Patent Publication Number: US-10323716-B2

Title: Damper device

Description:
This is a national phase application of PCT/JP2015/086404 filed Dec. 25, 2015, claiming priority to Japanese Patent Application No. JP2014-263006 filed Dec. 25, 2014, the contents of which are incorporated herein by reference. 
     TECHNICAL FIELD 
     The present disclosure relates to a damper device including an input element, an output element and an elastic body configured to transmit a torque between the input element and the output element. 
     BACKGROUND 
     A conventionally known configuration of a torsional vibration damper includes a damper spring device including a plurality of springs arrayed in a circumferential direction, and a rotary inertia mass damper (as shown in, for example, Patent Literature 1). In this torsional vibration damper, the damper spring device includes a plurality of sliding blocks that are respectively placed between adjacent springs and arranged such that an entire outer circumferential surface is supported by a drive-side damper element (input element) to be slidable. The rotary inertia mass damper includes the driven-side damper element arranged to rotate integrally with a turbine shell of a turbine wheel and configured to include a sun gear of a planetary gear; a driven-side damper element (output element) arranged to rotate integrally with a turbine hub of the turbine wheel and configured to serve as a carrier of the planetary gear; a pinion gear rotatably supported by the driven-side damper element and arranged to engage with the sun gear; and a ring gear configured to serve as a mass body engaging with the pinion gear. In the torsional vibration damper of this configuration, when the driven-side damper element is rotated (twisted) relative to the driven-side damper element, the springs of the damper spring device are deflected, and the ring gear as the mass body is rotated with relative rotation of the drive-side damper element to the driven-side damper element. This configuration causes a torque according to an inertia (moment of inertia) of the ring gear as the mass body and a difference in angular acceleration between the drive-side damper element and the driven-side damper element to be applied to the driven-side damper element and improves the vibration damping performance of the torsional vibration damper. 
     CITATION LIST 
     Patent Literature 
     PTL1: Japanese Patent No. 3476719 ( FIGS. 5 and 7 ) 
     SUMMARY 
     When a vibration transmitted to the drive-side damper element (input element) has a fixed amplitude, the torque applied from the rotary inertia mass damper to the driven-side damper element (output element) is gradually increased with an increase in frequency of the vibration (rotation speed of the drive-side damper element). As shown in FIG. 2 of Patent Literature 1, on the other hand, the amplitude of a vibration transmitted from the damper spring device to the output element is decreased with an increase in rotation speed (frequency) of the drive-side damper element, especially in a higher rotation speed range (higher frequency range) than a rotation speed corresponding to a natural frequency EF 4 . Accordingly, in the prior art torsional vibration damper, a large torque is applied from the rotary inertia mass damper to the output element in the state that the amplitude of the vibration transmitted from the damper spring device to the output element is lowered. This is likely to rather reduce the vibration damping performance. 
     A main object of the disclosure is to provide a damper device having better vibration damping performance. 
     The present disclosure is directed to a damper device. The damper device configured to include an input element to which a torque from an engine is transmitted and an output element, the damper device includes a torque transmission path including an intermediate element, a first elastic body configured to transmit a torque between the input element and the intermediate element, and a second elastic body configured to transmit a torque between the intermediate element and the output element, and a rotary inertia mass damper configured to include a mass body rotating with relative rotation of the input element to the output element and provided between the input element and the output element in parallel to the torque transmission path. A damping ratio of the intermediate element determined based on a moment of inertia of the intermediate element and rigidities of the first and the second elastic bodies is less than a value  1 . 
     In the damper device of this aspect, on the assumption that an input torque transmitted to the input element is periodically vibrated, the phase of a vibration transmitted from the input element to the output element via the torque transmission path is shifted by 180 degrees from the phase of a vibration transmitted from the input element to the output element via the rotary inertia mass damper. A plurality of natural frequencies (resonance frequencies) may be set in the torque transmission path including the intermediate element having the damping ratio of less than the value  1 , in the state that deflections of the first and the second elastic bodies are allowed. A resonance of the intermediate element may be caused in this torque transmission path when the rotation speed of the input element reaches a rotation speed corresponding to one of the plurality of natural frequencies. This configuration enables two antiresonance points, where the vibration transmitted from the input element to the output element via the torque transmission path and the vibration transmitted from the input element to the output element via the rotary inertia mass damper are theoretically cancelled out each other, to be set in the damper device of this aspect. This configuration thus remarkably effectively improves the vibration damping performance of the damper device by making the frequencies of the two antiresonance points equal to (closer to) the frequency of the vibration (resonance) to be damped by the damper device. Additionally, in the damper device of this aspect, a resonance of the intermediate element occurs when the rotation speed of the input element becomes higher than a rotation speed corresponding to the frequency of an antiresonance point of the lower rotation speed (lower frequency). The amplitude of the vibration transmitted from the second elastic body to the output element accordingly changes from decreasing to increasing before the rotation speed of the input element reaches a rotation speed corresponding to a relatively lower natural frequency of the intermediate element. Even when the amplitude of the vibration transmitted from the rotary inertia mass damper to the output element is gradually increased with an increase in rotation speed of the input element, this expands an area where the vibration transmitted from the rotary inertia mass damper to the output element cancels out at least part of the vibration transmitted from the second elastic body to the output element. This results in further improving the vibration damping performance of the damper device in a relatively low rotation speed range of the input element. 
    
    
     
       BRIEF DESCRIPTION OF DRAWINGS 
         FIG. 1  is a schematic configuration diagram illustrating a starting device including a damper device according to one embodiment of the disclosure; 
         FIG. 2  is a sectional view illustrating the starting device of  FIG. 1 ; 
         FIG. 3  is a front view illustrating the damper device of  FIGS. 1 and 2 ; 
         FIG. 4  is a diagram illustrating a relationship of rotation speed of an engine to torque variation T Fluc  at an output element in the damper device of  FIG. 1  and other drawings; 
         FIG. 5  is a schematic configuration diagram illustrating a starting device according to another embodiment of the disclosure; 
         FIG. 6  is a schematic configuration diagram illustrating a starting device according to yet another embodiment of the disclosure; 
         FIG. 7  is a schematic configuration diagram illustrating a starting device according to another embodiment of the disclosure; 
         FIG. 8  is a schematic configuration diagram illustrating a starting device according to yet another embodiment of the disclosure; 
         FIG. 9  is a schematic configuration diagram illustrating a starting device according to another embodiment of the disclosure; 
         FIG. 10  is a schematic configuration diagram illustrating a starting device according to yet another embodiment of the disclosure; and 
         FIGS. 11A, 11B, 11C and 11D  are schematic diagrams illustrating other rotary inertia mass dampers applicable to the damper devices of  FIG. 1  and other drawings. 
     
    
    
     DESCRIPTION OF EMBODIMENTS 
     The following describes some embodiments of the present disclosure with reference to drawings. 
       FIG. 1  is a schematic configuration diagram illustrating a starting device  1  including a damper device  10  according to one embodiment of the disclosure.  FIG. 2  is a sectional view illustrating the starting device  1 . The starting device  1  illustrated in these drawings is mounted on a vehicle equipped with an engine (internal combustion engine) EG as the prime mover and may include, in addition to the damper device  10 , for example, a front cover  3  serving as an input member connected with a crankshaft of the engine EG and configured to receive a torque transmitted from 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  arranged to be rotatable coaxially with the pump impeller  4 , a damper hub  7  serving as an input member connected with the damper device  10  and fixed to an input shaft IS of a transmission TM that is either an automatic transmission (AT) or a continuously variable transmission (CVT), and a lockup clutch  8 . 
     In the description below, a term “axial direction” basically means an extending direction of a central axis (axial center) of the starting device  1  or the damper device  10 , unless otherwise specified. A term “radial direction” basically means a radial direction of the starting device  1 , the damper device  10  or a rotational element of the damper device  10  and so on, i.e., an extending direction of a straight line extended in a direction perpendicular to the central axis (radial direction) from the central axis of the starting device  1  or the damper device  10 , unless otherwise specified. Additionally, a term “circumferential direction” basically means a circumferential direction of the starting device  1 , the damper device  10  or a rotational element of the damper device and so on, i.e., a direction along a rotation direction of the rotational element, unless otherwise specified. 
     As shown in  FIG. 2 , the pump impeller  4  includes a pump shell  40  closely fixed to the front cover  3  and a plurality of pump blades  41  provided on an inner surface of the pump shell  40 . As shown in  FIG. 2 , the turbine runner  5  includes a turbine shell  50  and a plurality of turbine blades  51  provided on an inner surface of the turbine shell  50 . An inner circumferential portion of the turbine shell  50  is fixed to the turbine hub  52  by means of a plurality of rivets. The turbine hub  52  is rotatably supported by the damper hub  7 . The motion of the turbine hub  52  (turbine runner  5 ) in the axial direction of the starting device  1  is restricted by the damper hub  7  and a snap ring mounted to the damper hub  7 . 
     The pump impeller  4  and the turbine runner  5  are opposed to each other, and a stator  6  is placed coaxially between the pump impeller  4  and the turbine runner  5  to straighten the flow of hydraulic oil (working fluid) from the turbine runner  5  to the pump impeller  4 . The stator  6  includes a plurality of stator blades  60 . The rotation direction of the stator  6  is set to only one direction by a one-way clutch  61 . The pump impeller  4 , the turbine runner  5  and the stator  6  form a torus (annular passage) to circulate the hydraulic oil and serves as a torque converter (fluid transmission device) having the torque amplification function. In the starting device  1 , however, the stator  6  and the one-way clutch  61  may be omitted, and the pump impeller  4  and the turbine runner  5  may serve as fluid coupling. 
     The lockup clutch  8  is configured to engage lockup that couples the front cover  3  with the damper hub  7  via the damper device  10  and to release the lockup. According to this embodiment, the lockup clutch  8  is configured as a single-disc hydraulic clutch and includes a lockup piston  80  that is arranged inside of the front cover  3  and close to an engine EG-side inner wall surface of the front cover  3  and that is fit in the damper hub  7  to be rotatable and movable in the axial direction. As shown in  FIG. 2 , friction members  81  are applied on an outer circumferential-side and front cover  3 -side surface of the lockup piston  80 . A lockup chamber  85  is defined between the lockup piston  80  and the front cover  3  to be connected with a non-illustrated hydraulic control device via a hydraulic oil supply passage and an oil passage formed in the input shaft IS. 
     The hydraulic oil from the hydraulic control device supplied radially outward from the axial center side of the pump impeller  4  and the turbine runner  5  (periphery of the one-way clutch  61 ) through, for example, the oil passage formed in the input shaft IS to the pump impeller  4  and the turbine runner (torus) is allowed to flow into the lockup chamber  85 . Accordingly, when the internal pressure of the lockup chamber  85  is maintained equal to the internal pressure of a fluid transmission chamber  9  defined by the front cover  3  and the pump shell of the pump impeller  4 , the lockup piston  80  does not move toward the front cover  3  and is not frictionally engaged with the front cover  3 . When the internal pressure of the lockup chamber  85  is reduced by the non-illustrated hydraulic control device, on the other hand, the pressure difference causes the lockup piston  80  to move toward the front cover  3  and to be frictionally engaged with the front cover  3 . The front cover  3  is accordingly coupled with the damper hub  7  via the damper device  10 . A multi-disc hydraulic clutch including at least one frictional engagement plate (a plurality of friction members) may be employed for the lockup clutch  8 . 
     As shown in  FIGS. 1 and 2 , the damper device  10  includes a drive member (input element)  11 , an intermediate member (intermediate element)  12  and a driven member (output element)  15 , as rotational elements. The damper device  10  also includes a plurality of (for example, three according to this embodiment) first inner springs (first elastic bodies) SP 1  configured to transmit the toque between the drive member  11  and the intermediate member  12 , a plurality of (for example, three according to this embodiment) second inner springs (second elastic bodies) SP 2  configured to respectively work in series with the corresponding first inner springs SP 1  and to transmit the torque between the intermediate member  12  and the driven member  15 , and a plurality of (for example, three according to this embodiment) outer springs SPo configured to transmit the torque between the drive member  11  and the driven member  15 , as torque transmission elements (torque transmission elastic bodies). 
     As shown in  FIG. 1 , the damper device  10  accordingly has a first torque transmission path TP 1  and a second torque transmission path TP 2  that are provided parallel to each other between the drive member  11  and the driven member  15 . The first torque transmission path TP 1  is formed from the plurality of first inner springs SP 1 , the intermediate member  12  and the plurality of second inner springs SP 2  and is configured to transmit the torque between the drive member  11  and the driven member  15  via these elements. The second torque transmission path TP 2  is formed from the plurality of outer springs SPo and is configured to transmit the torque between the drive member  11  and the driven member  15  via the plurality of outer springs SPo that work parallel to one another. 
     According to this embodiment, coil springs having an identical specification (spring constant) are employed for the first and the second inner springs SP 1  and SP 2 . Additionally, the plurality of outer springs SPo forming the second torque transmission path TP 2  are configured to work in parallel to the first and the second inner springs SP 1  and SP 2  forming the first torque transmission path TP 1 , after the input torque into the drive member  11  reaches a predetermined torque (first threshold value) T 1  that is smaller than a torque T 2  (second threshold value) corresponding to a maximum torsion angle θmax of the damper device  10  and the torsion angle of the drive member  11  relative to the driven member  15  becomes equal to or larger than a predetermined angle θref. The damper device  10  accordingly has two-step (two-stage) damping characteristics. 
     According to this embodiment, a linear coil spring formed from a metal material that is spirally wound to have a axial center extended linearly without application of a load is employed for the first and the second inner springs SP 1  and SP 2  and the outer springs SPo. Compared with employing an arc coil spring, this more appropriately expands and contracts the first and the second inner springs SP 1  and SP 2  and the outer springs SPo along their axial centers and reduces a hysteresis H (difference between a torque output from the driven member  15  in response to an increase of the input torque into the drive member  11  and a torque output from the driven member  15  in response to a decrease of this input torque). The arc coil spring may, however, be employed for at least any of the first and the second inner springs SP 1  and SP 2  and the outer springs SPo. 
     As shown in  FIG. 2 , the drive member  11  of the damper device  10  includes a short outer cylinder portion  11   a  and a plate-like annular  11   b  extended radially inward from one end of the outer cylinder portion  11   a . The outer cylinder portion  11   a  of the drive member  11  is coupled with an outer circumferential portion of the lockup piston  80  of the lockup clutch  8  via an interlocking engagement portion  811 . This configuration causes the drive member  11  to be rotatable integrally with the lockup piston  80 . The front cover  3  is coupled with the drive member  11  of the damper device  10  by engagement of the lockup clutch  8 . 
     As shown in  FIG. 3 , the drive member  11  includes a plurality of (for example, three at intervals of 120 degrees according to this embodiment) inner spring abutting portions  11   c  that are protruded radially inward (toward the axial center of the drive member  11 ) from an inner circumferential surface of the annular portion  11   b  and are arranged at intervals in the circumferential direction. Additionally, as shown in  FIG. 3 , a plurality of (for example, three at intervals of 120 degrees according to this embodiment) outer spring receiving portions  11   o  are formed in the annular portion  11   b  to be respectively located on the outer side in the radial direction of the corresponding inner spring abutting portions  11   c  and to be arranged at intervals in the circumferential direction. Each of the outer spring receiving portions  11   o  has a circumferential length according to the natural length of the outer spring SPo, and one outer spring abutting portion  11   f  is formed on each side in the circumferential direction of each outer spring receiving portion  11   o.    
     The intermediate member  12  is a plate-like annular member and includes a plurality of (for example, three at intervals of 120 degrees according to this embodiment) spring abutting portions  12   c  that are protruded radially outward from its outer circumferential surface and are arranged at intervals in the circumferential direction. As shown in  FIG. 2 , an inner circumferential portion of the intermediate member  12 , along with an inner circumferential portion of the turbine shell  50 , is fixed to the turbine hub  52  by means of a plurality of rivets. This configuration couples the intermediate member  12  and the turbine runner  5  with each other to be integrally rotated. Coupling of the turbine runner  5  (and the turbine hub  52 ) with the intermediate member  12  allows for a further increase of the substantial moment of inertia of the intermediate member  12  (total moment of inertia of the intermediate member  12 , the turbine runner  5  and the like). 
     The driven member  15  includes an annular first output plate member  16  that is arranged on the turbine runner  5 -side, and an annular second output plate member  17  that is arranged on the front cover  3 -side and is coupled with (fixed to) the first output plate member  16  by means of a plurality of rivets. The first output plate member  16  of the driven member  15  includes a plurality of (for example, three at intervals of 120 degrees according to this embodiment) arc-shaped spring support portions  16   a  that are arranged along its inner circumferential edge at intervals in the circumferential direction; a plurality of (for example, three at intervals of 120 degrees according to this embodiment) arc-shaped spring support portions  16   b  that are located on the outer side in the radial direction of the plurality of spring support portions  16   a , are arranged at intervals in the circumferential direction and are respectively opposed to the corresponding spring support portions  16   a  in the radial direction of the first output plate member  16 ; and a plurality of (for example, three according to this embodiment) inner spring abutting portions  16   c . Each of the plurality of inner spring abutting portions  16   c  is provided between each pair of the spring support portions  16   a  and  16   b  arranged adjacent to each other in the circumferential direction. 
     Additionally, the first output plate member  16  includes a plurality of (for example, three at intervals of 120 degrees according to this embodiment) arc-shaped spring support portions  16   d  that are located on the outer side in the radial direction of the spring support portions  16   b  and are arranged at intervals in the circumferential direction; and a plurality of (for example, three at intervals of 120 degrees according to this embodiment) arc-shaped spring support portions  16   e  that are located on the outer side in the radial direction of the plurality of spring support portions  16   d , are arranged at intervals in the circumferential direction and are respectively opposed to the corresponding spring support portions  16   d  in the radial direction of the first output plate member  16 . A central region of each pair of the spring support portions  16   d  and  16   e  is located on the outer side in the radial direction of the corresponding inner spring abutting portion  16   c . Each pair of the spring support portions  16   d  and  16   e  are arranged to overlap in the radial direction with both an adjacent pair of the spring support portions  16   b . One outer spring abutting portion  16   f  is formed on each side of the pair of spring support portions  16   d  and  16   e.    
     As shown in  FIG. 2 , an inner circumferential portion of the second output plate member  17  included in the driven member is fixed to the damper hub  7  by means of a plurality of rivets. The second output plate members  17  includes a plurality of (for example, three at intervals of 120 degrees according to this embodiment) arc-shaped spring receiving portions  17   a  that are arranged along its inner circumferential edge at intervals in the circumferential direction; a plurality of (for example, three at intervals of 120 degrees according to this embodiment) arc-shaped spring support portions  17   b  that are located on the outer side in the radial direction of the plurality of spring support portions  17   a , are arranged at intervals in the circumferential direction and are respectively opposed to the corresponding spring support portions  17   a  in the radial direction of the second output plate member  17 ; and a plurality of (for example, three according to this embodiment) inner spring abutting portions  17   c . Each of the plurality of inner spring abutting portions  17   c  is provided between each pair of the spring support portions  17   a  and  17   b  arranged adjacent to each other in the circumferential direction. 
     Additionally, the second output plate member  17  includes a plurality of (for example, three at intervals of 120 degrees according to this embodiment) arc-shaped spring support portions  17   d  that are located on the outer side in the radial direction of the spring support portions  17   b  and are arranged at intervals in the circumferential direction; and a plurality of (for example, three at intervals of 120 degrees according to this embodiment) arc-shaped spring support portions  17   e  that are located on the outer side in the radial direction of the plurality of spring support portions  17   d , are arranged at intervals in the circumferential direction and are respectively opposed to the corresponding spring support portions  17   d  in the radial direction of the second output plate member  17 . A central region of each pair of the spring support portions  17   d  and  17   e  is located on the outer side in the radial direction of the corresponding inner spring abutting portion  17   c . Each pair of the spring support portions  17   d  and  17   e  are arranged to overlap in the radial direction with both an adjacent pair of the spring support portions  17   b . One outer spring abutting portion  17   f  is formed on each side of the pair of spring support portions  17   d  and  17   e.    
     In the state that the first output plate member  16  and the second output plate member  17  are coupled with each other, the plurality of spring support portions  16   a  of the first output plate member  16  serve to respectively support (guide), from the inner circumferential side, turbine runner  5 -side lateral portions of the corresponding first and second inner springs SP 1  and SP 2 . The plurality of spring support portions  16   b  serve to respectively support (guide), from the outer circumferential side, the turbine runner  5 -side lateral portions of the corresponding first and second inner springs SP 1  and SP 2 . The plurality of spring support portions  17   a  of the second output plate member  17  are opposed in the axial direction to the corresponding spring support portions  16   a  of the first output plate member  16  and serve to respectively support (guide), from the inner circumferential side, lockup piston  80 -side lateral portions of the corresponding first and second inner springs SP 1  and SP 2 . The plurality of spring support portions  17   b  are opposed in the axial direction to the corresponding spring support portions  16   b  of the first output plate member  16  and serve to respectively support (guide), from the outer circumferential side, the lockup piston  80 -side lateral portions of the corresponding first and second inner springs SP 1  and SP 2 . 
     The first and the second inner springs SP 1  and SP 2  are accordingly supported by the spring support portions  16   a  and  16   b  of the first output plate member  16  and the spring support portions  17   a  and  17   b  of the second output plate member  17 , such that one first inner spring SP 1  and one second inner spring SP 2  form a pair (to act in series) and that the first inner springs SP 1  and the second inner springs SP 2  are alternately arranged in the circumferential direction of the driven member  15 . According to this embodiment, as shown in  FIG. 3 , the plurality of first inner springs SP 1  and the plurality of second inner springs SP 2  are arranged on an identical circumference, such that the distance between the axial center of the starting device  1  or the damper device  10  and the axial center of each first inner spring SP 1  is equal to the distance between the axial center of the starting device  1  and so on and the axial center of each second inner spring SP 2 . 
     In the mounting state of the damper device  10 , each inner spring abutting portion  11   c  of the drive member  11  is placed between the first inner spring SP 1  and the second inner spring SP 2  to abut on respective one ends the first inner spring SP 1  and the second inner spring SP 2  that are respectively supported by the different spring support portions  16   a ,  16   b ,  17   a  and  17   b  and do not form a pair (not to act in series). Each spring abutting portion  12   c  of the intermediate member  12  is placed between the first inner spring SP 1  and the second inner spring SP 2  to abut on respective one ends the first inner spring SP 1  and the second inner spring SP 2  that are supported by the same spring support portions  16   a ,  16   b ,  17   a  and  17   b  and form a pair. Each inner spring abutting portion  16   c  of the first output plate member  16  is placed between the first inner spring SP 1  and the second inner spring SP 2  to abut on respective one ends the first inner spring SP 1  and the second inner spring SP 2  that are respectively supported by the different spring support portions  16   a ,  16   b ,  17   a  and  17   b  and do not form a pair (not to act in series). Similarly each inner spring abutting portion  17   c  of the second output plate member  17  is placed between the first inner spring SP 1  and the second inner spring SP 2  to abut on respective one ends the first inner spring SP 1  and the second inner spring SP 2  that are respectively supported by the different spring support portions  16   a ,  16   b ,  17   a  and  17   b  and do not form a pair (not to act in series). 
     In the mounting state of the damper device  10 , one end of each first inner spring SP 1  abuts on a corresponding inner spring abutting portion  11   c  of the drive member  11 , and the other end of each first inner spring SP 1  abuts on a corresponding spring abutting portion  12   c  of the intermediate member  12 . One end of each second inner spring SP 2  abuts on a corresponding spring abutting portion  12   c  of the intermediate member  12 , and the other end of each second inner spring SP 2  abuts on corresponding inner spring abutting portions  16   c  and  17   c  of the driven member  15 . As a result, each pair of the first and the second inner springs SP 1  and SP 2  are coupled with each other in series via the spring abutting portion  12   c  of the intermediate member  12  between the drive member  11  and the driven member  15 . Accordingly this configuration of the damper device  10  further reduces the rigidity of the elastic bodies configured to transmit the torque between the drive member  11  and the driven member  15  or more specifically a combined spring constant of the first and the second inner springs SP 1  and SP 2 . 
     In the state that the first output plate member  16  and the second output plate member  17  are coupled with each other, the plurality of spring support portions  16   d  of the first output plate member  16  serve to respectively support (guide), from the inner circumferential side, turbine runner  5 -side lateral portions of the corresponding outer springs SPo. The plurality of spring support portions  16   e  serve to respectively support (guide), from the outer circumferential side, the turbine runner  5 -side lateral portions of the corresponding outer springs SPo. The plurality of spring support portions  17   d  of the second output plate member  17  are opposed in the axial direction to the corresponding spring support portions  16   d  of the first output plate member  16  and serve to respectively support (guide), from the inner circumferential side, lockup piston  80 -side lateral portions of the corresponding outer springs SPo. The plurality of spring support portions  17   e  are opposed in the axial direction to the corresponding spring support portions  16   e  of the first output plate member  16  and serve to respectively support (guide), from the outer circumferential side, the lockup piston  80 -side lateral portions of the corresponding outer springs SPo. 
     In the mounting state of the damper device  10 , each of the outer springs SPo is placed between the corresponding spring support portions  16 ,  16   e ,  17   d  and  17   e  in the circumferential direction. Each of the outer springs SPo abuts on one pair of the outer spring abutting portions  16   f  and  17   f  provided on the respective sides of the spring support portions  16   d ,  16   e ,  17   d  and  17   e  when the input torque into the drive member  11  or the drive torque (or the torque applied from the axle side to the driven member  15  (driven torque)) reaches the above torque T 1  and the torsion angle of the drive member  11  relative to the driven member  15  becomes equal to or larger than the predetermined angle θref. As shown in  FIGS. 2 and 3 , each of the outer springs SPo is arranged in an outer circumferential-side region in the fluid transmission chamber  9  such as to surround the first and the second inner springs SP 1  and SP 2 . This configuration further shortens the axial length of the damper device  10  and thereby the axial length of the starting device  1 . 
     As shown in  FIG. 1 , the damper device  10  includes a rotary inertia mass damper  20  that is connected with the drive member  11  and the driven member  15  and is arranged parallel to both the first torque transmission path TP 1  and the second torque transmission path TP 2 . According to this embodiment, the rotary inertia mass damper  20  is configured by a single pinion-type planetary gear  21  placed between the drive member  11  that is the input element of the damper device  10  and the driven member  15  that is the output element of the damper device  10 . The planetary gear  21  includes a sun gear (third element)  22  that is an external gear, a ring gear (second element)  23  that is an internal gear arranged concentrically with the sun gear  22 , and a plurality of (for example, three according to this embodiment) pinion gear  24  that are arranged to respectively engage with the sun gear  22  and the ring gear  23 . 
     The sun gear  22  of the planetary gear  21  has, for example, a slightly larger outer diameter than the inner diameter of the first output plate member  16  of the driven member  15  and includes a mass portion  22   m  that is arranged inside of a plurality of external teeth and is configured to increase the moment of inertia. The sun gear  22  is rotatably supported by the damper hub  7 . The motion of the sun gear  22  in the axial direction of the starting device  1  is restricted by the damper hub  7  and the snap ring mounted to the damper hub  7 . The ring gear  23  has, for example, a larger inner diameter than the spring support portions  16   b  and  17   b  of the first and the second output plate members  16  and  17  and is fixed to the driven member  15  by means of a plurality of rivets provided to couple the first output plate member  16  with the second output plate member  17 . This configuration enables the ring gear  23  to be rotated integrally with the driven member  15 . 
     The plurality of pinion gears  24  are rotatably supported by the lockup piston  80  via, for example, journal bearings  88  such that the pinion gears  24  are arranged at intervals (at equal intervals) in the circumferential direction and are opposed to regions between respective adjacent pairs of the first and the second inner springs SP 1  and SP 2  in the axial direction. As described above, the lockup piston  80  is rotatably supported by the damper hub  7  and is rotatable integrally with the drive member  11  that is the input element of the damper device  10 . Accordingly the lockup piston  80  serves as a planetary carrier (first element) of the planetary gear  21  that supports the plurality of pinion gears  24  to be rotatable (rotatable on their own axes) and revolvable about the sun gear  22  and the ring gear  23 . 
     When the lockup by the lockup clutch  8  is released in the starting device  1  having the configuration described above, as clearly understood from  FIG. 1 , the torque transmitted from the engine EG to the front cover  3  is transmitted to the input shaft IS of the transmission TM via the path of the pump impeller  4 , the turbine runner  5 , the intermediate member  12 , the second inner springs SP 2 , the driven member  15  and the damper hub  7 . When the lockup is engaged by the lockup clutch  8  in the starting device  1 , on the other hand, the torque transmitted from the engine EG to the drive member  11  via the front cover  3  and the lockup clutch  8  is transmitted to the driven member  15  and the damper hub  7  via the first torque transmission path TP 1  including the plurality of first inner springs SP 1 , the intermediate member  12  and the plurality of second springs SP 2 , and the rotary inertia mass damper  20  until the input torque reaches the above torque T 1 . When the input torque becomes equal to or higher than the above torque T 1 , however, the torque transmitted to the drive member  11  is transmitted to the driven member  15  and the damper hub  7  via the first torque transmission path TP 1 , the second torque transmission path TP 2  including the plurality of outer springs SPo, and the rotary inertia mass damper  20 . 
     When the drive member  11  is rotated (twisted) relative to the driven member  15  under engagement of the lockup (engagement of the lockup clutch  8 ), the first and the second inner springs SP 1  and SP 2  are deflected, and the sun gear  22  as the mass body is rotated with relative rotation of the drive member  11  to the driven member  15 . More specifically, when the drive member  11  is rotated relative to the driven member  15 , the rotation speed of the lockup piston  80  serving as the planetary carrier that is the input element of the planetary gear  21  (and the rotation speed of the drive member  11 ) is higher than the rotation speed of the driven member  15  that is rotated integrally with the ring gear  23 . In this state, the rotation speed of the sun gear  22  is increased by the action of the planetary gear  21 , so that the sun gear  22  is rotated at a higher rotation speed than the rotation speed of the lockup piston  80  and the drive member  11 . This causes the moment of inertia (the inertia) to be applied from the sun gear  22  that is the mass body of the rotary inertia mass damper  20  to the driven member that is the output element of the damper device  10  and thereby damps the vibration of the driven member  15 . 
     The rotary inertia mass damper  20  is configured to mainly transmit the inertia torque between the drive member  11  and the driven member  15  but not to transmit the average torque. In the damper device  10 , when the input torque reaches the above torque T 2 , a stopper ST restricts the relative rotation of the drive member  11  to the driven member  15  and thereby restricts the deflections of all the first and the second inner springs SP 1  and SP 2  and the outer springs SPo. According to this embodiment, the stopper ST includes collars  19  mounted to a plurality of rivets provided to clamp the first and the second output plate members  16  and  17  (and the ring gear  23 ), and a plurality of openings  11   s  formed in the drive member  11 , for example, in an arc shape. In the mounting state of the damper device  10 , each of the collars  19  is arranged in the corresponding opening  11   s  of the drive member  11  such as not to abut on respective inner wall surfaces defining the opening  11   s . When each of the collars  19  abuts on one of the inner wall surfaces of the corresponding opening  11   s  accompanied with relative rotation of the drive member  11  to the driven member  15 , this restricts the relative rotation of the drive member  11  to the driven member  15  and the deflections of the springs SP 1 , SP 2  and SPo. 
     The following describes a design procedure of the damper device  10 . 
     As described above, in the damper device  10 , until the input torque transmitted to the drive member  11  reaches the above torque T 1 , the first and the second inner springs SP 1  and SP 2  included in the first torque transmission path TP 1  work in parallel to the rotary inertia mass damper  20 . When the first and the second inner springs SP 1  and SP 2  work in parallel to the rotary inertia mass damper  20 , the torque transmitted from the first torque transmission path TP 1  including the intermediate member  12  and the first and the second inner springs SP 1  and SP 2  to the driven member  15  depends on (is proportional to) the displacement (amount of deflection or torsion angle) of the second inner springs SP 2  between the intermediate member  12  and the driven member  15 . The torque transmitted from the rotary inertia mass damper  20  to the driven member  15 , on the other hand, depends on (is proportional to) a difference in angular acceleration between the drive member  11  and the driven member  15 , i.e., a second order differential equation result of the displacement of the first and the second inner springs SP 1  and SP 2  between the drive member  11  and the driven member  15 . On the assumption that the input torque transmitted to the drive member  11  of the damper device  10  is periodically vibrated as shown by Equation (1) given below, the phase of the vibration transmitted from the drive member  11  to the driven member  15  via the first torque transmission path TP 1  is accordingly shifted by 180 degrees from the phase of the vibration transmitted from the drive member  11  to the driven member  15  via the rotary inertia mass damper  20 . 
     [Math. 1]
 
T=T 0  sin ωt  (1)
 
     Additionally, in the damper device  10  including the intermediate member  12 , two natural frequencies (resonance frequencies) may be set in the state that the deflections of the first and the second inner springs SP 1  and SP 2  are allowed and the outer springs SPo are not deflected. On the assumption that transmission of the torque from the engine EG to the drive member  11  is started under engagement of the lockup by the lockup clutch  8 , the resonance by the vibration of the drive member  11  and the driven member  15  in the opposite phases and the resonance mainly in the transmission (first resonance, shown as a resonance point R 1  in  FIG. 4 ) between the drive member  11  and a non-illustrated driveshaft occur in the first torque transmission path TP 1  in the state that the deflections of the first and the second inner springs SP 1  and SP 2  are allowed and the outer springs SPo are not deflected. 
     The intermediate member  12  of the first torque transmission path TP 1  is formed in an annular shape. This causes the inertia force applied to the intermediate member  12  to be greater than the resistance force of interfering with vibration of the intermediate member  12  (frictional force caused by the centrifugal force mainly applied to the rotating intermediate member  12 ) in the process of transmitting the torque from the engine EG to the drive member  11 . Accordingly, a damping ratio ζ of the intermediate member  12  that is vibrated with transmission of the torque from the engine EG to the drive member  11  becomes less than a value  1 . The damping ratio ζ of the intermediate member  12  in a single-degree-of-freedom system may be expressed as ζ=C/(2·√(J 2 ·(k 1 +k 2 )). Herein “J 2 ” denotes a moment of inertia of the intermediate member  12  (total moment of inertia of the intermediate member  12  and the turbine runner according to this embodiment), “k 1 ” denotes a combined spring constant of the plurality of first inner springs SP 1  working in parallel between the drive member  11  and the intermediate member  12 , “k 2 ” denotes a combined spring constant of the plurality of second inner springs SP 2  working in parallel between the intermediate member  12  and the driven member  15 , and “C” denotes a damping force (resistance fore) per unit speed of the intermediate member  12  that interferes with the vibration of the intermediate member  12 . Accordingly the damping ratio ζ of the intermediate member  12  is determined based on at least the moment of inertia J 2  of the intermediate member  12  and the rigidities k 1  and k 2  of the first and the second inner springs SP 1  and SP 2 . 
     The above damping force C may be determined by a procedure given below. When a displacement x of the intermediate member  12  is expressed as x=A·sin(ω 12 ·t), a loss energy Sc by the above damping force C may be expressed as Sc=π·C·A 2 ·ω 12  (where “A” denotes an amplitude and “ω 12 ” denotes a vibration frequency of the intermediate member  12 ). Additionally, when the displacement x of the intermediate member  12  is expressed as x=A·sin(ω 12 ·t), a loss energy Sh by the above hysteresis H in one cycle vibration of the intermediate member  12  may be expressed as Sh=2·H·A. On the assumption that the loss energy Sc by the above damping force C is equal to the loss energy Sh by the hysteresis H, the above damping force C may be expressed as C=(2·H)/(π·A·ω 12 ). 
     Additionally, a natural frequency f 12  of the intermediate member  12  in the single-degree-of-freedom system is expressed as f 12 =½π·√((k 1 +k 2 )/J 2 ). Forming the intermediate member  12  in an annular shape relatively increases the moment of inertia J 2 , so that the natural frequency f 12  of the intermediate member  12  relatively decreases. Accordingly, as shown in  FIG. 4 , in the state that the deflections of the first and the second inner springs SP 1  and SP 2  are allowed and the outer springs SPo are not deflected, the resonance of the intermediate member  12  (second resonance, shown as a resonance point R 2  in  FIG. 4 ) by the vibration of the intermediate member  12  in the opposite phase to those of the drive member  11  and the driven member  15  occurs in the first torque transmission path TP 1  at the stage when the rotation speed of the drive member  11  reaches a rotation speed corresponding to the greater between the two natural frequencies, i.e., at a higher rotation speed (higher frequency) than the first resonance. 
     In order to further improve the vibration damping effect of the damper device  10  having the above characteristics, as the result of intensive studies and analyses, the inventors have noted that the damper device  10  can damp the vibration of the driven member  15  by making the amplitude of the vibration of the first torque transmission path TP 1  equal to the amplitude of the vibration of the rotary inertia mass damper  20  in the opposite phase. The inventors have established an equation of motion as shown by Equation (2) given below in a vibration system including the damper device  10  in which the torque is transmitted from the engine EG to the drive member  11  under engagement of the lockup and the outer springs SPo are not deflected. In Equation (2), “J 1 ” denotes a moment of inertia of the drive member  11 , “J 2 ” denotes the moment of inertia of the intermediate member  12  as described above, “J 3 ” denotes a moment of inertia of the driven member  15 , and “J 1 ” denotes a moment of inertia of the sun gear  22  that is the mass body of the rotary inertia mass damper  20 . Additionally, “θ 1 ” denotes a torsion angle of the drive member  11 , “θ 2 ” denotes a torsion angle of the intermediate member  12 , “θ 3 ” denotes a torsion angle of the driven member  15 , and “λ” denotes a gear ratio of the planetary gear  21  (number of teeth of the sun gear  22 /number of teeth of the ring gear  23 ) included in the rotary inertia mass damper  20 . 
     
       
         
           
             
               
                 
                   
                       
                   
                   ⁢ 
                   
                     [ 
                     
                       Math 
                       . 
                       
                           
                       
                       ⁢ 
                       2 
                     
                     ] 
                   
                 
               
               
                 
                     
                 
               
             
             
               
                 
                   
                     [ 
                     
                       
                         
                           
                             
                               J 
                               1 
                             
                             + 
                             
                               
                                 J 
                                 i 
                               
                               · 
                               
                                 
                                   
                                     ( 
                                     
                                       1 
                                       + 
                                       λ 
                                     
                                     ) 
                                   
                                   2 
                                 
                                 
                                   λ 
                                   2 
                                 
                               
                             
                           
                         
                         
                           0 
                         
                         
                           
                             
                               - 
                               
                                 J 
                                 i 
                               
                             
                             · 
                             
                               
                                 1 
                                 + 
                                 λ 
                               
                               
                                 λ 
                                 2 
                               
                             
                           
                         
                       
                       
                         
                           0 
                         
                         
                           
                             J 
                             2 
                           
                         
                         
                           0 
                         
                       
                       
                         
                           
                             
                               - 
                               
                                 J 
                                 i 
                               
                             
                             · 
                             
                               
                                 1 
                                 + 
                                 λ 
                               
                               
                                 λ 
                                 2 
                               
                             
                           
                         
                         
                           0 
                         
                         
                           
                             
                               J 
                               3 
                             
                             + 
                             
                               
                                 J 
                                 i 
                               
                               · 
                               
                                 1 
                                 
                                   λ 
                                   2 
                                 
                               
                             
                           
                         
                       
                     
                     ] 
                   
                   ⁢ 
                   
                       
                     
                       
                         
                           [ 
                           
                             
                               
                                 
                                   
                                     θ 
                                     ¨ 
                                   
                                   1 
                                 
                               
                             
                             
                               
                                 
                                   
                                     θ 
                                     ¨ 
                                   
                                   2 
                                 
                               
                             
                             
                               
                                 
                                   
                                     θ 
                                     ¨ 
                                   
                                   3 
                                 
                               
                             
                           
                           ] 
                         
                         + 
                         
                           
                             [ 
                             
                               
                                 
                                   
                                     k 
                                     1 
                                   
                                 
                                 
                                   
                                     - 
                                     
                                       k 
                                       1 
                                     
                                   
                                 
                                 
                                   0 
                                 
                               
                               
                                 
                                   
                                     - 
                                     
                                       k 
                                       1 
                                     
                                   
                                 
                                 
                                   
                                     
                                       k 
                                       1 
                                     
                                     + 
                                     
                                       k 
                                       2 
                                     
                                   
                                 
                                 
                                   
                                     - 
                                     
                                       k 
                                       2 
                                     
                                   
                                 
                               
                               
                                 
                                   0 
                                 
                                 
                                   
                                     - 
                                     
                                       k 
                                       2 
                                     
                                   
                                 
                                 
                                   
                                     k 
                                     2 
                                   
                                 
                               
                             
                             ] 
                           
                           ⁡ 
                           
                             [ 
                             
                               
                                 
                                   
                                     θ 
                                     1 
                                   
                                 
                               
                               
                                 
                                   
                                     θ 
                                     2 
                                   
                                 
                               
                               
                                 
                                   
                                     θ 
                                     3 
                                   
                                 
                               
                             
                             ] 
                           
                         
                       
                       = 
                       
                         [ 
                         
                           
                             
                               T 
                             
                           
                           
                             
                               0 
                             
                           
                           
                             
                               0 
                             
                           
                         
                         ] 
                       
                     
                   
                 
               
               
                 
                   ( 
                   2 
                   ) 
                 
               
             
           
         
       
     
     Additionally, the inventors have assumed that the input torque T is periodically vibrated as shown by Equation (1) given above and have also assumed that the torsion angle θ 1  of the drive member  11 , the torsion angle θ 2  of the intermediate member  12  and the torsion angle θ 3  of the driven member  15  are periodically responded (vibrated) as shown by Equation (3) given below. In Equations (1) and (3), “ω” denotes an angular frequency in the periodical fluctuation (vibration) of the input torque T. In Equation (3), “Θ 1 ” denotes an amplitude of the vibration (vibration amplitude, i.e., maximum torsion angle) of the drive member  11  generated during transmission of the torque from the engine EG, “Θ 2 ” denotes an amplitude of vibration (vibration amplitude) of the intermediate member  12  generated during transmission of the torque from the engine EG to the drive member  11 , and “Θ 3 ” denotes an amplitude of vibration (vibration amplitude) of the driven member  15  generated during transmission of the torque from the engine EG to the drive member  11 . On such assumptions, an identity of Equation (4) given below is obtained by substituting Equations (1) and (3) into Equation (2) and eliminating “sin ωt” from both sides. 
     
       
         
           
             
               
                 
                   
                       
                   
                   ⁢ 
                   
                     [ 
                     
                       Math 
                       . 
                       
                           
                       
                       ⁢ 
                       3 
                     
                     ] 
                   
                 
               
               
                 
                     
                 
               
             
             
               
                 
                   
                       
                   
                   ⁢ 
                   
                     
                       [ 
                       
                         
                           
                             
                               θ 
                               1 
                             
                           
                         
                         
                           
                             
                               θ 
                               2 
                             
                           
                         
                         
                           
                             
                               θ 
                               3 
                             
                           
                         
                       
                       ] 
                     
                     = 
                     
                       
                         [ 
                         
                           
                             
                               
                                 Θ 
                                 1 
                               
                             
                           
                           
                             
                               
                                 Θ 
                                 2 
                               
                             
                           
                           
                             
                               
                                 Θ 
                                 3 
                               
                             
                           
                         
                         ] 
                       
                       ⁢ 
                       sin 
                       ⁢ 
                       
                           
                       
                       ⁢ 
                       wt 
                     
                   
                 
               
               
                 
                   ( 
                   3 
                   ) 
                 
               
             
             
               
                 
                   
                     [ 
                     
                       
                         
                           
                             T 
                             0 
                           
                         
                       
                       
                         
                           0 
                         
                       
                       
                         
                           0 
                         
                       
                     
                     ] 
                   
                   = 
                   
                       
                     
                       
                         [ 
                         
                           
                             
                               
                                 
                                   k 
                                   1 
                                 
                                 - 
                                 
                                   
                                     ω 
                                     2 
                                   
                                   ⁡ 
                                   
                                     ( 
                                     
                                       
                                         J 
                                         1 
                                       
                                       + 
                                       
                                         
                                           J 
                                           i 
                                         
                                         · 
                                         
                                           
                                             
                                               ( 
                                               
                                                 1 
                                                 + 
                                                 λ 
                                               
                                               ) 
                                             
                                             2 
                                           
                                           
                                             λ 
                                             2 
                                           
                                         
                                       
                                     
                                     ) 
                                   
                                 
                               
                             
                             
                               
                                 - 
                                 
                                   k 
                                   1 
                                 
                               
                             
                             
                               
                                 
                                   ω 
                                   2 
                                 
                                 ⁡ 
                                 
                                   ( 
                                   
                                     
                                       J 
                                       i 
                                     
                                     · 
                                     
                                       
                                         1 
                                         + 
                                         λ 
                                       
                                       
                                         λ 
                                         2 
                                       
                                     
                                   
                                   ) 
                                 
                               
                             
                           
                           
                             
                               
                                 - 
                                 
                                   k 
                                   1 
                                 
                               
                             
                             
                               
                                 
                                   
                                     k 
                                     1 
                                   
                                   + 
                                   
                                     k 
                                     2 
                                   
                                 
                                 = 
                                 
                                   
                                     ω 
                                     2 
                                   
                                   ⁢ 
                                   
                                     J 
                                     2 
                                   
                                 
                               
                             
                             
                               
                                 - 
                                 
                                   k 
                                   2 
                                 
                               
                             
                           
                           
                             
                               
                                 
                                   ω 
                                   2 
                                 
                                 ⁡ 
                                 
                                   ( 
                                   
                                     
                                       J 
                                       i 
                                     
                                     · 
                                     
                                       
                                         1 
                                         + 
                                         λ 
                                       
                                       
                                         λ 
                                         2 
                                       
                                     
                                   
                                   ) 
                                 
                               
                             
                             
                               
                                 - 
                                 
                                   k 
                                   2 
                                 
                               
                             
                             
                               
                                 
                                   k 
                                   2 
                                 
                                 - 
                                 
                                   
                                     ω 
                                     2 
                                   
                                   ⁡ 
                                   
                                     ( 
                                     
                                       
                                         J 
                                         3 
                                       
                                       + 
                                       
                                         
                                           J 
                                           i 
                                         
                                         · 
                                         
                                           1 
                                           
                                             λ 
                                             2 
                                           
                                         
                                       
                                     
                                     ) 
                                   
                                 
                               
                             
                           
                         
                         ] 
                       
                       ⁢ 
                       
                           
                         
                           [ 
                           
                             
                               
                                 
                                   Θ 
                                   1 
                                 
                               
                             
                             
                               
                                 
                                   Θ 
                                   2 
                                 
                               
                             
                             
                               
                                 
                                   Θ 
                                   3 
                                 
                               
                             
                           
                           ] 
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   4 
                   ) 
                 
               
             
           
         
       
     
     In Equation (4), when the vibration amplitude Θ 3  of the driven member  15  is zero, this means that the vibration from the engine EG is theoretically damped completely by the damper device  10  and that no vibration is theoretically transmitted to the transmission TM, the driveshaft and the like located downstream of the driven member  15 . From this point of view, the inventors have obtained a conditional expression of Equation (5) by solving the identity of Equation (4) with respect to the vibration amplitude Θ 3  and setting Θ 3 =0. Equation (5) is a quadratic equation with regard to the square of angular frequency ω 2  in the periodical fluctuation of the input torque T. When the square of angular frequency ω 2  is either of two real roots (or multiple root) of Equation (5), the vibration from the engine EG transmitted from the drive member  11  to the driven member  15  via the first torque transmission path TP 1  and the vibration transmitted from the drive member  11  to the driven member  15  via the rotary inertia mass damper  20  are cancelled out each other, and the vibration amplitude Θ 3  of the driven member  15  theoretically becomes equal to zero. 
     
       
         
           
             
               
                 
                   [ 
                   
                     Math 
                     . 
                     
                         
                     
                     ⁢ 
                     4 
                   
                   ] 
                 
               
               
                 
                     
                 
               
             
             
               
                 
                   
                     
                       
                         
                           J 
                           2 
                         
                         · 
                         
                           J 
                           i 
                         
                         · 
                         
                           
                             1 
                             + 
                             λ 
                           
                           
                             λ 
                             2 
                           
                         
                       
                       ⁢ 
                       
                         
                           ( 
                           
                             ω 
                             2 
                           
                           ) 
                         
                         2 
                       
                     
                     - 
                     
                       
                         J 
                         i 
                       
                       · 
                       
                         
                           1 
                           + 
                           λ 
                         
                         
                           λ 
                           2 
                         
                       
                       · 
                       
                         ( 
                         
                           
                             k 
                             1 
                           
                           + 
                           
                             k 
                             2 
                           
                         
                         ) 
                       
                       · 
                       
                         ω 
                         2 
                       
                     
                     + 
                     
                       
                         k 
                         1 
                       
                       · 
                       
                         k 
                         2 
                       
                     
                   
                   = 
                   0 
                 
               
               
                 
                   ( 
                   5 
                   ) 
                 
               
             
           
         
       
     
     This result of analysis indicates that a total of two antiresonance points providing theoretically zero vibration amplitude Θ 3  of the driven member  15  may be set in the damper device  10  that includes the intermediate member  12  and accordingly provides two peaks, i.e., a resonance in the torque transmitted via the first torque transmission path TP 1  as shown in  FIG. 4 . The damper device  10  can thus significantly effectively damp the vibration of the driven member  15  by making the amplitude of the vibration of the first torque transmission path TP 1  equal to the amplitude of the vibration of the rotary inertia mass damper  20  in the opposite phase at two points corresponding to the two resonances occurring in the first torque transmission path TP 1 . 
     Additionally, in the damper device  10 , the resonance of the intermediate member  12  occurs in the stage where the rotation speed of the drive member  11  becomes rather higher than a rotation speed corresponding to the frequency at an antiresonance point A 1  of the lower rotation speed (the lower frequency). The amplitude of the vibration transmitted from the second inner springs SP 2  to the driven member  15  changes from decreasing to increasing before the rotation speed of the drive member  11  (engine EG) reaches a rotation speed corresponding to the relatively low natural frequency of the intermediate member  12 , as shown by a one-dot chain line curve in  FIG. 4 . Even when the amplitude of the vibration transmitted from the rotary inertia mass damper  20  to the driven member  15  is gradually increased with an increase in rotation speed of the drive member  11  (as shown by a two-dot chain line curve in  FIG. 4 ), this expands an area where the vibration transmitted from the rotary inertia mass damper  20  to the driven member  15  cancels out at least part of the vibration transmitted from the second inner springs SP 2  to the driven member  15 . This results in further improving the vibration damping performance of the damper device  10  in a relatively low rotation speed range of the drive member  11 . 
     A vehicle equipped with the engine EG as the source of generating power for driving may be configured as to further decrease a lockup rotation speed Nlup of the lockup clutch  8  (rotation speed at the time of first coupling of the engine EG with the damper device  10  after a start of the engine EG and the lowest among a plurality of lockup rotation speeds; in other words, minimum rotation speed in a rotation speed range where the torque is transmitted from the drive member  11  through the torque transmission path TP 1  to the driven member  15 ) and mechanically transmit the torque from the engine EG to the transmission TM at an earlier timing, such as to improve the power transmission efficiency between the engine EG and the transmission TM and thereby further improve the fuel consumption of the engine EG. The vibration transmitted from the engine EG via the lockup clutch  8  to the drive member  11 , however, increase in a low rotation speed range of approximately 500 rpm to 1500 rpm that is likely to be set as a range of the lockup rotation speed Nlup. The vibration level significantly increases especially in a vehicle equipped with a smaller-number cylinder engine such as three-cylinder engine or four-cylinder engine. Accordingly, in order to suppress transmission of a large vibration to the transmission TM and so on during or immediately after engagement the lockup, there is a need to further reduce the vibration level in a rotation speed range of about the lockup rotation speed Nlup of the entire damper device  10  (driven member  15 ) configured to transmit the torque (vibration) from the engine EG to the transmission TM under engagement of the lockup. 
     By taking into account the foregoing, the inventors have configured the damper device  10  such as to form the antiresonance point A 1  of the lower rotation speed (the lower frequency) when the rotation speed Ne of the engine EG is in the range of 500 rpm to 1500 rpm (in the expected setting range of the lockup rotation speed Nlup), based on the predetermined lockup rotation speed Nlup of the lockup clutch  8 . Two solutions ω 1  and ω 2  of Equation (5) given above may be obtained as Equations (6) and (7) given below according to the quadratic formula, and ω 1 &lt;ω 2 . A frequency fa 1  at the antiresonance point A 1  of the lower rotation speed (the lower frequency) (hereinafter referred to as “minimum frequency”) is expressed by Equation (8) given below, and a frequency fa 2  at an antiresonance point A 2  of the higher rotation speed (the higher frequency) (fa 2 &gt;fa 1 ) is expressed by Equation (9) given below. A rotation speed Nea 1  of the engine EG corresponding to the minimum frequency fa 1  is expressed as Nea 1 =(120/n)·fa 1 , where “n” denotes the number of cylinders of the engine EG. 
     
       
         
           
             
               
                 
                   [ 
                   
                     Math 
                     . 
                     
                         
                     
                     ⁢ 
                     5 
                   
                   ] 
                 
               
               
                 
                     
                 
               
             
             
               
                 
                   
                     ω 
                     1 
                     2 
                   
                   = 
                   
                     
                       
                         ( 
                         
                           
                             k 
                             1 
                           
                           + 
                           
                             k 
                             2 
                           
                         
                         ) 
                       
                       - 
                       
                         
                           
                             
                               ( 
                               
                                 
                                   k 
                                   1 
                                 
                                 + 
                                 
                                   k 
                                   2 
                                 
                               
                               ) 
                             
                             2 
                           
                           - 
                           
                             4 
                             · 
                             
                               
                                 J 
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     Accordingly, the combined spring constant k 1  of the plurality of first inner springs SP 1 , the combined spring constant k 2  of the plurality of second inner springs SP 2 , the moment of inertia J 2  of the intermediate member  12  (determined by taking into account (summing up) the moments of inertia of the turbine runner  5  and the like coupled to be integrally rotated), and the moment of inertia J 1  of the sun gear  22  that is the mass body of the rotary inertia mass damper  20  are selected and set in the damper device  10 , in order to satisfy Expression (10) given below. More specifically, in the damper device  10 , the spring constants k 1  and k 2  of the first and the second inner springs SP 1  and SP 2  and the moment of inertia J 2  of the intermediate member  12  are determined, based on the above minimum frequency fa 1  (and the lockup rotation speed Nlup). 
     
       
         
           
             
               
                 
                   [ 
                   
                     Math 
                     . 
                     
                         
                     
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                     6 
                   
                   ] 
                 
               
               
                 
                     
                 
               
             
             
               
                 
                   
                     500 
                     ⁢ 
                     
                         
                     
                     ⁢ 
                     rpm 
                   
                   ≤ 
                   
                     
                       
                         120 
                         n 
                       
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                       f 
                     
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                     a 
                   
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                     1500 
                     ⁢ 
                     
                         
                     
                     ⁢ 
                     rpm 
                   
                 
               
               
                 
                   ( 
                   10 
                   ) 
                 
               
             
           
         
       
     
     As described above, the antiresonance point A 1  of the lower rotation speed that is likely to provide theoretically zero vibration amplitude Θ 3  of the driven member  15  (that is likely to further decrease the vibration amplitude Θ 3 ) may be set in the low rotation speed range of 500 rpm to 1500 rpm (in the expected setting range of the lockup rotation speed Nlup). As shown in  FIG. 4 , this shifts one resonance having the lower frequency (first resonance) out of the resonances occurring in the first torque transmission path TP 1  toward the lower rotation speed (toward the lower frequency), such as to be included in a non-lockup area of the lockup clutch  8  (shown by the two-dot chain line curve in  FIG. 4 ). According to this embodiment, the resonance at the resonance point R 1  (i.e., resonance at the lower natural frequency between two natural frequencies) is thus a virtual resonance that does not occur in the rotation speed range where the damper device  10  is used. This results in allowing for the lockup (coupling of the engine EG with the drive member  11 ) at the lower rotation speed. 
     When the damper device  10  is configured to satisfy Expression (10), it is preferable to select and set the spring constants k 1  and k 2  and the moments of inertia J 2  and J i , such as to minimize the frequency of the lower rotation-speed (lower-frequency) resonance (at the resonance point R 1  shown as resonance point R 1  in  FIG. 4 ) occurring in the first torque transmission path TP 1  to the minimum possible value that is lower than the above minimum frequency fa 1 . This further reduces the minimum frequency fa 1  and allows for the lockup at the further lower rotation speed. 
     Moreover, the configuration that the two antiresonance points A 1  and A 2  are set enables the antiresonance point A 1  having the minimum frequency (fa 1 ) between the two antiresonance points A 1  and A 2  to be shifted toward the lower frequency, compared with the configuration that only one antiresonance point is set (shown by a broken line curve in  FIG. 4 ). Additionally, as clearly understood from  FIG. 4 , the configuration that the two antiresonance points A 1  and A 2  are set enables the vibration from the engine EG transmitted from the drive member  11  to the driven member  15  via the first torque transmission path TP 1  (shown by the one-dot chain line curve in  FIG. 4 ) to be effectively damped by the vibration transmitted from the drive member  11  to the driven member  15  via the rotary inertia mass damper  20  (shown by the two-dot chain line curve in  FIG. 4 ) in a relatively wide rotation speed range between the two antiresonance points A 1  and A 2 . 
     This further improves the vibration damping effect of the damper device  10  in the lower rotation speed range of a lockup area that is likely to increase the vibration from the engine EG. In the damper device  10 , on the occurrence of the second resonance (second resonance as shown by the resonance point R 2  in  FIG. 4 ), the intermediate member  12  is vibrated in the opposite phase to that of the driven member  15 . As shown by the one-dot chain line curve in  FIG. 4 , the phase of the vibration transmitted from the drive member  11  to the driven member  15  via the first torque transmission path TP 1  becomes identical with the phase of the vibration transmitted from the drive member  11  to the driven member  15  via the rotary inertia mass damper  20 . 
     In the damper device  10  configured as described above, in order to further improve the vibration damping performance around the lockup rotation speed Nlup, there is a need to appropriately separate the lockup rotation speed Nlup and the rotation speed of the engine EG corresponding to the resonance point R 2 . Accordingly, when the damper device  10  is configured to satisfy Expression (10), it is preferable to select and set the spring constants k 1  and k 2  and the moments of inertia J 2  and J i , such as to satisfy Nlup≤(120/n)·fa 1  (=Nea 1 ). This engages the lockup by the lockup clutch  8 , while effectively suppressing transmission of the vibration to the input shaft IS of the transmission TM. This also enables the vibration from the engine EG to be remarkably effectively damped by the damper device  10 , immediately after engagement of the lockup. 
     Additionally, coupling the intermediate member  12  with the turbine runner  5  to be integrally rotated further increases the substantial moment of inertia J 2  of the intermediate member  12  (total moment of inertia of the intermediate member  12 , the turbine runner  5  and the like). This further decreases the frequency fa 1  at the antiresonance point A 1  and enables the antiresonance point A 1  to be set at the lower rotation speed (at the lower frequency) as clearly understood from Equation (8). 
     As described above, designing the damper device  10  based on the frequency (minimum frequency) fa 1  at the antiresonance point A 1  remarkably effectively improves the vibration damping performance of the damper device  10 . According to the inventors&#39; studies and analyses, it has been confirmed that when the lockup rotation speed Nlup is set to, for example, a value of about 1000 rpm, the damper device  10  configured to satisfy, for example, 900 rpm≤(120/n)·fa 1 ≤1200 rpm provides the remarkably effective results in practice. 
     The rotary inertia mass damper  20  included in the damper device  10  described above includes the lockup piston  80  that serves as the planetary carrier (first element) rotating integrally with the drive member  11 , and the planetary gear  21  that includes the ring gear  23  (second element) rotating integrally with the driven member  15  and the sun gear  22  (third element as the mass body) rotating integrally with the mass portion  22   m . This configuration increases the rotation speed of the sun gear  22  as the mass body to be higher than the rotation speed of the drive member  11 . Accordingly this reduces the mass body of the rotary inertia mass damper  20 , while effectively ensuring the moment of inertia applied from the rotary inertia mass damper  20  to the driven member  15 . This also enhances the flexibility in design of the rotary inertia mass damper  20  and the entire damper device  10 . The planetary gear  21  of the rotary inertia mass damper  20  may, however, be configured to decrease the rotation speed of the mass body (sun gear  22 ) to be lower than the rotation speed of the drive member  11 , according to the magnitude of the moment of inertia of the mass body. The rotary inertia mass damper  20  may include, for example, a differential with a link mechanism and so on, other than the planetary gear  21 . Additionally, depending on the magnitude of the moment of inertia of the mass body, the rotary inertia mass damper  20  may be a constant speed rotation mechanism configured to rotate the mass body at the same speed as that of the drive member  11 . The rotary inertia mass damper  20  may include a speed change mechanism configured to change the rotation speed of the mass body according to the rotation speed of relative rotation of the drive member  11  to the driven member  15 . 
     As described above, the configuration that two antiresonance points A 1  and A 2  are set enables the antiresonance point A 1  to be shifted toward the lower frequency. Depending on the specification of the vehicle, the motor and so on equipped with the damper device  10 , the multiple root of Equation (5) (=½π·√{(k 1 +k 2 )/(2·J 2 )} may be set to the above minimum frequency fa 1 . Determining the spring constants k 1  and k 2  of the first and the second inner springs SP 1  and SP 2  and the moment of inertia J 2  of the intermediate member  12  based on the multiple root of Equation (5) also improves the vibration damping effect of the damper device  10  in the lower rotation speed range of the lockup area that is likely to increase the vibration from the engine EG as shown by the broken line curve in  FIG. 4 . 
     In the damper device  10  described above, springs having the identical specification (spring constant) are employed for the first and the second inner springs SP 1  and SP 2 . This is, however, not restrictive. The spring constants k 1  and k 2  of the first and the second inner springs SP 1  and SP 2  may be different from each other (k 1 &gt;k 2  or k 1 &lt;k 2 ). This further increases the value of the √ term (discriminant) in Equations (6) and (8) and further increases the interval between the two antiresonance points A 1  and A 2 , thus further improving the vibration damping effect of the damper device in the low frequency range (low rotation speed range). In this case, the damper device  10  may be provided with a stopper configured to restrict the deflection of one of the first and the second inner springs SP 1  and SP 2  (for example, one having the lower rigidity). 
     Moreover, in the damper device  10  described above, the plurality of pinion gears  24  of the planetary gear  21  configuring the rotary inertia mass damper  20  are supported by the lockup piston  80  that serves as the planetary carrier. The plurality of pinion gears  24  of the planetary gear  21  may be rotatably supported by the drive member  11  of the damper device  10 . In other words, the drive member  11  itself may be used as the planetary carrier of the planetary gear  21 . As shown by the two-dot chain line in  FIG. 1 , the turbine runner  5  may be coupled with either one of the drive member  11  and the driven member  15 . Even when the turbine runner  5  is not coupled with the intermediate member  12 , formation of the intermediate member  12  in the annular shape suppresses the motion of the intermediate member  12  in the radial direction by the centrifugal force. The damping ratio ζ of the intermediate member  12  that is vibrated with transmission of the torque from the engine EG to the drive member  11  thus basically becomes less than the value  1 . 
     Additionally, the rotary inertia mass damper  20  (planetary gear  21 ) may be configured such that the ring gear  23  (first element) is rotated integrally with the drive member  11  (lockup piston  80 ) and that the planetary carrier (second element) is rotated integrally with the driven member  15 . The rotary inertia mass damper  20  (planetary gear  21 ) may be configured such that the ring gear  23  (first element) is rotated integrally with the drive member  11  (lockup piston  80 ), that the sun gear  22  (second element) is rotated integrally with the driven member  15  and that the planetary carrier (third element) is rotated integrally with the mass portion (mass body). In the latter configuration, the mass portion may be arranged on at least one of the outer side in the radial direction of the ring gear  23  and the inner side in the radial direction of the sun gear  22 . Furthermore, the rotary inertia mass damper  20  (planetary gear  21 ) may be configured such that the sun gear  22  (first element) is rotated integrally with the drive member  11  (lockup piston  80 ), that the ring gear  23  (second element) is rotated integrally with the driven member  15  and that the planetary carrier (third element) is rotated integrally with the mass portion (mass body). In this configuration, the mass portion may be arranged on at least one of the outer side in the radial direction of the ring gear  23  and the inner side in the radial direction of the sun gear  22 . 
       FIG. 5  is a schematic configuration diagram illustrating a starting device  1 X including a damper device  10 X according to another embodiment of the disclosure. Among the components of the starting device  1 X and the damper device  10 X, the same components to those of the starting device  1  and the damper device  10  described above are expressed by the same reference signs and their repeated description is omitted. 
     In the damper device  10 X shown in  FIG. 5 , a turbine runner  5  that configures, in combination with a pump impeller  4 , a torque converter (fluid transmission device) is coupled via a coupling member with a sun gear  22  of a planetary gear  21  configuring a rotary inertia mass damper  20 X. In other words, the rotary inertia mass damper  20 X includes the turbine runner  5  as the mass body that is rotated with relative rotation of the drive member  11  to the driven member  15 . This configuration further increases the moment of inertia of the mass body of the rotary inertia mass damper  20 X and further increases the moment of inertia applied from the rotary inertia mass damper  20 X to the driven member  15 , while suppressing overall size expansion of the damper device  10 , compared with a configuration equipped with a dedicated mass body (mass portion). 
       FIG. 6  is a schematic configuration diagram illustrating a starting device  1 X′ including a damper device  10 X′ according to yet another embodiment of the disclosure. Among the components of the starting device  1 X′ and the damper device  10 X′, the same components to those of the starting device  1  and the damper device  10  described above are expressed by the same reference signs and their repeated description is omitted. 
     The damper device  10 X′ shown in  FIG. 6  includes a drive member (input element)  11 ′, an intermediate member (intermediate element)  12 ′ and a driven member (output element)  15 ′, as rotational elements. The damper device  10 X′ also includes a plurality of first springs (first elastic bodies) SP 1 ′ configured to transmit the torque between the drive member  11 ′ and the intermediate member  12 ′ and a plurality of second springs (second elastic bodies) SP 2 ′ configured to respectively work in series with the corresponding first springs SP 1 ′ and to transmit the torque between the intermediate member  12 ′ and the driven member  15 ′, as torque transmission elements (torque transmission elastic bodies). The plurality of first springs (first elastic bodies) SP 1 ′, the intermediate member  12 ′ and the plurality of second springs (second elastic bodies) SP 2 ′ form a torque transmission path TP 1  between the drive member  11 ′ and the driven member  15 ′. The intermediate member  12 ′ is an annular member. A damping ratio (of the intermediate member  12 ′ that is vibrated with transmission of the torque from the engine EG to the drive member  11 ′ is less than the value  1 . 
     The damper device  10 X′ further includes a plurality of inner springs SPi configured to transmit the torque between the drive member  11 ′ and the driven member  15 ′. Like the outer springs SPo of the damper device  10  described above, the plurality of inner springs SPi are configured to work in parallel to the first and the second springs SP 1 ′ and SP 2 ′ forming the torque transmission path TP 1 , after the input torque into the drive member  11 ′ reaches the above torque (first threshold value) T 1  and the torsion angle of the drive member  11 ′ relative to the driven member  15 ′ becomes equal to or larger than the above predetermined angle θref. In the illustrated example of  FIG. 6 , the springs SPi are arranged on the inner side in the radial direction of the first and the second springs SP 1 ′ and SP 2 ′ to be surrounded by the first and the second springs SP 1 ′ and SP 2 ′. 
     In the damper device  10 X′, the drive member  11 ′ supports a plurality of pinion gears  24  to be rotatable (rotatable on their own axes) and revolvable about a sun gear  22  and a ring gear  23 , and serves as a planetary carrier (first element) of a planetary gear  21  configuring a rotary inertia mass damper  20 X′. Furthermore, the sun gear (second element)  22  is coupled with the driven member  15 ′ to be integrally rotated. Accordingly the ring gear  23  and a mass portion  23   m  as the mass body (third element) are rotated with relative rotation of the drive member  11 ′ to the driven member  15 ′. This damper device  10 X′ provides the similar functions and advantageous effects to those of the damper device  10  and so on described above. The rotary inertia mass damper  20 X′ (planetary gear  21 ) may be configured such that the sun gear  22  (first element) is rotated integrally with the drive member  11 ′ and that the planetary carrier (second element) is rotated integrally with the driven member  15 ′, although not being specifically illustrated. 
       FIG. 7  is a schematic configuration diagram illustrating a starting device  1 Y including a damper device  10 Y according to another embodiment of the disclosure. Among the components of the starting device  1 Y and the damper device  10 Y, the same components to those of the starting device  1  and the damper device  10  described above are expressed by the same reference signs and their repeated description is omitted. 
     The damper device  10 Y shown in  FIG. 7  includes a drive member (input element)  11 Y, an intermediate member (intermediate element)  12 Y and a driven member (output element)  15 Y, as rotational elements. The damper device  10 Y also includes a plurality of first springs (first elastic bodies) SP 1 ′ configured to transmit the torque between the drive member  11 Y and the intermediate member  12 Y and a plurality of second springs (second elastic bodies) SP 2 ′ configured to respectively work in series with the corresponding first springs SP 1 ′ and to transmit the torque between the intermediate member  12 Y and the driven member  15 Y, as torque transmission elements (torque transmission elastic bodies). The plurality of first springs (first elastic bodies) SP 1 ′, the intermediate member  12 Y and the plurality of second springs (second elastic bodies) SP 2 ′ form a torque transmission path TP between the drive member  11 Y and the driven member  15 Y. The intermediate member  12 Y is an annular member. A damping ratio ζ of the intermediate member  12 Y that is vibrated with transmission of the torque from the engine EG to the drive member  11 Y is less than the value  1 . Like the rotary inertia mass damper  20 , a rotary inertia mass damper  20 Y is configured by a single pinion-type planetary gear  21  and is connected with the drive member  11 Y and the driven member  15 Y and arranged in parallel to the torque transmission path TP to mainly transmit an inertia torque to the driven member  15 Y. Furthermore, as illustrated, the intermediate member  12 Y is coupled with the turbine runner  5  to be integrally rotated. As shown by a two-dot chain line in  FIG. 7 , however, the turbine runner  5  may be coupled with either one of the drive member  11 Y and the driven member  15 Y. 
     The damper device  10 Y further includes a first stopper ST 1  configured to restrict the relative rotation of the drive member  11 Y to the intermediate member  12 Y, i.e., deflection of the first springs SP 1 ′ and a second stopper ST 2  configured to restrict the relative rotation of the intermediate member  12 Y to the driven member  15 Y, i.e., deflection of the second springs SP 2 ′. One of the first stopper ST 1  and the second stopper ST 2  is configured to restrict the relative rotation of the drive member  11 Y to the intermediate member  12 Y or the relative rotation of the intermediate member  12 Y to the driven member  15 Y when the input torque into the drive member  11 Y reaches a predetermined torque T 1  that is smaller than a torque T 2  corresponding to a maximum torsion angle θmax of the damper device  10 Y and the torsion angle of the drive member  11 Y relative to the driven member  15 Y becomes equal to or larger than a predetermined angle θref. The other of the first stopper ST 1  and the second stopper ST 2  is configured to restrict the relative rotation of the intermediate member  12 Y to the driven member  15 Y or the relative rotation of the drive member  11 Y to the intermediate member  12 Y when the input torque into the drive member  11 Y reaches the torque T 2 . 
     This configuration allows for the deflections of the first and the second springs SP 1 ′ and SP 2 ′ until one of the first and the second stoppers ST 1  an ST 2  operates. When one of the first and the second stoppers ST 1  and ST 2  operates, the deflection of one of the first and the second springs SP 1 ′ and SP 2 ′ is restricted. When both the first and the second stoppers ST 1  and ST 2  operate, the deflections of both the first and the second springs SP 1 ′ and SP 2 ′ are restricted. The damper device  10 Y accordingly has the two-step (two-stage) damping characteristics. The first stopper ST 1  or the second stopper ST 2  corresponding to the torque T 2  may be configured such as to restrict the relative rotation of the drive member  11 Y to the driven member  15 Y. 
     The damper device  10 Y configured as described above provides the similar functions and advantageous effects to those of the damper device  10  described above. In the damper device  10 Y, one of the first and the second springs SP 1 ′ and SP 2 ′ may be arranged on the outer side in the radial direction of the other to be arrayed at intervals in the circumferential direction. More specifically, for example, the plurality of first springs SP 1 ′ may be arranged in an outer circumferential-side area in the fluid transmission chamber  9  to be arrayed at intervals in the circumferential direction. The plurality of second springs SP 2 ′ may be arranged on the inner side in the radial direction of the plurality of first springs SP 1 ′ to be arrayed at intervals in the circumferential direction. In this configuration, the first and the second springs SP 1 ′ and SP 2 ′ may be arranged to at least partially overlap with each other in the radial direction. 
       FIG. 8  is a schematic configuration diagram illustrating a starting device  1 Z including a damper device  10 Z according to yet another embodiment of the disclosure. Among the components of the starting device  1 Z and the damper device  10 Z, the same components to those of the starting device  1  and the damper device  10  described above are expressed by the same reference signs and their repeated description is omitted. 
     The damper device  10 Z shown in  FIG. 8  includes a drive member (input element)  11 Z, a first intermediate member (first intermediate element)  13 , a second intermediate member (second intermediate element)  14  and a driven member (output element)  15 Z, as rotational elements. Both the first and the second intermediate members  13  and  14  are annular members. A damping ratio  4  of at least one of the first and the second intermediate members  13  and  14  that are vibrated with transmission of the torque from the engine EG to the drive member  11 Z is less than the value  1 . The damper device  10 Z also includes a plurality of first springs (first elastic bodies) SP 1 ′ configured to transmit the torque between the drive member  11 Z and the first intermediate member  13 , a plurality of second springs (second elastic bodies) SP 2 ′ configured to transmit the torque between the first intermediate member  13  and the second intermediate member  14 , and a plurality of third springs (third elastic bodies) SP 3  configured to transmit the torque between the second intermediate member  14  and the driven member  15 Z, as torque transmission elements (torque transmission elastic bodies). The plurality of first springs SP 1 ′, the first intermediate member  13 , the plurality of second springs SP 2 ′, the second intermediate member  14  and the plurality of third springs SP 3  form a torque transmission path TP between the drive member  11 Z and the driven member  15 Z. A rotary inertia mass damper  20 Z is connected with the drive member  11 Z and the driven member  15 Z and is arranged in parallel to the torque transmission path TP to mainly transmit an inertia torque to the driven member  15 Z. Furthermore, as illustrated, the first intermediate member  13  is coupled with the turbine runner  5  to be integrally rotated. 
     In the damper device  10 Z including the first and the second intermediate members  13  and  14 , three resonances occur in the torque transmission path TP when the deflections of all the first to the third springs SP 1 ′, SP 2 ′ and SP 3  are allowed. More specifically, a resonance of the entire damper device  10 Z occurs in the torque transmission path TP by the vibrations of the drive member  11 Z and the driven member  15 Z in the opposite phases when the deflections of the first to the third springs SP 1 ′, SP 2 ′ and SP 3  are allowed. A resonance also occurs in the torque transmission path TP by the vibrations of the first and the second intermediate members  13  and  14  in the opposite phase to both the drive member  11 Z and the driven member  15 Z when the deflections of the first to the third springs SP 1 ′, SP 2 ′ and SP 3  are allowed. A resonance further occurs in the torque transmission path TP by the vibration of the first intermediate member  13  in the opposite phase to the drive member  11 Z, the vibration of the second intermediate member  14  in the opposite phase to the first intermediate member  13  and the vibration of the driven member  15 Z in the opposite phase to the second intermediate member  14  when the deflections of the first to the third springs SP 1 ′, SP 2 ′ and SP 3  are allowed. This configuration thus enables a total of three antiresonance points, where the vibration transmitted from the drive member  11 Z to the driven member  15 Z via the torque transmission path TP and the vibration transmitted from the drive member  11 Z to the driven member  15 Z via the rotary inertia mass damper  20 Z are theoretically cancelled out each other, to be set in the damper device  10 Z. 
     Among the three antiresonance points that are likely to provide theoretically zero vibration amplitude of the driven member  15 Z (that are likely to further decrease the vibration amplitude), a first antiresonance point of the lowest rotation speed may be set in the low rotation speed range of 500 rpm to 1500 rpm (in the expected setting range of the lockup rotation speed Nlup). This shifts one resonance having the minimum frequency of the resonances occurring in the torque transmission path TP toward the lower rotation speed (toward the lower frequency), such as to be included in a non-lockup area of the lockup clutch  8 . This results in allowing for the lockup at the lower rotation speed and remarkably effectively improving the vibration damping performance of the damper device  10 Z in the low rotation speed range that is likely to increase the vibration from the engine EG. The damper device  10 Z may make a second antiresonance point of the higher rotation speed (the higher frequency) than the first antiresonance point equal to (closer to), for example, (the frequency) of a resonance point of the input shaft IS of the transmission TM or may make a third antiresonance point of the higher rotation speed (the higher frequency) than the second antiresonance point equal to (closer to), for example, (the frequency) of a resonance point in the damper device  10 Z, such as to effectively suppress the occurrence of such resonances. The damper device  10 Z may be configured such as to include three or more intermediate members in the torque transmission path TP. The turbine runner  5  may be coupled with the second intermediate member  14  or may be coupled with one of the drive member  11 Z and the driven member  15 Z as shown by a two-dot chain line in  FIG. 8 . 
       FIG. 9  is a schematic configuration diagram illustrating a starting device  1 V including a damper device  10 V according to another embodiment of the disclosure. Among the components of the starting device  1 V and the damper device  10 V, the same components to those of the starting device  1  and the damper device  10  described above are expressed by the same reference signs and their repeated description is omitted. 
     The damper device  10 V shown in  FIG. 9  includes a drive member (input element)  11 V, a first intermediate member (first intermediate element)  13 , a second intermediate member (second intermediate element)  14  and a driven member (output element)  15 V, as rotational elements. With regard to the first and the second intermediate members  13  and  14  that are vibrated with transmission of the torque from the engine EG to the drive member  11 V, at least the first intermediate member  13  is an annular member, and a damping ratio ζ of at least the first intermediate member  13  is less than the value  1 . The damper device  10 V also includes a plurality of first springs (first elastic bodies) SP 1 ′ configured to transmit the torque between the drive member  11 V and the first intermediate member  13 , a plurality of second springs (second elastic bodies) SP 2 ′ configured to transmit the torque between the first intermediate member  13  and the second intermediate member  14  (driven member  15 V), and a plurality of third springs (third elastic bodies) SP 3  configured to transmit the torque between the second intermediate member  14  and the driven member  15 V, as torque transmission elements (torque transmission elastic bodies). The plurality of first springs SP 1 ′, the first intermediate member  13 , the plurality of second springs SP 2 ′, the second intermediate member  14  and the plurality of third springs SP 3  form a torque transmission path TP between the drive member  11 V and the driven member  15 V. In the illustrated example of  FIG. 9 , the third springs SP 3  are configured to have a larger spring constant (higher rigidity) than the spring constants (rigidities) of the first and the second springs SP 1 ′ and SP 2 ′. 
     A rotary inertia mass damper  20 V is connected with the drive member  11 V and the second intermediate member  14  and is arranged in parallel to the plurality of first springs (first elastic bodies) SP 1 ′, the first intermediate member  13  and the plurality of second springs (second elastic bodies) SP 2 ′ of the torque transmission path TP to mainly transmit an inertia torque to the driven member  15 V via the second intermediate member  14  and the third springs SP 3 . Accordingly the drive member  11 V serves as a planetary carrier (first element) of a planetary gear  21  that supports a plurality of pinion gears  24  to be rotatable (rotatable on their own axes) and revolvable about a sun gear  22  and a ring gear  23 . Additionally, the ring gear (second element)  23  of the planetary gear  21  is fixed to the second intermediate member  14  to be rotatable integrally with the second intermediate member  14 . This configuration causes the sun gear  22  and a mass portion  22   m  as the mass body (third element) to be rotated with relative rotation of the drive member  11 V to the driven member  15 V or more specifically with relative rotation of the drive member  11 V to the second intermediate member  14 . In the damper device  10 V, the drive member  11 V is coupled with the turbine runner  5  to be rotated integrally. 
     This configuration of the damper device  10 V is substantially equivalent to the configuration that the plurality of third springs SP 3  working in parallel are placed between the driven member  15 Y and the input shaft IS of the transmission TM in the damper device  10 Y shown in  FIG. 7 . In the damper device  10 V, the rotary inertia mass damper  20 V is provided in parallel to the first and the second springs SP 1 ′ and SP 2 ′ and the first intermediate member  13 . Accordingly, in the damper device  10 V, two (multiple) natural frequencies may be set for the torque transmission path from the drive member  11 V to the second intermediate member  14  in the state that the deflections of at least the first and the second springs SP 1 ′ and SP 2 ′ are allowed, and a resonance of the first intermediate member  13  (second resonance) may occur at the higher rotation speed (the higher frequency) than a first resonance. As a result, this enables a total of two antiresonance points that provide theoretically zero vibration amplitude of the driven member  15 V to be set in the damper device  10 V. 
     The damper device  10 V is especially suitable to be used in combination with a transmission TM for rear-wheel drive. In the transmission TM for rear-wheel drive having a long length from one end of an input shaft IS (starting device  1 V-side end) to one end of a non-illustrated output shaft (wheel side-end), the rigidities of the input shaft IS coupled with the driven member  15 V of the damper device  10 V and of the output shaft (and additionally an intermediate shaft) are decreased. Accordingly the natural frequency (resonance frequency) determined by the moments of inertia of these shaft members is decreased (lowered) by the effect of the moment of inertia of the entire rotary inertia mass damper  20 V. This may obviously cause a resonance, which is supposed to occur at the high rotation speed of the drive member  11  (engine EG), even in a low rotation speed range. The configuration that the rotary inertia mass damper  20 V is coupled with the drive member  11 V and with the second intermediate member  14  of the damper device  10 V, on the other hand, causes the third springs SP 3  to be placed between the rotary inertia mass damper  20 V and the input shaft IS of the transmission TM coupled with the driven member  15 V and thereby substantially separates the rotary inertia mass damper  20 V from the input shaft IS. This configuration enables two antiresonance points to be set and remarkably effectively reduces the effect of the moment of inertia of the entire rotary inertia mass damper  20 V on the natural frequency determined by the moment of inertia of the shaft member coupled with the driven member  15 V and so on. 
     The damper device  10 V may, however, be used in combination with a transmission TM for front-wheel device. In the case where the damper device  10 V is combined with the transmission TM for front-wheel vehicle, the configuration of the damper device  10 V also remarkably effectively reduces the effect of the moment of inertia of the entire rotary inertia mass damper  20 V on the natural frequency determined by the moment of inertia of the shaft member coupled with the driven member  15 V and so on, and additionally improves the vibration damping performance of the damper device  10 V by a further decrease of the rigidity. The damper device  10 V may be configured such as to include any additional intermediate member and springs (elastic bodies) between the first intermediate member  13  and the second intermediate member  14 . Furthermore, the turbine runner  5  may be coupled with one of the first and the second intermediate members  13  and  14  as shown by a two-dot chain line in  FIG. 9  or may be coupled with the driven member  15 V. The rotary inertia mass damper  20 V may be configured such as to include the turbine runner  5  as the mass body that rotates with relative rotation of the drive member  11  to the driven member  15  (to the second intermediate member  14 ). 
       FIG. 10  is a schematic configuration diagram illustrating a starting device  1 W including a damper device  10 W according to yet another embodiment of the disclosure. Among the components of the starting device  1 W and the damper device  10 W, the same components to those of the starting device  1  and the damper device  10  described above are expressed by the same reference signs and their repeated description is omitted. 
     The damper device  10 W shown in  FIG. 10  is equivalent to the configuration that the rotary inertia mass damper  20 V is replaced with a different rotary inertia mass damper  20 W in the damper device  10 V described above. The rotary inertia mass damper  20 W is connected with a drive member  11 W and a second intermediate member  14  and is arranged in parallel to a plurality of first springs (first elastic bodies) SP 1 ′, a first intermediate member  13  and a plurality of second springs (second elastic bodies) SP 2 ′ of a torque transmission path TP to mainly transmit an inertia torque to the driven member  15 W via the second intermediate member  14  and third springs SP 3 . Accordingly the drive member  11 W serves as a planetary carrier (first element) of a planetary gear  21  that supports a plurality of pinion gears  24  to be rotatable (rotatable on their own axes) and revolvable about a sun gear  22  and a ring gear  23 . Additionally, the sun gear (second element)  22  of the planetary gear  21  is fixed to the second intermediate member  14  to be rotatable integrally with the second intermediate member  14 . This configuration causes the ring gear  23  and a mass portion  23   m  as the mass body (third element) to be rotated with relative rotation of the drive member  11 W to the driven member  15 W or more specifically with relative rotation of the drive member  11 W to the second intermediate member  14 . This damper device  10 W provides the similar functions and advantageous effects to those of the damper device  10 V described above. 
       FIGS. 11A, 11B, 11C and 11D  are schematic diagrams respectively illustrating rotary inertia mass dampers  20 D to  20 G that are applicable to the above damper device  10 ,  10 X,  10 Y,  10 Z,  10 V or  10 W. 
     The rotary inertia mass damper  20 D shown in  FIG. 11A  is configured by a double pinion-type planetary gear  21 D. The planetary gear  21 D includes a sun gear (third element)  22 D that is an external gear, a ring gear (first element)  23 D that is an internal gear arranged concentrically with the sun gear  22 , and a planetary carrier (second element)  25  that is configured to hold multiple sets of two pinion gears  24   a  and  24   b  such as to be rotatable (rotatable on their own axes) and revolvable. The two pinion gears  24   a  and  24   b  are arranged such that they engage with each other and that one engages with the sun gear  22 D and the other engages with the ring gear  23 D. 
     The sun gear  22 D of the planetary gear  21 D is used as the mass body of the rotary inertia mass damper  20 D, includes a mass portion  22   m  to increase the moment of inertia, and is supported rotatably by a non-illustrated damper hub and so on. The ring gear  23 D is coupled with a drive member  11  or a non-illustrated lockup piston to be rotatable integrally with the drive member  11 . The planetary carrier  25  is coupled with a driven member  15  to be rotatable integrally with the driven member  15 . According to a modification, the plurality of pinion gears  24   a  and  24   b  may be supported by the driven member  15 . In other words, the driven member  15  itself may be used as the planetary carrier of the planetary gear  21 D. 
     The rotary inertia mass damper  20 E shown in  FIG. 11B  is configured by a planetary gear  21 E that includes a plurality of step pinion gears  26 , each including a smaller-diameter pinion gear  26   a  and a larger-diameter pinion gear  26   b  coaxially and integrally connected with each other. The planetary gear  21 E includes, in addition to the plurality of step pinion gears  26 , a first sun gear (first element)  27   a  arranged to engage with the smaller-diameter pinion gear  26   a  of each step pinion gear  26 , a second sun gear (second element)  27   b  arranged to engage with the larger-diameter pinion gear  26   b  of each step pinion gear  26 , and a planetary carrier (third element)  28  arranged to support the plurality of step pinion gears  26  such as to be rotatable and revolvable. 
     The number of teeth of the larger-diameter pinion gear  26   b  of the step pinion gear  26  is determined to be larger than the number of teeth of the smaller-diameter pinion gear  26   a . The first sun gear  27   a  is coupled with a drive member  11  or a non-illustrated lockup piston to be rotatable integrally with the drive member  11 . The second sun gear  27   b  is coupled with a driven member  15  to be rotatable integrally with the driven member  15 . The planetary carrier  28  is used as the mass body of the rotary inertia mass damper  20 E, includes a mass portion  28   m  to increase the moment of inertia, and is supported rotatably by, for example, a non-illustrated damper hub and so on. 
     The rotary inertia mass damper  20 F shown in  FIG. 11C  is configured by a planetary gear  21 F that includes a plurality of step pinion gears  26 , each including a smaller-diameter pinion gear  26   a  and a larger-diameter pinion gear  26   b  coaxially and integrally connected with each other. The planetary gear  21 F includes, in addition to the plurality of step pinion gears  26 , a first ring gear (first element)  29   a  arranged to engage with the smaller-diameter pinion gear  26   a  of each step pinion gear  26 , a second ring gear (second element)  29   b  arranged to engage with the larger-diameter pinion gear  26   b  of each step pinion gear  26 , and a planetary carrier (third element)  28  arranged to support the plurality of step pinion gears  26  such as to be rotatable and revolvable. 
     The number of teeth of the larger-diameter pinion gear  26   b  of the step pinion gear  26  is determined to be larger than the number of teeth of the smaller-diameter pinion gear  26   a . The first ring gear  29   a  is coupled with a drive member  11  or a non-illustrated lockup piston to be rotatable integrally with the drive member  11 . The second ring gear  29   b  is coupled with a driven member  15  to be rotatable integrally with the driven member  15 . The planetary carrier  28  is used as the mass body of the rotary inertia mass damper  20 F, includes a mass portion  28   m  to increase the moment of inertia, and is supported rotatably by, for example, a non-illustrated damper hub and so on. 
     The rotary inertia mass damper  20 G shown in  FIG. 11D  is configured by a planetary gear  21 G that includes a plurality of step pinion gears  26 , each including a smaller-diameter pinion gear  26   a  and a larger-diameter pinion gear  26   b  coaxially and integrally connected with each other. The planetary gear  21 G includes, in addition to the plurality of step pinion gears  26 , a planetary carrier (first element)  28  arranged to support the plurality of step pinion gears  26  such as to be rotatable and revolvable, a sun gear (second element)  27 G arranged to engage with the larger-diameter pinion gear  26   b  of each step pinion gear  26  and a ring gear (third element)  29 G arranged to engage with the smaller-diameter pinion gear  26   a  of each step pinion gear  26 . 
     The number of teeth of the larger-diameter pinion gear  26   b  of the step pinion gear  26  is determined to be larger than the number of teeth of the smaller-diameter pinion gear  26   a . The planetary carrier  28  is coupled with a drive member  11  or a non-illustrated lockup piston to be rotatable integrally with the drive member  11 . The sun gear  27 G is coupled with a driven member  15  to be rotatable integrally with the driven member  15 . The ring gear  29 G is used as the mass body of the rotary inertia mass damper  20 G, includes a mass portion  29   m  to increase the moment of inertia, and is supported rotatably by, for example, a non-illustrated damper hub and so on. 
     As described above, a damper device ( 10 ,  10 X,  10 X′,  10 Y,  10 Z,  10 V,  10 W) according to one aspect of the disclosure is configured to include an input element ( 11 ,  11 ′,  11 Y,  11 Z,  11 V,  11 W) to which a torque from an engine (EG) is transmitted and an output element ( 15 ,  15 ′,  15 Y,  15 Z,  15 V,  15 W). The damper device ( 10 ,  10 X,  10 X′,  10 Y,  10 Z,  10 V,  10 W) includes: a torque transmission path (TP 1 , TP) including an intermediate element ( 12 ,  12 ′,  13 ,  14 ), a first elastic body (SP 1 , SP 1 ′) configured to transmit a torque between the input element ( 11 ,  11 ′,  11 Y,  11 Z,  11 V,  11 W) and the intermediate element ( 12 ,  12 ′  13 ), and a second elastic body (SP 2 , SP 2 ′) configured to transmit a torque between the intermediate element ( 12 ,  12 ′,  14 ) and the output element ( 15 ,  15 ′,  15 Y,  15 Z,  15 V,  15 W); and a rotary inertia mass damper ( 20 ,  20 X,  20 Y,  20 Z,  20 V,  20 W,  20 D,  20 E,  20 F,  20 G) configured to include a mass body ( 5 ,  22 ,  22   m ,  23 ,  23   m ,  28 ,  28   m ,  29 G,  29   m ) rotating with relative rotation of the input element ( 11 ,  11 ′,  11 Y,  11 Z,  11 V,  11 W) to the output element ( 15 ,  15 ′,  15 Y,  15 Z,  15 V,  15 W) and provided between the input element ( 11 ,  11 ′,  11 Y,  11 Z,  11 V,  11 W) and the output element ( 15 ,  15 ′,  15 Y,  15 Z,  15 V,  15 W) in parallel to the torque transmission path (TP 1 , TP). A damping ratio (ζ) of the intermediate element ( 12 ,  12 ′,  13 ,  14 ) determined based on a moment of inertia (J 2 ) of the intermediate element ( 12 ,  12 ′,  13 ,  14 ) and rigidities (k 1 , k 2 ) of the first elastic body (SP 1 , SP 1 ′) and the second elastic body (SP 2 , SP 2 ′) is less than a value  1 . 
     In the damper device of this aspect, the intermediate element, the first elastic body and the second elastic body form the torque transmission path, and the rotary inertia mass damper including the mass body that is rotated with relative rotation of the input element to the output element is provided between the input element and the output element in parallel to the torque transmission path. The damping ratio of the intermediate element determined based on the moment of inertia of the intermediate element and the rigidities of the first and the second elastic bodies is less than the value  1 . 
     In the damper device of this aspect, the torque transmitted from the torque transmission path including the intermediate element and the first and the second elastic bodies to the output element depends on (is proportional to) the displacement of the second elastic body between the intermediate element and the output element. The torque transmitted from the rotary inertia mass damper to the output element, on the other hand, depends on (is proportional to) a difference in angular acceleration between the input element and the output element, i.e., a second order differential equation result of the displacement of the elastic body placed between the input element and the output element. On the assumption that an input torque transmitted to the input element  26  of the damper device is periodically vibrated, the phase of the vibration transmitted from the input element to the output element via the torque transmission path is accordingly shifted by 180 degrees from the phase of the vibration transmitted from the input element to the output element via the rotary inertia mass damper. A plurality of natural frequencies (resonance frequencies) may be set in the torque transmission path including the intermediate element having the damping ratio of less than the value  1 , in the state that deflections of the first and the second elastic bodies are allowed. A resonance of the intermediate element may be caused in this torque transmission path when the rotation speed of the input element reaches a rotation speed corresponding to one of the plurality of natural frequencies. 
     This configuration enables two antiresonance points, where the vibration transmitted from the input element to the output element via the torque transmission path and the vibration transmitted from the input element to the output element via the rotary inertia mass damper are theoretically cancelled out each other, to be set in the damper device of this aspect. This configuration thus remarkably effectively improves the vibration damping performance of the damper device by making the frequencies of the two antiresonance points equal to (closer to) the frequency of the vibration (resonance) to be damped by the damper device. Additionally, in the damper device of this aspect, a resonance of the intermediate element occurs when the rotation speed of the input element becomes higher than a rotation speed corresponding to the frequency of an antiresonance point of the lower rotation speed (lower frequency). The amplitude of the vibration transmitted from the second elastic body to the output element accordingly changes from decreasing to increasing before the rotation speed of the input element reaches a rotation speed corresponding to a relatively lower natural frequency of the intermediate element. Even when the amplitude of the vibration transmitted from the rotary inertia mass damper to the output element is gradually increased with an increase in rotation speed of the input element, this expands an area where the vibration transmitted from the rotary inertia mass damper to the output element cancels out at least part of the vibration transmitted from the second elastic body to the output element. This results in further improving the vibration damping performance of the damper device in a relatively low rotation speed range of the input element. 
     The intermediate element ( 12 ,  12 ′,  13 ) may be an annular member. This configuration provides a relatively low natural frequency of the intermediate element and enables the amplitude of the vibration transmitted to the output element via the second elastic body to be changed from decreasing to increasing at a relatively early timing. 
     The rotation speed corresponding to the natural frequency of the intermediate element ( 12 ,  12 ′,  13 ) may be higher than a minimum rotation speed (Nlup) in a rotation speed range where a torque is transmitted from the input element ( 11 ,  11 ′,  11 Y,  11 Z,  11 V,  11 W) to the output element ( 15 ,  15 ′,  15 Y,  15 Z,  15 V,  15 W) via the torque transmission path (TP 1 , TP). The amplitude of the vibration transmitted to the output element ( 15 ,  15 ′,  15 Y,  15 Z,  15 V,  15 W) via the second elastic body (SP 2 , SP 2 ′) may change from decreasing to increasing before the rotation speed of the input element ( 11 ,  11 ′,  11 Y,  11 Z,  11 V,  11 W) that is equal to or higher than the minimum rotation speed (Nlup) reaches the rotation speed corresponding to the natural frequency of the intermediate element ( 12 ,  12 ′,  13 ). 
     Additionally, the rotary inertia mass damper ( 20 ,  20 X,  20 Y,  20 Z,  20 D,  20 E,  20 F,  20 G) may include a differential ( 21 ,  21 D,  21 E,  21 F  21 G) including a first element ( 80 ,  11 ,  11 Y,  11 Z,  23 D,  27   a ,  28 ,  29   a ) arranged to rotate integrally with the input element, a second element ( 23 ,  25 ,  27   b ,  27 G  29   b ) arranged to rotate integrally with the output element ( 15 ,  15 Y,  15 Z) and a third element ( 22 ,  22 D,  28 ,  29 G) arranged to integrally with the mass body ( 22 ,  22   m ,  5 ,  28   m ,  29   m ). This configuration can thus increase the rotation speed of the third element, i.e., the mass body, to be higher than the rotation speed of the input element or decrease the rotation speed of the third element to be lower than the rotation speed of the input element, according to, for example, the magnitude of the moment of inertia of the mass body. This results in enhancing the flexibility in design of the mass body and the entire damper device, while effectively ensuring the moment of inertia applied from the rotary inertia mass damper to the output element. Depending on the magnitude of the moment of inertia of the mass body, the rotary inertia mass damper may be a constant speed rotation mechanism configured to rotate the mass body at the same speed as that of the input element or may include a speed change mechanism configured to change the rotation speed of the mass body according to the relative rotation speed of the input element to output element. 
     The differential may be a planetary gear ( 21 ,  21 W,  21 D,  21 E,  21 F,  21 G). The differential is, however, not limited to the planetary gear but may be configured such as to, for example, include a link mechanism and so on. 
     Moreover, the torque transmission path (TP) may include a first intermediate element ( 13 ) and a second intermediate element ( 14 ) as the intermediate element and may further include a third elastic body (SP 3 ). The first elastic body (SP 1 ′) may be arranged to transmit a torque between the input element ( 11 Z) and the first intermediate element ( 13 ). The second elastic body (SP 2 ′) may be arranged to transmit a torque between the first intermediate element ( 13 ) and the second intermediate element ( 14 ). The third elastic body (SP 3 ) may be arranged to transmit a torque between the second intermediate element ( 14 ) and the output element ( 15 Z). The first and the second intermediate elements ( 13 ,  14 ) may be annular members. The damping ratio (ζ) of at least one of the first and the second intermediate elements ( 13 ,  14 ) may be less than the value  1 . The damper device of this configuration can set a total of three antiresonance points, where the vibration transmitted from the input element to the output element via the torque transmission path and the vibration transmitted from the input element to the output element via the rotary inertia mass damper are theoretically cancelled out each other and thereby further improve the vibration damping performance. 
     The intermediate element ( 12 ,  12 ′,  13 ,  14 ) may be coupled with a turbine runner ( 5 ) of a fluid transmission device to be rotated integrally. This configuration substantially increases the moment of inertia of the intermediate element and shifts an antiresonance point having a minimum frequency to the lower frequency. 
     Additionally, the mass body of the rotary inertia mass damper ( 20 X) may be configured to include a turbine runner ( 5 ) of a fluid transmission device. This configuration further increases the moment of inertia of the mass body of the rotary inertia mass damper and further increases the moment of inertia applied from the rotary inertia mass damper to the output element, while suppressing overall size expansion of the damper device, compared with a configuration equipped with a dedicated mass body. 
     The first elastic body (SP 1 , SP 1 ′) may have a spring constant that is identical with a spring constant of the second elastic body (SP 2 , SP 2 ′). 
     The first elastic body (SP 1 , SP 1 ′) may have a spring constant that is different from a spring constant of the second elastic body (SP 2 , SP 2 ′). This further increases an interval between the two antiresonance points and thereby further improves the vibration damping effect of the damper device in a low frequency range (low rotation speed range). 
     The spring constants of the first and the second elastic bodies and the moments of inertia of the intermediate element and the mass body may be determined, based on a minimum frequency among antiresonance points that provide zero vibration amplitude of the output element. 
     The spring constants of the first and the second elastic bodies and the moments of inertia of the intermediate element and the mass body may be determined, based on the minimum frequency of the antiresonance point and the number of cylinders of the engine. 
     The damper device may be configured to satisfy 500 rpm≤(120/n)·fa 1 ≤1500 rpm, where “fa 1 ” denotes the minimum frequency of the antiresonance point and “n” denotes the number of cylinders of the engine. 
     Setting the antiresonance point that is likely to further decrease the vibration amplitude of the output element in the low rotation speed range of 500 rpm to 1500 rpm allows for coupling of the engine with the input element at the lower rotation speed and further improves the vibration damping effect of the damper device in a low rotation speed range where the vibration from the engine is likely to be increased. Configuration of the damper device such that the frequency of a resonance (R 1 ) occurring in the torque transmission path is minimized to a minimum possible value that is lower than the frequency fa 1  of the antiresonance point further reduces the frequency fa 1  of the antiresonance point and allows for coupling of an internal combustion engine with the input element at the further lower rotation speed. 
     The damper device may be configured to satisfy Nlup≤(120/n)·fa 1 , where “Nlup” denotes a lockup rotation speed of a lockup clutch arranged to couple the engine with the input element. This configuration enables the vibration from an internal combustion engine to be remarkably effectively damped by the damper device during coupling of the internal combustion engine with the input element by the lockup clutch or immediately after such coupling. 
     The damper device may be configured to satisfy 900 rpm≤(120/n)·fa 1 ≤1200 rpm. 
     The differential may be a planetary gear, and the minimum frequency fa 1  of the antiresonance point may be expressed by Equation (8) given above. When the rotary inertia mass damper is configured by a non-single pinion-type planetary gear, “γ=λ2/(1+λ)” in Equation (8), and the constant λ may be determined according to the connection configuration of rotational elements of the planetary gear with the input element and the output element and the gear ratio of the planetary gear. 
     The damper device may be configured not to restrict deflections of the first and the second elastic bodies until an input torque transmitted to the input element becomes equal to or larger than a predetermined threshold value. In this aspect, the threshold value may be a torque value corresponding to a maximum torsion angle of the damper device or a smaller torque value. 
     The output element may be operably coupled with an input shaft of a transmission. 
     The damper device ( 10 V,  10 W) configured to include an input element ( 11 V,  11 W) to which a torque from an engine (EG) is transmitted and an output element ( 15 V,  15 W). The damper device includes a torque transmission path (TP) including first and second intermediate elements ( 13 ,  14 ), a first elastic body (SP 1 ′) configured to transmit a torque between the input element ( 11 V,  11 W) and the first intermediate element ( 13 ), a second elastic body (SP 2 ′) configured to transmit a torque between the first intermediate element ( 13 ) and the second intermediate element ( 14 ), and a third elastic body (SP 3 ) configured to transmit a torque between the second intermediate element ( 14 ) and the output element ( 15 V,  15 W), and a rotary inertia mass damper ( 20 V,  20 W) configured to include a mass body ( 22 ,  22   m ,  23 ,  23   m ) rotating with relative rotation of the input element ( 11 V,  11 W) to the second intermediate element ( 14 ) and provided in parallel to the first elastic body (SP 1 ′), the first intermediate element ( 13 ) and the second elastic body (SP 1 ′) of the torque transmission path (TP). A damping ratio of the first intermediate element ( 13 ) determined based on at least a moment of inertia of the first intermediate element ( 13 ) and rigidities of the first and the second elastic bodies (SP 1 ′, SP 2 ′) is less than a value  1 . 
     In the damper device of this aspect, the rotary inertia mass damper is coupled with the input element and the second intermediate element of the damper device. This configuration causes the third elastic body to be placed between the rotary inertia mass damper and a member coupled with the output element and thereby substantially separates the rotary inertia mass damper from the member. This enables two antiresonance points to be set and remarkably effectively reduces the effect of the moment of inertia of the entire rotary inertia mass damper on the natural frequency determined by the moment of inertia of the member coupled with the output element. Even when the member coupled with the output element of the damper device has a low rigidity and the natural frequency (resonance frequency) determined by the moment of inertia of the member is decreased by the effect of the moment of inertia of the entire rotary inertia mass damper, this results in effectively suppressing a resonance which is supposed to occur at the high rotation speed of the input element from being obviously caused even in a low rotation range. 
     The rotary inertia mass damper may configured to include a planetary gear that includes a first element arranged to rotate integrally with the input element, a second element arranged to rotate integrally with the second intermediate element and a third element arranged to rotate integrally with the mass body. 
     The disclosure is not limited to the above embodiments in any sense but may be changed, altered or modified in various ways within the scope of extension of the disclosure. Additionally, the embodiments described above are only concrete examples of some aspect of the disclosure described in Summary and are not intended to limit the elements of the disclosure described in Summary. 
     INDUSTRIAL APPLICABILITY 
     The present disclosure is applicable to, for example, the field of manufacture of the damper device.