10  and the like, that is, a direction along a rotation direction of the rotational elements unless otherwise specified. 
     The pump impeller  4  includes a pump shell  40  closely fixed to the front cover  3 , and a plurality of pump blades  41  disposed on the inner surface of the pump shell  40 . The turbine runner  5  includes a turbine shell  50  (see  FIG. 2 ), and a plurality of turbine blades  51  disposed on the inner surface of the turbine shell  50 . The inner peripheral portion of the turbine shell  50  is fixed to a turbine hub  52  via a plurality of rivets, and the turbine hub  52  is supported by the damper hub  7  in a freely rotatable manner. Movement of the turbine hub  52  (turbine runner  5 ) in the axial direction of the starting apparatus  1  is restricted by the damper hub  7  and a snap ring attached to the damper hub  7 . 
     The pump impeller  4  and the turbine runner  5  face each other. A stator  6  configured to adjust a flow of hydraulic oil (fluid) from the turbine runner  5  to the pump impeller  4  is coaxially disposed between the pump impeller  4  and the turbine runner  5 . The stator  6  includes a plurality of stator blades  60 . The rotation direction of the stator  6  is set only to one direction by a one-way clutch  61 . The pump impeller  4 , the turbine runner  5 , and the stator  6  form a torus (toric flow path) configured to circulate the hydraulic oil, and function as a torque converter (fluid transmission apparatus) having a torque amplifying function. In the starting apparatus  1 , the stator  6  and the one-way clutch  61  may be omitted, and the pump impeller  4  and the turbine runner  5  may function as a fluid coupling. 
     The lock-up clutch  8  is a multi-plate hydraulic clutch, which executes lock-up for coupling the front cover  3  and the damper hub  7  to each other via the damper apparatus  10  and terminates the lock-up. The lock-up clutch  8  includes a lock-up piston  80  supported by a center piece  3   c  fixed to the front cover  3  so as to be movable in the axial direction, a clutch drum  81 , an annular clutch hub  82  fixed to the inner surface of a lateral wall portion  3   w  of the front cover  3  so as to face the lock-up piston  80 , a plurality of first friction engagement plates (friction plates having friction materials on both sides)  83  fitted to splines formed on the inner periphery of the clutch drum  81 , and a plurality of second friction engagement plates  84  (separator plates) fitted to splined formed on the outer periphery of the clutch hub  82 . 
     The lock-up clutch  8  further includes an annular flange member (oil chamber defining member)  85  attached to the center piece  3   c  of the front cover  3  so as to be located opposite to the lock-up piston  80  from the front cover  3  with respect, that is, located closer to the damper apparatus  10  and the turbine runner  5  than the lock-up piston  80 , and a plurality of return springs  86  disposed between the front cover  3  and the lock-up piston  80 . As in the illustration, the lock-up piston  80  and the flange member  85  define an engagement oil chamber  87 , and hydraulic oil (engagement hydraulic pressure) is supplied from an unillustrated hydraulic controller to the engagement oil chamber  87 . By increasing the engagement hydraulic pressure for the engagement oil chamber  87 , the lock-up piston  80  is moved in the axial direction so as to press the first and second friction engagement plates  83  and  84  toward the front cover  3 . Thus, the lock-up clutch  8  can be engaged (complete engagement or slip engagement). 
     The damper apparatus  10  damps vibrations between the engine EG and the transmission TM. As illustrated in  FIG. 1 , the damper apparatus  10  includes a driving member (input element)  11 , a first intermediate member (first intermediate element)  12 , a second intermediate member (second intermediate element)  14 , and a driven member (output element)  16  as rotational elements (rotational members, that is, rotational masses) configured to coaxially rotate relative to each other. The damper apparatus  10  further includes, as torque transfer elements (torque transfer elastic bodies), a plurality of (for example, three in this embodiment) first inner springs (first elastic bodies) SP 11  disposed between the driving member  11  and the first intermediate member  12  and configured to transfer a rotational torque (torque in the rotation direction), a plurality of (for example, three in this embodiment) second inner springs (second elastic bodies) SP 12  disposed between the first intermediate member  12  and the driven member  16  and configured to transfer the rotational torque, a plurality of (for example, three in this embodiment) first outer springs (third elastic bodies) SP 21  disposed between the driving member  11  and the second intermediate member  14  and configured to transfer the rotational torque, a plurality of (for example, three in this embodiment) second outer springs (fourth elastic bodies) SP 22  disposed between the second intermediate member  14  and the driven member  16  and configured to transfer the rotational torque, and a plurality of (for example, three or six in this embodiment) intermediate springs (fifth elastic bodies) SPm disposed between the first intermediate member  12  and the second intermediate member  14  and configured to transfer the rotational torque. 
     In this embodiment, a linear coil spring formed of a metal material that is helically would so as to have an axis center extending straight when no load is applied is employed as each of the first and second inner springs SP 11  and SP 12 , the first and second outer springs SP 21  and SP 22 , and the intermediate springs SPm. Thus, each of the springs SP 11  to SPm is extended and contracted along the axis center more appropriately than in a case where an arc coil spring is used. Accordingly, it is possible to reduce a hysteresis caused by a frictional force generated between the spring that transfers the torque and the rotational element, that is, a difference between a torque output when the torque input to the driving member  11  increases and a torque output when the torque input to the driving member  11  decreases. The hysteresis may be quantified by a difference between a torque output from the driven member  16  when the torsion angle of the damper apparatus  10  is a predetermined angle in a state in which the torque input to the driving member  11  increases and a torque output from the driven member  16  when the torsion angle of the damper apparatus  10  is the predetermined angle in a state in which the torque input to the driving member  11  decreases. At least one of the springs SP 11  to SPm may be an arc coil spring. 
     In this embodiment, the first and second inner springs SP 11  and SP 12  are disposed in a fluid chamber  9  defined by the front cover  3  and the pump shell  40  of the pump impeller  4  so as to be alternately arranged along the circumferential direction of the damper apparatus  10  (first intermediate member  12 ). The first and second outer springs SP 21  and SP 22  are disposed in an outer peripheral region of the fluid chamber  9  so as to be alternately arranged along the circumferential direction of the damper apparatus  10  (second intermediate member  14 ). That is, the first and second outer springs SP 21  and SP 22  are disposed on a radially outer side of the first and second inner springs SP 11  and SP 12  so as to be close to the outer periphery of the starting apparatus  1 . 
     Thus, in the damper apparatus  10 , an average attachment radius ro of the first and second outer springs SP 21  and SP 22  is larger than an average attachment radius ri of the first and second inner springs SP 11  and SP 12 . As illustrated in  FIG. 3 , the average attachment radius ro of the first and second outer springs SP 21  and SP 22  is an average (=(r SP21 +r SP22 )/2) of an attachment radius r SP21  of the first outer spring (third elastic body) SP 21  that is a distance from the central axis CA of the damper apparatus  10  to the axis center of the first outer spring SP 21  and an attachment radius r SP2  of the second outer spring (fourth elastic body) SP 22  that is a distance from the central axis CA to the axis center of the second outer spring SP 22 . As illustrated in  FIG. 3 , the average attachment radius ri of the first and second inner springs SP 11  and SP 12  is an average (=(r SP11 +r SP12 )/2) of an attachment radius r SP1  of the first inner spring (first elastic body) SP 11  that is a distance from the central axis CA to the axis center of the first inner spring SP 11  and an attachment radius r SP12  of the second inner spring (second elastic body) SP 12  that is a distance from the central axis CA to the axis center of the second inner spring SP 12 . The attachment radius r SP11 , r SP12 , r SP21 , or r SP22  may be a distance between the central axis CA and a predetermined point on the axis center of each spring SP 11 , SP 12 , SP 21 , or SP 22  (for example, a center or end in the axial direction). 
     In this embodiment, the first and second outer springs SP 21  and SP 22  (and the intermediate springs SPm) are arranged on the same circumference (first circumference) so that the attachment radius r SP21  and the attachment radius r SP22  are equal to each other, and the axis center of the first outer spring SP 21  and the axis center of the second outer spring SP 22  are included in one plane orthogonal to the central axis CA. In this embodiment, the first and second inner springs SP 11  and SP 12  are arranged on the same circumference (second circumference having a diameter larger than that of the first circumference) so that the attachment radius r SP11  and the attachment radius r SP12  are equal to each other, and the axis center of the first inner spring SP 11  and the axis center of the second inner spring SP 12  are included in one plane orthogonal to the central axis CA. In addition, in the damper apparatus  10 , the first and second inner springs SP 11  and SP 12  are disposed on a radially inner side of the first and second outer springs SP 21  and SP 22  so as to overlap the first and second outer springs SP 21  and SP 22  in the axial direction as viewed in the radial direction. Thus, the damper apparatus  10  can be made compact in the radial direction, and the axial length of the damper apparatus  10  can further be reduced. 
     As illustrated in  FIG. 3 , the attachment radius r SP21  from the central axis CA to the axis center of the first outer spring SP 21  and the attachment radius r SP22  from the central axis CA to the axis center of the second outer spring SP 22  may be different from each other. The attachment radius r SP11  from the central axis CA to the axis center of the first inner spring SP 11  and the attachment radius r SP12  from the central axis CA to the axis center of the second inner spring SP 12  may be different from each other. That is, the attachment radius r SP21  or r SP22  of at least one of the first and second outer springs SP 21  and SP 22  may be larger than the attachment radius r SP11  or r SP12  of at least one of the first and second inner springs SP 11  and SP 12 . The axis center of the first outer spring SP 21  and the axis center of the second outer spring SP 22  need not be included in one plane orthogonal to the central axis CA. The axis center of the first inner spring SP 11  and the axis center of the second inner spring SP 12  need not be included in one plane orthogonal to the central axis CA. The axis centers of the springs SP 11 , SP 12 , SP 21 , and SP 22  may be included in one plane orthogonal to the central axis CA. The axis center of at least one of the springs SP 11 , SP 12 , SP 21 , and SP 22  need not be included in the one plane. 
     In this embodiment, when “k 11 ” represents a stiffness, that is, a spring rate of the first inner spring SP 11 , “k 12 ” represents a stiffness, that is, a spring rate of the second inner spring SP 12 , “k 21 ” represents a stiffness, that is, a spring rate of the first outer spring SP 21 , and “k 22 ” represents a stiffness, that is, a spring rate of the second outer spring SP 22 , the spring rates k 11 , k 12 , k 21 , and k 22  are selected so as to satisfy relationships of k 11 ≠k 21  and k 11 /k 21 ≠k 12 /k 22 . More specifically, the spring rates k 11 , k 12 , k 21 , and k 22  satisfy relationships of k 11 /k 21 &lt;k 12 /k 22  and k 1 &lt;k 12 &lt;k 22 &lt;k 21 . That is, a smaller one (k 11 ) of the spring rates k 11  and k 12  of the first and second inner springs SP 11  and SP 12  is smaller than a smaller one (k 22 ) of the spring rates k 21  and k 22  of the first and second outer springs SP 21  and SP 22 . When “k m ” represents a stiffness, that is, a spring rate of the intermediate spring SPm, the spring rates k 11 , k 12 , k 21 , k 22 , and k m  satisfy a relationship of k 11 &lt;k m &lt;k 12 &lt;k 22 &lt;k 21 . 
     As illustrated in  FIG. 2 , the driving member  11  includes the clutch drum  81  (first input member) of the lock-up clutch  8  described above to which a torque from the engine EG is transferred, and an annular input plate  111  (second input member) coupled (fixed) to the clutch drum  81  side by side in the axial direction via a plurality of rivets. Thus, the front cover  3  (engine EG) and the driving member  11  of the damper apparatus  10  are coupled to each other through the engagement of the lock-up clutch  8 . The clutch drum  81  includes an annular spring support portion  81   a  formed on a radially outer side with respect to the splines described above, and a plurality of (for example, three in this embodiment) spring abutment portions (elastic body abutment portions)  81   c  each extending in the axial direction. The spring support portion  81   a  is formed so as to support (guide) outer portions of the plurality of first and second outer springs SP 21  and SP 22  in the radial direction, front-cover-3-side (engine side) lateral portions (lateral portions on the left side in  FIG. 2 ) thereof, inner sides of the lateral portions in the radial direction, and outer sides (shoulder portions) of turbine-runner-5-side (transmission side) lateral portions thereof in the radial direction. The clutch drum  81  is disposed in the fluid chamber  9  so that the spring support portion  81   a  is close to the outer periphery of the starting apparatus  1 . 
     The input plate  111  is a plate-shaped annular member including a plurality of (for example, three in this embodiment) spring support portions  111   a , a plurality of (for example, three in this embodiment) outer spring abutment portions (elastic body abutment portions)  111   co , and a plurality of (for example, three in this embodiment) inner spring abutment portions (elastic body abutment portions)  111   ci . The plurality of spring support portions  111   a  are formed on an outer peripheral portion of the input plate  111  at intervals (at regular intervals) in the circumferential direction. Every single inner spring abutment portion  111   ci  is provided between the spring support portions  111   a  adjacent to each other along the circumferential direction. The inner spring abutment portions  111   ci  extend radially inward from an inner peripheral portion of the input plate  111  at intervals (at regular intervals) in the circumferential direction. In this embodiment, the plurality of inner spring abutment portions  111   ci  are offset in the axial direction of the damper apparatus  10  so as to be closer to the turbine runner  5  than the plurality of outer spring abutment portions  111   co.    
     As illustrated in  FIG. 2 , the first intermediate member  12  includes an annular first plate member (first member)  121  supported (aligned) by the damper hub  7  in a freely rotatable manner, and an annular second plate member  122  (second member) coupled (fixed) to the turbine runner  5  that is a mass so as to rotate together with the turbine runner  5 . The first plate member  121  of the first intermediate member  12  includes a plurality of (for example, three in this embodiment) spring abutment portions  121   c  that protrude radially outward at intervals (at regular intervals) in the circumferential direction. As illustrated in  FIG. 2 , a rectangular or elongated hole-shaped opening portion  121   h  extending through each spring abutment portion  121   c  is formed in the spring abutment portion  121   c.    
     The second plate member  122  of the first intermediate member  12  includes a plurality of (for example, three in this embodiment) coupling abutment portions  122   c , and a plurality of (for example, six in this embodiment) outer abutment portions (torque transfer portions)  122   d  disposed on a radially outer side with respect to the coupling abutment portions  122   c . As in the illustration, the inner peripheral portion of the second plate member  122  is fixed to the turbine hub  52  together with the turbine shell  50  of the turbine runner  5 . The coupling abutment portions  122   c  extend in the axial direction from a body of the second plate member  122  at intervals (at regular intervals) in the circumferential direction. A protruding portion  122   p  fitted to the opening portion  121   h  of the first plate member  121  is formed at the distal end of each coupling abutment portion  122   c . The protruding portion  122   p  has a width slightly smaller than the width of the opening portion  121   h  of the first intermediate member  12  in the circumferential direction, and also has a thickness sufficiently smaller than the length of the opening portion  121   h  (opening length) of the first intermediate member  12  in the radial direction. The outer abutment portions  122   d  are formed symmetrically with respect to the axis center of the second plate member  122  so that every two (pair of) outer abutment portions  122   d  are close to each other. The two paired outer abutment portions  122   d  are arranged in the circumferential direction with a distance in accordance with, for example, the equilibrium length of the intermediate spring SPm. A plurality of circular arc guide holes (elongated holes)  122   g  are formed in an outer peripheral portion of the second plate member  122  at intervals (at regular intervals) in the circumferential direction. 
     The second intermediate member  14  includes a first annular member (single member)  141 , and a second annular member (second member)  142  coupled (fixed) to the first annular member  141  side by side in the axial direction via a plurality of rivets. The second intermediate member  14  has a moment of inertia smaller than that of the first intermediate member  12 . As illustrated in  FIG. 2 , spacers  145  each having a thickness slightly larger than that of the second plate member  122  of the first intermediate member  12  are disposed between the first and second annular members  141  and  142  in the axial direction. The first and second annular members  141  and  142  are fastened to each other with a plurality of rivets passing through the first and second annular members  141  and  142  and the spacers  145 . 
     As illustrated in  FIG. 2 , the spacers  145  (and the rivets) are disposed in the guide holes  122   g  of the second plate member  122  of the first intermediate member  12 . Thus, the second intermediate member  14  is supported by the second plate member  122  disposed between the first and second annular members  141  and  142  in the axial direction so as to be rotatable relative to the first intermediate member  12 . By disposing the spacers  145  described above between the first and second annular members  141  and  142  in the axial direction, clearances are provided between the inner surfaces of the first and second annular members  141  and  142  and the surfaces of the second plate member  122 . Thus, the second intermediate member  14  can smoothly be moved relative to the second plate member  122  (first intermediate member  12 ). 
     The first annular member  141  includes a plurality of (for example, three in this embodiment) first spring abutment portions (first torque transfer portions)  141   c , and a plurality of (for example, six in this embodiment) second spring abutment portions (second torque transfer portions)  141   d . The plurality of first spring abutment portions  141   c  extend from a body of the first annular member  141  to the radially outer side and to one side in the axial direction (left side in  FIG. 2 ; front cover  3  side) at intervals in the circumferential direction. The plurality of second spring abutment portions  141   d  extend from the body of the first annular member  141  to the radially outer side and to the other side in the axial direction, that is, away from the first spring abutment portions  141   c  (right side in  FIG. 2 ; turbine runner  5  side) at intervals in the circumferential direction. The second spring abutment portions  141   d  are formed symmetrically with respect to the axis center of the first annular member  141  so that every two (pair of) second spring abutment portions  141   d  are closer to each other. The two paired second spring abutment portions  141   d  are arranged in the circumferential direction with a distance in accordance with, for example, the equilibrium length of the intermediate spring SPm. 
     The second annular member  142  includes an annular spring support portion  142   a . The spring support portion  142   a  is formed so as to support (guide) outer portions of the plurality of intermediate springs SPm in the radial direction, turbine-runner-side (transmission side) lateral portions (lateral portions on the right side in  FIG. 2 ) thereof, inner sides of the lateral portions in the radial direction, and outer sides (shoulder portions) of front-cover-3-side (engine side) lateral portions thereof in the radial direction. The second annular member  142  may include a plurality of spring support portions  142   a  formed at intervals (at regular intervals) in the circumferential direction. In this case, it is appropriate that each of the plurality of spring support portions  142   a  be formed so as to have a circumferential length sufficiently larger than the circumferential length of the intermediate spring SPm. 
     The driven member  16  includes a first output plate (first output member)  161 , and an annular second output plate (second output member)  162  disposed so as to be closer to the turbine runner  5  than the first output plate  161  and coupled (fixed) to the first output plate  161  side by side in the axial direction via a plurality of rivets. The first output plate  161  of the driven member  16  is a plate-shaped annular member, and the inner peripheral portion of the first output plate  161  is fixed to the damper hub  7  via a plurality of rivets. As in the illustration, the first output plate  161  includes a plurality of (for example, three) spring housing windows  161   w  disposed at intervals (at regular intervals) in the circumferential direction, a plurality of (for example, three) spring support portions  161   a  extending along the inner peripheral edges of the respective spring housing windows  161   w , a plurality of (for example, three) spring support portions  161   b  extending along the outer peripheral edges of the respective spring housing windows  161   w , a plurality of (for example, three) inner spring abutment portions  161   ci , and a plurality of (for example, three) outer spring abutment portions  161   co.    
     The plurality of inner spring abutment portions  161   ci  are provided so that every single inner spring abutment portion  161   ci  extends in the radial direction between the spring housing windows  161   w  (spring support portions  161   a  and  161   b ) adjacent to each other along the circumferential direction. The plurality of outer spring abutment portions  161   co  extend radially outward from an outer peripheral portion of the first output plate  161  at intervals (at regular intervals) in the circumferential direction. In this embodiment, the plurality of outer spring abutment portions  161   co  are offset in the axial direction of the damper apparatus  10  so as to be closer to the front cover  3  than the plurality of inner spring abutment portions  161   ci . The first output plate  161  further includes a short tubular supporting portion  161   s  extending in the axial direction between the plurality of inner spring abutment portions  161   ci  and the plurality of outer spring abutment portions  161   co  in the radial direction. 
     The second output plate  162  of the driven member  16  is a plate-shaped annular member including a plurality of (for example, three) spring housing windows  162   w  disposed at intervals (at regular intervals) in the circumferential direction, a plurality of (for example, three) spring support portions  162   a  extending along the inner peripheral edges of the respective spring housing windows  162   w , a plurality of (for example, three) spring support portions  162   b  extending along the outer peripheral edges of the respective spring housing windows  162   w , and a plurality of (for example, three) spring abutment portions  162   c . The plurality of spring abutment portions  162   c  are provided so that every single spring abutment portion  162   c  extends in the radial direction between the spring housing windows  162   w  (spring support portions  162   a  and  162   b ) adjacent to each other along the circumferential direction. 
     As illustrated in  FIG. 2 , the first and second output plates  161  and  162  are coupled to each other so that the associated spring support portions  161   a  and  162   a  face each other and the associated spring support portions  161   b  and  162   b  face each other. An inner peripheral half segment of the input plate  111  of the driving member  11  is disposed between the first and second output plates  161  and  162  in the axial direction, and a supported portion  111   s  formed on the input plate  111  is supported by the short tubular supporting portion  161   s  of the first output plate  161 . Thus, the input plate  111  is supported (aligned) by the driven member  16  (first output plate  161 ) in a freely rotatable manner, and the outer spring abutment portions  111   co  of the input plate  111  extend radially outward over the supporting portion  161   s.    
     The first plate member  121  of the first intermediate member  12  is disposed between the first and second output plates  161  and  162  so as to be surrounded by the annular portion of the input plate  111 . The inner spring abutment portions  111   ci  of the input plate  111  and the spring abutment portions  121   c  of the first plate member  121  are arranged in the axial direction between the first and second output plates  161  and  162 , and overlap each other in the axial direction (located substantially flush with each other) as viewed in the radial direction. The second plate member  122  of the first intermediate member  12  is fixed to the turbine hub  52  via a plurality of rivets so as to extend in the radial direction between the turbine runner  5  and the second output plate  162  in the axial direction. The second intermediate member  14  is supported by the second plate member  122 , and is disposed in the outer peripheral region of the fluid chamber  9  so that the spring support portion  142   a  overlaps the spring support portion  81   a  of the clutch drum  81  in the radial direction of the damper apparatus  10  as viewed in the axial direction of the damper apparatus  10 . The outer spring abutment portions  161   co  of the driven member  16  extend in the radial direction between the outer spring abutment portions  111   co  of the input plate  111  and the clutch drum  81  (spring abutment portions  81   c ) in the axial direction. 
     The first and second inner springs SP 1   l  and SP 12  are supported by the driven member  16 , that is, the associated spring support portions  161   a ,  161   b ,  162   a , and  162   b  of the first and second output plates  161  and  162  so that every single first inner spring SP 11  and every single second inner spring SP 12  are paired with each other (act in series) and the first and second inner springs SP 1  and SP 12  are alternately arranged in the circumferential direction (circumferential direction of the first intermediate member  12 ). That is, as illustrated in  FIG. 2 , the plurality of spring support portions  161   a  of the first output plate  161  support (guide), from an inner peripheral side, front-cover-3-side lateral portions of the respective first and second inner springs SP 11  and SP 12  (each spring support portion corresponds to one inner spring). The plurality of spring support portions  161   b  of the first output plate  161  support (guide), from an outer peripheral side, the front-cover-3-side lateral portions of the respective first and second inner springs SP 11  and SP 12  (each spring support portion corresponds to one inner spring). The plurality of spring support portions  162   a  of the second output plate  162  support (guide), from the inner peripheral side, the-turbine-runner-5-side lateral portions of the respective first and second inner springs SP 11  and SP 12  (each spring support portion corresponds to one inner spring). The plurality of spring support portions  162   b  of the second output plate  162  support (guide), from the outer peripheral side, the-turbine-runner-5-side lateral portions of the respective first and second inner springs SP 11  and SP 12  (each spring support portion corresponds to one inner spring). 
     In a state in which the damper apparatus  10  is attached, between the first and second inner springs SP 11  and SP 12  that are disposed in different spring housing windows  161   w  and  162   w  and are not paired with each other (do not act in series), the driving member  11 , that is, each inner spring abutment portion  111   ci  of the input plate  111  abuts against the ends of those first and second inner springs SP 11  and SP 12 . In the state in which the damper apparatus  10  is attached, between the first and second inner springs SP 11  and SP 12  that are not paired with each other (do not act in series), each inner spring abutment portion  161   ci  of the first output plate  161  abuts against the ends of those first and second inner springs SP 11  and SP 12  similarly to the inner spring abutment portion  111   ci  of the input plate  111 . In the state in which the damper apparatus  10  is attached, between the first and second inner springs SP 11  and SP 12  that are not paired with each other (do not act in series), each spring abutment portion  162   c  of the second output plate  162  similarly abuts against the ends of those first and second inner springs SP 11  and SP 12 . 
     Each spring abutment portion  121   c  of the first plate member  121  of the first intermediate member  12  extends in the radial direction between the first and second inner springs SP 11  and SP 12  that are paired with each other (act in series), and abuts against the ends of those first and second inner springs SP 11  and SP 12 . In this embodiment, as illustrated in  FIG. 2 , the protruding portion  122   p  of the coupling abutment portion  122   c  of the second plate member  122  is fitted (coupled) to the opening portion  121   h  of the spring abutment portion  121   c  of the first plate member  121 . As illustrated in  FIG. 4  and  FIG. 5 , each coupling abutment portion  122   c  extends in the axial direction between the first and second inner springs SP 11  and SP 12 , and abuts against the ends of those first and second inner springs SP 11  and SP 12 . That is, the lateral surfaces of each coupling abutment portion  122   c  on both sides in the circumferential direction each abut against the end of the first or second inner spring SP 11  or SP 12 . 
     Thus, in the state in which the damper apparatus  10  is attached, one end of the first inner spring SP 11  and the other end of the second inner spring SP 12  that are not paired with the first inner spring SP 11  abut against the associated inner spring abutment portions  111   ci  of the driving member  11  and the associated spring abutment portions  161   ci  and  162   c  of the driven member  16 . In the state in which the damper apparatus  10  is attached, the other end of the first inner spring SP 11  and one end of the second inner spring SP 12  paired with the first inner spring SP 11  abut against the first intermediate member  12 , that is, the spring abutment portion  121   c  of the first plate member  121  and the coupling abutment portion  122   c  of the second plate member  122 . As a result, the driven member  16  is coupled to the driving member  11  via the plurality of first inner springs SP 11 , the first intermediate member  12  (first plate member  121  and second plate member  122 ), and the plurality of second inner springs SP 12 . 
     The first and second outer springs SP 21  and SP 22  are supported by the driving member  11 , that is, the spring support portion  81   a  of the clutch drum  81  and the spring support portions  111   a  of the input plate  111  so that every single first outer spring SP 21  and every single second outer spring SP 22  are paired with each other (act in series) and the first and second outer springs SP 21  and SP 22  are alternately arranged in the circumferential direction (circumferential direction of the second intermediate member  14 ). In the state in which the damper apparatus  10  is attached, between the first and second outer springs SP 21  and SP 22  that are not paired with each other (do not act in series), the driving member  11 , that is, each of the spring abutment portion  81   c  of the clutch drum  81  and the outer spring abutment portion  111   co  of the input plate  111  abuts against the ends of those first and second outer springs SP 21  and SP 22 . Each first spring abutment portion  141   c  of the first annular member  141  of the second intermediate member  14  is inserted into an opening defined between the spring support portion  81   a  and the input plate  111 . Between the first and second outer springs SP 21  and SP 22  that are paired with each other (act in series), each first spring abutment portion  141   c  abuts against the ends of those first and second outer springs SP 21  and SP 22 . In the state in which the damper apparatus  10  is attached, between the first and second outer springs SP 21  and SP 22  that are not paired with each other (do not act in series), each outer spring abutment portion  161   co  of the first output plate  161  abuts against the ends of those first and second outer springs SP 21  and SP 22 . 
     Thus, in the state in which the damper apparatus  10  is attached, one end of the first outer spring SP 21  and the other end of the second outer spring SP 22  that are not paired with the first outer spring SP 21  abut against the associated spring abutment portions  81   c  and  111   co  of the driving member  11  and the associated spring abutment portions  161   co  of the driven member  16 . In the state in which the damper apparatus  10  is attached, the other end of the first outer spring SP 21  and one end of the second outer spring SP 22  paired with the first outer spring SP 21  abut against the second intermediate member  14 , that is, the first spring abutment portion  141   c  of the first annular member  141 . As a result, the driven member  16  is coupled to the driving member  11  via the plurality of first outer springs SP 21 , the second intermediate member  14  (first annular member  141  and second annular member  142 ), and the plurality of second outer springs SP 22 . 
     The intermediate springs SPm are supported by the spring support portion  142   a  of the second annular member  142  of the second intermediate member  14 . In the state in which the damper apparatus  10  is attached, a pair of outer abutment portions  122   d  of the second plate member  122  abut against the respective ends of the intermediate spring SPm, and a pair of second spring abutment portions  141   d  of the first annular member  141  abut against the respective ends of the intermediate spring SPm. Thus, in the state in which the damper apparatus  10  is attached, each intermediate spring SPm is supported from both sides in the circumferential direction by the first intermediate member  12 , that is, the pair of outer abutment portions  122   d  of the second plate member  122 , and is also supported from both sides in the circumferential direction by the second intermediate member  14 , that is, the pair of second spring abutment portions  141   d  of the first annular member  141 . Accordingly, the first intermediate member  12  and the second intermediate member  14  are coupled to each other via the plurality of intermediate springs SPm. As illustrated in  FIG. 1 , spring seats Ss each abutting against the outer abutment portion  122   d  or the second spring abutment portion  141   d  may be attached to the ends of the intermediate spring SPm. 
     As illustrated in  FIG. 1 , the damper apparatus  10  includes a first stopper  21  configured to restrict a relative rotation between the first intermediate member  12  and the driven member  16  and a deflection of the second inner springs SP 12 , a second stopper  22  configured to restrict a relative rotation between the second intermediate member  14  and the driven member  16  and a deflection of the second outer springs SP 22 , and a third stopper  23  configured to restrict a relative rotation between the driving member  11  and the driven member  16 . The first and second stoppers  21  and  22  are structured to restrict the relative rotations between the respective rotational elements and the deflections of the springs substantially simultaneously in a phase in which the input torque transferred from the engine EG to the driving member  11  reaches a predetermined torque T 1  (first threshold) smaller than a torque T 2  (second threshold) corresponding to a maximum torsion angle θmax of the damper apparatus  10 . The third stopper  23  is structured to restrict the relative rotation between the driving member  11  and the driven member  16  in a phase in which the torque input to the driving member  11  reaches the torque T 2  corresponding to the maximum torsion angle θmax. Thus, the damper apparatus  10  has a two-phase (two-stage) damping characteristic. The arrangement positions of the plurality of stoppers in the damper apparatus  10  are not limited to the positions illustrated in  FIG. 1 . That is, the plurality of stoppers may be disposed at any positions as long as the stoppers can appropriately restrict the deflections of the first and second inner springs SP 11  and SP 12 , the first and second outer springs SP 21  and SP 22 , and the intermediate springs SPm. 
     In the damper apparatus  10  structured as described above, the average attachment radius ro of the first and second outer springs SP 21  and SP 22  having larger spring rates (higher stiffnesses) than the first and second inner springs SP 11  and SP 12  is defined so as to be larger than the average attachment radius ri of the first and second inner springs SP 11  and SP 12 . Therefore, the torsion angles (strokes) of the first and second outer springs SP 21  and SP 22  can further be increased. Thus, the stiffnesses of the first and second outer springs SP 21  and SP 22  can be reduced while transfer of a large torque to the driving member  11  is permitted. 
     In the damper apparatus  10 , the first outer springs SP 21  (third elastic bodies) and the second outer springs SP 22  (fourth elastic bodies) are disposed on the outer side of the first and second inner springs SP 11  and SP 12  in the radial direction of the damper apparatus  10 . As illustrated in  FIG. 2 , the intermediate springs SPm are disposed on the outer side of the first and second inner springs SP 11  and SP 12  in the radial direction so as to be closer to the turbine runner  5  than the first and second inner springs SP 11  and SP 12  and the first and second outer springs SP 21  and SP 22  in the axial direction of the damper apparatus  10 . That is, the intermediate springs SPm are disposed on the radially outer side of the first and second inner springs SP 11  and SP 12  with distances from the first and second outer springs SP 21  and SP 22  in the axial direction. Thus, it is possible to increase the degrees of freedom in terms of setting of the stiffnesses, the numbers of arrangement, the torsion angles (strokes), and the like of the first and second inner springs SP 11  and SP 12 , the first and second outer springs SP 21  and SP 22 , and the intermediate springs SPm. 
     According to the damper apparatus  10 , an increase in the size of the starting apparatus  1  along with the arrangement of the intermediate springs SPm can be suppressed by effectively using the space. More specifically, the first and second outer springs SP 21  and SP 22  are disposed so as to partially overlap at least one of the first and second inner springs SP 11  and SP 12  in the axial direction of the damper apparatus  10  (see a dotted line arrow in  FIG. 2 ) as viewed in the radial direction of the damper apparatus  10  (see a wide line arrow in  FIG. 2 ). The intermediate springs SPm are disposed so as to partially overlap at least one of the first and second inner springs SP 11  and SP 12  in the axial direction as viewed in the radial direction. Thus, the axial length of the damper apparatus  10  and furthermore the axial length of the starting apparatus  1  can further be reduced. The intermediate springs SPm are disposed so as to partially overlap at least one of the first and second outer springs SP 21  and SP 22  in the radial direction as viewed in the axial direction. Thus, it is possible to increase the degrees of freedom in terms of setting of the spring rates k 21 , k 22 , and k m , the numbers of arrangement, the torsion angles (strokes), and the like of the first and second outer springs SP 21  and SP 22  and the intermediate springs SPm. 
     The first and second outer springs SP 21  and SP 22  are disposed so as to partially overlap a part of the lock-up clutch  8  (for example, the clutch drum  81 , the lock-up piston  80 , the flange member  85 , and the return spring  86 ) in the axial direction as viewed in the radial direction. Thus, the axial length of the damper apparatus  10  and furthermore the axial length of the starting apparatus  1  can further be reduced. In addition, the first and second inner springs SP 11  and SP 12  are disposed so as to partially overlap the friction engagement portions of the lock-up clutch  8 , that is, the first and second friction engagement plates  83  and  84  in the radial direction as viewed in the axial direction, and the first and second outer springs SP 21  and SP 22  are disposed on the outer side in the radial direction with respect to the first and second friction engagement plates  83  and  84 . Thus, the vibration damping performance of the damper apparatus  10  can further be improved by reducing the hysteresis of the first and second inner springs SP 11  and SP 12  while the axial length of the damper apparatus  10  and furthermore the axial length of the starting apparatus  1  are further reduced. 
     The first and second inner springs SP 11  and SP 12  are disposed on an inner side in the radial direction with respect to a maximally bulging portion  5   x  (see  FIG. 2 ) of the turbine runner  5  in the axial direction, and the first and second outer springs SP 21  and SP 22  are disposed on an outer side in the radial direction with respect to the maximally bulging portion  5   x  of the turbine runner  5 . Thus, the axial length of the damper apparatus  10  and furthermore the axial length of the starting apparatus  1  can further be reduced. In addition, the intermediate springs SPm are disposed so as to partially overlap the turbine runner  5  in the radial direction as viewed in the axial direction. Thus, the space usage of the entire starting apparatus  1  can be improved by effectively using, as an arrangement space for the intermediate springs SPm, the region in the vicinity of the outer peripheral portion of the turbine runner  5  that is likely to be a dead space. 
     In the damper apparatus  10 , the second intermediate member  14  includes the first and second annular members  141  and  142 . The second intermediate member  14  is supported by the second plate member  122  of the first intermediate member  12  so as to be rotatable relative to the first intermediate member  12 . The second intermediate member  14  is disposed between the outer peripheral portion of the turbine runner  5  and the clutch drum  81  in the axial direction. The first spring abutment portions  141   c  each abutting against the ends of the first and second outer springs SP 21  and SP 22  between the first and second outer springs SP 21  and SP 22  and the second spring abutment portions  141   d  each abutting against the end of the intermediate spring SPm are both formed on the first annular member  141  on one side. The plurality of intermediate springs SPm are supported by the second annular member  142  on the other side. The first spring abutment portion  141   c  extends from the first annular member  141  to one side in the axial direction of the damper apparatus  10  so as to abut against the ends of the first and second outer springs SP 21  and SP 22 . The second spring abutment portion  141   d  extends from the first annular member  141  to the other side in the axial direction so as to abut against the end of the intermediate spring SPm. Thus, the second intermediate member  14  can be coupled to the first and second outer springs SP 21  and SP 22  and the intermediate springs SPm while the increase in the size of the starting apparatus  1  along with the arrangement of the intermediate springs SPm is suppressed by effectively using the space in the starting apparatus  1 . 
     In the damper apparatus  10 , in addition to the spring abutment portion  121   c  of the first plate member  121 , the coupling abutment portion  122   c  of the second plate member  122  that is fitted to the spring abutment portion  121   c  abuts against the ends of the first and second inner springs SP 11  and SP 12  between the first and second inner springs SP 11  and SP 12 . By causing both of the spring abutment portion  121   c  extending in the radial direction of the damper apparatus  10  and the coupling abutment portion  122   c  extending in the axial direction of the damper apparatus  10  to abut against the first and second inner springs SP 11  and SP 12 , the first and second inner springs SP 11  and SP 12  can appropriately be pressed by the first intermediate member  12  so as to extend and contract along the axis center. As a result, the vibration damping performance of the damper apparatus  10  can further be improved. 
     By causing the coupling abutment portion  122   c  fitted to the spring abutment portion  121   c  to abut against the ends of the first and second inner springs SP 11  and SP 12  between the first and second inner springs SP 11  and SP 12 , the second plate member  122  can be supported from both sides in the circumferential direction by the first and second inner springs SP 11  and SP 12 . Thus, the first plate member  121  and the second plate member  122  can loosely be fitted to each other, and the coupling abutment portion  122   c  can easily be fitted to the spring abutment portion  121   c . That is, in the damper apparatus  10 , as described above, the opening length of the opening portion  121   h  of the spring abutment portion  121   c  in the radial direction is defined as being larger than the thickness of the protruding portion  122   p  of the coupling abutment portion  122   c  in the radial direction. Thus, the protruding portion  122   p  of the coupling abutment portion  122   c  of the second plate member  122  can easily be fitted to the opening portion  121   h  of the spring abutment portion  121   c  of the first plate member  121 . Accordingly, the assembling workability of the damper apparatus  10  can be secured satisfactorily. 
     The coupling abutment portion  122   c  of the second plate member  122  is retained by the first and second inner springs SP 11  and SP 12 , and therefore the turbine runner  5  that is a mass and the turbine hub  52  are coupled to the first intermediate member  12 . Thus, a substantial moment of inertia of the first intermediate member  12  (sum of the moments of inertia of the first and second plate members  121  and  122 , the turbine runner  5 , the turbine hub  52 , and the like) can further be increased. In addition, by coupling the inner peripheral portion of the second plate member  122  to the turbine runner  5 , the first intermediate member  12  and the turbine runner  5  can be coupled to each other while the mountability is improved by suppressing an increase in the size of the damper apparatus  10 . 
     In the damper apparatus  10 , as illustrated in  FIG. 2 , the inner and outer spring abutment portions  111   ci  and  111   co  of the driving member  11 , the spring abutment portions  121   c  of the first intermediate member  12 , and the inner spring abutment portions  161   ci , the spring abutment portions  162   c , and the outer spring abutment portions  161   co  of the driven member  16  extend in the radial direction of the damper apparatus  10 . Thus, the associated spring SP 11 , SP 12 , SP 21 , or SP 22  can be pressed by the spring abutment portion  111   ci ,  111   co ,  161   ci ,  162   c , or  161   co  so as to appropriately extend and contract along the axis center. As a result, the vibration damping performance of the damper apparatus  10  can further be improved. 
     In the damper apparatus  10 , the coupling abutment portions  122   c  each abutting against the ends of the first and second inner springs SP 11  and SP 12  between the first and second inner springs SP 11  and SP 12  and the outer abutment portions  122   d  each abutting against the end of the intermediate spring SPm are both formed on the second plate member  122  (single member) included in the first intermediate member  12 . Further, the first spring abutment portions  141   c  each abutting against the ends of the first and second outer springs SP 21  and SP 22  between the first and second outer springs SP 21  and SP 22  and the second spring abutment portions  141   d  each abutting against the end of the intermediate spring SPm are both formed on the first annular member  141  (single member) included in the second intermediate member  14 . Thus, an increase in the number of components and the increase in the size of the damper apparatus  10  can be suppressed. 
     Next, an operation of the damper apparatus  10  is described. In the starting apparatus  1 , when the lock-up is not executed by the lock-up clutch  8 , for example, a rotational torque (power) transferred from the engine EG to the front cover  3  is transferred to the input shaft IS of the transmission TM via a path including the pump impeller  4 , the turbine runner  5 , the first intermediate member  12 , the second inner springs SP 12 , the driven member  16 , and the damper hub  7  and via a path including the pump impeller  4 , the turbine runner  5 , the first intermediate member  12 , the intermediate springs SPm, the second intermediate member  14 , the second outer springs SP 22 , the driven member  16 , and the damper hub  7 . When the lock-up is executed by the lock-up clutch  8  of the starting apparatus  1 , the rotational torque (input torque) transferred from the engine EG to the driving member  11  via the front cover  3  and the lock-up clutch  8  (lock-up piston  80 ) is transferred to the driven member  16  and the damper hub  7  via all the springs SP 11  to SPm until the torque input to the driving member  11  reaches the torque T 1  described above, that is, while the deflections of all of the first and second inner springs SP 11  and SP 12 , the first and second outer springs SP 21  and SP 22 , and the intermediate springs SPm are permitted. 
     That is, until the input torque reaches the torque T 1  during the execution of the lock-up, the first inner springs (first elastic bodies) SP 11  transfer the rotational torque from the driving member  11  to the first intermediate member  12 , and the second inner springs (second elastic bodies) SP 12  transfer the rotational torque from the first intermediate member  12  to the driven member  16 . Further, the first outer springs (third elastic bodies) SP 21  transfer the rotational torque from the driving member  11  to the second intermediate member  14 , and the second outer springs (fourth elastic bodies) SP 22  transfer the rotational torque from the second intermediate member  14  to the driven member  16 . Thus, as illustrated in  FIG. 6 , the damper apparatus  10  has a first torque transfer path P 1  including the first inner springs SP 11 , the first intermediate member  12 , and the second inner springs SP 12 , and a second torque transfer path P 2  including the first outer springs SP 21 , the second intermediate member  14 , and the second outer springs SP 22  as torque transfer paths between the driving member  11  and the driven member  16 . 
     In the damper apparatus  10 , as described above, the spring rates k 11 , k 12 , k 21 , and k 22  of the first and second inner springs SP 11  and SP 12  and the first and second outer springs SP 21  and SP 22  satisfy the relationship of k 11 &lt;k 12 &lt;k 22 &lt;k 21 . Therefore, when the torque is transferred to the driving member  11  within a period until the input torque reaches the torque T 1  during the execution of the lock-up, as illustrated in  FIG. 6 , the second intermediate member  14  is (slightly) twisted to an advancing side (downstream side) in the rotation direction (rotation direction when the vehicle travels forward) relative to the first intermediate member  12 . Thus, each intermediate spring SPm is pressed toward one of the paired outer abutment portions  122   d  of the first intermediate member  12  on the advancing side in the rotation direction by one of the paired second spring abutment portions  141   d  of the second intermediate member  14  on a side opposite to the advancing side in the rotation direction. That is, until the input torque reaches the torque T 1  during the execution of the lock-up, each intermediate spring SPm transfers, to the first intermediate member  12 , a part of the torque (part of an average torque) transferred from the driving member  11  to the second intermediate member  14  via the first outer spring SP 21 . Thus, the damper apparatus  10  has a third torque transfer path P 3  including the first outer springs SP 21 , the second intermediate member  14 , the intermediate springs SPm, the first intermediate member  12 , and the second inner springs SP 12 . 
     As a result, until the torque input to the driving member  11  reaches the torque T 1  described above during the execution of the lock-up, the torque is transferred from the driving member  11  to the driven member  16  via the first, second, and third torque transfer paths P 1 , P 2 , and P 3 . More specifically, while the deflections of all of the first and second inner springs SP 11  and SP 12 , the first and second outer springs SP 21  and SP 22 , and the intermediate springs SPm are permitted, the rotational torque from the first inner springs SP 11  and the rotational torque from the first outer springs SP 21 , the second intermediate member  14 , and the intermediate springs SPm are transferred to the second inner springs SP 12 . Further, the rotational torque from the first outer springs SP 21  is transferred to the second outer springs SP 22 . While the deflections of all the springs SP 11  to SPm are permitted, the springs SP 11  to SPm damp (absorb) a fluctuation of the torque transferred to the driving member  11 . Thus, the vibration damping performance of the damper apparatus  10  when the rotation speed of the driving member  11  is low can satisfactorily be improved. 
     When the torque input to the driving member  11  reaches the torque T 1  described above and the first and second stoppers  21  and  22  are actuated, the first stopper  21  restricts the relative rotation between the first intermediate member  12  and the driven member  16  and the deflection of the second inner springs SP 12 , and the second stopper  22  restricts the relative rotation between the second intermediate member  14  and the driven member  16  and the deflection of the second outer springs SP 22 . By restricting the relative rotations of the first and second intermediate members  12  and  14  to the driven member  16 , the deflection of the intermediate springs SPm is restricted as well. Thus, during a period from the time when the torque input to the driving member  11  reaches the torque T 1  described above to the time when the input torque reaches the torque T 2  described above and the third stopper  23  is actuated, the first inner springs SP 11  and the first outer springs SP 21  act in parallel to damp (absorb) the fluctuation of the torque transferred to the driving member  11 . 
     In the damper apparatus  10 , while the deflections of all the springs SP 11  to SPm are permitted, the force applied to the first spring abutment portions  141   c  of the second intermediate member  14  from the first and second outer springs SP 21  and SP 22  may be opposite to the force applied to the second spring abutment portions  141   d  of the second intermediate member  14  from the intermediate springs SPm. Thus, when the first spring abutment portions are formed on one of the first and second annular members  141  and  142  of the second intermediate member  14  and the second spring abutment portions are formed on the other, shear forces acting on coupling portions of the first and second annular members  141  and  142  increase, and the durability of the second intermediate member  14  may decrease. When the first and second spring abutment portions  141   c  and  141   d  are provided on the first annular member  141  (single member) of the second intermediate member  14  as described above, the first annular member  141  can receive the two forces acting in opposite directions. Thus, the shear forces acting on the coupling portions (around the rivets) of the first and second annular members  141  and  142  can be reduced as compared to the case where the first spring abutment portions are formed on one of the first and second annular members  141  and  142  and the second spring abutment portions are formed on the other. As a result, it is possible to further improve the durability of the coupling portions of the first and second annular members  141  and  142  and furthermore the durability of the second intermediate member  14  to which the torque is transferred from the first outer springs SP 21  having a larger share of the torque than the first inner springs SP 11 . 
     Similarly, in the damper apparatus  10 , while the deflections of all the springs SP 11  to SPm are permitted, the force applied to the first intermediate member  12  from the first and second inner springs SP 11  and SP 12  may be opposite to the force applied to the first intermediate member  12 , that is, the second plate member  122  from the intermediate springs SPm. When the outer abutment portions  122   d  are provided on the second plate member  122  including the coupling abutment portions  122   c , the second plate member  122  (single member) can substantially receive the two forces acting in opposite directions. Thus, it is possible to reduce the shear forces acting on fitting portions (opening portions  121   h  and protruding portions  122   p ) of the first and second plate members  121  and  122 . Accordingly, it is possible to further improve the durability of the fitting portions of the spring abutment portions  121   c  of the first plate member  121  and the coupling abutment portions  122   c  of the second plate member  122  and furthermore the durability of the first intermediate member  12 . 
     The description of the designing procedure of the damper apparatus  10  is continued. 
     As described above, in the damper apparatus  10 , when the deflections of all of the first and second inner springs SP 11  and SP 12 , the first and second outer springs SP 21  and SP 22 , and the intermediate springs SPm are permitted, the torque (average torque) is transferred between the driving member  11  and the driven member  16  via all the springs SP 11  to SPm. The inventors have extensively conducted researches and analyses into the damper apparatus  10  having the complicated torque transfer paths that are not provided in series or in parallel. As a result, the inventors have found that the damper apparatus  10  has two natural frequencies as a whole when the deflections of all the springs SP 11  to SPm are permitted. According to the researches and analyses conducted by the inventors, also in the damper apparatus  10 , when resonance occurs at a smaller one of the two natural frequencies (natural frequency on a low speed rotation side (low frequency side)) in accordance with the frequency of the vibration transferred to the driving member  11  (in this embodiment, resonance of the first intermediate member  12  when the first and second intermediate members  12  and  14  vibrate in the same phase), a phase of the vibration transferred from the second inner springs SP 12  to the driven member  16  deviates from a phase of the vibration transferred from the second outer springs SP 22  to the driven member  16 . Therefore, as the rotation speed of the driving member  11  increases after the resonance occurs at the smaller one of the two natural frequencies, one of the vibration transferred from the second inner springs SP 12  to the driven member  16  and the vibration transferred from the second outer springs SP 22  to the driven member  16  cancels out at least a part of the other. 
     Based on the findings described above, the inventors have formulated an equation of motion as represented by Expression (1) below regarding a vibration system including the damper apparatus  10  in a state in which the torque is transferred from the engine (internal combustion engine) EG to the driving member  11  through the execution of the lock-up. In Expression (1), “J 1 ” represents a moment of inertia of the driving member  11 , “J 21 ” represents a moment of inertia of the first intermediate member  12 , “J 22 ” represents a moment of inertia of the second intermediate member  14 , and “J 3 ” represents a moment of inertia of the driven member  16 . Further, “θ 1 ” represents a torsion angle of the driving member  11 , “θ 21 ” represents a torsion angle of the first intermediate member  12 , “θ 22 ” represents a torsion angle of the second intermediate member  14 , and “θ 3 ” represents a torsion angle of the driven member  16 . Still further, “k 1 ” represents a combined spring rate of the plurality of first inner springs SP 11  acting in parallel between the driving member  11  and the first intermediate member  12 , “k 2 ” represents a combined spring rate of the plurality of second inner springs SP 12  acting in parallel between the first intermediate member  12  and the driven member  16 , “k 3 ” represents a combined spring rate of the plurality of first outer springs SP 21  acting in parallel between the driving member  11  and the second intermediate member  14 , “k 4 ” represents a combined spring rate of the plurality of second outer springs SP 22  acting in parallel between the second intermediate member  14  and the driven member  16 , “k 5 ” represents a combined spring rate (stiffness) of the plurality of intermediate springs SPm acting in parallel between the first intermediate member  12  and the second intermediate member  14 , “k R ” represents a stiffness, that is, a spring rate in the transmission TM, drive shafts, and the like that are disposed in a range from the driven member  16  to wheels of the vehicle, and “T” represents an input torque transferred from the engine EG to the driving member  11 . 
     
       
         
           
             
               [ 
               
                 Math 
                 . 
                 
                     
                 
                  
                 1 
               
               ] 
             
              
             
                 
             
           
         
       
       
         
           
             
               
                 
                   
                     
                       
                         ( 
                         
                           
                             
                               
                                 J 
                                 1 
                               
                             
                             
                               0 
                             
                             
                               0 
                             
                             
                               0 
                             
                           
                           
                             
                               0 
                             
                             
                               
                                 J 
                                 21 
                               
                             
                             
                               0 
                             
                             
                               0 
                             
                           
                           
                             
                               0 
                             
                             
                               0 
                             
                             
                               
                                 J 
                                 22 
                               
                             
                             
                               0 
                             
                           
                           
                             
                               0 
                             
                             
                               0 
                             
                             
                               0 
                             
                             
                               
                                 J 
                                 1 
                               
                             
                           
                         
                         ) 
                       
                        
                       
                         ( 
                         
                           
                             
                               
                                 
                                   θ 
                                   ¨ 
                                 
                                 1 
                               
                             
                           
                           
                             
                               
                                 
                                   θ 
                                   ¨ 
                                 
                                 31 
                               
                             
                           
                           
                             
                               
                                 
                                   θ 
                                   ¨ 
                                 
                                 32 
                               
                             
                           
                           
                             
                               
                                 
                                   θ 
                                   ¨ 
                                 
                                 3 
                               
                             
                           
                         
                         ) 
                       
                     
                     + 
                     
                       
                         ( 
                         
                           
                             
                               
                                 
                                   k 
                                   1 
                                 
                                 + 
                                 
                                   k 
                                   2 
                                 
                               
                             
                             
                               
                                 - 
                                 
                                   k 
                                   1 
                                 
                               
                             
                             
                               
                                 - 
                                 
                                   k 
                                   1 
                                 
                               
                             
                             
                               0 
                             
                           
                           
                             
                               
                                 - 
                                 
                                   k 
                                   1 
                                 
                               
                             
                             
                               
                                 
                                   k 
                                   1 
                                 
                                 + 
                                 
                                   k 
                                   2 
                                 
                                 + 
                                 
                                   k 
                                   3 
                                 
                               
                             
                             
                               
                                 - 
                                 
                                   k 
                                   3 
                                 
                               
                             
                             
                               
                                 - 
                                 
                                   k 
                                   2 
                                 
                               
                             
                           
                           
                             
                               
                                 - 
                                 
                                   k 
                                   3 
                                 
                               
                             
                             
                               
                                 - 
                                 
                                   k 
                                   3 
                                 
                               
                             
                             
                               
                                 
                                   k 
                                   3 
                                 
                                 + 
                                 
                                   k 
                                   4 
                                 
                                 + 
                                 
                                   k 
                                   4 
                                 
                               
                             
                             
                               
                                 - 
                                 
                                   k 
                                   4 
                                 
                               
                             
                           
                           
                             
                               0 
                             
                             
                               
                                 - 
                                 
                                   k 
                                   1 
                                 
                               
                             
                             
                               
                                 - 
                                 
                                   k 
                                   4 
                                 
                               
                             
                             
                               
                                 
                                   k 
                                   3 
                                 
                                 + 
                                 
                                   k 
                                   4 
                                 
                                 + 
                                 
                                   k 
                                   R 
                                 
                               
                             
                           
                         
                         ) 
                       
                        
                       
                         ( 
                         
                           
                             
                               
                                 θ 
                                 1 
                               
                             
                           
                           
                             
                               
                                 θ 
                                 21 
                               
                             
                           
                           
                             
                               
                                 θ 
                                 22 
                               
                             
                           
                           
                             
                               
                                 θ 
                                 3 
                               
                             
                           
                         
                         ) 
                       
                     
                   
                   = 
                   
                     ( 
                     
                       
                         
                           0 
                         
                       
                       
                         
                           T 
                         
                       
                       
                         
                           0 
                         
                       
                       
                         
                           0 
                         
                       
                     
                     ) 
                   
                 
               
               
                 
                   ( 
                   1 
                   ) 
                 
               
             
           
         
       
     
     The inventors have assumed that the input torque T periodically vibrates as represented by Expression (2) below, and also assumed that the torsion angle θ 1  of the driving member  11 , the torsion angle θ 21  of the first intermediate member  12 , the torsion angle θ 22  of the second intermediate member  14 , and the torsion angle θ 3  of the driven member  16  periodically respond (vibrate) as represented by Expression (3) below. In Expressions (2) and (3), “ω” represents an angular frequency in the periodic fluctuation (vibration) of the input torque T. In Expression (3), “Θ 1 ” represents an amplitude of the vibration (vibration amplitude, that is, maximum torsion angle) of the driving member  11  that occurs along with the transfer of the torque from the engine EQ “Θ 2 ,” represents an amplitude of the vibration (vibration amplitude) of the first intermediate member  12  that occurs along with the transfer of the torque from the engine EG to the driving member  11 , “Θ 22 ” represents an amplitude of the vibration (vibration amplitude) of the second intermediate member  14  that occurs along with the transfer of the torque from the engine EG to the driving member  11 , and “Θ 3 ” represents an amplitude of the vibration (vibration amplitude) of the driven member  16  that occurs along with the transfer of the torque from the engine EG to the driving member  11 . Under those assumptions, Expressions (2) and (3) are substituted into Expression (1), and “sin ωt” is cleared from both sides. Accordingly, an identity of Expression (4) below can be yielded. 
     
       
         
           
             [ 
             
               Math 
               . 
               
                   
               
                
               2 
             
             ] 
           
         
       
       
         
           
             
               
                 
                   T 
                   = 
                   
                     
                       T 
                       0 
                     
                      
                     sin 
                      
                     
                         
                     
                      
                     ω 
                      
                     
                         
                     
                      
                     t 
                   
                 
               
               
                 
                   ( 
                   2 
                   ) 
                 
               
             
             
               
                 
                   
                     [ 
                     
                       
                         
                           
                             θ 
                             1 
                           
                         
                       
                       
                         
                           
                             θ 
                             21 
                           
                         
                       
                       
                         
                           
                             θ 
                             22 
                           
                         
                       
                       
                         
                           
                             θ 
                             3 
                           
                         
                       
                     
                     ] 
                   
                   = 
                   
                     
                       [ 
                       
                         
                           
                             
                               Θ 
                               1 
                             
                           
                         
                         
                           
                             
                               Θ 
                               21 
                             
                           
                         
                         
                           
                             
                               Θ 
                               22 
                             
                           
                         
                         
                           
                             
                               Θ 
                               3 
                             
                           
                         
                       
                       ] 
                     
                      
                     sin 
                      
                     
                         
                     
                      
                     ω 
                      
                     
                         
                     
                      
                     t 
                   
                 
               
               
                 
                   ( 
                   3 
                   ) 
                 
               
             
             
               
                 
                   
                     
                       ( 
                       
                         
                           
                             
                               
                                 
                                   - 
                                   
                                     ω 
                                     2 
                                   
                                 
                                  
                                 
                                   J 
                                   1 
                                 
                               
                               + 
                               
                                 k 
                                 1 
                               
                               + 
                               
                                 k 
                                 3 
                               
                             
                           
                           
                             
                               - 
                               
                                 k 
                                 3 
                               
                             
                           
                           
                             
                               - 
                               
                                 k 
                                 3 
                               
                             
                           
                           
                             0 
                           
                         
                         
                           
                             
                               - 
                               
                                 k 
                                 3 
                               
                             
                           
                           
                             
                               
                                 
                                   - 
                                   
                                     ω 
                                     2 
                                   
                                 
                                  
                                 
                                   J 
                                   21 
                                 
                               
                               + 
                               
                                 k 
                                 1 
                               
                               + 
                               
                                 k 
                                 2 
                               
                               + 
                               
                                 k 
                                 3 
                               
                             
                           
                           
                             
                               - 
                               
                                 k 
                                 5 
                               
                             
                           
                           
                             
                               - 
                               
                                 k 
                                 7 
                               
                             
                           
                         
                         
                           
                             
                               - 
                               
                                 k 
                                 3 
                               
                             
                           
                           
                             
                               - 
                               
                                 k 
                                 3 
                               
                             
                           
                           
                             
                               
                                 
                                   - 
                                   
                                     ω 
                                     2 
                                   
                                 
                                  
                                 
                                   J 
                                   22 
                                 
                               
                               + 
                               
                                 k 
                                 3 
                               
                               + 
                               
                                 k 
                                 4 
                               
                               + 
                               
                                 k 
                                 5 
                               
                             
                           
                           
                             
                               - 
                               
                                 k 
                                 4 
                               
                             
                           
                         
                         
                           
                             0 
                           
                           
                             
                               - 
                               
                                 k 
                                 7 
                               
                             
                           
                           
                             
                               - 
                               
                                 k 
                                 4 
                               
                             
                           
                           
                             
                               
                                 
                                   - 
                                   
                                     ω 
                                     2 
                                   
                                 
                                  
                                 
                                   J 
                                   1 
                                 
                               
                               + 
                               
                                 k 
                                 2 
                               
                               + 
                               
                                 k 
                                 4 
                               
                               + 
                               
                                 k 
                                 8 
                               
                             
                           
                         
                       
                       ) 
                     
                      
                     
                       ( 
                       
                         
                           
                             
                               Θ 
                               1 
                             
                           
                         
                         
                           
                             
                               Θ 
                               21 
                             
                           
                         
                         
                           
                             
                               Θ 
                               22 
                             
                           
                         
                         
                           
                             
                               Θ 
                               3 
                             
                           
                         
                       
                       ) 
                     
                   
                   = 
                   
                     ( 
                     
                       
                         
                           0 
                         
                       
                       
                         
                           T 
                         
                       
                       
                         
                           0 
                         
                       
                       
                         
                           0 
                         
                       
                     
                     ) 
                   
                 
               
               
                 
                   ( 
                   4 
                   ) 
                 
               
             
           
         
       
     
     The inventors have focused attention on the fact that, when the vibration amplitude Θ 3  of the driven member  16  in Expression (4) is zero, the vibration from the engine EG is damped by the damper apparatus  10  and therefore the vibration is not theoretically transferred to the transmission TM, the drive shafts, and the like that are provided at a stage subsequent to the driven member  16 . From this viewpoint, the inventors have solved the identity of Expression (4) in terms of the vibration amplitude Θ 3 , and have set Θ 3 =0, thereby yielding a conditional expression represented by Expression (5) below. When the relationship of Expression (5) holds, the vibrations from the engine EG that are transferred from the driving member  11  to the driven member  16  via the first, second, and third torque transfer paths P 1 , P 2 , and P 3  are canceled out, and the vibration amplitude Θ 3  of the driven member  16  is theoretically zero. 
     
       
         
           
             
               [ 
               
                 Math 
                 . 
                 
                     
                 
                  
                 3 
               
               ] 
             
              
             
                 
             
           
         
       
       
         
           
             
               
                 
                   
                     ω 
                     2 
                   
                   = 
                   
                     
                       
                         
                           
                             
                               k 
                               3 
                             
                             + 
                             
                               ( 
                               
                                 
                                   k 
                                   1 
                                 
                                 + 
                                 
                                   k 
                                   3 
                                 
                               
                               ) 
                             
                             + 
                             
                               ( 
                               
                                 
                                   k 
                                   1 
                                 
                                 + 
                                 
                                   k 
                                   4 
                                 
                               
                               ) 
                             
                             + 
                             
                               
                                 k 
                                 1 
                               
                                
                               
                                 k 
                                 2 
                               
                                
                               
                                 k 
                                 3 
                               
                             
                             + 
                           
                         
                       
                       
                         
                           
                             
                               
                                 k 
                                 1 
                               
                                
                               
                                 k 
                                 2 
                               
                                
                               
                                 k 
                                 4 
                               
                             
                             + 
                             
                               
                                 k 
                                 1 
                               
                                
                               
                                 k 
                                 3 
                               
                                
                               
                                 k 
                                 4 
                               
                             
                             + 
                             
                               
                                 k 
                                 2 
                               
                                
                               
                                 k 
                                 3 
                               
                                
                               
                                 k 
                                 4 
                               
                             
                           
                         
                       
                     
                     
                       
                         
                           J 
                           21 
                         
                          
                         
                           k 
                           3 
                         
                          
                         
                           k 
                           4 
                         
                       
                       + 
                       
                         
                           J 
                           22 
                         
                          
                         
                           k 
                           1 
                         
                          
                         
                           k 
                           2 
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   5 
                   ) 
                 
               
             
           
         
       
     
     The analysis results described above demonstrate that, in the damper apparatus  10  having the structure described above, an anti-resonance point A at which the vibration amplitude Θ 3  (torque fluctuation) of the driven member  16  is theoretically zero can be set as illustrated in  FIG. 7  such that the phase of the vibration transferred from the second inner springs SP 12  to the driven member  16  deviates (is inverted) by 180 degrees from the phase of the vibration transferred from the second outer springs SP 22  to the driven member  16  and both the vibrations are canceled out through the occurrence of the resonance at the smaller one of the two natural frequencies. When the frequency at the anti-resonance point A is represented by “fa” and “ω=2πfa” is substituted into Expression (5) above, the frequency fa at the anti-resonance point A is represented by Expression (6) below.  FIG. 7  exemplifies a relationship between the rotation speed of the engine EG and theoretical (under the assumption that no hysteresis is present) vibration amplitudes (torque fluctuations) of the driven member of the damper apparatus disclosed herein and a driven member of a damper apparatus from which the intermediate springs SPm are omitted (damper apparatus disclosed in Patent Document 1; hereinafter referred to as “damper apparatus of comparative example”). 
     
       
         
           
             
               [ 
               
                 Math 
                 . 
                 
                     
                 
                  
                 4 
               
               ] 
             
              
             
                 
             
           
         
       
       
         
           
             
               
                 
                   fa 
                   = 
                   
                     
                       1 
                       
                         2 
                          
                         π 
                       
                     
                      
                     
                       
                         
                           
                             
                               
                                 
                                   k 
                                   3 
                                 
                                 + 
                                 
                                   ( 
                                   
                                     
                                       k 
                                       1 
                                     
                                     + 
                                     
                                       k 
                                       3 
                                     
                                   
                                   ) 
                                 
                                 + 
                                 
                                   ( 
                                   
                                     
                                       k 
                                       1 
                                     
                                     + 
                                     
                                       k 
                                       4 
                                     
                                   
                                   ) 
                                 
                                 + 
                                 
                                   
                                     k 
                                     1 
                                   
                                    
                                   
                                     k 
                                     2 
                                   
                                    
                                   
                                     k 
                                     3 
                                   
                                 
                                 + 
                               
                             
                           
                           
                             
                               
                                 
                                   
                                     k 
                                     1 
                                   
                                    
                                   
                                     k 
                                     2 
                                   
                                    
                                   
                                     k 
                                     4 
                                   
                                 
                                 + 
                                 
                                   
                                     k 
                                     1 
                                   
                                    
                                   
                                     k 
                                     3 
                                   
                                    
                                   
                                     k 
                                     4 
                                   
                                 
                                 + 
                                 
                                   
                                     k 
                                     2 
                                   
                                    
                                   
                                     k 
                                     3 
                                   
                                    
                                   
                                     k 
                                     4 
                                   
                                 
                               
                             
                           
                         
                         
                           
                             
                               J 
                               21 
                             
                              
                             
                               k 
                               3 
                             
                              
                             
                               k 
                               4 
                             
                           
                           + 
                           
                             
                               J 
                               22 
                             
                              
                             
                               k 
                               1 
                             
                              
                             
                               k 
                               2 
                             
                           
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   6 
                   ) 
                 
               
             
           
         
       
     
     Under the assumption that the torsion angle θ 1  of the driving member  11  and the torsion angle θ 3  of the driven member  16  are zero and displacements of the driving member  11  and the driven member  16  are both zero, Expression (1) can be transformed into Expression (7) below. Under the assumption that the first and second intermediate members  12  and  14  harmonically vibrate as represented by Expression (8) below, Expression (8) is substituted into Expression (7), and “sin ωt” is cleared from both sides. Accordingly, an identity of Expression (9) below can be yielded. 
     
       
         
           
             
               [ 
               
                 Math 
                 . 
                 
                     
                 
                  
                 5 
               
               ] 
             
              
             
                 
             
           
         
       
       
         
           
             
               
                 
                   
                     
                       
                         ( 
                         
                           
                             
                               
                                 J 
                                 21 
                               
                             
                             
                               0 
                             
                           
                           
                             
                               0 
                             
                             
                               
                                 J 
                                 22 
                               
                             
                           
                         
                         ) 
                       
                        
                       
                         ( 
                         
                           
                             
                               
                                 
                                   θ 
                                   ¨ 
                                 
                                 21 
                               
                             
                           
                           
                             
                               
                                 
                                   θ 
                                   ¨ 
                                 
                                 22 
                               
                             
                           
                         
                         ) 
                       
                     
                     + 
                     
                       
                         ( 
                         
                           
                             
                               
                                 
                                   k 
                                   1 
                                 
                                 + 
                                 
                                   k 
                                   2 
                                 
                                 + 
                                 
                                   k 
                                   3 
                                 
                               
                             
                             
                               
                                 - 
                                 
                                   k 
                                   5 
                                 
                               
                             
                           
                           
                             
                               
                                 - 
                                 
                                   k 
                                   5 
                                 
                               
                             
                             
                               
                                 
                                   k 
                                   3 
                                 
                                 + 
                                 
                                   k 
                                   4 
                                 
                                 + 
                                 
                                   k 
                                   5 
                                 
                               
                             
                           
                         
                         ) 
                       
                        
                       
                         ( 
                         
                           
                             
                               
                                 θ 
                                 21 
                               
                             
                           
                           
                             
                               
                                 θ 
                                 22 
                               
                             
                           
                         
                         ) 
                       
                     
                   
                   = 
                   
                     ( 
                     
                       
                         
                           0 
                         
                       
                       
                         
                           0 
                         
                       
                     
                     ) 
                   
                 
               
               
                 
                   ( 
                   7 
                   ) 
                 
               
             
             
               
                 
                   
                     ( 
                     
                       
                         
                           
                             θ 
                             21 
                           
                         
                       
                       
                         
                           
                             θ 
                             22 
                           
                         
                       
                     
                     ) 
                   
                   = 
                   
                     
                       ( 
                       
                         
                           
                             
                               Θ 
                               21 
                             
                           
                         
                         
                           
                             
                               Θ 
                               22 
                             
                           
                         
                       
                       ) 
                     
                      
                     sin 
                      
                     
                         
                     
                      
                     ω 
                      
                     
                         
                     
                      
                     t 
                   
                 
               
               
                 
                   ( 
                   8 
                   ) 
                 
               
             
             
               
                 
                   
                     
                       ( 
                       
                         
                           
                             
                               
                                 
                                   - 
                                   
                                     ω 
                                     2 
                                   
                                 
                                  
                                 
                                   J 
                                   21 
                                 
                               
                               + 
                               
                                 k 
                                 1 
                               
                               + 
                               
                                 k 
                                 2 
                               
                               + 
                               
                                 k 
                                 3 
                               
                             
                           
                           
                             
                               - 
                               
                                 k 
                                 5 
                               
                             
                           
                         
                         
                           
                             
                               - 
                               
                                 k 
                                 5 
                               
                             
                           
                           
                             
                               
                                 
                                   - 
                                   
                                     ω 
                                     2 
                                   
                                 
                                  
                                 
                                   J 
                                   22 
                                 
                               
                               + 
                               
                                 k 
                                 3 
                               
                               + 
                               
                                 k 
                                 4 
                               
                               + 
                               
                                 k 
                                 5 
                               
                             
                           
                         
                       
                       ) 
                     
                      
                     
                       ( 
                       
                         
                           
                             
                               Θ 
                               2 
                             
                           
                         
                         
                           
                             
                               Θ 
                               22 
                             
                           
                         
                       
                       ) 
                     
                   
                   = 
                   
                     ( 
                     
                       
                         
                           0 
                         
                       
                       
                         
                           0 
                         
                       
                     
                     ) 
                   
                 
               
               
                 
                   ( 
                   9 
                   ) 
                 
               
             
           
         
       
     
     When the first and second intermediate members  12  and  14  harmonically vibrate, the amplitudes Θ 21  and Θ 22  are not both zero. Therefore, the determinant of the square matrix on the left-hand side of Expression (9) is zero, and a conditional expression of Expression (10) below should hold. Expression (10) is a quadratic equation regarding square values ω 2  of two natural angular frequencies of the damper apparatus  10 . Thus, two natural angular frequencies ω 1  and ω 2  of the damper apparatus  10  are represented as in Expressions (11) and (12) below, and ω 1 &lt;ω 2  holds. As a result, when “f 21 ” represents a frequency of resonance that generates the resonance point A (resonance point R 1 ), that is, a natural frequency of the first intermediate member  12 , and “f 22 ” represents a frequency of resonance that occurs on a higher speed rotation side than the anti-resonance point A (resonance point R 2 ), that is, a natural frequency of the second intermediate member  14 , the natural frequency f 21  on the low speed rotation side (low frequency side) is represented by Expression (13) below, and the natural frequency f 22  (f 22 &gt;f 21 ) on the high speed rotation side (high frequency side) is represented by Expression (14) below. 
     
       
         
           
             [ 
             
               Math 
               . 
               
                   
               
                
               6 
             
             ] 
           
         
       
       
         
           
             
               
                 
                   
                     
                       
                         ( 
                         
                           
                             
                               - 
                               
                                 ω 
                                 2 
                               
                             
                              
                             
                               J 
                               21 
                             
                           
                           + 
                           
                             k 
                             1 
                           
                           + 
                           
                             k 
                             2 
                           
                           + 
                           
                             k 
                             3 
                           
                         
                         ) 
                       
                        
                       
                         ( 
                         
                           
                             
                               - 
                               
                                 ω 
                                 2 
                               
                             
                              
                             
                               J 
                               22 
                             
                           
                           + 
                           
                             k 
                             3 
                           
                           + 
                           
                             k 
                             4 
                           
                           + 
                           
                             k 
                             5 
                           
                         
                         ) 
                       
                     
                     - 
                     
                       k 
                       5 
                       2 
                     
                   
                   = 
                   0 
                 
               
               
                 
                   ( 
                   10 
                   ) 
                 
               
             
             
               
                 
                   
                     ω 
                     1 
                   
                   = 
                   
                     
                       
                         1 
                         2 
                       
                        
                       
                         { 
                         
                           
                             
                               
                                 k 
                                 1 
                               
                               + 
                               
                                 k 
                                 2 
                               
                               + 
                               
                                 k 
                                 3 
                               
                             
                             
                               J 
                               21 
                             
                           
                           + 
                           
                             
                               
                                 k 
                                 3 
                               
                               + 
                               
                                 k 
                                 4 
                               
                               + 
                               
                                 k 
                                 5 
                               
                             
                             
                               J 
                               22 
                             
                           
                           - 
                           
                             
                               
                                 
                                   ( 
                                   
                                     
                                       
                                         
                                           k 
                                           3 
                                         
                                         + 
                                         
                                           k 
                                           4 
                                         
                                         + 
                                         
                                           k 
                                           5 
                                         
                                       
                                       
                                         J 
                                         22 
                                       
                                     
                                     - 
                                     
                                       
                                         
                                           k 
                                           1 
                                         
                                         + 
                                         
                                           k 
                                           2 
                                         
                                         + 
                                         
                                           k 
                                           3 
                                         
                                       
                                       
                                         J 
                                         21 
                                       
                                     
                                   
                                   ) 
                                 
                                 2 
                               
                               + 
                               
                                 
                                   4 
                                    
                                   
                                     k 
                                     5 
                                     2 
                                   
                                 
                                 
                                   
                                     J 
                                     21 
                                   
                                    
                                   
                                     J 
                                     22 
                                   
                                 
                               
                             
                           
                         
                         } 
                       
                     
                   
                 
               
               
                 
                   ( 
                   11 
                   ) 
                 
               
             
             
               
                 
                   
                     ω 
                     2 
                   
                   = 
                   
                     
                       
                         1 
                         2 
                       
                        
                       
                         { 
                         
                           
                             
                               
                                 k 
                                 1 
                               
                               + 
                               
                                 k 
                                 2 
                               
                               + 
                               
                                 k 
                                 3 
                               
                             
                             
                               J 
                               21 
                             
                           
                           + 
                           
                             
                               
                                 k 
                                 3 
                               
                               + 
                               
                                 k 
                                 4 
                               
                               + 
                               
                                 k 
                                 5 
                               
                             
                             
                               J 
                               22 
                             
                           
                           + 
                           
                             
                               
                                 
                                   ( 
                                   
                                     
                                       
                                         
                                           k 
                                           3 
                                         
                                         + 
                                         
                                           k 
                                           4 
                                         
                                         + 
                                         
                                           k 
                                           5 
                                         
                                       
                                       
                                         J 
                                         22 
                                       
                                     
                                     - 
                                     
                                       
                                         
                                           k 
                                           1 
                                         
                                         + 
                                         
                                           k 
                                           2 
                                         
                                         + 
                                         
                                           k 
                                           3 
                                         
                                       
                                       
                                         J 
                                         21 
                                       
                                     
                                   
                                   ) 
                                 
                                 2 
                               
                               + 
                               
                                 
                                   4 
                                    
                                   
                                     k 
                                     5 
                                     2 
                                   
                                 
                                 
                                   
                                     J 
                                     21 
                                   
                                    
                                   
                                     J 
                                     22 
                                   
                                 
                               
                             
                           
                         
                         } 
                       
                     
                   
                 
               
               
                 
                   ( 
                   12 
                   ) 
                 
               
             
             
               
                 
                   
                     f 
                     21 
                   
                   = 
                   
                     
                       1 
                       
                         2 
                          
                         π 
                       
                     
                      
                     
                       
                         
                           
                             
                               k 
                               1 
                             
                             + 
                             
                               k 
                               2 
                             
                             + 
                             
                               k 
                               3 
                             
                           
                           
                             2 
                              
                             
                               J 
                               21 
                             
                           
                         
                         + 
                         
                           
                             
                               k 
                               3 
                             
                             + 
                             
                               k 
                               4 
                             
                             + 
                             
                               k 
                               5 
                             
                           
                           
                             2 
                              
                             
                               J 
                               22 
                             
                           
                         
                         - 
                         
                           
                             
                               
                                 ( 
                                 
                                   
                                     
                                       
                                         k 
                                         3 
                                       
                                       + 
                                       
                                         k 
                                         4 
                                       
                                       + 
                                       
                                         k 
                                         5 
                                       
                                     
                                     
                                       2 
                                        
                                       
                                         J 
                                         22 
                                       
                                     
                                   
                                   - 
                                   
                                     
                                       
                                         k 
                                         1 
                                       
                                       + 
                                       
                                         k 
                                         2 
                                       
                                       + 
                                       
                                         k 
                                         3 
                                       
                                     
                                     
                                       2 
                                        
                                       
                                         J 
                                         21 
                                       
                                     
                                   
                                 
                                 ) 
                               
                               2 
                             
                             + 
                             
                               
                                 4 
                                  
                                 
                                   k 
                                   5 
                                   2 
                                 
                               
                               
                                 
                                   J 
                                   21 
                                 
                                  
                                 
                                   J 
                                   22 
                                 
                               
                             
                           
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   13 
                   ) 
                 
               
             
             
               
                 
                   
                     f 
                     22 
                   
                   = 
                   
                     
                       1 
                       
                         2 
                          
                         π 
                       
                     
                      
                     
                       
                         
                           
                             
                               k 
                               1 
                             
                             + 
                             
                               k 
                               2 
                             
                             + 
                             
                               k 
                               3 
                             
                           
                           
                             2 
                              
                             
                               J 
                               21 
                             
                           
                         
                         + 
                         
                           
                             
                               k 
                               3 
                             
                             + 
                             
                               k 
                               4 
                             
                             + 
                             
                               k 
                               5 
                             
                           
                           
                             2 
                              
                             
                               J 
                               22 
                             
                           
                         
                         + 
                         
                           
                             
                               
                                 ( 
                                 
                                   
                                     
                                       
                                         k 
                                         3 
                                       
                                       + 
                                       
                                         k 
                                         4 
                                       
                                       + 
                                       
                                         k 
                                         5 
                                       
                                     
                                     
                                       2 
                                        
                                       
                                         J 
                                         22 
                                       
                                     
                                   
                                   - 
                                   
                                     
                                       
                                         k 
                                         1 
                                       
                                       + 
                                       
                                         k 
                                         2 
                                       
                                       + 
                                       
                                         k 
                                         3 
                                       
                                     
                                     
                                       2 
                                        
                                       
                                         J 
                                         21 
                                       
                                     
                                   
                                 
                                 ) 
                               
                               2 
                             
                             + 
                             
                               
                                 4 
                                  
                                 
                                   k 
                                   5 
                                   2 
                                 
                               
                               
                                 
                                   J 
                                   21 
                                 
                                  
                                 
                                   J 
                                   22 
                                 
                               
                             
                           
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   14 
                   ) 
                 
               
             
           
         
       
     
     An equivalent stiffness k eq  of the damper apparatus  10  when the deflections of all the springs SP 11  to SPm are permitted can be determined as follows. That is, under the assumption that a constant input torque (static external force) represented by T=T 0  is transferred to the driving member  11  and a relationship of balance as represented by Expression (15) below holds, T=T 0  and Expression (15) are substituted into Expression (1), and accordingly an identity of Expression (16) below can be yielded. 
     
       
         
           
             
               [ 
               
                 Math 
                 . 
                 
                     
                 
                  
                 7 
               
               ] 
             
              
             
                 
             
           
         
       
       
         
           
             
               
                 
                   
                     [ 
                     
                       
                         
                           
                             θ 
                             1 
                           
                         
                       
                       
                         
                           
                             θ 
                             21 
                           
                         
                       
                       
                         
                           
                             θ 
                             22 
                           
                         
                       
                       
                         
                           
                             θ 
                             3 
                           
                         
                       
                     
                     ] 
                   
                   = 
                   
                     [ 
                     
                       
                         
                           
                             Θ 
                             1 
                           
                         
                       
                       
                         
                           
                             Θ 
                             21 
                           
                         
                       
                       
                         
                           
                             Θ 
                             22 
                           
                         
                       
                       
                         
                           
                             Θ 
                             3 
                           
                         
                       
                     
                     ] 
                   
                 
               
               
                 
                   ( 
                   15 
                   ) 
                 
               
             
             
               
                 
                   
                     
                       ( 
                       
                         
                           
                             
                               
                                 k 
                                 1 
                               
                               + 
                               
                                 k 
                                 2 
                               
                             
                           
                           
                             
                               - 
                               
                                 k 
                                 1 
                               
                             
                           
                           
                             
                               - 
                               
                                 k 
                                 1 
                               
                             
                           
                           
                             0 
                           
                         
                         
                           
                             
                               - 
                               
                                 k 
                                 1 
                               
                             
                           
                           
                             
                               
                                 k 
                                 1 
                               
                               + 
                               
                                 k 
                                 2 
                               
                               + 
                               
                                 k 
                                 3 
                               
                             
                           
                           
                             
                               - 
                               
                                 k 
                                 3 
                               
                             
                           
                           
                             
                               - 
                               
                                 k 
                                 2 
                               
                             
                           
                         
                         
                           
                             
                               - 
                               
                                 k 
                                 3 
                               
                             
                           
                           
                             
                               - 
                               
                                 k 
                                 3 
                               
                             
                           
                           
                             
                               
                                 k 
                                 3 
                               
                               + 
                               
                                 k 
                                 4 
                               
                               + 
                               
                                 k 
                                 4 
                               
                             
                           
                           
                             
                               - 
                               
                                 k 
                                 4 
                               
                             
                           
                         
                         
                           
                             0 
                           
                           
                             
                               - 
                               
                                 k 
                                 1 
                               
                             
                           
                           
                             
                               - 
                               
                                 k 
                                 4 
                               
                             
                           
                           
                             
                               
                                 k 
                                 3 
                               
                               + 
                               
                                 k 
                                 4 
                               
                               + 
                               
                                 k 
                                 R 
                               
                             
                           
                         
                       
                       ) 
                     
                      
                     
                       ( 
                       
                         
                           
                             
                               Θ 
                               1 
                             
                           
                         
                         
                           
                             
                               Θ 
                               21 
                             
                           
                         
                         
                           
                             
                               Θ 
                               22 
                             
                           
                         
                         
                           
                             
                               Θ 
                               3 
                             
                           
                         
                       
                       ) 
                     
                   
                   = 
                   
                     ( 
                     
                       
                         
                           
                             T 
                             0 
                           
                         
                       
                       
                         
                           0 
                         
                       
                       
                         
                           0 
                         
                       
                       
                         
                           0 
                         
                       
                     
                     ) 
                   
                 
               
               
                 
                   ( 
                   16 
                   ) 
                 
               
             
           
         
       
     
     Among the torque T 0 , the equivalent stiffness k eq , of the damper apparatus  10 , the vibration amplitude (torsion angle) Θ 1  of the driving member  11 , and the vibration amplitude (torsion angle) Θ 3  of the driven member  16 , a relationship of T 0 =k eq ·(Θ 1 −Θ 3 ) holds. When the identity of Expression (16) is solved in terms of the vibration amplitudes (torsion angles) Θ 1  and Θ 3 , “Θ 1  . . . Θ 3 ” is represented by Expression (17) below. Based on T 0 =k eq ·(Θ 1 -Θ 3 ) and Expression (17), the equivalent stiffness k eq  of the damper apparatus  10  is represented by Expression (18) below. 
     
       
         
           
             
               [ 
               
                 Math 
                 . 
                 
                     
                 
                  
                 8 
               
               ] 
             
              
             
                 
             
           
         
       
       
         
           
             
               
                 
                   
                     
                       Θ 
                       2 
                     
                     - 
                     
                       Θ 
                       3 
                     
                   
                   = 
                   
                     
                       
                         { 
                         
                           
                             
                               k 
                               5 
                             
                             · 
                             
                               ( 
                               
                                 
                                   k 
                                   1 
                                 
                                 + 
                                 
                                   k 
                                   2 
                                 
                                 + 
                                 
                                   k 
                                   3 
                                 
                                 + 
                                 
                                   k 
                                   4 
                                 
                               
                               ) 
                             
                           
                           + 
                           
                             
                               ( 
                               
                                 
                                   k 
                                   1 
                                 
                                 + 
                                 
                                   k 
                                   2 
                                 
                               
                               ) 
                             
                              
                             
                               ( 
                               
                                 
                                   k 
                                   3 
                                 
                                 + 
                                 
                                   k 
                                   4 
                                 
                               
                               ) 
                             
                           
                         
                         } 
                       
                        
                       
                         T 
                         0 
                       
                     
                     
                       
                         
                           
                             
                               
                                 
                                   k 
                                   5 
                                 
                                  
                                 
                                   ( 
                                   
                                     
                                       k 
                                       1 
                                     
                                     + 
                                     
                                       k 
                                       3 
                                     
                                   
                                   ) 
                                 
                               
                                
                               
                                 ( 
                                 
                                   
                                     k 
                                     2 
                                   
                                   + 
                                   
                                     k 
                                     4 
                                   
                                 
                                 ) 
                               
                             
                             + 
                           
                         
                       
                       
                         
                           
                             ( 
                             
                               
                                 
                                   k 
                                   1 
                                 
                                  
                                 
                                   k 
                                   2 
                                 
                                  
                                 
                                   k 
                                   3 
                                 
                               
                               + 
                               
                                 
                                   k 
                                   1 
                                 
                                  
                                 
                                   k 
                                   2 
                                 
                                  
                                 
                                   k 
                                   4 
                                 
                               
                               + 
                               
                                 
                                   k 
                                   1 
                                 
                                  
                                 
                                   k 
                                   3 
                                 
                                  
                                 
                                   k 
                                   4 
                                 
                               
                               + 
                               
                                 
                                   k 
                                   2 
                                 
                                  
                                 
                                   k 
                                   3 
                                 
                                  
                                 
                                   k 
                                   4 
                                 
                               
                             
                             ) 
                           
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   17 
                   ) 
                 
               
             
             
               
                 
                   
                     K 
                     eq 
                   
                   = 
                   
                     
                       
                         
                           
                             
                               
                                 
                                   k 
                                   5 
                                 
                                  
                                 
                                   ( 
                                   
                                     
                                       k 
                                       1 
                                     
                                     + 
                                     
                                       k 
                                       3 
                                     
                                   
                                   ) 
                                 
                               
                                
                               
                                 ( 
                                 
                                   
                                     k 
                                     2 
                                   
                                   + 
                                   
                                     k 
                                     4 
                                   
                                 
                                 ) 
                               
                             
                             + 
                           
                         
                       
                       
                         
                           
                             ( 
                             
                               
                                 
                                   k 
                                   1 
                                 
                                  
                                 
                                   k 
                                   2 
                                 
                                  
                                 
                                   k 
                                   3 
                                 
                               
                               + 
                               
                                 
                                   k 
                                   1 
                                 
                                  
                                 
                                   k 
                                   2 
                                 
                                  
                                 
                                   k 
                                   4 
                                 
                               
                               + 
                               
                                 
                                   k 
                                   1 
                                 
                                  
                                 
                                   k 
                                   3 
                                 
                                  
                                 
                                   k 
                                   4 
                                 
                               
                               + 
                               
                                 
                                   k 
                                   2 
                                 
                                  
                                 
                                   k 
                                   3 
                                 
                                  
                                 
                                   k 
                                   4 
                                 
                               
                             
                             ) 
                           
                         
                       
                     
                     
                       
                         
                           k 
                           5 
                         
                         · 
                         
                           ( 
                           
                             
                               k 
                               1 
                             
                             + 
                             
                               k 
                               2 
                             
                             + 
                             
                               k 
                               3 
                             
                             + 
                             
                               k 
                               4 
                             
                           
                           ) 
                         
                       
                       + 
                       
                         
                           ( 
                           
                             
                               k 
                               1 
                             
                             + 
                             
                               k 
                               2 
                             
                           
                           ) 
                         
                          
                         
                           ( 
                           
                             
                               k 
                               3 
                             
                             + 
                             
                               k 
                               4 
                             
                           
                           ) 
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   18 
                   ) 
                 
               
             
           
         
       
     
       FIG. 8  to  FIG. 13  illustrate analysis results obtained by the inventors for the natural frequency f 21 , on the low speed rotation side, the frequency fa at the anti-resonance point A, and the equivalent stiffness k eq  of the damper apparatus  10  that are obtained as described above.  FIG. 8  to  FIG. 13  illustrate how the natural frequency f 21 , the frequency fa at the anti-resonance point A, and the equivalent stiffness k eq  are changed when only one parameter out of the combined spring rates k 1 , k 2 , k 3 , k 4 , and k 5  and the moments of inertia J 21  and J 22  of the first and second intermediate members  12  and  14  is changed while the parameters other than the one parameter are set to constant values (fixed values). 
     When only the combined spring rate (stiffness) k 1  of the first inner springs (first elastic bodies) SP 11  is changed while the combined spring rates k 2 , k 3 , k 4 , and k 5  and the moments of inertia J 21  and J 22  of the damper apparatus  10  are set to constant values, as illustrated in  FIG. 8 , the natural frequency f 21  and the frequency fa at the anti-resonance point A increase as the combined spring rate k 1  increases, and gradually decrease as the combined spring rate k 1  decreases. As illustrated in  FIG. 8 , the equivalent stiffness k eq  steeply increases when the combined spring rate k 1  slightly increases from a preadapted value, and steeply decreases when the combined spring rate k 1  slightly decreases from the adapted value. That is, the change (gradient of change) in the equivalent stiffness k eq  is significantly large relative to the change in the combined spring rate k 1  of the first inner springs SP 11 . 
     Also when only the combined spring rate (stiffness) k 2  of the second inner springs (second elastic bodies) SP 12  is changed while the combined spring rates k 1 , k 3 , k 4 , and k 5  and the moments of inertia J 21  and J 22  of the damper apparatus  10  are set to constant values, as illustrated in  FIG. 9 , the natural frequency f 2 , and the frequency fa at the anti-resonance point A increase as the combined spring rate k 2  increases, and gradually decrease as the combined spring rate k 2  decreases. As illustrated in  FIG. 9 , the equivalent stiffness k e  steeply increases when the combined spring rate k 2  slightly increases from a preadapted value, and steeply decreases when the combined spring rate k 2  slightly decreases from the adapted value. That is, the change (gradient of change) in the equivalent stiffness k eq  is also significantly large relative to the change in the combined spring rate k 2  of the second inner springs SP 12 . 
     When only the combined spring rate (stiffness) k 3  of the first outer springs (third elastic bodies) SP 21  is changed while the combined spring rates k 1 , k 2 , k 4 , and k 5  and the moments of inertia J 21  and J 22  of the damper apparatus  10  are set to constant values, as illustrated in  FIG. 10 , the natural frequency f 21  slightly increases (is kept substantially constant) as the combined spring rate k 3  increases, and the frequency fa at the anti-resonance point A increases as the combined spring rate k 3  decreases, and gradually decreases as the combined spring rate k 3  increases. As illustrated in  FIG. 10 , the equivalent stiffness k eq  steeply decreases when the combined spring rate k 3  slightly decreases from a preadapted value, and steeply increases when the combined spring rate k 3  slightly increases from the adapted value. That is, the change (gradient of change) in the equivalent stiffness k eq  is also significantly large relative to the change in the combined spring rate k 3  of the first outer springs SP 21 . 
     Also when only the combined spring rate (stiffness) k 4  of the second outer springs (fourth elastic bodies) SP 22  is changed while the combined spring rates k 1 , k 2 , k 3 , and k 5  and the moments of inertia J 21  and J 22  of the damper apparatus  10  are set to constant values, as illustrated in  FIG. 11 , the natural frequency f 21  slightly increases (is kept substantially constant) as the combined spring rate k 4  increases, and the frequency fa at the anti-resonance point A increases as the combined spring rate k 4  decreases, and gradually decreases as the combined spring rate k 4  increases. As illustrated in  FIG. 11 , the equivalent stiffness k eq  steeply decreases when the combined spring rate k 4  slightly decreases from a preadapted value, and steeply increases when the combined spring rate k 4  slightly increases from the adapted value. That is, the change (gradient of change) in the equivalent stiffness k eq  is also significantly large relative to the change in the combined spring rate k 4  of the second outer springs SP 22 . 
     When only the combined spring rate (stiffness) k 5  of the intermediate springs (fifth elastic bodies) SPm is changed while the combined spring rates k 1 , k 2 , k 3 , and k 4  and the moments of inertia J 21  and J 22  of the damper apparatus  10  are set to constant values, as illustrated in  FIG. 12 , the natural frequency f 21  and the frequency fa at the anti-resonance point A increase as the combined spring rate k 5  increases, and gradually decrease as the combined spring rate k 5  decreases. As illustrated in  FIG. 12 , a difference between the natural frequency f 21  and the frequency fa at the anti-resonance point A (fa−f 21 ) corresponding to the certain combined spring rate k 5  gradually increases as the combined spring rate k 5  increases. When only the combined spring rate k 5  of the intermediate springs SPm is changed, as illustrated in  FIG. 12 , the equivalent stiffness k 4  increases as the combined spring rate k 5  increases, and gradually decreases as the combined spring rate k 5  decreases. That is, the change (gradient of change) in the equivalent stiffness k eq  relative to the change in the combined spring rate (stiffness) k 5  of the intermediate springs SPm is significantly smaller than the changes (gradients of changes) in the equivalent stiffness k eq  relative to the changes in the combined spring rates (stiffnesses) k 1 , k 2 , k 3 , and k 4 . 
     When only the moment of inertia J 21  of the first intermediate member  12  is changed while the combined spring rates k 1 , k 2 , k 3 , k 4 , and k 5  and the moment of inertia J 22  of the second intermediate member  14  of the damper apparatus  10  are set to constant values, as illustrated in  FIG. 13 , the natural frequency f 21  and the frequency fa at the anti-resonance point A increase as the moment of inertia J 21  decreases, and gradually decrease as the moment of inertia J 21  increases. Even when only the moment of inertia J 21  of the first intermediate member  12  is changed, as illustrated in  FIG. 13 , the equivalent stiffness k q  is kept substantially constant. Although illustration is omitted, also when only the moment of inertia J 22  of the second intermediate member  14  is changed while the combined spring rates k 1 , k 2 , k 3 , k 4 , and k 5  and the moment of inertia J 21  of the first intermediate member  12  of the damper apparatus  10  are set to constant values, similar results are obtained as in the case where only the moment of inertia J 21  of the first intermediate member  12  is changed. 
     As can be understood from the analysis results described above, the natural frequency f 21  on the low speed rotation side (see Expression (13)) and the frequency fa at the anti-resonance point A (see Expression (6)) can further be reduced by reducing the stiffness of the intermediate spring SPm (reducing the spring rate k m  and the combined spring rate k 5 ). The difference between the natural frequency f 21  on the low speed rotation side and the frequency fa at the anti-resonance point A (fa−f 21 ) can further be increased by conversely increasing the stiffness of the intermediate spring SPm (increasing the spring rate k m  and the combined spring rate k 5 ). Even when the stiffness of the intermediate spring SPm is reduced (the spring rate k m  and the combined spring rate k 5  are reduced), the equivalent stiffness k eq  does not decrease significantly. Thus, in the damper apparatus  10 , the natural frequency f 21  on the low speed rotation side and the frequency fa at the anti-resonance point A can appropriately be set by adjusting the stiffness of the intermediate spring SPm (spring rate k m  and combined spring rate k 5 ) while the equivalent stiffness k eq  is appropriately kept in accordance with the maximum torque input to the driving member  11  and an increase in the weights, that is, the moments of inertia J 21  and J 22  of the first and second intermediate members  12  and  14  is suppressed. Further, the natural frequency f 21  on the low speed rotation side and the frequency fa at the anti-resonance point A can further be reduced by reducing the stiffnesses of the first and second inner springs SP 11  and SP 12  (reducing the spring rates k 11  and k 12  and the combined spring rates K 1  and K 2 ). Still further, the frequency fa at the anti-resonance point A can further be reduced by increasing the stiffnesses of the first and second outer springs SP 21  and SP 22  (increasing the spring rates k 21  and k 22  and the combined spring rates K 3  and K 4 ). 
     In the vehicle on which the engine (internal combustion engine) EG is mounted as a source of traveling power, the power transfer efficiency between the engine EG and the transmission TM is improved by further reducing a lock-up rotation speed Nlup and mechanically transferring the torque from the engine EG to the transmission TM at an early stage. Thus, the fuel efficiency of the engine EG can further be improved. In a low rotation speed range that is about 500 rpm to 1500 rpm and may be a range in which the lock-up rotation speed Nlup is set, the vibration transferred from the engine EG to the driving member  11  via the lock-up clutch increases. In particular, the vibration level remarkably increases in a vehicle on which an engine is mounted with a small number of cylinders, such as a three-cylinder or four-cylinder engine. In order to prevent a significant vibration from being transferred to the transmission TM or the like during or immediately after the execution of the lock-up, it is necessary to further reduce, in a rotation speed range in the vicinity of the lock-up rotation speed Nlup, the vibration level of the entire damper apparatus  10  (driven member  16 ) that transfers the torque (vibration) from the engine EG to the transmission TM in a state in which the lock-up is executed. 
     In view of the above, the inventors have structured the damper apparatus  10  such that the anti-resonance point A described above is formed when the rotation speed of the engine EG falls within the range of 500 rpm to 1500 rpm (expected range in which the lock-up rotation speed Nlup is set) based on the lock-up rotation speed Nlup defined for the lock-up clutch  8 . A rotation speed Nea of the engine EG corresponding to the frequency fa at the anti-resonance point A is represented by Nea=(120/n)·fa, where “n” represents the number of cylinders of the engine (internal combustion engine) EG. Thus, in the damper apparatus  10 , the combined spring rate k 1  of the plurality of first inner springs SP 11 , the combined spring rate k 2  of the plurality of second inner springs SP 12 , the combined spring rate k 3  of the plurality of first outer springs SP 21 , the combined spring rate k 4  of the plurality of second outer springs SP 22 , the combined spring rate k 5  of the plurality of intermediate springs SPm, the moment of inertia J 21  of the first intermediate member  12  (the moment of inertia of the turbine runner  5  or the like that is coupled so as to rotate together is taken into consideration (added as a sum); the same applies hereinafter), and the moment of inertia J 22  of the second intermediate member  14  are selected and set so as to satisfy Expression (19) below. That is, in the damper apparatus  10 , the spring rates k 11 , k 12 , k 21 , k 22 , and k m  of the springs SP 11  to SPm and the moments of inertia J 21  and J 22  of the first and second intermediate members  12  and  14  are selected and set based on the frequency fa at the anti-resonance point A (and the lock-up rotation speed Nlup). 
     
       
         
           
             
               [ 
               
                 Math 
                 . 
                 
                     
                 
                  
                 9 
               
               ] 
             
              
             
                 
             
           
         
       
       
         
           
             
               
                 
                   
                     500 
                      
                     
                         
                     
                      
                     rpm 
                   
                   ≤ 
                   
                     
                       120 
                       n 
                     
                      
                     fa 
                   
                   ≤ 
                   
                     1500 
                      
                     
                         
                     
                      
                     rpm 
                   
                 
               
               
                 
                   ( 
                   19 
                   ) 
                 
               
             
           
         
       
     
     As described above, the anti-resonance point A at which the vibration amplitude Θ 3  of the driven member  16  can theoretically be set to zero (the vibration can further be reduced) is set within the low rotation speed range of 500 rpm to 1500 rpm (expected range in which the lock-up rotation speed Nlup is set). Thus, as illustrated in  FIG. 7 , the resonance that generates the anti-resonance point A (resonance that needs to be caused in order to form the anti-resonance point A, in this embodiment, the resonance of the first intermediate member  12 ; see the resonance point R 1  in  FIG. 7 ) can be shifted to a lower speed rotation side (lower frequency side) so as to be included in a non-lock-up range of the lock-up clutch  8  (see a long dashed double-short dashed line in  FIG. 7 ). That is, in this embodiment, the resonance of the first intermediate member  12  (resonance at the smaller one of the two natural frequencies) is imaginary resonance that does not occur in the rotation speed range in which the damper apparatus  10  is used. As illustrated in  FIG. 7 , the rotation speed corresponding to the smaller one of the two natural frequencies of the damper apparatus  10  (natural frequency of the first intermediate member  12 ) is lower than the lock-up rotation speed Nlup of the lock-up clutch  8 , and the rotation speed corresponding to a larger one of the two natural frequencies of the damper apparatus  10  (natural frequency of the second intermediate member  14 ) is higher than the lock-up rotation speed Nlup. Thus, one of the vibration transferred from the second inner springs SP 12  to the driven member  16  and the vibration transferred from the second outer springs SP 22  to the driven member  16  can cancel out at least a part of the other from the time when the lock-up is executed by the lock-up clutch  8 . 
     When the damper apparatus  10  is structured so as to satisfy Expression (19) above, the spring rates k 11 , k 12 , k 21 , k 22 , and k m  and the moments of inertia J 21  and J 22  are preferably selected and set so that the frequency of the resonance that generates the anti-resonance point A (see the resonance point R 1  in  FIG. 7 ) is smaller than the frequency fa at the anti-resonance point A and is as smaller as possible. Therefore, in the damper apparatus  10  of this embodiment, the values of the spring rates k 11 , k 12 , k 21 , k 22 , and k m  are defined so as to satisfy the relationship of k 11 &lt;k m &lt;k 12 &lt;k 22 &lt;k 21  described above. 
     That is, in the damper apparatus  10 , the spring rate k m  of the intermediate spring SPm and the spring rates k 11  and k 12  of the first and second inner springs SP 11  and SP 12  are defined as small values so that the natural frequency f 21  on the low speed rotation side and the frequency fa at the anti-resonance point A are smaller. Further, the spring rates k 21  and k 22  of the first and second outer springs SP 21  and SP 22  are defined as large values so that the natural frequency f 21  on the low speed rotation side is smaller. Thus, the natural frequency f 21  on the low speed rotation side and the frequency fa at the anti-resonance point A are smaller. Accordingly, the start point of a rotation speed band (frequency band) in which one of the vibration transferred from the second inner springs SP 12  to the driven member  16  and the vibration transferred from the second outer springs SP 22  to the driven member  16  cancels out at least a part of the other can be set to a lower speed rotation side (lower frequency side). By setting the start point of the rotation speed band to the low speed rotation side, the rotation speed (frequency) at which the phase of the vibration transferred from the second inner springs SP 12  to the driven member  16  deviates by 180 degrees from the phase of the vibration transferred from the second outer springs SP 22  to the driven member  16  can also be set to the low speed rotation side. As a result, the lock-up is permitted at an even lower rotation speed, and the vibration damping performance in the low rotation speed range can further be improved. 
     In the damper apparatus  10 , as illustrated in  FIG. 7 , when the rotation speed of the engine EG further increases after a vibration damping peak of the driven member  16  occurs in the vicinity of the anti-resonance point A, resonance occurs at the larger one of the two natural frequencies (in this embodiment, the resonance of the second intermediate member  14 ; see the resonance point R 2  in  FIG. 7 ). Therefore, the phase of the vibration transferred from the second inner springs SP 12  to the driven member  16  is equal to the phase of the vibration transferred from the second outer springs SP 22  to the driven member  16 . That is, in the damper apparatus  10  of this embodiment, during a period from the time when the resonance occurs at the smaller one of the two natural frequencies described above (resonance of the first intermediate member  12 ) to the time when the resonance occurs at the larger one of the two natural frequencies (resonance of the second intermediate member  14 ), one of the vibration transferred from the second inner springs SP 12  to the driven member  16  and the vibration transferred from the second outer springs SP 22  to the driven member  16  cancels out at least a part of the other. Thus, the spring rates (combined spring rates) k 1 , k 2 , k 3 , k 4 , and k 5  and the moments of inertia J 21  and J 22  are preferably selected and set so that the frequency of the resonance that occurs on the higher speed rotation side (higher frequency side) than the anti-resonance point A is larger. Thus, the resonance (resonance point R 2 ) can be caused on a high rotation speed range side on which the vibration is difficult to be conspicuous. Accordingly, the vibration damping performance of the damper apparatus  10  in the low rotation speed range can further be improved. 
     In order to further improve the vibration damping performance of the damper apparatus  10  in the vicinity of the lock-up rotation speed Nlup, it is necessary to separate the lock-up rotation speed Nlup from the rotation speed of the engine EG corresponding to the resonance point R 2  to the extent possible. Therefore, when the damper apparatus  10  is structured so as to satisfy Expression (19), the spring rates k 1 , k 2 , k 3 , k 4 , and k 5  and the moments of inertia J 21  and J 22  are preferably selected and set so as to satisfy Nlup≤(120/n)·fa (=Nea). Thus, the lock-up can be executed by the lock-up clutch  8  while the transfer of the vibration to the input shaft IS of the transmission TM is satisfactorily suppressed. In addition, the vibration from the engine EG can excellently be damped by the damper apparatus  10  immediately after the execution of the lock-up. 
     By designing the damper apparatus  10  based on the frequency fa at the anti-resonance point A as described above, the vibration damping performance of the damper apparatus  10  can excellently be improved. The researches and analyses conducted by the inventors demonstrate that, when the lock-up rotation speed Nlup is defined as a value of, for example, around 1000 rpm, excellent results are obtained in practical use by structuring the damper apparatus  10  so as to satisfy, for example, 900 rpm≤(120/n)·fa≤1200 rpm. 
     As can be understood from Expressions (13) and (14), the two natural frequencies f 21  and f 22  of the damper apparatus  10  are influenced by the moments of inertia J 21  and J 22  of both of the first and second intermediate members  12  and  14 . That is, in the damper apparatus  10 , the first intermediate member  12  and the second intermediate member  14  are coupled to each other via the intermediate springs SPm. Therefore, forces from the intermediate springs SPm (see outline arrows in  FIG. 6 ) act on both of the first and second intermediate members  12  and  14 . Thus, the vibration of the first intermediate member  12  and the vibration of the second intermediate member  14  interact with each other (both the vibrations influence each other). Through the interaction between the vibration of the first intermediate member  12  and the vibration of the second intermediate member  14 , the natural frequencies f 21  and f 22  are influenced by the moments of inertia J 21  and J 22  of both of the first and second intermediate members  12  and  14 . Thus, in the damper apparatus  10 , the natural frequencies f 21  and f 22  and the frequency fa at the anti-resonance point A can be set while the increase in the weights, that is, the moments of inertia J 21  and J 22  of the first and second intermediate members  12  and  14  is suppressed so that the resonance at the smaller one of the two natural frequencies f 21  and f 22  is easily shifted to the low speed rotation side, that is, the non-lock-up range and the vibrations are canceled out more satisfactorily in the driven member  16  in a state in which the rotation speed of the driving member  11  is lower. 
     In the damper apparatus  10 , the two natural frequencies f 21  and f 22  are influenced by the moments of inertia J 21  and J 22  of both of the first and second intermediate members  12  and  14 . By adjusting the moments of inertia J 21  and J 22  of the first and second intermediate members  12  and  14 , the natural frequency f 21  on the low speed rotation side (resonance point R 1 ) can easily be shifted to a lower speed rotation side of the non-lock-up range as compared to the damper apparatus of the comparative example described above while the frequency fa at the anti-resonance point A is set approximately equal to a frequency fa′ at an anti-resonance point of the damper apparatus of the comparative example as illustrated in  FIG. 7 . Thus, in the damper apparatus  10 , the vibration level in the vicinity of the anti-resonance point A can further be reduced as compared to the damper apparatus of the comparative example (see a dashed line in  FIG. 7 ). By further reducing the natural frequency f 21  on the low speed rotation side to further reduce the vibration level in the vicinity of the anti-resonance point A as described above, the lock-up rotation speed Nlup can be kept lower even when the order of the vibration from the engine EG decreases along with execution of a fewer-cylinder operation of the engine EG having a cylinder halting function. 
     The analyses conducted by the inventors prove that the vibrations transferred from the first, second, and third torque transfer paths P 1 , P 2 , and P 3  described above to the driven member  16  are easily canceled out by coupling the first and second intermediate members  12  and  14  to each other via the intermediate springs SPm and causing both the vibrations to interact with each other, whereby an actual vibration amplitude of the driven member  16  in the vicinity of the anti-resonance point A can further be reduced and a difference in the torque amplitude (torque fluctuation) between the second inner springs SP 12  and the second outer springs SP 22  can be reduced (both the torque amplitudes can be made closer). Thus, in the damper apparatus  10 , the lock-up (coupling between the engine EG and the driving member  11 ) can be permitted at a lower rotation speed, and the vibration damping performance in the low rotation speed range in which the vibration from the engine EG is likely to increase can further be improved. 
     When k 5 =0 is set in Expression (13) above, a natural frequency f 21 ′ of a first intermediate member of the damper apparatus of the comparative example from which the intermediate springs SPm are omitted is represented by Expression (20) below. When k 5 =0 is set in Expression (14) above, a natural frequency f 22 ′ of a second intermediate member of the damper apparatus of the comparative example is represented by Expression (21) below. As can be understood from Expressions (20) and (21), in the damper apparatus of the comparative example, the natural frequency f 21 ′ of the first intermediate member is not influenced by the moment of inertia J 22  of the second intermediate member, and the natural frequency f 22 ′ of the second intermediate member is not influenced by the moment of inertia J 21  of the first intermediate member. In this regard, it is understood that, in the damper apparatus  10 , the degrees of freedom in terms of setting of the natural frequencies f 21  and f 22  of the first and second intermediate members  12  and  14  can be improved as compared to the damper apparatus of the comparative example. 
     
       
         
           
             
               [ 
               
                 Math 
                 . 
                 
                     
                 
                  
                 10 
               
               ] 
             
              
             
                 
             
           
         
       
       
         
           
             
               
                 
                   
                     f 
                     21 
                     ′ 
                   
                   = 
                   
                     
                       1 
                       
                         2 
                          
                         π 
                       
                     
                      
                     
                       
                         
                           
                             k 
                             1 
                           
                           + 
                           
                             k 
                             2 
                           
                         
                         
                           J 
                           21 
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   20 
                   ) 
                 
               
             
             
               
                 
                   
                     f 
                     22 
                     ′ 
                   
                   = 
                   
                     
                       1 
                       
                         2 
                          
                         π 
                       
                     
                      
                     
                       
                         
                           
                             k 
                             1 
                           
                           + 
                           
                             k 
                             2 
                           
                         
                         
                           J 
                           22 
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   21 
                   ) 
                 
               
             
           
         
       
     
     When k 5 =0 is set in Expression (6) above, the frequency fa′ at the anti-resonance point of the damper apparatus of the comparative example is represented by Expression (22) below. Comparing Expression (6) and Expression (22), when the spring rates k 1 , k 2 , k 3 , and k 4  and the moments of inertia J 21  and J 22  are the same, the frequency fa′ at the anti-resonance point of the damper apparatus of the comparative example is smaller than the frequency fa at the anti-resonance point A of the damper apparatus  10 . In the damper apparatus  10 , the frequency fa can easily be set approximately equal to the frequency fa′ at the anti-resonance point of the damper apparatus of the comparative example (see the dashed line in  FIG. 7 ) by mainly selecting the moments of inertia J 21  and J 22  of the first and second intermediate members  12  and  14  as appropriate. 
     
       
         
           
             
               [ 
               
                 Math 
                 . 
                 
                     
                 
                  
                 11 
               
               ] 
             
              
             
                 
             
           
         
       
       
         
           
             
               
                 
                   
                     fa 
                     ′ 
                   
                   = 
                   
                     
                       1 
                       
                         2 
                          
                         π 
                       
                     
                      
                     
                       
                         
                           
                             
                               k 
                               1 
                             
                              
                             
                               k 
                               2 
                             
                              
                             
                               k 
                               3 
                             
                           
                           + 
                           
                             
                               k 
                               1 
                             
                              
                             
                               k 
                               2 
                             
                              
                             
                               k 
                               4 
                             
                           
                           + 
                           
                             
                               k 
                               2 
                             
                              
                             
                               k 
                               3 
                             
                              
                             
                               k 
                               4 
                             
                           
                         
                         
                           
                             
                               J 
                               21 
                             
                              
                             
                               k 
                               3 
                             
                              
                             
                               k 
                               4 
                             
                           
                           + 
                           
                             
                               J 
                               22 
                             
                              
                             
                               k 
                               1 
                             
                              
                             
                               k 
                               2 
                             
                           
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   22 
                   ) 
                 
               
             
           
         
       
     
     In the damper apparatus  10  described above, the first and second outer springs SP 21  and SP 22  corresponding to the second intermediate member  14  having a natural frequency larger than that of the first intermediate member  12  are disposed on the radially outer side of the first and second inner springs SP 11  and SP 12  corresponding to the first intermediate member  12 . That is, the average attachment radius ro of the first and second outer springs SP 21  and SP 22  is larger than the average attachment radius ri of the first and second inner springs SP 11  and SP 12  corresponding to the first intermediate member  12 . Thus, the torsion angles (strokes) of the first and second outer springs SP 21  and SP 22  having high stiffnesses can further be increased. Accordingly, the stiffnesses of the first and second outer springs SP 21  and SP 22  can be reduced while the transfer of a large torque to the driving member  11  is permitted. As a result, the equivalent stiffness k eq  of the damper apparatus  10  can further be reduced, and the resonance of the entire vibration system including the damper apparatus  10 , that is, the resonance caused by the vibration between the entire damper apparatus  10  and the drive shafts of the vehicle (resonance caused by the vibration that occurs between the driving member and the drive shafts) can be shifted to a lower speed rotation side (lower frequency side). Thus, in the damper apparatus  10 , the vibration damping performance can excellently be improved such that the frequency at the anti-resonance point A described above is made closer to the frequency of the resonance of the entire vibration system. 
     In the damper apparatus  10  of the starting apparatus  1 , the first and second outer springs SP 21  and SP 22  (third and fourth elastic bodies) are disposed on the outer side of the first and second inner springs SP 11  and SP 12  (first and second elastic bodies) in the radial direction of the damper apparatus  10 . The intermediate springs SPm are disposed on the radially outer side of the first and second inner springs SP 11  and SP 12  with distances from the first and second outer springs SP 21  and SP 22  in the axial direction (so as to be close to the turbine runner  5 ). That is, when the starting apparatus  1  is cut along a plane including the central axis CA, the first and second inner springs SP 11  and SP 12 , the first and second outer springs SP 21  and SP 22 , and the intermediate springs SPm are included in a region having a shape of a triangle (inverted triangle) defined between the turbine runner  5  and the lock-up clutch  8  (first and second friction engagement plates  83  and  84  serving as the friction engagement portions) in the axial direction such that a vertex opposite to the shortest side is located on the central axis CA side. More specifically, as illustrated in  FIG. 2 , the first and second outer springs SP 21  and SP 22  are disposed in the vicinity of one vertex on the shortest side of the triangle, the intermediate springs SPm are disposed in the vicinity of the other vertex on the shortest side, and the first and second inner springs SP 11  and SP 12  are disposed in the vicinity of the vertex opposite to the shortest side. 
     Thus, it is possible to increase the degrees of freedom in terms of setting of the stiffnesses, the numbers of arrangement, the torsion angles (strokes), and the like of the first and second inner springs SP 11  and SP 12 , the first and second outer springs SP 21  and SP 22 , and the intermediate springs SPm, and to suppress the increase in the size of the starting apparatus  1  along with the arrangement of the intermediate springs SPm by effectively using the space in the starting apparatus  1 . As a result, the vibration damping performance of the damper apparatus  10  can further be improved by easily and appropriately setting the two natural frequencies described above (natural frequencies f 21  and f 22  of the first and second intermediate members  12  and  14 ) while the increase in the size of the starting apparatus  1  is suppressed. In the damper apparatus  10 , the intermediate springs SPm may be disposed between the first and second outer springs SP 21  and SP 22  and the first and second inner springs SP 11  and SP 12  in the radial direction of the damper apparatus  10 . In this case, the intermediate springs SPm may be disposed so as to partially overlap at least one of the first and second outer springs SP 21  and SP 22  and at least one of the first and second inner springs SP 11  and SP 12  in the axial direction as viewed in the radial direction. 
     In the damper apparatus  10 , the first intermediate member  12  includes the first and second plate members  121  and  122  that are two members coupled to each other. The coupling abutment portions  122   c  (first abutment portions) each abutting against the ends of the first and second inner springs SP 11  and SP 12  between the first and second inner springs SP 11  and SP 12  and the outer abutment portions  122   d  (second abutment portions) each abutting against the end of the intermediate spring SPm are both formed on the second plate member  122  that is one of the two members. Similarly, the second intermediate member  14  of the damper apparatus  10  includes the first and second annular members  141  and  142  that are two members coupled to each other. The first spring abutment portions  141   c  (first abutment portions) each abutting against the ends of the first and second outer springs SP 21  and SP 22  between the first and second outer springs SP 21  and SP 22  and the second spring abutment portions  141   d  (second abutment portions) each abutting against the end of the intermediate spring SPm are both formed on the first annular member  141  that is one of the two members. Thus, it is possible to reduce the shear forces acting on the fitting portions of the spring abutment portions  121   c  of the first plate member  121  and the coupling abutment portions  122   c  of the second plate member  122  and on the coupling portions of the first and second annular members  141  and  142 . Accordingly, it is possible to further improve the durability of the fitting portions and the coupling portions and furthermore the durability of the first and second intermediate members  12  and  14 . 
     In the damper apparatus  10 , both of the spring abutment portion  121   c  of the first plate member  121  that extends in the radial direction of the damper apparatus  10  and the coupling abutment portion  122   c  of the second plate member  122  that extends in the axial direction of the damper apparatus  10  abut against the ends of the first and second inner springs SP 11  and SP 12  between the first and second inner springs SP 11  and SP 12 . Thus, the first and second inner springs SP 11  and SP 12  can appropriately be pressed by the first intermediate member  12  so as to extend and contract along the axis center. Accordingly, the vibration damping performance of the damper apparatus  10  can further be improved. By causing both of the spring abutment portion  121   c  and the coupling abutment portion  122   c  to abut against the ends of the first and second inner springs SP 11  and SP 12  between the first and second inner springs SP 11  and SP 12 , the spring abutment portion  121   c  and the coupling abutment portion  122   c  are supported from both sides by the first and second inner springs SP 11  and SP 12 . Therefore, there is no need to tighten the fitting of the first plate member  121  and the second plate member  122 . Thus, the coupling abutment portion  122   c  can easily be fitted to the spring abutment portion  121   c . Accordingly, the assembling workability of the damper apparatus  10  can be secured satisfactorily. 
     In the damper apparatus  10 , the moment of inertia J 21  of the first intermediate member  12  (first and second plate members  121  and  122 ) is set larger than the moment of inertia J 22  of the second intermediate member  14 . Thus, the natural frequency f 21  on the low frequency side is further reduced. Accordingly, the resonance point of the first intermediate member  12  can be set to a lower speed rotation side (lower frequency side). In addition, the first intermediate member  12  is coupled to the turbine runner  5  so as to rotate together with the turbine runner  5 . Thus, a substantial moment of inertia of the first intermediate member  12  (sum of the moments of inertia of the first and second plate members  121  and  122 , the turbine runner  5 , the turbine hub  52 , and the like) can further be increased. Instead of coupling the turbine runner  5  to the first intermediate member  12 , that is, the second plate member  122 , a weight (dedicated weight) other than the turbine runner may be coupled. 
     In the damper apparatus  10  described above, the natural frequency of the second intermediate member  14  corresponding to the first and second outer springs SP 21  and SP 22  disposed on the radially outer side of the first and second inner springs SP 11  and SP 12  may be set smaller than the natural frequency of the first intermediate member  12 . That is, the natural frequency of the second intermediate member  14  may be defined based on Expression (13) above, and the natural frequency of the first intermediate member  12  may be defined based on Expression (14) above. In this case, it is appropriate that the smaller one of the spring rates k 21  and k 22  of the first and second outer springs SP 21  and SP 22  be set smaller than the smaller one of the spring rates k 11  and k 12  of the first and second inner springs SP 11  and SP 12 . That is, in this case, it is appropriate that the spring rates k 11 , k 12 , k 21 , and k 22  be selected so as to satisfy relationships of k 21 ≠k 11  and k 21 /k 11 ≠k 22 /k 12 . More specifically, it is appropriate that the spring rates k 11 , k 12 , k 21 , k 22 , and k m  be selected so as to satisfy relationships of k 21 /k 11 &lt;k 22 /k 12  and k 21 &lt;k m &lt;k&lt;k 22 &lt;k 12 &lt;k 11 . 
     In the damper apparatus  10  structured as described above, the average attachment radius ro of the first and second outer springs SP 21  and SP 22  corresponding to the second intermediate member  14  having a natural frequency smaller than that of the first intermediate member  12  is larger than the average attachment radius ri of the first and second inner springs SP 11  and SP 12 . Thus, it is possible to further increase the moment of inertia J 22  of the second intermediate member  14 , and to further reduce the stiffnesses of the first and second outer springs SP 21  and SP 22 . In this case, the first and second outer springs SP 21  and SP 22  having low stiffnesses and relatively small weights are disposed on the outer peripheral side of the damper apparatus  10 , and the first and second inner springs SP 11  and SP 12  having high stiffnesses and relatively large weights are disposed on the central axis CA side of the damper apparatus  10 . Thus, the hysteresis of the first and second outer springs SP 21  and SP 22  on the outer peripheral side can be reduced by reducing the weights of the first and second outer springs SP 21  and SP 22  along with the reduction in the stiffnesses. Further, the hysteresis of the first and second inner springs SP 11  and SP 12  on the inner peripheral side can be reduced by reducing a centrifugal force acting on the first and second inner springs SP 11  and SP 12 . Thus, in the damper apparatus  10 , the frictional force generated between the springs SP 11 , SP 12 , SP 21 , and SP 22  and the respective rotational elements due to the centrifugal force is reduced. Accordingly, the hysteresis of the entire damper apparatus  10  can further be reduced. As a result, in the damper apparatus  10 , the vibration damping performance can excellently be improved such that the frequency at the anti-resonance point A described above is made closer to the frequency of the vibration (resonance) to be damped. 
     In the damper apparatus  10  described above, the spring rate K 21  of the first outer spring SP 21  is larger than the spring rate K 22  of the second outer spring SP 22  (k 22 &lt;k 21 ). The present disclosure is not limited to this case. That is, in order to facilitate the designing of the damper apparatus  10 , specifications such as the spring rate K 21 , the coil diameter, and the axial length of the first outer spring SP 21  may be set identical to specifications such as the spring rate K 22 , the coil diameter, and the axial length of the second outer spring SP 22  (k 21 =k 22 ). Similarly, specifications such as the spring rate K 11 , the coil diameter, and the axial length of the first inner spring SP 11  may be set identical to specifications such as the spring rate K 12 , the coil diameter, and the axial length of the second inner spring SP 12  (k 11 =k 12 ). When the natural frequency of the second intermediate member  14  is smaller than the natural frequency of the first intermediate member  12 , the spring rates k 11 , k 12 , k 21 , and k 22  may be selected so as to satisfy a relationship of k 21 &lt;k 22 &lt;k 12 =k 11 . 
     In the damper apparatus  10 , the spring rate k m  of the intermediate spring SPm may be defined as being smaller than the spring rates k 11 , k 12 , k 21 , and k 22  of the first and second inner springs SP 11  and SP 12  and the first and second outer springs SP 21  and SP 22 . That is, as described above, the natural frequency f 21  on the low speed rotation side (low frequency side) and the frequency fa at the anti-resonance point A decrease as the combined spring rate k 5  of the intermediate springs SPm decreases (see  FIG. 12 ). Thus, when the spring rate (stiffness) k m  of the intermediate spring SPm is set smaller than the spring rates k 11 , k 12 , k 21 , and k 22 , the natural frequency f 21  and the frequency fa can further be reduced. Even when this structure is employed, the start point of the rotation speed band in which one of the vibration transferred from the second inner springs SP 12  to the driven member  16  and the vibration transferred from the second outer springs SP 22  to the driven member  16  cancels out at least a part of the other can be set to a lower speed rotation side. By setting the start point of the rotation speed band to the low speed rotation side, the rotation speed (frequency) at which the phase of the vibration transferred from the second inner springs SP 12  to the driven member  16  deviates by 180 degrees from the phase of the vibration transferred from the second outer springs SP 22  to the driven member  16  can also be set to the low speed rotation side (low frequency side). In this case, it is appropriate that the spring rates k 11 , k 12 , k 21 , and k 22  of the first and second inner springs SP 11  and SP 12  and the first and second outer springs SP 21  and SP 22  satisfy at least the relationships of k 11 ≠k 21  and k 11 /k 21 ≠k 12 /k 22 . 
     In the damper apparatus  10 , the spring rate k m  of the intermediate spring SPm may be defined as being larger than the spring rates k 11 , k 12 , k 2t , and k 22  of the first and second inner springs SP 11  and SP 12  and the first and second outer springs SP 21  and SP 22 . That is, as described above, the difference between the natural frequency f 21  on the low speed rotation side (low frequency side) and the frequency fa at the anti-resonance point A (fa−f 21 ) increases as the combined spring rate k 5  of the intermediate springs SPm increases (see  FIG. 12 ). Thus, when the spring rate (stiffness) k m  of the intermediate spring SPm is set larger than the spring rates k 11 , k 12 , k 21 , and k 22 , the difference between the natural frequency f 21  and the frequency fa (fa−f 21 ) is increased. Accordingly, it is possible to further extend the rotation speed band in which one of the vibration transferred from the second inner springs SP 12  to the driven member  16  and the vibration transferred from the second outer springs SP 22  to the driven member  16  cancels out at least a part of the other, that is, the range in which the vibration level of the driven member  16  can be reduced satisfactorily. 
     In this case, it is appropriate that the spring rates k 11 , k 12 , k 21 , and k 22  of the first and second inner springs SP 11  and SP 12  and the first and second outer springs SP 21  and SP 22  be adjusted so that the natural frequency f 21  and the frequency fa at the anti-resonance point A are further reduced and the difference therebetween (fa−f 21 ) is further increased. It is advantageous that this structure be applied to a damper apparatus in which the maximum torque input to the driving member  11  is relatively small and the required equivalent stiffness k eq  is relatively low from the viewpoint of ease of numerical value setting for the spring rates k 11 , k 12 , k 21 , and k 22  that is made in order to further reduce the natural frequency f 21  and the frequency fa at the anti-resonance point A. In this case as well, it is appropriate that the spring rates k 11 , k 12 , k 21 , and k 22  of the first and second inner springs SP 11  and SP 12  and the first and second outer springs SP 21  and SP 22  satisfy at least the relationships of k 11 ≠k 21  and k 11 /k 21 ≠k 2 /k 22 . 
     When the damper apparatus  10  includes an even number of intermediate springs SPm, two intermediate springs SPm may be supported from both sides in the circumferential direction by a pair of abutment portions provided on one of the first and second intermediate members  12  and  14 , and an abutment portion provided on the other one of the first and second intermediate members  12  and  14  may abut against the ends of the two intermediate springs SPm between the two intermediate springs SPm. 
     In addition to the first, second, and third torque transfer paths P 1 , P 2 , and P 3 , the damper apparatus  10  may further include, for example, at least one torque transfer path provided in parallel to the first and second torque transfer paths P 1  and P 2 . Further, at least one set of an intermediate member and springs (elastic bodies) may be added to, for example, at least one of the first and second torque transfer paths P 1  and P 2  of the damper apparatus  10 . 
     In the starting apparatus  1 , when slip control is executed so that an actual slip speed (actual rotation speed difference) between the engine EG and the input shaft of the transmission TM (driving member  11 ) is caused to coincide with a target slip speed, the frequency fa at the anti-resonance point A described above may be caused to coincide with a frequency fs of a shudder that occurs when the slip control is executed, or may be set to a value in the vicinity of the frequency fs of the shudder. Thus, it is possible to further reduce the shudder that occurs when the slip control is executed. When “J pd ” represents a moment of inertia of the lock-up piston  80  and the driving member  11  that rotate together, the frequency fs of the shudder can be represented by fs=½π·√(k eq /J pd ) by using the moment of inertia J pd  and the equivalent stiffness k eq  of the damper apparatus  10 . 
     Unillustrated spring seats may be attached to the ends of the springs SP 11  to SPm described above. That is, the “abutment portion (spring abutment portion)” of the damper apparatus  10  may be a portion abutting against the spring seat that is substantially a part of the springs SP 11  to SPm. The “abutment portion” of the damper apparatus  10  may also be a “torque transfer portion” configured to exchange a torque with a corresponding spring (elastic body) (the same applies hereinafter). 
       FIG. 14  is a sectional view illustrating a starting apparatus  1 B including another damper apparatus  10 B disclosed herein. The same components of the starting apparatus  1 B and the damper apparatus  10 B as the components of the starting apparatus  1  and the damper apparatus  10  described above are represented by the same reference symbols to omit redundant description. 
     The starting apparatus  1 B illustrated in  FIG. 14  includes a lock-up clutch  8 B structured as a single-plate hydraulic clutch. The lock-up clutch  8 B includes a lock-up piston  80 B disposed inside the front cover  3  and in the vicinity of the inner wall surface of the front cover  3  on the engine EG side and fitted to the damper hub  7  in a freely rotatable and axially movable manner. A friction material  88  is attached to a surface of the lock-up piston  80 B on the outer peripheral side and on the front cover  3  side. A lock-up chamber  89  connected to the unillustrated hydraulic controller via a hydraulic oil supply passage and an oil passage formed in the input shaft IS is defined between the lock-up piston  80 B and the front cover  3 . In the starting apparatus  1 B, the hydraulic pressure in the fluid chamber  9  is set higher than the hydraulic pressure in the lock-up chamber  89  by the unillustrated hydraulic controller to engage the lock-up clutch  8 B. Thus, the front cover  3  and the damper hub  7  can be coupled to each other via the damper apparatus  10 . Further, the hydraulic pressure in the lock-up chamber  89  is set higher than the hydraulic pressure in the fluid chamber  9  by the unillustrated hydraulic controller to release the lock-up clutch  8 B. Thus, the front cover  3  and the damper hub  7  can be decoupled from each other. 
     As illustrated in  FIG. 14 , a driving member  11 B of the damper apparatus  10 B includes the lock-up piston  80 B (first input member) of the lock-up clutch  8 B to which the torque from the engine EG is transferred, and an annular input plate  111 B (second input member) coupled to the lock-up piston  80 B via a plurality of rivets. Thus, the front cover  3  (engine EG) and the driving member  11 B of the damper apparatus  10 B are coupled to each other through the engagement of the lock-up clutch  8 B. 
     The lock-up piston  80 B includes a spring support portion  80   a  formed on an outer peripheral portion, and a plurality of (for example, three in this embodiment) unillustrated spring abutment portions (elastic body abutment portions). As in the illustration, the spring support portion  80   a  is disposed in the outer peripheral region of the fluid chamber  9 , and supports (guides) the outer portions of the plurality of first and second outer springs SP 21  and SP 22  in the radial direction, front-cover-3-side (engine side) lateral portions (lateral portions on the left side in  FIG. 4 ) thereof, and outer sides (shoulder portions) of turbine-runner-5-side (transmission side) lateral portions thereof in the radial direction. The input plate  111 B is a plate-shaped annular member including the plurality of (for example, three in this embodiment) spring support portions  111   a , the plurality of (for example, three in this embodiment) outer spring abutment portions (elastic body abutment portions)  111   co , and the plurality of (for example, three in this embodiment) inner spring abutment portions (elastic body abutment portions)  111   ci . As in the illustration, the input plate  111 B further includes an annular coupling portion formed so as to protrude toward the lock-up piston  80 B. A plurality of rivets are inserted through the coupling portion. 
     The first intermediate member  12  of the damper apparatus  10 B is basically structured similarly to the first intermediate member  12  of the damper apparatus  10  described above, and includes the first plate member  121  and the second plate member  122 . The second intermediate member  14  of the damper apparatus  10 B is basically structured similarly to the second intermediate member  14  of the damper apparatus  10  described above, and includes the first and second annular members  141  and  142 . The second intermediate member  14  of the damper apparatus  10 B also has a moment of inertia smaller than that of the first intermediate member  12 . The driven member  16  of the damper apparatus  10 B is basically structured similarly to the driven member  16  of the damper apparatus  10  described above, and includes the first and second output plates  161  and  162 . As in the illustration, the driven member  16  of the damper apparatus  10 B is formed so as not to interfere with the plurality of rivets that couple the lock-up piston  80 B and the input plate  111 B to each other. 
     Also in the damper apparatus  10 B structured as described above, the first and second inner springs SP 11  and SP 12 , the first and second outer springs SP 21  and SP 22 , and the intermediate springs SPm are basically disposed in the fluid chamber  9  similarly to the damper apparatus  10 . Further, the first and second intermediate members  12  and  14  of the damper apparatus  10 B are basically structured similarly to those of the damper apparatus  10  described above. Thus, the starting apparatus  1 B and the damper apparatus  10 B can also attain actions and effects similar to those of the starting apparatus  1  and the damper apparatus  10 . 
     In the damper apparatus  10 B, the first and second inner springs SP 11  and SP 12  are disposed on an inner side in the radial direction with respect to the friction engagement portion, that is, the friction material  88  of the lock-up clutch  8 B, and the first and second outer springs SP 21  and SP 22  are disposed so as to at least partially overlap the friction material  88  (friction engagement portion) in the radial direction as viewed in the axial direction. Thus, the vibration damping performance of the damper apparatus  10 B can further be improved by increasing the degrees of freedom in terms of setting of the spring rates k 21  and k 22 , the numbers of arrangement, the torsion angles (strokes), and the like of the first and second outer springs SP 21  and SP 22  while the axial length of the damper apparatus  10 B and furthermore the axial length of the starting apparatus  1 B are further reduced. 
       FIG. 15  and  FIG. 16  are a sectional view and a front-side elevation illustrating a starting apparatus  1 X including still another damper apparatus  10 X disclosed herein. The same components of the starting apparatus  1 X and the damper apparatus  10 X as the components of the starting apparatus  1  and the damper apparatus  10  described above are represented by the same reference symbols to omit redundant description. 
     As illustrated in  FIG. 15 , a first intermediate member  12 X of the damper apparatus  10 X includes an annular first plate member (first member)  121 X supported (aligned) by the damper hub  7  in a freely rotatable manner, an annular second plate member (second member)  122 X coupled (fixed) to the turbine runner  5  that is a mass so as to rotate together with the turbine runner  5  and also coupled to the first plate member  121 X, and an annular third plate member (third member)  123 X disposed so as to be closer to the turbine runner  5  than the second plate member  122 X and coupled (fixed) to the second plate member  122 X via a plurality of rivets. Similarly to the first plate member  121  described above, the first plate member  121 X of the first intermediate member  12 X includes the plurality of (for example, three in this embodiment) spring abutment portions  121   c  that protrude radially outward at intervals (at regular intervals) in the circumferential direction. As illustrated in  FIG. 15 , the rectangular or elongated hole-shaped opening portion  121   h  extending through each spring abutment portion  121   c  is formed in the spring abutment portion  121   c.    
     As illustrated in  FIG. 15  and  FIG. 16 , the second plate member  122 X of the first intermediate member  12 X includes a plurality of (for example, three) spring housing windows  122   w  (see  FIG. 16 ) disposed at intervals (at regular intervals) in the circumferential direction, a plurality of (for example, three) spring support portions  122   a  extending along the inner peripheral edges of the respective spring housing windows  122   w , a plurality of (for example, three) spring support portions  122   b  extending along the outer peripheral edges of the respective spring housing windows  122   w , the plurality of (for example, three in this embodiment) coupling abutment portions  122   c , and the plurality of (for example, twice as many as the intermediate springs SPm) outer abutment portions (torque transfer portions)  122   d.    
     As in the illustration, the inner peripheral portion of the second plate member  122 X is fixed to the turbine hub  52  together with the turbine shell  50  of the turbine runner  5 . The coupling abutment portions  122   c  extend, on a radially inner side with respect to the spring support portions  122   a , from a main body of the second plate member  122 X to one side in the axial direction of the damper apparatus  10 X (left side in  FIG. 15 ; front cover  3  side) at intervals (at regular intervals) in the circumferential direction. The tapered protruding portion  122   p  fitted to the opening portion  121   h  of the first plate member  121 X is formed at the distal end of each coupling abutment portion  122   c . The protruding portion  122   p  has a width slightly smaller than the width of the opening portion  121   h  of the first plate member  121 X in the circumferential direction, and also has a thickness sufficiently smaller than the length of the opening portion  121   h  (opening length) of the first plate member  121 X in the radial direction. The spring housing window  122   w  has a circumferential length in accordance with the equilibrium length of the intermediate spring SPm. Every single outer abutment portion  122   d  is provided on each side of the spring housing window  122   w  in the circumferential direction so as to be located on a radially outer side with respect to the coupling abutment portion  122   c.    
     As illustrated in  FIG. 15 , the third plate member  123 X of the first intermediate member  12 X is an annular member having a bore diameter larger than the bore diameter of the second plate member  122 X, and also having an outside diameter larger than the outside diameter of the second plate member  122 X. The third plate member  123 X includes a plurality of (for example, three) spring housing windows  123   w  disposed at intervals (at regular intervals) in the circumferential direction, a plurality of (for example, three) spring support portions  123   a  extending along the inner peripheral edges of the respective spring housing windows  122   w , a plurality of (for example, three) spring support portions  123   b  extending along the outer peripheral edges of the respective spring housing windows  122   w , a plurality of (for example, twice as many as the intermediate springs SPm) spring abutment portions (torque transfer portions)  123   d , and a cylindrical annular extending portion  123   m . As in the illustration, the inner peripheral portion of the third plate member  123 X is fixed to the second plate member  122 X via a plurality of rivets. The spring housing window  123   w  has a circumferential length in accordance with the equilibrium length of the intermediate spring SPm. Every single spring abutment portion  123   d  is provided on each side of the spring housing window  123   w  in the circumferential direction. The annular extending portion  123   m  extends from the outer periphery of the third plate member  123 X to one side in the axial direction of the damper apparatus  10 X (left side in  FIG. 15 ; front cover  3  side). 
     A second intermediate member  14 X is a single annular member having a moment of inertia smaller than that of the first intermediate member  12 X. The second intermediate member  14 X includes a base portion  14   a  substantially having an L-shape in cross section, a plurality of (for example, three in this embodiment) first spring abutment portions (torque transfer portions)  14   c , and a plurality of (for example, twice as many as the intermediate springs SPm) second spring abutment portions (torque transfer portions)  14   d . The plurality of first spring abutment portions  14   c  extend from the base portion  14   a  to one side in the axial direction of the damper apparatus  10 X (left side in  FIG. 15 ; front cover  3  side) at intervals in the circumferential direction. The plurality of second spring abutment portions  14   d  extend from the base portion  14   a  to an inner side in the radial direction of the damper apparatus  10 X toward the axis center of the second intermediate member  14 X, and are arranged in the circumferential direction with a distance in accordance with the equilibrium length of the intermediate spring SPm. The second intermediate member  14 X may include twice as many second spring abutment portions  14   d  as the intermediate springs SPm. In this case, it is appropriate that the plurality of second spring abutment portions  14   d  be formed symmetrically with respect to the axis center of the second intermediate member  14 X so that every two (pair of) second spring abutment portions  14   d  are closer to each other in the circumferential direction with a distance in accordance with, for example, the equilibrium length of the intermediate spring SPm. 
     As illustrated in  FIG. 15 , the first plate member  121 X of the first intermediate member  12 X is disposed between the first and second output plates  161  and  162  so as to be surrounded by the annular portion of the input plate  111 . An assembly of the first intermediate member  12 X, the second intermediate member  14 X, the plurality of intermediate springs SPm, the turbine hub  52 , and the turbine runner  5  is disposed on a lateral side of the second output plate  162 . That is, prior to fixing the second plate member  122 X to the turbine hub  52 , the second and third plate members  122 X and  123 X are coupled (fixed) to each other via the plurality of rivets so that the plurality of intermediate springs SPm and the plurality of second spring abutment portions  14   d  of the second intermediate member  14 X are interposed therebetween. 
     The plurality of spring support portions  122   a  of the second plate member  122 X support (guide), from the inner peripheral side, front-cover-3-side lateral portions of the respective intermediate springs SPm (each spring support portion corresponds to one intermediate spring). The spring support portions  122   b  support (guide), from the outer peripheral side, the front-cover-3-side lateral portions of the respective intermediate springs SPm (each spring support portion corresponds to one intermediate spring). The plurality of spring support portions  123   a  of the third plate member  123 X support (guide), from the inner peripheral side, turbine-runner-5-side lateral portions of the respective intermediate springs SPm (each spring support portion corresponds to one intermediate spring). The plurality of spring support portions  123   b  support (guide), from the outer peripheral side, the turbine-runner-5-side lateral portions of the respective intermediate springs SPm (each spring support portion corresponds to one intermediate spring). As illustrated in  FIG. 15 , the inner peripheral surface of the base portion  14   a  of the second intermediate member  14 X is supported by the outer peripheral surface of the second plate member  122 X, and the second intermediate member  14 X is supported (aligned) by the second plate member  122 X in a freely rotatable manner. 
     By fitting the turbine hub  52  to the damper hub  7 , the second plate member  122 X of the first intermediate member  12 X that is fixed to the turbine hub  52  extends in the radial direction between the turbine runner  5  and the second output plate  162  in the axial direction, and the third plate member  123 X is closer to the turbine runner  5  than the second plate member  122 X. As illustrated in  FIG. 15 , the annular extending portion  123   m  of the third plate member  123 X surrounds a part of each first spring abutment portion  14   c  of the second intermediate member  14 X and the second plate member  122 X. That is, the annular extending portion  123   m  of the first intermediate member  12 X (third plate member  123 X) is located on an outer side in the radial direction of the damper apparatus  10 X with respect to the second intermediate member  14 X. 
     Each spring abutment portion  121   c  of the first plate member  121 X of the first intermediate member  12 X extends in the radial direction between the first and second inner springs SP 11  and SP 12  that are paired with each other (act in series), and abuts against the ends of those first and second inner springs SP 11  and SP 12 . The protruding portion  122 P of the coupling abutment portion  122   c  of the second plate member  122 X is fitted (coupled) to the opening portion  121   h  of the spring abutment portion  121   c  of the first plate member  121 X. Each coupling abutment portion  122   c  extends in the axial direction between the first and second inner springs SP 11  and SP 12 , and abuts against the ends of those first and second inner springs SP 11  and SP 12 . Thus, the driven member  16  is coupled to the driving member  11  via the plurality of first inner springs SP 11 , the first intermediate member  12 X (first plate member  121 X and second plate member  122 X), and the plurality of second inner springs SP 12 . Each first spring abutment portion  14   c  of the second intermediate member  14 X is inserted into the opening defined between the spring support portion  81   a  and the input plate  111 . Each first spring abutment portion  14   c  extends in the axial direction between the first and second outer springs SP 21  and SP 22  that are paired with each other (act in series), and abuts against the ends of those first and second outer springs SP 21  and SP 22 . Thus, the driven member  16  is coupled to the driving member  11  via the plurality of first outer springs SP 21 , the second intermediate member  14 X, and the plurality of second outer springs SP 22 . 
     The intermediate springs SPm are supported by the associated spring support portions  122   a ,  122   b ,  123   a , and  123   b  of the second and third plate members  122 X and  123 X of the first intermediate member  12 X, and are disposed on the radially outer side of the first and second inner springs SP 11  and SP 12  and on the radially inner side of the first and second outer springs SP 21  and SP 22  with distances from the first and second outer springs SP 21  and SP 22  in the axial direction of the damper apparatus  10 X. In the damper apparatus  10 X, the intermediate springs SPm partially overlap at least one of the first and second outer springs SP 21  and SP 22  in the radial direction as viewed in the axial direction, and partially overlap at least one of the first and second inner springs SP 11  and SP 12  in the axial direction as viewed in the radial direction. 
     In a state in which the damper apparatus  10 X is attached, a pair of outer abutment portions  122   d  located on both sides of the spring housing window  122   w  of the second plate member  122 X abut against the respective ends of the intermediate spring SPm, and a pair of spring abutment portions  123   d  located on both sides of the spring housing window  123   w  of the third plate member  123 X abut against the respective ends of the intermediate spring SPm. Similarly, a pair of second spring abutment portions  144  of the second intermediate member  14 X abut against the respective ends of the intermediate spring SPm between the second and third plate members  122 X and  123 X in the axial direction. Thus, in the state in which the damper apparatus  10 X is attached, each intermediate spring SPm is supported from both sides in the circumferential direction by the first intermediate member  12 X, that is, the pair of outer abutment portions  122   d  of the second plate member  122 X and the pair of spring abutment portions  123   d  of the third plate member  123 X, and is also supported from both sides in the circumferential direction by the pair of second spring abutment portions  14   d  of the second intermediate member  14 X. Accordingly, the first intermediate member  12 X and the second intermediate member  14 X are coupled to each other via the plurality of intermediate springs SPm. 
     In the damper apparatus  10 X, the first stopper  21  is structured by a plurality of stopper portions  122   z  (see  FIG. 16 ) each extending in the axial direction toward the front cover  3  from a central portion of the spring support portion  122   a  of the second plate member  122 X in the circumferential direction, and by a plurality of slits (cutout portions)  162   zi  (see  FIG. 15 ) formed in the second output plate  162  at intervals in the circumferential direction so as to extend in a circular arc shape. In the state in which the damper apparatus  10 X is attached, each stopper portion  122   z  of the first intermediate member  12 X (second plate member  122 X) is inserted through the corresponding slit  162   zi  of the driven member  16  (second output plate  162 ) so as not to abut against the wall surfaces of the second output plate  162  that define the ends of the slit  162   zi  on both sides. Thus, when each stopper portion  122   z  of the second plate member  122 X and one of the wall surfaces that define the ends of the slit  162   zi  on both sides abut against each other along with a relative rotation between the first intermediate member  12 X and the driven member  16 , the relative rotation between the first intermediate member  12 X and the driven member  16  and the deflection of the second inner springs SP 12  are restricted. 
     In the damper apparatus  10 X, the second stopper  22  is structured by the plurality of first spring abutment portions  14   c  of the second intermediate member  14 X, and by a plurality of cutout portions  162   zo  formed on the outer periphery of the second output plate  162  at intervals in the circumferential direction. In the state in which the damper apparatus  10 X is attached, each first spring abutment portion  14   c  of the second intermediate member  14 X is inserted through the corresponding cutout portion  162   zo  of the driven member  16  (second output plate  162 ) so as not to abut against the wall surfaces of the second output plate  162  that define the ends of the cutout portion  162   zo  on both sides. Thus, when each first spring abutment portion  14   c  of the second intermediate member  14 X and one of the wall surfaces that define the ends of the cutout portion  162   zo  on both sides abut against each other along with a relative rotation between the second intermediate member  14 X and the driven member  16 , the relative rotation between the second intermediate member  14 X and the driven member  16  and the deflection of the second outer springs SP 22  are restricted. 
     In the damper apparatus  10 X, the third stopper  23  is structured by a plurality of rivets  115  that couple the clutch drum  81  and the input plate  111  to each other, and by a plurality of slits (cutout portions)  161   z  formed in the first output plate  161  at intervals in the circumferential direction so as to extend in a circular arc shape. In the state in which the damper apparatus  10 X is attached, each of the plurality of rivets  115  is disposed in the corresponding slit  161   z  of the driven member  16  so as not to abut against the wall surfaces of the first output plate  161  that define the ends of the slit  161   z  on both sides. Thus, when each rivet  115  and one of the wall surfaces that define the ends of the slit  161   z  on both sides abut against each other along with the relative rotation between the driving member  11  and the driven member  16 , the relative rotation between the driving member  11  and the driven member  16  is restricted. 
     Also in the damper apparatus  10 X structured as described above, the second plate member  122 X includes the outer abutment portions  122   d  each abutting against the end of the intermediate spring SPm on the outer side in the radial direction with respect to the coupling abutment portion  122   c . By providing the outer abutment portions  122   d  each abutting against the intermediate spring SPm on the second plate member  122 X including the coupling abutment portions  122   c , even when the force applied to the coupling abutment portions  122   c  from the first and second inner springs SP 11  and SP 12  is opposite to the force applied to the outer abutment portions  122   d  from the intermediate springs SPm, the single second plate member  122 X can receive the two forces acting in opposite directions. Thus, it is possible to facilitate designing in terms of the durability of the first intermediate member  12 X by reducing the shear forces acting on fitting portions of the first and second plate members  121 X and  122 X, coupling portions of the second and third plate members  122 X and  123 X, and the third plate member  123 X. By using, as the second intermediate member  14 X, the single member including the first and second spring abutment portions  14   c  and  14   d  as described above, even when the force applied to the first spring abutment portions  14   c  from the first and second outer springs SP 21  and SP 22  is opposite to the force applied to the second spring abutment portions  14   d  from the intermediate springs SPm, the second intermediate member  14 X that is the single member can receive the two forces acting in opposite directions. Thus, the durability of the second intermediate member  14 X can further be improved. 
     In the damper apparatus  10 X, both of the coupling abutment portion  122   c  of the second plate member  122 X that extends in the axial direction and the spring abutment portion  121   c  of the first plate member  121 X that extends in the radial direction abut against the first and second inner springs SP 11  and SP 12 . Thus, the first and second inner springs SP 11  and SP 12  can be pressed by the first intermediate member  12 X so as to extend and contract along the axis center. Further, the coupling abutment portion  122   c  of the second plate member  122 X and the spring abutment portion  121   c  of the first plate member  121 X are supported from both sides by the first and second inner springs SP 11  and SP 12 . Therefore, there is no need to tighten the fitting of the second plate member  122 X and the first plate member  121 X. Thus, the coupling abutment portion  122   c  of the second plate member  122 X can easily be fitted to the spring abutment portion  121   c  of the first plate member  121 X. 
     The first intermediate member  12 X of the damper apparatus  10 X includes the annular extending portion  123   m  that extends so as to be located on the outer side in the radial direction with respect to the second intermediate member  14 X. Thus, the moment of inertia of the first intermediate member  12 X included in the torque transfer path P 1  on the radially inner side and on the low stiffness side can further be increased. As a result, the vibration damping performance of the damper apparatus  10 X can further be improved by further reducing the natural frequency f 21  of the first intermediate member  12 X. 
     The first intermediate member  12 X of the damper apparatus  10 X includes the annular second plate member  122 X including the coupling abutment portions  122   c  each abutting against the ends of the first and second inner springs SP 11  and SP 12  between the first and second inner springs SP 11  and SP 12 , and the annular third plate member  123 X coupled to the second plate member  122 X and including the annular extending portion  123   m  extending from the outer periphery in the axial direction of the damper apparatus  10 X so as to be located on the outer side in the radial direction with respect to the second intermediate member  14 X. Thus, the annular extending portion  123   m  located on the radially outer side with respect to the second intermediate member  14 X can easily be provided on the first intermediate member  12 X coupled to the first and second inner springs SP 11  and SP 12  on the radially inner side, and the moment of inertia of the first intermediate member  12 X can easily be increased while an increase in the size of the damper apparatus  10 X is suppressed by adjusting the dimensions and shapes of the second and third plate members  122 X and  123 X. In the damper apparatus  10 X, the annular extending portion  123   m  of the third plate member  123 X is located on the outer side in the radial direction with respect to the first spring abutment portions  14   c  of the second intermediate member  14 X each abutting against the ends of the first and second outer springs SP 21  and SP 22  on the radially outer side with respect to the intermediate springs SPm. Thus, the required moment of inertia of the first intermediate member  12 X can easily be obtained by sufficiently securing the dimensions of the annular extending portion  123   m  of the third plate member  123 X. 
     Also in the damper apparatus  10 X, the inner peripheral portion of the second plate member  122 X of the first intermediate member  12 X is coupled to the turbine runner  5 . Thus, the first intermediate member  12 X and the turbine runner  5  are coupled to each other while the mountability is improved by suppressing the increase in the size of the damper apparatus  10 X. Accordingly, a substantial moment of inertia of the first intermediate member  12 X (sum of the moments of inertia of the first to third plate members  121 X,  122 X, and  123 X, the turbine runner  5 , the turbine hub  52 , and the like) can further be increased. As a result, the natural frequency f 21  of the first intermediate member  12 X is further reduced, and the resonance point of the first intermediate member  12 X can be set to a lower speed rotation side (lower frequency side). Further, the difference between the natural frequencies f 21  and f of the first and second intermediate elements can further be increased. 
     In the damper apparatus  10 X, as illustrated in  FIG. 15 , the inner and outer spring abutment portions  111   ci  and  111   co  of the driving member  11 , the outer spring abutment portions  122   d  and the spring abutment portions  123   d  and  121   c  of the first intermediate member  12 X, the second spring abutment portions  14   d  of the second intermediate member  14 X, and the inner spring abutment portions  161   ci , the spring abutment portions  162   c , and the outer spring abutment portions  161   co  of the driven member  16  extend in the radial direction of the damper apparatus  10 X. Thus, the associated spring SP 11 , SP 12 , SP 21 , SP 22 , or SPm can be pressed by the spring abutment portion  11   ci ,  11   co ,  122   d ,  123   d ,  121   c ,  14   d ,  161   ci ,  162   c , or  161   co  so as to appropriately extend and contract along the axis center. As a result, the vibration damping performance of the damper apparatus  10 X can further be improved. 
     As described above, the damper apparatus disclosed herein is the damper apparatus ( 10 ,  10 B,  10 X) including the input element ( 11 ,  11 B) to which the torque from the engine (EG) is transferred, and the output element ( 16 ). The damper apparatus includes the first intermediate element ( 12 ,  12 X), the second intermediate element ( 14 ,  14 X), the first elastic body (SP 11 ) configured to transfer the torque between the input element ( 11 ,  11 B) and the first intermediate element ( 12 ,  12 X), the second elastic body (SP 12 ) configured to transfer the torque between the first intermediate element ( 12 ,  12 X) and the output element ( 16 ), the third elastic body (SP 21 ) configured to transfer the torque between the input element ( 11 ,  11 B) and the second intermediate element ( 14 ,  14 X), the fourth elastic body (SP 22 ) configured to transfer the torque between the second intermediate element ( 14 ,  14 X) and the output element ( 16 ), and the fifth elastic body (SPm) configured to transfer the torque between the first intermediate element ( 12 ,  12 X) and the second intermediate element ( 14 ,  14 X). At least one of the first and second intermediate elements ( 12 ,  12 X,  14 ,  14 X) includes the single member ( 122 ,  141 ,  14 X) on which the first torque transfer portion ( 122   c ,  141   c ,  14   c ) disposed between the first and second elastic bodies (SP 11 , SP 12 ) or between the third and fourth elastic bodies (SP 21 , SP 22 ) and the second torque transfer portion ( 122   d ,  141   d ,  14   d ) configured to exchange the torque with the fifth elastic body are both formed. 
     In the damper apparatus disclosed herein, two natural frequencies can be set as a whole in the state in which the deflections of all of the first to fifth elastic bodies are permitted. The two natural frequencies are appropriately set by adjusting the stiffness of the fifth elastic body. Thus, the vibration damping performance of the damper apparatus can further be improved. At least one of the first and second intermediate elements includes the single member on which the first torque transfer portion disposed between the first and second elastic bodies or between the third and fourth elastic bodies and the second torque transfer portion configured to exchange the torque with the fifth elastic body are both formed. Thus, the increase in the number of components and the increase in the size of the damper apparatus can be suppressed. In the damper apparatus disclosed herein, the force applied to the first torque transfer portion from the first and second elastic bodies or from the third and fourth elastic bodies may be opposite to the force applied to the second torque transfer portion from the fifth elastic body. Thus, when at least one of the first and second intermediate elements includes two members coupled to each other and the first torque transfer portion is formed on one of the two members while the second torque transfer portion is formed on the other, shear forces acting on coupling portions of the two members increase, and the durability of at least one of the first and second intermediate elements may decrease. When the first and second torque transfer portions are provided on the single member described above, the single member can receive the two forces acting in opposite directions. Thus, the durability of at least one of the first and second intermediate elements can further be improved. As a result, in the damper apparatus disclosed herein, the increase in the number of components and the increase in the size can be suppressed while the durability of at least one of the first and second intermediate elements is improved. 
     The stiffness of the third elastic body (SP 21 ) may be higher than the stiffness of the first elastic body (SP 11 ), and the second intermediate element ( 14 X) may be the single member. Thus, it is possible to further improve the durability of the second intermediate element to which the torque is transferred from the third elastic body having a larger share of the torque than the first elastic body. 
     The fifth elastic body (SPm) may be disposed with a distance from the third and fourth elastic bodies (SP 21 , SP 22 ) in the axial direction of the damper apparatus ( 10 ,  10 B,  10 X), and may at least partially overlap at least one of the third and fourth elastic bodies (SP 21 , SP 22 ) in the radial direction of the damper apparatus ( 10 ,  10 B,  10 X) as viewed in the axial direction. The first torque transfer portion ( 14   c ) may extend from the single member ( 14 X) to one side in the axial direction toward the ends of the third and fourth elastic bodies (SP 21 , SP 22 ), and the second torque transfer portion ( 14   d ) may extend from the single member ( 14 X) to the other side in the axial direction toward the end of the fifth elastic body (SPm). Thus, the second intermediate element can be coupled to the third, fourth, and fifth elastic bodies while the increase in the size of the damper apparatus is suppressed. 
     The first intermediate element ( 12 X) may include the first member ( 121 X) including the torque transfer portion ( 121   c ) disposed between the first and second elastic bodies (SP 11 , SP 12 ), the second member ( 122 X) coupled to the first member ( 121 X), and the third member ( 123 X) coupled to the second member ( 122 X). The fifth elastic body (SPm) may be supported by the second and third members ( 122 X,  123 X). At least one of the second and third members ( 122 X,  123 X) may include the torque transfer portion ( 122   d ,  123   d ) configured to exchange the torque with the fifth elastic body (SPm). In this manner, the first intermediate element may be structured by the three members. Thus, it is possible to increase the degrees of freedom in terms of setting of the stiffnesses, the numbers of arrangement, the torsion angles (strokes), and the like of the first to fifth elastic bodies by appropriately disposing the fifth elastic body. Accordingly, the vibration damping performance can further be improved by setting the natural frequencies of the first and second intermediate elements more appropriately. 
     The torque transfer portion ( 121   c ) of the first member ( 121 X) of the first intermediate element ( 12 X) may extend in the radial direction of the damper apparatus ( 10 X) between the first and second elastic bodies (SP 11 , SP 12 ), and may abut against the ends of the first and second elastic bodies (SP 11 , SP 12 ). The second member ( 122 X) of the first intermediate element ( 12 X) may include the coupling abutment portion ( 122   c ) fitted to the torque transfer portion ( 121   c ) of the first member ( 121 X) and abutting against the ends of the first and second elastic bodies (SP 11 , SP 12 ) between the first and second elastic bodies (SP 11 , SP 12 ), and the torque transfer portion ( 122   d ) abutting against the end of the fifth elastic body (SPm) on the outer side in the radial direction with respect to the coupling abutment portion ( 122   c ). By providing the torque transfer portion abutting against the fifth elastic body on the second member including the coupling abutment portion, even when the force applied to the coupling abutment portion from the first and second elastic bodies is opposite to the force applied to the torque transfer portion from the fifth elastic body (SPm), the single second member can receive the two forces acting in opposite directions. Thus, it is possible to facilitate designing in terms of the durability of the first intermediate element by reducing the forces applied to the fitting portions of the first and second members. 
     The second intermediate element ( 14 X) may be supported by the second member ( 122 X) of the first intermediate element ( 12 X) so as to be rotatable relative to the first intermediate element ( 12 X). 
     The inner peripheral portion of the second member ( 122 X) of the first intermediate element ( 12 X) may be coupled to the turbine runner ( 5 ) of the fluid transmission apparatus. Thus, the first intermediate element and the turbine runner can be coupled to each other while the mountability is improved by suppressing the increase in the size of the damper apparatus. 
     The stiffness of the third elastic body (SP 21 ) may be higher than the stiffness of the first elastic body (SP 11 ), and the second intermediate element ( 14 ) may include the single member ( 141 ), and the second member ( 142 ) coupled to the single member ( 141 ) and configured to support the fifth elastic body (SPm). Thus, it is possible to further improve the durability of the second intermediate element to which the torque is transferred from the third elastic body having a larger share of the torque than the first elastic body. 
     The first intermediate element ( 12 ) may include the first member ( 121 ) including the torque transfer portion ( 121   c ) disposed between the first and second elastic bodies (SP 11 , SP 12 ), and the second member ( 122 ) coupled to the first member ( 121 ) and including the torque transfer portion ( 122   d ) configured to exchange the torque with the fifth elastic body (SPm). That is, when the second intermediate element is structured by the two members, the first intermediate element may be structured by the two members. 
     The fifth elastic body (SPm) may be disposed with a distance from the third and fourth elastic bodies (SP 21 , SP 22 ) in the axial direction of the damper apparatus ( 10 ,  10 B,  10 X), and may at least partially overlap at least one of the third and fourth elastic bodies (SP 21 , SP 22 ) in the radial direction of the damper apparatus ( 10 ,  10 B,  10 X) as viewed in the axial direction. The first torque transfer portion ( 141   c ) may extend from the single member ( 141 ) to one side in the axial direction toward the ends of the third and fourth elastic bodies (SP 21 , SP 22 ), and the second torque transfer portion ( 141   d ) may extend from the single member ( 141 ) to the other side in the axial direction toward the end of the fifth elastic body (SPm). Thus, the second intermediate element can be coupled to the third, fourth, and fifth elastic bodies while the increase in the size of the damper apparatus is suppressed. 
     The torque transfer portion ( 121   c ) of the first member ( 121 ) of the first intermediate element ( 12 ) may extend in the radial direction of the damper apparatus ( 10 ,  10 B) between the first and second elastic bodies (SP 11 , SP 12 ), and may abut against the ends of the first and second elastic bodies (SP 11 , SP 12 ). The second member ( 122 ) of the first intermediate element ( 12 ) may include the coupling abutment portion ( 122   c ) fitted to the torque transfer portion ( 121   c ) of the first member ( 121 ) and abutting against the ends of the first and second elastic bodies (SP 11 , SP 12 ) between the first and second elastic bodies (SP 11 , SP 12 ), and the torque transfer portion ( 122   d ) abutting against the end of the fifth elastic body (SPm) on the outer side in the radial direction with respect to the coupling abutment portion ( 122   c ). By providing the torque transfer portion abutting against the fifth elastic body on the second member including the coupling abutment portion, even when the force applied to the coupling abutment portion from the first and second elastic bodies is opposite to the force applied to the torque transfer portion from the fifth elastic body (SPm), the single second member can receive the two forces acting in opposite directions. Thus, it is possible to facilitate designing in terms of the durability of the first intermediate element by reducing the forces applied to the fitting portions of the first and second members. 
     The second intermediate element ( 14 ) may be supported by the second member ( 122 ) of the first intermediate element ( 12 ) so as to be rotatable relative to the first intermediate element ( 12 ). 
     The inner peripheral portion of the second member ( 122 ) of the first intermediate element ( 12 ) may be coupled to the turbine runner ( 5 ) of the fluid transmission apparatus. Thus, the first intermediate element and the turbine runner can be coupled to each other while the mountability is improved by suppressing the increase in the size of the damper apparatus. 
     The input element ( 11 ,  11 B) may include the two members ( 81 ,  111 ,  80 B,  111 B) coupled to each other, at least one of the two members being configured to support the third and fourth elastic bodies or the first and second elastic bodies (SP 21 , SP 22 ), and at least one of the two members including the torque transfer portion ( 81   c ,  111   co ,  111   ci ) configured to exchange the torque with the third or first elastic body (SP 11 , SP 21 ). The output element ( 16 ) may include the two members ( 161 ,  162 ) coupled to each other, at least one of the two members being configured to support the first and second elastic bodies or the third and fourth elastic bodies (SP 11 , SP 12 ), and at least one of the two members including the torque transfer portion ( 161   ci ,  161   co ,  162   c ) configured to exchange the torque with the second or fourth elastic body (SP 12 , SP 22 ). That is, the input element, the output element, the first intermediate element, and the second intermediate element of the damper apparatus disclosed herein may be structured by eight members in total. 
     The third and fourth elastic bodies (SP 21 , SP 22 ) may be disposed on the outer side of the first and second elastic bodies (SP 11 , SP 12 ) in the radial direction of the damper apparatus ( 10 ,  10 B,  10 X). By disposing the third elastic body having a stiffness higher than that of the first elastic body on the radially outer side of the first elastic body, the torsion angle (stroke) of the third elastic body can further be increased. Thus, the stiffness of the third elastic body is reduced while the transfer of a large torque to the input element is permitted. Accordingly, the equivalent stiffness of the damper apparatus can further be reduced. 
     The natural frequency of the second intermediate element ( 14 ,  14   x ) may be larger than the natural frequency of the first intermediate element ( 12 ,  12 X). 
     It is appropriate to prevent the deflections of the first to fifth elastic bodies (SP 11 , SP 12 , SP 21 , SP 22 , SPm) from being restricted until the input torque (T) transferred to the input element ( 11 ,  11 B) is equal to or larger than the predetermined threshold (T 1 ). Thus, the vibration damping performance of the damper apparatus when the input torque transferred to the input element is relatively small and the rotation speed of the input element is low can satisfactorily be improved. 
     The output element ( 16 ) may actively (directly or indirectly) be coupled to the input shaft (IS) of the transmission (TM). 
     It is understood that the invention disclosed herein is not limited to the embodiment described above and various modifications may be made within the extensive scope of the disclosure. The embodiment described above is merely one specific mode of the invention described in the “SUMMARY” section, and is not intended to limit the elements of the invention described in the “SUMMARY” section. 
     INDUSTRIAL APPLICABILITY 
     The invention disclosed herein is applicable to, for example, the field of manufacture of a damper apparatus.