Patent Publication Number: US-2019178323-A1

Title: Vibration damping apparatus

Description:
BACKGROUND 
     The present disclosure relates to a vibration damping apparatus. 
     Hitherto, as this type of vibration damping apparatus, there is proposed a vibration damping apparatus including a first spring that is arranged so as to extend in a circumferential direction between an input element to which a torque from an engine is transferred and an output element and has a positive constant stiffness (spring rate), and a second spring that is arranged so as to extend in a radial direction between the input element and the output element and has a negative constant stiffness (spring rate) (see, for example, DE 102010053250 A1). In a state in which the second spring is attached, the second spring is compressed shorter than its equilibrium length, and is stretched in response to relative rotation (displacement) of the two rotational elements coupled via the second spring. In this vibration damping apparatus, the first spring and the second spring are arranged in parallel between the input element and the output element, thereby reducing the stiffness of the vibration damping apparatus. 
     SUMMARY 
     The vibration damping apparatus described above uses the first spring having the positive constant torsional stiffness (spring rate) and the second spring having the negative constant torsional stiffness (spring rate). Therefore, the overall torsional stiffness (combined spring rate) of the first spring and the second spring is constant irrespective of the rotation speed of the input element. This limits a rotation speed range in which high vibration damping performance can be exerted. 
     An exemplary aspect of the disclosure extends a rotation speed range in which high vibration damping performance can be exerted for a rotational element to which a torque from an engine is transferred. 
     A first vibration damping apparatus disclosed herein is summarized as follows. The first vibration damping apparatus is a vibration damping apparatus having a plurality of rotational elements including an input to which a torque from an engine is transferred, and an output. The vibration damping apparatus includes a first torsional stiffness mechanism arranged between the input and the output and having a positive torsional stiffness, and a second torsional stiffness mechanism configured to act in parallel to the first torsional stiffness mechanism between the input and the output and having a negative torsional stiffness. The negative torsional stiffness of the second torsional stiffness mechanism increases on a negative side as a rotation speed of the engine increases. 
     In the first vibration damping apparatus disclosed herein, the first torsional stiffness mechanism having the positive torsional stiffness and the second torsional stiffness mechanism having the negative torsional stiffness act in parallel between the input to which the torque from the engine is transferred and the output. Thus, the overall torsional stiffness of the plurality of torsional stiffness mechanisms including the first torsional stiffness mechanism and the second torsional stiffness mechanism (corresponding to a combined spring rate in a case of springs) can be reduced. Further, the negative torsional stiffness of the second torsional stiffness mechanism increases on the negative side as the rotation speed of the engine increases. Thus, the overall torsional stiffness of the plurality of torsional stiffness mechanisms can appropriately change in response to the rotation speed of the engine. As a result, it is possible to extend the rotation speed range in which high vibration damping performance can be exerted for the input to which the torque from the engine is transferred. 
     A second vibration damping apparatus disclosed herein is summarized as follows. The second vibration damping apparatus is a vibration damping apparatus configured to damp a vibration of a rotational element to which a torque from an engine is transferred. The vibration damping apparatus includes a first torsional stiffness mechanism coupled to the rotational element in a freely rotatable manner and having a positive torsional stiffness, a second torsional stiffness mechanism coupled to the rotational element in a freely rotatable manner and having a negative torsional stiffness, and a coupling mechanism that couples the first torsional stiffness mechanism and the second torsional stiffness mechanism to each other. The torsional stiffness of the second torsional stiffness mechanism increases on a negative side as a rotation speed of the engine increases. 
     In the second vibration damping apparatus disclosed herein, the first torsional stiffness mechanism coupled in a freely rotatable manner to the rotational element to which the torque from the engine is transferred and having the positive torsional stiffness and the second torsional stiffness mechanism coupled to the rotational element in a freely rotatable manner and having the negative torsional stiffness are coupled to each other via the coupling mechanism. In this structure, it can be considered that the first torsional stiffness mechanism and the second torsional stiffness mechanism act on the rotational element in parallel. Therefore, the overall torsional stiffness of the plurality of torsional stiffness mechanisms including the first torsional stiffness mechanism and the second torsional stiffness mechanism can be reduced. Further, in this structure, when the first torsional stiffness mechanism and the second torsional stiffness mechanism deviate from their positions in a stationary state due to the occurrence of fluctuation in the rotation of the rotational element, a vibration having a phase opposite to that of the vibration transferred from the engine to the rotational element is applied to the rotational element from the vibration damping apparatus so that the first torsional stiffness mechanism may return to its position in the stationary state and the second torsional stiffness mechanism may increase the amount of the deviation. Thus, the vibration of the rotational element can be absorbed (damped). Further, the torsional stiffness of the second torsional stiffness mechanism increases on the negative side as the rotation speed of the engine increases. Thus, the overall torsional stiffness of the plurality of torsional stiffness mechanisms including the first torsional stiffness mechanism and the second torsional stiffness mechanism can appropriately change in response to the rotation speed of the engine. As a result, it is possible to extend the rotation speed range in which high vibration damping performance can be exerted for the rotational element to which the torque from the engine is transferred. 
    
    
     
       BRIEF DESCRIPTION OF THE DRAWINGS 
         FIG. 1  is a schematic structural diagram of a starting apparatus including a damper apparatus disclosed herein. 
         FIG. 2  is a schematic structural diagram of a main part of the damper apparatus. 
         FIG. 3  is an explanatory drawing for describing an operation of a torsional stiffness mechanism. 
         FIG. 4  is an explanatory drawing for describing an operation of a torsional stiffness mechanism. 
         FIG. 5  is a schematic structural diagram of another damper apparatus disclosed herein. 
         FIG. 6  is a schematic structural diagram of a starting apparatus including another damper apparatus disclosed herein. 
         FIG. 7  is a sectional view of the damper apparatus. 
         FIG. 8  is a front-side elevation of the damper apparatus. 
         FIG. 9  is an explanatory drawing for describing an operation of a torsional stiffness mechanism. 
         FIG. 10  is an explanatory drawing illustrating an example of a relationship between a distance and a torsional stiffness. 
         FIG. 11  is a sectional view of another damper apparatus disclosed herein. 
         FIG. 12  is a front-side elevation of the damper apparatus. 
         FIG. 13  is a sectional view of another damper apparatus disclosed herein. 
         FIG. 14  is a front-side elevation of the damper apparatus. 
         FIG. 15  is an explanatory drawing illustrating a state in which an angular velocity (rotation speed) of an engine is small and a relative torsion angle between an input-side rotational member and a driven member is zero. 
         FIG. 16  is an explanatory drawing illustrating a state in which the angular velocity (rotation speed) of the engine is large and the relative torsion angle between the input-side rotational member and the driven member is zero. 
         FIG. 17  is a schematic structural diagram of an inner spring. 
         FIG. 18  is a schematic structural diagram of another damper apparatus disclosed herein. 
         FIG. 19  is a schematic structural diagram of another damper apparatus disclosed herein. 
         FIG. 20  is a schematic structural diagram of a centrifugal pendulum vibration absorbing apparatus. 
         FIG. 21  is a schematic structural diagram of the centrifugal pendulum vibration absorbing apparatus. 
         FIG. 22  is a sectional view of the centrifugal pendulum vibration absorbing apparatus. 
         FIG. 23  is a schematic structural diagram of a centrifugal pendulum vibration absorbing apparatus. 
     
    
    
     DETAILED DESCRIPTION OF EMBODIMENTS 
     Next, modes for carrying out the disclosure disclosed herein are described with reference to the drawings. 
       FIG. 1  is a schematic structural diagram of a starting apparatus  1  including a damper apparatus  10  disclosed herein. In the starting apparatus  1  of  FIG. 1 , the damper apparatus  10  corresponds to a “vibration damping apparatus” disclosed herein. As in the illustration, the starting apparatus  1  is mounted on, for example, a vehicle including an engine (internal combustion engine) EG serving as a drive apparatus. In addition to the damper apparatus  10 , the starting apparatus  1  includes a front cover  3  serving as an input member coupled to a crankshaft of the engine EG, a torque converter (fluid transmission apparatus) TC, a damper hub  7  serving as an output member fixed to an input shaft IS of a transmission (power transfer apparatus) TM, a lock-up clutch  8 , and the like. The torque converter TC includes a pump impeller (input-side fluid transmission element)  4  fixed to the front cover  3  and configured to rotate together with the front cover  3 , a turbine runner (output-side fluid transmission element)  5  rotatable coaxially with the pump impeller  4  and fixed to a driven member  15  of the damper apparatus  10  and the damper hub  7 , a stator  6  configured to adjust a flow of hydraulic oil (fluid) from the turbine runner  5  to the pump impeller  4 , and a one-way clutch  61  configured to regulate a rotation direction of the stator  6 . In place of the torque converter TC, there may be used a structure without the stator  6  and the one-way clutch  61 , that is, a structure in which the pump impeller  4  and the turbine runner  5  function as a fluid coupling. Examples of the transmission TM include an automatic transmission (AT), a continuously variable transmission (CVT), a dual clutch transmission (DCT), a hybrid transmission, and a speed reducer. The lock-up clutch  8  executes lock-up for coupling the front cover  3  and the damper hub  7  to each other via the damper apparatus  10 , and terminates the lock-up. 
     In the following description, an “axial direction” is basically an extending direction of a central axis (axis center) of the starting apparatus  1  and the damper apparatus  10  unless otherwise specified. A “radial direction” is basically a radial direction of the starting apparatus  1 , the damper apparatus  10 , and their rotational elements, that is, an extending direction of a straight line extending from the central axis in a direction orthogonal to the central axis (direction of a radius) unless otherwise specified. A “circumferential direction” is basically a circumferential direction of the starting apparatus  1 , the damper apparatus  10 , and their rotational elements, that is, a direction along a rotation direction unless otherwise specified. 
     The damper apparatus  10  includes a driving member (input element)  11 , an intermediate member (intermediate element)  12 , and the driven member (output element)  15  as the rotational elements. The damper apparatus  10  further includes, as torque transfer elements, a plurality of (for example, two) torsional stiffness mechanisms  20  arranged between the driving member  11  and the driven member  15 , a plurality of (for example, two) torsional stiffness mechanisms  30  arranged between the driving member  11  and the intermediate member  12 , and a plurality of (for example, two) torsional stiffness mechanisms  40  arranged between the intermediate member  12  and the driven member  15 . 
     As illustrated in  FIG. 2 , the driving member  11  is a plate-shaped annular member, and is coupled (fixed) to a lock-up piston of the lock-up clutch  8 . Thus, when the lock-up is executed by the lock-up clutch  8 , the front cover  3  (engine EG) and the driving member  11  are coupled to each other. The intermediate member  12  is a plate-shaped annular member having a diameter smaller than that of the driving member  11 . The driven member  15  is a plate-shaped annular member having a diameter smaller than those of the driving member  11  and the intermediate member  12 , and is fixed to the damper hub  7  and the turbine runner  5 . The driving member  11 , the intermediate member  12 , and the driven member  15  are arranged concentrically. 
     The plurality of torsional stiffness mechanisms  20  are arranged away from each other by 180°, and each include a coupling member  21 , a rivet  23  for coupling the coupling member  21  and the driven member  15  to each other in a freely rotatable manner, and a pin  24  for coupling the coupling member  21  and the driving member  11  to each other. The coupling member  21  is formed so as to extend in a given direction, and has a hole  22  extending in the extending direction of the coupling member  21  over a range from a substantial center to one end side. The coupling member  21  is supported on the driven member  15  via the rivet  23  in a freely rotatable manner, and is also supported by the driving member  11  so as to freely rotate and to freely move in the extending direction of the hole  22  (coupling member  21 ) such that the pin  24  fixed to the driving member  11  is located in the hole  22  of the coupling member  21 . Thus, the coupling member  21  has a relationship of a revolute pair with the driven member  15  and a sliding pair with the driving member  11 . The coupling member  21  extends in the radial direction when a relative torsion angle (relative displacement) between the driving member  11  and the driven member  15  is zero. A center of gravity  21   g  of the coupling member  21  is located on a radially outer side with respect to the rivet  23  and the pin  24  on a straight line (in the radial direction when the relative torsion angle between the driving member  11  and the driven member  15  is zero) passing through the rivet  23  (position of the revolute pair with the driven member  15 ) and the pin  24  (position of the sliding pair with the driving member  11 ). 
     The plurality of torsional stiffness mechanisms  30  are arranged away from each other by 180° at positions different from those of the plurality of torsional stiffness mechanisms  20  in the circumferential direction, and each include a coupling member  31 , a rivet  33  for coupling the coupling member  31  and the intermediate member  12  to each other in a freely rotatable manner, and a pin  34  for coupling the coupling member  31  and the driving member  11  to each other. The coupling member  31  is formed so as to extend in a given direction, and has a hole  32  extending in the extending direction of the coupling member  31  over a range from a substantial center to one end side. The coupling member  31  is supported on the intermediate member  12  via the rivet  33  in a freely rotatable manner, and is also supported by the driving member  11  so as to freely rotate and to freely move in the extending direction of the hole  32  (coupling member  31 ) such that the pin  34  fixed to the driving member  11  is located in the hole  32  of the coupling member  31 . Thus, the coupling member  31  has a relationship of a revolute pair with the intermediate member  12  and a sliding pair with the driving member  11 . The coupling member  31  extends in the radial direction when a relative torsion angle between the driving member  11  and the intermediate member  12  is zero. A center of gravity  31   g  of the coupling member  31  is located on a radially outer side with respect to the rivet  33  and the pin  34  on a straight line (in the radial direction when the relative torsion angle between the driving member  11  and the intermediate member  12  is zero) passing through the rivet  33  (position of the revolute pair with the intermediate member  12 ) and the pin  34  (position of the sliding pair with the driving member  11 ). 
     The plurality of torsional stiffness mechanisms  40  are arranged away from each other by 180° at positions different from those of the plurality of torsional stiffness mechanisms  20  and the plurality of torsional stiffness mechanisms  30  in the circumferential direction, and each include a coupling member  41 , a rivet  43  for coupling the coupling member  41  and the driven member  15  to each other in a freely rotatable manner, and a pin  44  for coupling the coupling member  41  and the intermediate member  12  to each other. The coupling member  41  is formed so as to extend in a given direction, and has a hole  42  extending in the extending direction of the coupling member  41  over a range from a substantial center to one end side. The coupling member  41  is supported on the driven member  15  via the rivet  43  in a freely rotatable manner, and is also supported by the intermediate member  12  so as to freely rotate and to freely move in the extending direction of the hole  42  (coupling member  41 ) such that the pin  44  fixed to the intermediate member  12  is located in the hole  42  of the coupling member  41 . Thus, the coupling member  41  has a relationship of a revolute pair with the driven member  15  and a sliding pair with the intermediate member  12 . The coupling member  41  extends in the radial direction when a relative torsion angle between the intermediate member  12  and the driven member  15  is zero. A center of gravity  41   g  of the coupling member  41  is located on a radially outer side with respect to the rivet  43  and the pin  44  on a straight line (in the radial direction when the relative torsion angle between the intermediate member  12  and the driven member  15  is zero) passing through the rivet  43  (position of the revolute pair with the driven member  15 ) and the pin  44  (position of the sliding pair with the intermediate member  12 ). 
     Next, an operation of the starting apparatus  1  including the damper apparatus  10  is described. As understood from  FIG. 1 , in the starting apparatus  1 , when the lock-up is not executed by the lock-up clutch  8 , a torque (power) transferred from the engine EG to the front cover  3  is transferred to the input shaft IS of the transmission TM via a path including the pump impeller  4 , the turbine runner  5 , and the damper hub  7 . When the lock-up is executed by the lock-up clutch  8 , the torque (power) transferred from the engine EG to the driving member  11  via the front cover  3  and the lock-up clutch  8  is transferred to the driven member  15 , the damper hub  7 , and the input shaft IS of the transmission TM via a first torque transfer path including the plurality of torsional stiffness mechanisms  20  and via a second torque transfer path including the plurality of torsional stiffness mechanisms  30 , the intermediate member  12 , and the plurality of torsional stiffness mechanisms  40 . 
     It is assumed that the lock-up is executed by the lock-up clutch  8  and the damper apparatus  10  coupled to the front cover  3  is rotated by the lock-up clutch  8  along with the rotation of the engine EG. When a relative torsion angle is formed between the driving member  11  and the driven member  15 , the torsional stiffness mechanisms  20  operate so as to reduce the relative torsion angle. When a relative torsion angle is formed between the driving member  11  and the intermediate member  12 , the torsional stiffness mechanisms  30  operate so as to reduce the relative torsion angle. When a relative torsion angle is formed between the intermediate member  12  and the driven member  15 , the torsional stiffness mechanisms  40  operate so as to increase the relative torsion angle. Operations of the torsional stiffness mechanisms  20 ,  30 , and  40  and torsional stiffnesses k 1 , k 2 , and k 3  are described below. 
     First, the operation of the torsional stiffness mechanism  20  and the torsional stiffness k 1  are described with reference to  FIG. 3 . When the engine EG (damper apparatus  10 ) rotates, a centrifugal force F 11  is applied to the center of gravity  21   g  of the coupling member  21 . The centrifugal force F 11  can be represented by Expression (1). In Expression (1), “m 1 ” represents a mass of the coupling member  21 , “D 11 ” represents a distance between a rotation center RC of the damper apparatus  10  (driving member  11 , intermediate member  12 , and driven member  15 ) and the center of gravity  21   g  of the coupling member  21 , and “Ω” represents an angular velocity of the engine EG. The direction of the centrifugal force F 11  is a radially outward direction in a direction of a straight line L 11  passing through the rotation center RC of the damper apparatus  10  and the center of gravity  21   g  of the coupling member  21 . 
       [Math. 1] 
         F 11= m 1· D 11·Ω′  (1)
 
     When the relative torsion angle between the driving member  11  and the driven member  15  is zero, the coupling member  21  extends in the radial direction (see  FIG. 2 ). Thus, all of the straight line L 11  described above, a straight line L 12  in the extending direction of the coupling member  21  (straight line passing through the rivet  23  and the pin  24 ), a straight line L 13  passing through the rotation center RC of the damper apparatus  10  and the rivet  23 , and a straight line L 14  passing through the rotation center RC of the damper apparatus  10  and the pin  24  coincide with each other. Therefore, a component force F 12  that is a part of the centrifugal force F 11  applied to the center of gravity  21   g  of the coupling member  21  and is applied in a direction orthogonal to the straight line L 12  has a value “0”. 
     When the relative torsion angle between the driving member  11  and the driven member  15  is not zero, the straight lines L 11  to L 14  deviate from each other as illustrated in  FIG. 3 . Thus, the component force F 12  that is a part of the centrifugal force F 11  applied to the center of gravity  21   g  of the coupling member  21  and is applied in the direction orthogonal to the straight line L 12  can be represented by Expression (2). In Expression (2), “α 1 ” represents an angle between the straight line L 11  and the straight line L 12 . The direction of the component force F 12  is a direction in which the relative torsion angle between the driving member  11  and the driven member  15  is reduced (upper right direction in  FIG. 3 ) in the direction orthogonal to the straight line L 12 . As understood from  FIG. 2  and  FIG. 3 , the center of gravity  21   g  of the coupling member  21  is located on a radially outermost side when the relative torsion angle between the driving member  11  and the driven member  15  is zero, shifted radially inward as the relative torsion angle between the driving member  11  and the driven member  15  increases, and shifted radially outward as the relative torsion angle between the driving member  11  and the driven member  15  decreases. When the relative torsion angle between the driving member  11  and the driven member  15  is not zero, the component force F 12  in the direction in which the relative torsion angle between the driving member  11  and the driven member  15  is reduced in the direction orthogonal to the straight line L 12  is generated at the center of gravity  21   g  of the coupling member  21 . Therefore, it can be considered that the torsional stiffness mechanism  20  operates so as to reduce the relative torsion angle between the driving member  11  and the driven member  15  (has a positive restoration force). 
       [Math. 2] 
         F 12= F 11·sin α1  (2)
 
     A force F 13  received by the coupling member  21  from the driving member  11  at the position of the pin  24  (position of the sliding pair of the driving member  11  and the coupling member  21 ) can be represented by Expression (3). In Expression (3), “D 12 ” represents a distance between the rivet  23  and the center of gravity  21   g  of the coupling member  21 , and “D 13 ” represents a distance between the rivet  23  and the pin  24 . The direction of the force F 13  is a direction in which the relative torsion angle between the driving member  11  and the driven member  15  is increased (direction opposite to that of the component force F 12  in  FIG. 3 ) in the direction orthogonal to the straight line L 12 . A component force F 14  that is a part of the force F 13  received by the coupling member  21  from the driving member  11  at the position of the pin  24  and is applied in the rotation direction of the damper apparatus  10  can be represented by Expression (4). In Expression (4), “β 1 ” represents an angle between the straight line L 12  and the straight line L 14 . The direction of the force F 14  is a direction in which the relative torsion angle between the driving member  11  and the driven member  15  is increased (counterclockwise direction in  FIG. 3 ) in the rotation direction of the damper apparatus  10 . 
     
       
         
           
             
               
                 
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     Thus, a torque T 1  transferred to the driving member  11  (the side where the relative torsion angle between the driving member  11  and the driven member  15  is reduced is a positive side) can be represented by Expression (5). In Expression (5), “D 14 ” represents a distance between the rotation center RC of the damper apparatus  10  and the pin  24 . In Expression (5), a value “1” is used as a coefficient of a right-hand side for the following reason. The direction of the force F 13  received by the coupling member  21  from the driving member  11  is determined by a relational expression for a balance of moment on the coupling member  21 . Based on the law of reaction, the driving member  11  receives, from the coupling member  21 , a force in a direction opposite to that of the force F 13  (force in a direction in which the relative torsion angle between the driving member  11  and the driven member  15  is reduced), that is, a positive restoration force. For this reason, the value “1” is used as the coefficient of the right-hand side of Expression (5). 
     When Expressions (1) to (5) are integrated, the torque T 1  can be represented by Expression (6). Assuming that an angle θ 1  between the straight line L 13  and the straight line L 14  is infinitesimal, that is, “sin θ 1 ≈θ 1  and cos θ 1 =1”, the torque T 1  can approximate Expression (7). As represented by Expression (7), the torque T 1  is proportional to the square of the angular velocity Ω of the engine EG. Therefore, as represented by Expression (8) that is based on Expression (7), it can be considered that the torsional stiffness mechanism  20  has the positive torsional stiffness k 1  proportional to the square of the angular velocity Ω of the engine EG. 
     Next, the operation of the torsional stiffness mechanism  30  and the torsional stiffness k 2  is described. The torsional stiffness mechanism  30  is structured similarly to the torsional stiffness mechanism  20  except that the torsional stiffness mechanism  30  is arranged between the driving member  11  and the intermediate member  12  whereas the torsional stiffness mechanism  20  is arranged between the driving member  11  and the driven member  15 . Therefore, the torsional stiffness mechanism  30  operates similarly to the torsional stiffness mechanism  20 . Thus, it can also be considered that the torsional stiffness mechanism  30  has the positive torsional stiffness k 2  proportional to the square of the angular velocity Ω of the engine EG 
     
       
         
           
             
               
                 
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     The operation of the torsional stiffness mechanism  40  and the torsional stiffness k 3  are described with reference to  FIG. 4 . When the engine EG (damper apparatus  10 ) rotates, a centrifugal force F 31  is applied to the center of gravity  41   g  of the coupling member  41 . The centrifugal force F 31  can be represented by Expression (9). In Expression (9), “m 3 ” represents a mass of the coupling member  41 , “D 31 ” represents a distance between the rotation center RC of the damper apparatus  10  and the center of gravity  41   g  of the coupling member  41 , and “Ω” represents the angular velocity of the engine EG as described above. The direction of the centrifugal force F 31  is a radially outward direction in a direction of a straight line L 31  passing through the rotation center RC of the damper apparatus  10  and the center of gravity  41   g  of the coupling member  41 . 
       [Math. 5] 
         F 31= m 3· D 31·Ω 3   (9)
 
     When the relative torsion angle between the intermediate member  12  and the driven member  15  is zero, the coupling member  41  extends in the radial direction (see  FIG. 2 ). Thus, all of the straight line L 31  described above, a straight line L 32  in the extending direction of the coupling member  41  (straight line passing through the rivet  43  and the pin  44 ), a straight line L 33  passing through the rotation center RC of the damper apparatus  10  and the rivet  43 , and a straight line L 34  passing through the rotation center RC of the damper apparatus  10  and the pin  44  coincide with each other. Therefore, a component force F 32  that is a part of the centrifugal force F 31  applied to the center of gravity  41   g  of the coupling member  41  and is applied in a direction orthogonal to the straight line L 32  has a value “0”. 
     When the relative torsion angle between the intermediate member  12  and the driven member  15  is not zero, the straight lines L 31  to L 34  deviate from each other as illustrated in  FIG. 4 . Thus, the component force F 32  that is a part of the centrifugal force F 31  applied to the center of gravity  41   g  of the coupling member  41  and is applied in the direction orthogonal to the straight line L 32  can be represented by Expression (10). In Expression (10), “α 3 ” represents an angle between the straight line L 31  and the straight line L 32 . The direction of the component force F 32  is a direction in which the relative torsion angle between the intermediate member  12  and the driven member  15  is increased (upper right direction in  FIG. 4 ) in the direction orthogonal to the straight line L 32 . As understood from  FIG. 2  and  FIG. 4 , the center of gravity  41   g  of the coupling member  41  is located on a radially innermost side when the relative torsion angle between the intermediate member  12  and the driven member  15  is zero, shifted radially outward as the relative torsion angle between the intermediate member  12  and the driven member  15  increases, and shifted radially inward as the relative torsion angle between the intermediate member  12  and the driven member  15  decreases. When the relative torsion angle between the intermediate member  12  and the driven member  15  is not zero, the component force F 32  in the direction in which the relative torsion angle between the intermediate member  12  and the driven member  15  is increased in the direction orthogonal to the straight line L 32  is generated at the center of gravity  41   g  of the coupling member  41 . Therefore, it can be considered that the torsional stiffness mechanism  40  operates so as to increase the relative torsion angle between the intermediate member  12  and the driven member  15  (has a negative restoration force). 
       [Math. 6] 
         F 32= F 31·sin α3  (10)
 
     A force F 33  received by the coupling member  41  from the intermediate member  12  at the position of the pin  44  (position of the sliding pair of the intermediate member  12  and the coupling member  41 ) can be represented by Expression (11). In Expression (11), “D 32 ” represents a distance between the rivet  43  and the center of gravity  41   g  of the coupling member  41 , and “D 33 ” represents a distance between the rivet  43  and the pin  44 . The direction of the force F 33  is a direction in which the relative torsion angle between the intermediate member  12  and the driven member  15  is reduced (same direction as that of the component force F 32  in  FIG. 4 ) in the direction orthogonal to the straight line L 32 . A component force F 34  that is a part of the force F 33  received by the coupling member  41  from the intermediate member  12  at the position of the pin  44  and is applied in the rotation direction of the damper apparatus  10  can be represented by Expression (12). In Expression (12), “β 3 ” represents an angle between the straight line L 32  and the straight line L 34 . The direction of the force F 34  is a direction in which the relative torsion angle between the intermediate member  12  and the driven member  15  is reduced (clockwise direction in  FIG. 4 ) in the rotation direction of the damper apparatus  10 . 
     
       
         
           
             
               
                 
                   [ 
                   
                     Math 
                     . 
                     
                         
                     
                      
                     7 
                   
                   ] 
                 
               
               
                 
                     
                 
               
             
             
               
                 
                   
                     F 
                      
                     
                         
                     
                      
                     33 
                   
                   = 
                   
                     F 
                      
                     
                         
                     
                      
                     
                       32 
                       · 
                       
                         
                           D 
                            
                           
                               
                           
                            
                           32 
                         
                         
                           D 
                            
                           
                               
                           
                            
                           33 
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   11 
                   ) 
                 
               
             
             
               
                 
                   
                     F 
                      
                     
                         
                     
                      
                     34 
                   
                   = 
                   
                     F 
                      
                     
                         
                     
                      
                     
                       33 
                       · 
                       cos 
                     
                      
                     
                         
                     
                      
                     β 
                      
                     
                         
                     
                      
                     3 
                   
                 
               
               
                 
                   ( 
                   12 
                   ) 
                 
               
             
           
         
       
     
     Thus, a torque T 3  transferred to the intermediate member  12  (the side where the relative torsion angle between the intermediate member  12  and the driven member  15  is reduced is a positive side) can be represented by Expression (13). In Expression (13), “D 34 ” represents a distance between the rotation center RC of the damper apparatus  10  and the pin  44 . In Expression (13), a value (−1) is used as a coefficient of a right-hand side for the following reason. The direction of the force F 33  received by the coupling member  41  from the intermediate member  12  is determined by a relational expression for a balance of moment on the coupling member  41 . Based on the law of reaction, the intermediate member  12  receives, from the coupling member  41 , a force in a direction opposite to that of the force F 33  (force in a direction in which the relative torsion angle between the intermediate member  12  and the driven member  15  is increased), that is, a negative restoration force. For this reason, the value (−1) is used as the coefficient of the right-hand side of Expression (13). 
     When Expressions (9) to (13) are integrated, the torque T 3  can be represented by Expression (14). Assuming that an angle θ 3  between the straight line L 33  and the straight line L 34  is infinitesimal, that is, “sin θ 3 ≈θ 3  and cos θ 3 =1”, Expression (14) is transformed by using a distance D 35  between the rotation center RC of the damper apparatus  10  and the rivet  43  and the distance D 34  between the rotation center RC of the damper apparatus  10  and the pin  44 . Then, the torque T 3  can approximate Expression (15). As represented by Expression (15), the torque T 3  decreases in proportion to the square of the angular velocity Ω of the engine EG (increases as a negative value). Therefore, as represented by Expression (16) that is based on Expression (15), it can be considered that the torsional stiffness mechanism  40  has the negative torsional stiffness k 3  proportional to the square of the angular velocity n of the engine EG. 
     
       
         
           
             
               
                 
                   [ 
                   
                     Math 
                     . 
                     
                         
                     
                      
                     8 
                   
                   ] 
                 
               
               
                 
                     
                 
               
             
             
               
                 
                   
                     T 
                      
                     
                         
                     
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                     3 
                   
                   = 
                   
                     
                       - 
                       F 
                     
                      
                     
                         
                     
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                       34 
                       · 
                       D 
                     
                      
                     
                         
                     
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                     34 
                   
                 
               
               
                 
                   ( 
                   13 
                   ) 
                 
               
             
             
               
                 
                   
                     T 
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                       3 
                       · 
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                       31 
                       · 
                       
                         Ω 
                         2 
                       
                       · 
                       
                         
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                             32 
                             · 
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                      
                     
                         
                     
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                       3 
                       · 
                       cos 
                     
                      
                     
                         
                     
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                   ( 
                   14 
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                     T 
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                       · 
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                       35 
                       · 
                       
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                         2 
                       
                       · 
                       
                         
                           ( 
                           
                             
                               D 
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                               34 
                             
                             
                               
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                                 35 
                               
                             
                           
                           ) 
                         
                         2 
                       
                       · 
                       θ 
                     
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                   ( 
                   15 
                   ) 
                 
               
             
             
               
                 
                   
                     k 
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                           ( 
                           
                             
                               D 
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                               34 
                             
                             
                               
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                         2 
                       
                     
                   
                 
               
               
                 
                   ( 
                   16 
                   ) 
                 
               
             
           
         
       
     
     The inventors have found the following matters as described in, for example, International Publication No. 2016/021669 (WO 2016/021669) in the structure of the damper apparatus  10 , that is, the structure including the first torque transfer path (torsional stiffness mechanism  20 ) and the second torque transfer path (torsional stiffness mechanism  30 , intermediate member  12 , and torsional stiffness mechanism  40 ) between the driving member  11  and the driven member  15 . In the structure of the damper apparatus  10 , there is an angular frequency ω of a vibration from the engine EG at an anti-resonance point at which the vibration from the engine EG that is transferred from the driving member  11  to the driven member  15  via the first torque transfer path and the vibration from the engine EG that is transferred from the driving member  11  to the driven member  15  via the second torque transfer path are canceled out and the vibration amplitude of the driven member  15  is theoretically zero. The angular frequency ω at the anti-resonance point can be represented by Expression (17). In Expression (17), “k 1 ”, “k 2 ”, and “k 3 ” represent the torsional stiffnesses of the torsional stiffness mechanisms  20 ,  30 , and  40 , respectively, and “J′” represents a value calculated from a moment of inertia J of the intermediate member  12 , masses m 1 , m 2 , and m 3  of the coupling members  21 ,  31 , and  41 , and distances from the rotation center to the rivets  23 ,  33 , and  43  and the pins  24 ,  34 , and  44 . The torsional stiffnesses k 1 , k 2 , and k 3  of the torsional stiffness mechanisms  20 ,  30 , and  40  and the value J′ are set so that a right-hand side of Expression (17) (specifically, the numerator in the radical sign) is a positive value. 
     
       
         
           
             
               
                 
                   [ 
                   
                     Math 
                     . 
                     
                         
                     
                      
                     9 
                   
                   ] 
                 
               
               
                 
                     
                 
               
             
             
               
                 
                   ω 
                   = 
                   
                     
                       
                         
                           ( 
                           
                             
                               k 
                                
                               
                                   
                               
                                
                               
                                 1 
                                 · 
                                 k 
                               
                                
                               
                                   
                               
                                
                               2 
                             
                             + 
                             
                               k 
                                
                               
                                   
                               
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                                 2 
                                 · 
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                                
                               
                                   
                               
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                             + 
                             
                               k 
                                
                               
                                   
                               
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                                 · 
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                                
                               
                                   
                               
                                
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                           ) 
                         
                         · 
                         2 
                       
                       
                         
                           
                             J 
                             ′ 
                           
                           · 
                           k 
                         
                          
                         
                             
                         
                          
                         1 
                       
                     
                   
                 
               
               
                 
                   ( 
                   17 
                   ) 
                 
               
             
           
         
       
     
     As described above, the torsional stiffness k 3  of the torsional stiffness mechanism  40  is a negative value. Therefore, in Expression (17), the value “k 2 ·k 3 ” and the value “k 3 ·k 1 ” in the radical sign of the right-hand side are negative values. Thus, the numerator in the radical sign of the right-hand side and furthermore the entire right-hand side can be reduced as compared to a case where the torsional stiffness k 3  of the torsional stiffness mechanism  40  is a positive value. Considering a case where the angular frequency ω at the anti-resonance point is set to a constant value, the denominator in the radical sign of the right-hand side of Expression (17) can be reduced, that is, the moment of inertia of the intermediate member  12  can be reduced. As a result, the damper apparatus  10  can be downsized, and the vibration damping performance can be improved. 
     As described above, the torsional stiffness mechanism  20  has the positive torsional stiffness k 1  proportional to the square of the angular velocity Ω of the engine EG, the torsional stiffness mechanism  30  has the positive torsional stiffness k 2  proportional to the square of the angular velocity Ω of the engine EG, and the torsional stiffness mechanism  40  has the negative torsional stiffness k 3  proportional to the square of the angular velocity Ω of the engine EG. Based on those facts, the rotation speed range of the engine EG in which high vibration damping performance can be exerted can be extended by setting the torsional stiffnesses k 1 , k 2 , and k 3  of the torsional stiffness mechanisms  20 ,  30 , and  40  and the value J′ so as to increase the angular frequency ω at the anti-resonance point such that the torsional stiffnesses k 1 , k 2 , and k 3  of the torsional stiffness mechanisms  20 ,  30 , and  40  (and furthermore the overall torsional stiffness) appropriately change as the angular velocity Ω (rotation speed) of the engine EG increases. In particular, the rotation speed range of the engine EG in which the anti-resonance occurs can further be extended by setting the torsional stiffnesses k 1 , k 2 , and k 3  of the torsional stiffness mechanisms  20 ,  30 , and  40  and the value J′ so that the angular frequency ω at the anti-resonance point substantially equals an angular frequency of the vibration from the engine EG on each occasion. 
     In the damper apparatus  10  described above, the coupling member  21  of the torsional stiffness mechanism  20  has the relationship of the revolute pair with the driven member  15  and the sliding pair with the driving member  11 , but may have a relationship of a sliding pair with the driven member  15  and a revolute pair with the driving member  11 . The coupling member  31  of the torsional stiffness mechanism  30  has the relationship of the revolute pair with the intermediate member  12  and the sliding pair with the driving member  11 , but may have a relationship of a sliding pair with the intermediate member  12  and a revolute pair with the driving member  11 . The coupling member  41  of the torsional stiffness mechanism  40  may have the relationship of the revolute pair with the driven member  15  and the sliding pair with the intermediate member  12 . 
     In the damper apparatus  10  described above, the torsional stiffness mechanism  20  has the positive torsional stiffness k 1  proportional to the square of the angular velocity Ω of the engine EG, the torsional stiffness mechanism  30  has the positive torsional stiffness k 2  proportional to the square of the angular velocity Ω of the engine EG, and the torsional stiffness mechanism  40  has the negative torsional stiffness k 3  proportional to the square of the angular velocity Ω of the engine EG. At least one of the torsional stiffness mechanism  20  and the torsional stiffness mechanism  30  may have a positive torsional stiffness that is constant irrespective of the rotation speed of the engine EG. When the torsional stiffness mechanism  20  or the torsional stiffness mechanism  30  has the constant positive torsional stiffness, an arc coil spring, a straight coil spring, or the like may be used as the torsional stiffness mechanism  20  or the torsional stiffness mechanism  30 . 
     In the damper apparatus  10  described above, the torsional stiffness mechanism  30  having the positive torsional stiffness is arranged between the driving member  11  and the intermediate member  12 , and the torsional stiffness mechanism  40  having the negative torsional stiffness is arranged between the intermediate member  12  and the driven member  15 . The torsional stiffness mechanism  40  may be arranged between the driving member  11  and the intermediate member  12 , and the torsional stiffness mechanism  30  may be arranged between the intermediate member  12  and the driven member  15 . 
     In the damper apparatus  10  described above, the turbine runner  5  of the torque converter TC is fixed to the driven member  15  and the damper hub  7 . As indicated by long dashed double-short dashed lines in  FIG. 1 , the turbine runner  5  may be fixed to the driving member  11  or the intermediate member  12 . 
       FIG. 5  is a schematic structural diagram of another damper apparatus  110  disclosed herein. The damper apparatus  110  of  FIG. 5  corresponds to an apparatus in which the intermediate member  12  is omitted from the damper apparatus  10  described above. The same components of the damper apparatus  110  of  FIG. 5  as the components of the damper apparatus  10  are represented by the same reference symbols to omit their detailed description. The damper apparatus  110  of  FIG. 5  includes the driving member (input element)  11  and the driven member (output element)  15  as the rotational elements. The damper apparatus  110  further includes, as the torque transfer elements, the plurality of (for example, two) torsional stiffness mechanisms  20  arranged between the driving member  11  and the driven member  15 , and a plurality of (for example, two) torsional stiffness mechanisms  140  arranged between the driving member  11  and the driven member  15  in parallel to (configured to act in parallel to) the torsional stiffness mechanisms  20 . In the damper apparatus  110 , the turbine runner  5  of the torque converter TC may be fixed to the driven member  15  and the damper hub  7  as indicated by a continuous line in  FIG. 5 , or may be fixed to the driving member  11  as indicated by a long dashed double-short dashed line in  FIG. 5 . The torsional stiffness mechanism  140  is structured similarly to the torsional stiffness mechanism  40  of the damper apparatus  10 , and has a negative torsional stiffness k 4  proportional to the square of the angular velocity Ω of the engine EG In the damper apparatus  110 , the torsional stiffness mechanism  140  functions similarly to the torsional stiffness mechanism  40  of the damper apparatus  10  of  FIG. 1 , whereby effects similar to those of the damper apparatus  10  of  FIG. 1  can be attained. 
       FIG. 6  is a schematic structural diagram of a starting apparatus  201  including another damper apparatus  210  disclosed herein.  FIG. 7  is a sectional view of the damper apparatus  210 .  FIG. 8  is a front-side elevation of the damper apparatus  210 . The same components of the starting apparatus  201  and the damper apparatus  210  of  FIG. 6  to  FIG. 8  as the components of the starting apparatus  1  and the damper apparatus  10  are represented by the same reference symbols to omit their detailed description. 
     The damper apparatus  210  includes, as the rotational elements, a driving member (input element)  211 , an input-side rotational member  212  coupled to the driving member  211 , an intermediate member (intermediate element)  213 , a driven member (output element)  215 , and an output-side rotational member  217  coupled to the driven member  215 . The damper apparatus  110  further includes, as the torque transfer elements, a plurality of (for example, four) outer springs (third torsional stiffness mechanisms)  220  arranged between the driving member  211  and the intermediate member  213 , a plurality of (for example, four) outer springs (fourth torsional stiffness mechanisms)  230  arranged between the intermediate member  213  and the output-side rotational member  217 , a plurality of (for example, four) inner springs (first torsional stiffness mechanisms)  240  arranged between the driving member  211  and the output-side rotational member  217 , and a plurality of (for example, four) torsional stiffness mechanisms (second torsional stiffness mechanisms)  250  arranged between the input-side rotational member  216  and the driven member  215 . 
     In this embodiment, a constant pitch straight coil spring, which is formed of a metal material helically wound so as to have an axis center extending straight when no load is applied and in which the pitch of an active coil portion (portion except seats) is a constant pitch, is employed as each of the outer springs  220  and  230  and the inner spring  240 . A constant pitch arc coil spring may be employed as at least one of the outer springs  220  and  230  and the inner spring  240 . 
     All of the plurality of outer springs  220  and  230  extend along a circumferential direction of the damper apparatus  210 , and are arranged in an outer peripheral region of a fluid chamber defined by the front cover  3  and the pump impeller  4  so that the outer springs  220  and the outer springs  230  are alternately arrayed along the circumferential direction to make pairs (act in series). The plurality of inner springs  240  extend along the circumferential direction of the damper apparatus  210 , and are arranged in an inner peripheral region of the fluid chamber so as to be arrayed at intervals along the circumferential direction. In a state in which the damper apparatus  210  is attached (when the relative torsion angle between the two rotational elements coupled via each spring is zero), all of the outer springs  220  and  230  and the inner spring  240  have their equilibrium lengths or are compressed slightly shorter than their equilibrium lengths. 
     The driving member  211  is coupled to a lock-up piston  81  of the lock-up clutch  8  via a plurality of rivets  211   r  at intervals in the circumferential direction. The driving member  211  is a plate-shaped annular member, and includes a plurality of (for example, four) outer abutment portions  211   co  and a plurality of (for example, four) inner abutment portions  211   ci . The plurality of outer abutment portions  211   co  are provided on an outer peripheral portion of the driving member  211  at intervals in the circumferential direction. The plurality of inner abutment portions  211   ci  are provided on an inner peripheral portion of the driving member  211  with intervals in the circumferential direction. 
     The input-side rotational member  212  has two plate-shaped annular members  212   a  and  212   b , which are coupled to each other via a plurality of rivets  253  at intervals in the circumferential direction. The input-side rotational member  212  is coupled to the driving member  211  by being coupled to the lock-up piston  81 . 
     The intermediate member  213  is a plate-shaped annular member, and includes a plurality of (for example, four) abutment portions  212   c  that protrude radially outward at intervals in the circumferential direction. The driven member  215  is a plate-shaped annular member. A plurality of openings  2150  extending along the circumferential direction and a plurality of (for example, four) guide holes  215   h  extending along the radial direction are formed in the driven member  215  at intervals in the circumferential direction. 
     The output-side rotational member  217  includes a bottomed tubular member  218  coupled to the driven member  215  and having a bottomed tubular shape, and a plate member  219  coupled to the bottomed tubular member  218 . The bottomed tubular member  218  includes protrusions  218   p  that protrude in the axial direction toward the driven member  215  at intervals in the circumferential direction. The bottomed tubular member  218  and the driven member  215  are coupled to each other such that the protrusions  218   p  of the bottomed tubular member  218  are fitted to the openings  2150  of the driven member  215 . The plate member  219  is coupled to the bottomed tubular member  218  via a plurality of rivets  217   r  at intervals in the circumferential direction. The plate member  219  includes a plurality of (for example, four) outer abutment portions  219   co  and a plurality of (for example, four) inner abutment portions  219   ci . The plurality of outer abutment portions  219   co  are provided on an outer peripheral portion of the plate member  219  at intervals in the circumferential direction. The plurality of inner abutment portions  219   ci  are provided on an inner peripheral portion of the plate member  219  at intervals in the circumferential direction. 
     In the state in which the damper apparatus  210  is attached (when the relative torsion angle between the two rotational elements coupled via each spring is zero), each outer abutment portion  211   co  of the driving member  211  abuts against the ends of the outer springs  220  and  230  that are not paired with each other (do not act in series) between those outer springs  220  and  230 . Similarly, each outer abutment portion  219   co  of the plate member  219  of the output-side rotational member  217  abuts against the ends of the outer springs  220  and  230  that are not paired with each other (do not act in series) between those outer springs  220  and  230 . Each abutment portion  213   c  of the intermediate member  213  abuts against the ends of the outer springs  220  and  230  that are paired with each other (act in series) between those outer springs  220  and  230 . 
     Thus, in the state in which the damper apparatus  210  is attached, one end of each outer spring  220  abuts against the corresponding outer abutment portion  211   co  of the driving member  211  and the corresponding outer abutment portion  219   co  of the plate member  219 , and the other end of each outer spring  220  abuts against the corresponding abutment portion  213   c  of the intermediate member  213 . One end of each outer spring  230  abuts against the corresponding abutment portion  213   c  of the intermediate member  213 , and the other end of each outer spring  230  abuts against the corresponding outer abutment portion  211   co  of the driving member  211  and the corresponding outer abutment portion  219   co  of the plate member  219 . 
     In the state in which the damper apparatus  210  is attached, each inner abutment portion  211   ci  of the driving member  211  abuts against the ends of two inner springs  240  that are adjacent to each other in the circumferential direction between those two inner springs  240 . The inner abutment portion  219   ci  of the plate member  219  is arranged between the two inner springs  240  that are adjacent to each other in the circumferential direction. The inner abutment portion  219   ci  does not abut against the inner spring  240  when a relative torsion angle between the driving member  211  and the output-side rotational member  217  (plate member  219 ) is smaller than a predetermined torsion angle as in the state in which the damper apparatus  210  is attached. The inner abutment portion  219   ci  abuts against the inner spring  240  when the relative torsion angle between the driving member  211  and the output-side rotational member  217  (plate member  219 ) is equal to or larger than the predetermined torsion angle. 
     The plurality of torsional stiffness mechanisms  250  are arranged at intervals in the circumferential direction, and are coupled to the input-side rotational member  212  and the driven member  215  so as to extend in the radial direction in the state in which the damper apparatus  210  is attached (when a relative torsion angle between the input-side rotational member  212  and the driven member  215  is zero). 
     The torsional stiffness mechanism  250  includes a spring (elastic body)  251 , an outer retaining member  252  that retains the radially outer end of the spring  251 , the rivet  253  described above for coupling the input-side rotational member  212  and the outer retaining member  252  to each other, an inner retaining member  254  that retains the radially inner end of the spring  251 , and a rivet  255  for coupling the driven member  215  and the inner retaining member  254  to each other. 
     A constant pitch straight coil spring, which is formed of a metal material helically wound so as to have an axis center extending straight when no load is applied and in which the pitch of an active coil portion (portion except seats) is a constant pitch, is employed as the spring  251 . In the state in which the damper apparatus  210  is attached (when the relative torsion angle between the input-side rotational member  212  and the driven member  215  is zero), the spring  251  is compressed sufficiently shorter than its equilibrium length. 
     The outer retaining member  252  includes a retaining portion  252   a  that retains the spring  251 , and a protrusion  252   b  extending from the opposite side of the retaining portion  252   a  from the spring  251 . In a state in which the protrusion  252   b  of the outer retaining member  252  is inserted between the pair of annular members  212   a  and  212   b  of the input-side rotation speed member  212 , the rivet  253  couples the pair of annular members  212   a  and  212   b  and the protrusion  252   b  to each other in a freely rotatable manner. 
     The inner retaining member  254  includes a retaining portion  254   a  that retains the spring  251 , and a pair of protrusions  254   b  and  254   c  extending from the opposite side of the retaining portion  254   a  from the spring  251  with a distance therebetween in the axial direction. In a state in which the driven member  215  is inserted between the pair of protrusions  254   b  and  254   c  of the inner retaining member  254 , the rivet  255  is inserted into the guide hole  215   h  of the driven member  215 , and couples the driven member  215  and the pair of protrusions  254   b  and  254   c  to each other in a freely rotatable manner. The rivet  255  also functions as a mass body, and is movable along the guide hole  215   h.    
     In the damper apparatus  210  structured as described above, when the relative torsion angle between the driving member  211  and the output rotational member  217  (plate member  219 ) is smaller than the predetermined torsion angle, the driven member  215  is coupled to the driving member  211  via the plurality of outer springs  220 , the intermediate member  213 , the plurality of outer springs  230 , and the output-side rotational member  217 , and is also coupled to the driving member  211  via the input-side rotational member  212  and the plurality of torsional stiffness mechanisms  250 . When the relative torsion angle between the driving member  211  and the output-side rotational member  217  (plate member  219 ) is equal to or larger than the predetermined torsion angle, the driven member  215  is coupled to the driving member  211  via the plurality of outer springs  220 , the intermediate member  213 , the plurality of outer springs  230 , and the output-side rotational member  217 , also coupled to the driving member  211  via the input-side rotational member  212  and the plurality of torsional stiffness mechanisms  250 , and further coupled to the driving member  211  via the plurality of inner springs  240  and the output-side rotational member  217 . 
     Next, an operation of the starting apparatus  201  including the damper apparatus  210  is described. As understood from  FIG. 6 , in the starting apparatus  201 , when the lock-up is not executed by the lock-up clutch  8 , the torque (power) transferred from the engine EG to the front cover  3  is transferred to the input shaft IS of the transmission TM via the path including the pump impeller  4 , the turbine runner  5 , and the damper hub  7 . 
     When the lock-up is executed by the lock-up clutch  8  and when the relative torsion angle between the driving member  211  and the plate member  219  is smaller than the predetermined torsion angle, the torque (power) transferred from the engine EG to the driving member  211  via the front cover  3  and the lock-up clutch  8  is transferred to the driven member  215 , the damper hub  7 , and the input shaft IS of the transmission TM via a first torque transfer path including the plurality of outer springs  220 , the intermediate member  213 , the plurality of outer springs  230 , and the output-side rotational member  217  and via a second torque transfer path including the input-side rotational member  212  and the plurality of torsional stiffness mechanisms  250 . When the lock-up is executed and when the relative torsion angle between the driving member  211  and the plate member  219  is equal to or larger than the predetermined torsion angle, the torque (power) transferred to the driving member  211  is transferred to the driven member  215  via the first torque transfer path, the second torque transfer path, and a third torque transfer path including the plurality of inner springs  240  and the output-side rotational member  217 . 
     In the state in which the damper apparatus  210  is attached (when the relative torsion angle between the two rotational elements coupled via each spring is zero), all of the outer springs  220  and  230  and the inner spring  240  extend along the circumferential direction of the damper apparatus  210 , and have their equilibrium lengths or are compressed slightly shorter than their equilibrium lengths. Thus, when the damper apparatus  10  is rotating along with the rotation of the engine EG through the execution of the lock-up and when a relative torsion angle is formed between the two rotational elements on both sides of each of the outer springs  220  and  230  and the inner spring  240 , each of the outer springs  220  and  230  and the inner spring  240  operates so as to reduce the relative torsion angle (has a positive restoration force). At this time, each of the outer springs  220  and  230  and the inner spring  240  functions as a spring having a constant spring rate, that is, a positive constant torsional stiffness. 
     In the state in which the damper apparatus  210  is attached (when the relative torsion angle between the input-side rotational member  212  and the driven member  215  is zero), the torsional stiffness mechanism  250  extends in the radial direction of the damper apparatus  210 . In the state in which the damper apparatus  210  is attached, the spring  251  of the torsional stiffness mechanism  250  is compressed sufficiently shorter than its equilibrium length. Thus, when the damper apparatus  10  is rotating along with the rotation of the engine EG through the execution of the lock-up and when a relative torsion angle is formed between the input-side rotational member  212  and the driven member  215 , the torsional stiffness mechanism  250  operates so as to increase the relative torsion angle (has a negative restoration force). An operation of the torsional stiffness mechanism  250  and a stiffness k 5  are described below with reference to  FIG. 9 . 
     In the torsional stiffness mechanism  250 , the rivet  253  is held in the rotation direction and the radial direction relative to the input-side rotational member  212 , and the rivet  255  is held in the rotation direction but movable in the radial direction relative to the driven member  215 . 
     In the torsional stiffness mechanism  250 , when the relative torsion angle between the input-side rotational member  212  and the driven member  215  is zero, the torsional stiffness mechanism  250  extends in the radial direction (see  FIG. 8 ). Thus, all of a straight line L 51  passing through the rotation center RC of the damper apparatus  210  and the rivet  253 , a straight line L 52  in the extending direction of the torsional stiffness mechanism  250  (straight line passing through the rivet  253  and the rivet  255 ), and a straight line L 53  passing through the rotation center RC of the damper apparatus  210  and the rivet  255  coincide with each other. When the relative torsion angle between the input-side rotational member  212  and the driven member  215  is not zero, the extending direction of the torsional stiffness mechanism  250  deviates from the radial direction as illustrated in  FIG. 9 . Thus, the straight lines L 51  to L 53  deviate from each other. 
     A force F 51  generated by the spring  251  can be represented by Expression (18) based on the Hooke&#39;s law. In Expression (18), “ks 5 ” represents a spring rate of the spring  251 , “Ls 50 ” represents an equilibrium length of the spring  251 , and “Ls 51 ” represents a current length of the spring  251 . A component force F 52  that is a part of the force F 51  and is applied in the rotation direction at the rivet  255  can be represented by Expression (19). In Expression (19), “ϕ 5 ” represents an angle between the straight line L 52  and the straight line L 53 . Thus, a torque T 5  transferred by the spring  251  can be represented by Expression (20). In Expression (20), “r 5 ” represents a distance between the rotation center RC of the damper apparatus  210  and the rivet  255 . As described above, the rivet  255  is movable in the radial direction relative to the driven member  215 . Therefore, the distance r 5  is variable. Specifically, a centrifugal force proportional to the square of the angular velocity Ω (rotation speed) of the engine EG is applied to the rivet  255  that functions as the mass body. Thus, the distance r 5  increases as the angular velocity Ω of the engine EG increases. The component force F 52  and the torque T 5  are a force and a torque in a direction in which the relative torsion angle between the input-side rotational member  212  and the driven member  215  is increased. Thus, it can be said that the torsional stiffness mechanism  250  has a negative restoration force. 
       [Math. 10] 
         F 51= ks 5·( L 51− Ls 50)  (18)
 
         F 52= ks 5·( Ls 51− Ls 50)·sin ϕ5  (19)
 
         T 5= ks 5·( Ls 51− Ls 50)·sin ϕ5· r 5  (20)
 
     When the law of sines and the law of cosines are applied to a triangle having vertices at the rotation center RC of the damper apparatus  210 , the rivet  253 , and the rivet  255 , Expression (21) and Expression (22) are obtained. In Expression (21) and Expression (22), “R 5 ” represents a distance between the rotation center RC of the damper apparatus  210  and the rivet  253 , and “05” represents the relative torsion angle between the input-side rotational member  212  and the driven member  215 . 
     
       
         
           
             
               
                 
                   [ 
                   
                     Math 
                     . 
                     
                         
                     
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     When Expression (21) and Expression (22) are substituted into Expression (20) to erase the current length Ls 51  of the spring  251  and the angle+5 between the straight line L 52  and the straight line L 53 , a relationship between the torque T 5  transferred by the spring  251  and the relative torsion angle θ 5  between the input-side rotational member  212  and the driven member  215  is obtained. In particular, when the relative torsion angle θ 5  is infinitesimal, the relationship between the torque T 5  and the relative torsion angle θ 5  can be represented by Expression (23). Thus, the overall torsional stiffness k 5  of the torsional stiffness mechanism  250  can be represented by Expression (24). 
     
       
         
           
             
               
                 
                   [ 
                   
                     Math 
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                     12 
                   
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       FIG. 10  is an explanatory drawing illustrating an example of a relationship between the distance r 5  and the torsional stiffness k 5  in Expression (24). As illustrated in  FIG. 10 , the torsional stiffness k 5  has a value “0” when the distance r 5  is equal to a difference (R 5 −Ls 50 ) between the distance R 5  and the equilibrium length Ls 50  of the spring  251 . The torsional stiffness k 5  decreases (increases as a negative value) as the distance r 5  increases within a range larger than the difference (R 5 −Ls 50 ) and smaller than the distance R 5 . Thus, it is demonstrated that the torsional stiffness mechanisms  250  and the guide holes  215   h  of the driven member  215  should be designed so that the distance r 5  is larger than the difference (R 5 −Ls 50 ) and smaller than the distance R 5 . 
     As described above, the distance r 5  increases as the angular velocity n (rotation speed) of the engine EG increases. Thus, it can be said that the overall torsional stiffness k 5  of the torsional stiffness mechanism  250  decreases (increases on the negative side) as the angular velocity Ω of the engine EG increases. As a result, effects similar to those of the damper apparatus  10  described above can be attained. 
       FIG. 11  is a sectional view of another damper apparatus  310  disclosed herein.  FIG. 12  is a front-side elevation of the damper apparatus  310 . The damper apparatus  310  of  FIG. 11  and  FIG. 12  corresponds to an apparatus in which the torsional stiffness mechanism  250  of the damper apparatus  210  described above is replaced with a torsional stiffness mechanism  350 . The same components of the damper apparatus  310  of  FIG. 11  and  FIG. 12  as the components of the damper apparatus  210  are represented by the same reference symbols to omit their detailed description. 
     As illustrated in  FIG. 11  and  FIG. 12 , the torsional stiffness mechanism  350  includes a positional adjuster  360  (see  FIG. 11 ) configured to adjust the position of the rivet  255  (distance r 5  described above) by adjusting the position of the inner retaining member  254  in addition to the spring  251 , the outer retaining member  252 , the rivet  253 , the inner retaining member  254 , and the rivet  255  that are similar to those of the torsional stiffness mechanism  250 . The rivet  255  of the torsional stiffness mechanism  350  only needs to be movable along the guide hole  215  of the driven member  215 , and may be lighter in the weight than the rivet  255  of the torsional stiffness mechanism  250 . 
     As illustrated in  FIG. 11 , the positional adjuster  360  includes a coupling member  361  coupled to the inner retaining member  252 , an actuator  362  configured to move the rivet  255  in the radial direction via the coupling member  361  and the inner retaining member  254 , a rotation speed sensor  363  configured to detect the rotation speed of the engine EG; and an electronic controller  353  configured to receive the angular velocity Ω (rotation speed) of the engine EG that is input from the rotation speed sensor  351  and to control the actuator  352 . A protrusion  254   d  that protrudes in the axial direction is formed on the outer wall surface of the protrusion  254   c  of the inner retaining member  254 . An opening  361   o  is formed in the coupling member  361 . The inner retaining member  254  and the coupling member  361  are coupled to each other such that the protrusion  254   d  of the inner retaining member  254  is fitted to the opening  361   o  of the coupling member  361 . 
     In the positional adjuster  360 , the electronic controller  353  controls the actuator  352  so that the inner retaining member  254  and the rivet  255  move radially outward as the angular velocity Ω of the engine EG increases. 
     An overall stiffness k 6  of the torsional stiffness mechanism  360  can be represented by using the spring rate ks 5  of the spring  251  (see Expression (24)) similarly to the overall torsional stiffness k 5  of the torsional stiffness mechanism  250  of the damper apparatus  210 . Thus, when the position of the rivet  255  is adjusted by the positional adjuster  360  as described above, the overall structure k 6  of the torsional stiffness mechanism  360  can be decreased (increased on the negative side) as the angular velocity Ω of the engine EG increases similarly to the overall torsional stiffness k 5  of the torsional stiffness mechanism  250 . As a result, effects similar to those of the damper apparatus  210  can be attained. 
       FIG. 13  is a sectional view of another damper apparatus  410  disclosed herein.  FIG. 14  is a front-side elevation of the damper apparatus  410 . The damper apparatus  310  of  FIG. 14  corresponds to an apparatus in which the driven member  215  and the torsional stiffness mechanism  250  of the damper apparatus  210  described above are replaced with a driven member  415  and a torsional stiffness mechanism  450 . The same components of the damper apparatus  410  of  FIG. 14  as the components of the damper apparatus  210  are represented by the same reference symbols to omit their detailed description. 
     As illustrated in  FIG. 13  and  FIG. 14 , the driven member  415  is identical to the driven member  215  of the damper apparatus  210  except that the guide hole  215   h  is not provided. Similarly to the torsional stiffness mechanism  250 , the torsional stiffness mechanism  450  is coupled to the input-side rotational member  212  and the driven member  215  so as to extend in the radial direction when a relative torsion angle between the input-side rotational member  212  and the driven member  215  is zero. The torsional stiffness mechanism  450  includes a spring  451 , the outer retaining member  252 , the rivet  253 , the inner retaining member  254 , and a rivet  455 . 
     A variable pitch straight coil spring, which is formed of a metal material helically wound so as to have an axis center extending straight when no load is applied and in which the pitch of an active coil portion (portion except seats) is a variable pitch, is employed as the spring  451 . In a state in which the damper apparatus  410  is attached, the pitch of the active coil portion of the spring  451  gradually decreases toward the radially outer side, and the spring  451  is compressed shorter than its equilibrium length. The rivet  455  couples the driven member  415  and the pair of protrusions  254   b  and  254   c  of the inner retaining member  254  to each other in a freely rotatable manner. 
     Next, an operation of the torsional stiffness mechanism  450  is described.  FIG. 15  is an explanatory drawing illustrating a state in which the angular velocity Ω (rotation speed) of the engine EG is small and the relative torsion angle between the input-side rotational member  212  and the driven member  215  is zero.  FIG. 16  is an explanatory drawing illustrating a state in which the angular velocity Ω (rotation speed) of the engine EG is large and the relative torsion angle between the input-side rotational member  212  and the driven member  215  is zero. 
     As described above, in the state in which the damper apparatus  410  is attached, the pitch of the active coil portion of the spring  451  gradually decreases toward the radially outer side. When the angular velocity Ω of the engine EG is small, a centrifugal force applied to the spring  451  of the torsional stiffness mechanism  450  is small. Therefore, as illustrated in  FIG. 15 , the degree of closeness of coils to the radially outer side is small in the spring  451  as a whole, and the number of close contact portions of the coils is zero or small in the spring  451  as a whole. Thus, the number of active coils of the spring  451  is large. When the angular velocity Ω of the engine EG is large, the centrifugal force applied to the spring  451  is large. Therefore, as illustrated in  FIG. 16 , the degree of closeness of the coils to the radially outer side is large in the spring  451  as a whole, and the coils are brought into close contact with each other or the amount of close contact increases in the radially outer portion of the spring  451 . Thus, the number of active coils of the spring  451  decreases. That is, as the angular velocity Ω of the engine EG increases, the amount of close contact (number of close contact coils) increases in the spring  451 , the number of active coils of the spring  451  decreases, a spring rate ks 7  of the spring  451  increases, and an overall torsional stiffness k 7  of the torsional stiffness mechanism  450  decreases (increases on the negative side). As a result, effects similar to those of the damper apparatus  210  can be attained. 
     In the damper apparatuses  210 ,  310 , and  410  described above, the constant pitch straight coil spring is employed as the outer spring  240  serving as the torsional stiffness mechanism. As illustrated in  FIG. 17 , a variable pitch straight coil spring, in which the pitch of an active coil portion (portion except seats) is a variable pitch, may be employed as an inner spring  240 B. In this case, the pitch of the active coil portion of the inner spring  240 B may gradually decrease toward the center of the inner spring  240 B from both ends in its extending direction. If the variable pitch straight coil spring is employed as the outer spring  220  and both ends of the outer spring  220  are supported in the radial direction, when the angular velocity Ω of the engine EG is small, a centrifugal force applied to the outer spring  220  is small. Therefore, the degree of radially outward bulging is small in the vicinity of the center of the outer spring  220  in its extending direction, and the number of close contact portions of coils is zero or small in the inner spring  240 B. Thus, the number of active coils of the spring  240 B is large. When the angular velocity Ω of the engine EG is large, the centrifugal force applied to the spring  240 B is large. Therefore, the degree of radially outward bulging increases in the vicinity of the center of the spring  240 B, and its curvature radius decreases. Further, the coils are brought into close contact with each other or the amount of close contact increases in the radially inner portion of the inner spring  240 B in the vicinity of the center. Thus, the number of active coils of the inner spring  240 B decreases. That is, as the angular velocity Ω of the engine EG increases, the amount of close contact (number of close contact coils) increases in the inner spring  240 B, the number of active coils of  240 B decreases, and the spring rate of the inner spring  240 B increases (the overall torsional stiffness of the first torsional stiffness mechanism increases on the positive side). Although the description is given of the inner spring  240 B, the same may apply to the outer springs  220  and  230 . 
     In the damper apparatuses  210 ,  310 , and  410  described above, the turbine runner  5  of the torque converter TC is fixed to the driven member  15  and the damper hub  7 . As indicated by long dashed double-short dashed lines in  FIG. 6 , the turbine runner  5  may be fixed to the driving member  211  or the intermediate member  213 . 
       FIG. 18  is a schematic structural diagram of another damper apparatus  510  disclosed herein. The damper apparatus  210 C of  FIG. 18  corresponds to an apparatus in which the outer springs  220  and  230  and the intermediate member  213  are omitted from the damper apparatus  210  described above and the inner spring  240  constantly operates (functions) irrespective of the relative torsion angle between the driving member  211  and the output-side rotational member  217  (plate member  219 ). The same components of the damper apparatus  210 C of  FIG. 18  as the components of the damper apparatus  210  are represented by the same reference symbols to omit their detailed description. The damper apparatus  210 C of  FIG. 18  includes, as the rotational elements, the driving member (input element)  211 , the input-side rotational member  212  coupled to the driving member  211 , the driven member (output element)  215 , and the output-side rotational member  217  coupled to the driven member  215 . The damper apparatus  110  further includes, as the torque transfer elements, the plurality of (for example, four) inner springs (first torsional stiffness mechanisms)  240  arranged between the driving member  211  and the output-side rotational member  217 , and the plurality of (for example, four) torsional stiffness mechanisms (second torsional stiffness mechanisms)  250  arranged between the input-side rotational member  216  and the driven member  215 . In the damper apparatus  210 C, the turbine runner  5  of the torque converter TC may be fixed to the driven member  15  and the damper hub  7  as indicated by a continuous line in  FIG. 18 , or may be fixed to the driving member  11  as indicated by a long dashed double-short dashed line in  FIG. 18 . Also in the damper apparatus  210 , effects similar to those of the damper apparatus  210  can be attained. 
       FIG. 19  is a schematic structural diagram of another damper apparatus  510  disclosed herein.  FIG. 20  and  FIG. 21  are schematic structural diagrams of a centrifugal pendulum vibration absorbing apparatus  520 .  FIG. 22  is a sectional view of the centrifugal pendulum vibration absorbing apparatus  520  of  FIG. 20  that is taken along a line AA.  FIG. 20  illustrates a stationary state of the centrifugal pendulum vibration absorbing apparatus  520 .  FIG. 21  illustrates a swinging state of the centrifugal pendulum vibration absorbing apparatus  520 . The same components of the damper apparatus  510  of  FIG. 19  as the components of the damper apparatus  10  described above are represented by the same reference symbols to omit their detailed description. The damper apparatus  510  of  FIG. 19  includes a driving member (input element)  511  and the driven member (output element)  15  as the rotational elements, and also includes, as the torque transfer element, a spring SP arranged between the driving member  511  and the driven member  15 . The damper apparatus  510  further includes the centrifugal pendulum vibration absorbing apparatus  520  coupled to the driving member  511 . In the damper apparatus  510  of  FIG. 19 , the centrifugal pendulum vibration absorbing apparatus  520  corresponds to the vibration damping apparatus disclosed herein instead of the damper apparatus  510 . 
     As illustrated in  FIG. 20  to  FIG. 22 , the centrifugal pendulum vibration absorbing apparatus  520  includes a torsional stiffness mechanism  530  coupled to the driving member  511 , a torsional stiffness mechanism  540  coupled to the driving member  511 , and a coupling mechanism  550  that couples the torsional stiffness mechanism  530  and the torsional stiffness mechanism  540  to each other. 
     The driving member  511  is identical to the damper apparatus  10  except that a plurality of (for example, four) guide holes  511   h  are provided at intervals in the circumferential direction. The guide hole  511   h  is an opening extending in a predetermined direction (upper right/lower left direction in  FIG. 20  and  FIG. 21 ). The guide hole  511   h  is formed symmetrically across a straight line passing through the rotation center RC of the driving member  511  and extending in a direction orthogonal to the extending direction of the guide hole  511   h  (hereinafter referred to as “reference line L 81 ”; see a straight line indicated by a long dashed short dashed line in  FIG. 20  and  FIG. 21 ). 
     The torsional stiffness mechanism  530  includes a mass body  531 , and a rivet  534  for coupling the mass body  531  and the driving member  511  to each other in a freely rotatable manner. The mass body  531  includes a columnar mass body base  532 , and an arm  533  extending from the outer periphery of the mass body base  532  in a given direction (radially inward when the centrifugal pendulum vibration absorbing apparatus  520  is in the stationary state). The distal end of the arm  533  is coupled to the driving member  511  via the rivet  534  in a freely rotatable manner at a position on the reference line L 81  that is spaced radially outward from the rotation center RC by a distance R 8  and radially inward from the guide hole  511   h  by a distance (r 8 /2). Thus, the mass body  531  (arm  533 ) has a relationship of a revolute pair with the driving member  511 . A center of gravity  531   g  of the mass body  531  is located at a position corresponding to the center of the mass body base  532  when viewed in the axial direction and spaced away from the rivet  534  (position of the revolute pair of the driving member  511  and the mass body  531 ) by a distance r 8 . The center of gravity  531   g  of the mass body  531  is located on a radially outermost side and at a position spaced radially outward from the guide hole  511   h  by the distance (r 8 /2) on the reference line L 81  when the centrifugal pendulum vibration absorbing apparatus  520  is in the stationary state, shifted radially inward as the swing amount of the centrifugal pendulum vibration absorbing apparatus  520  (displacement from the stationary state) increases, and shifted radially outward as the swing amount of the centrifugal pendulum vibration absorbing apparatus  520  decreases. The mass body base  532  and the arm  533  are formed integrally, but may be formed separately and coupled to each other with a rivet or the like. 
     The torsional stiffness mechanism  540  includes a mass body  541 , and a rivet  544  for coupling the mass body  541  and the driving member  511  to each other in a freely rotatable manner. The mass body  541  includes a columnar mass body base  542 , and an arm  543  extending from the outer periphery of the mass body base  542  in a given direction (radially outward when the centrifugal pendulum vibration absorbing apparatus  520  is in the stationary state). The distal end of the arm  543  is coupled to the driving member  511  via the rivet  544  in a freely rotatable manner at a position on the reference line L 81  that is spaced radially outward from the guide hole  511   h  by the distance (r 8 /2) (position spaced away from the rotation center RC by a distance (R 8 +r 8 )). Thus, the mass body  541  (arm  543 ) has a relationship of a revolute pair with the driving member  511 . A center of gravity  541   g  of the mass body  541  is located at a position corresponding to the center of the mass body base  542  when viewed in the axial direction and spaced away from the rivet  544  (position of the revolute pair of the driving member  511  and the mass body  541 ) by the distance r 8 . The center of gravity  541   g  of the mass body  541  is located on a radially innermost side and at a position spaced radially inward from the guide hole  511   h  by the distance (r 8 /2) on the reference line L 81  when the centrifugal pendulum vibration absorbing apparatus  520  is in the stationary state, shifted radially outward as the swing amount of the centrifugal pendulum vibration absorbing apparatus  520  (displacement from the stationary state) increases, and shifted radially inward as the swing amount of the centrifugal pendulum vibration absorbing apparatus  520  decreases. The mass body base  542  and the arm  543  are formed integrally, but may be formed separately and coupled to each other with a rivet or the like. 
     The coupling mechanism  550  includes a guide link  551 , a guide link  552 , a rivet  553  for coupling the guide link  551  and the mass body  531  to each other in a freely rotatable manner, a rivet  554  for coupling the guide link  551  and the mass body  541  to each other in a freely rotatable manner, and a pivot (rivet)  555  configured to move along the guide hole  511   h  formed in the driving member  511  and to couple the guide links  551  and  552  to each other in a freely rotatable manner. 
     The guide link  551  is formed so as to extend in a given direction. One end of the guide link  551  is coupled to the center of gravity  531   g  of the mass body  531  with the rivet  553  in a freely rotatable manner, and the other end of the guide link  551  is coupled to the guide link  552  and the pivot  555  with the pivot  555  in a freely rotatable manner. Thus, the guide link  551  has a relationship of a revolute pair with the mass body  531  at one end and a revolute pair with the guide link  552  and the pivot  555  at the other end. 
     The guide link  552  is formed so as to extend in a given direction. One end of the guide link  552  is coupled to the center of gravity  541   g  of the mass body  541  with the rivet  554  in a freely rotatable manner, and the other end of the guide link  552  is coupled to the guide link  551  and the pivot  555  with the pivot  555  in a freely rotatable manner. Thus, the guide link  552  has a relationship of a revolute pair with the mass body  541  at one end and a revolute pair with the guide link  551  and the pivot  555  at the other end. 
     As understood from  FIG. 20  and  FIG. 22 , when the centrifugal pendulum vibration absorbing apparatus  520  is in the stationary state, the rivet  534  (fulcrum of the mass body  531 ), the center of gravity  541   g  of the mass body  541 , and the rivet  554  are located at positions spaced away from the rotation center RC by the distance R 8  on the reference line L 81  when viewed in the axial direction. Further, the pivot  555  is located at a position spaced away from the rotation center RC by a distance (R 8 +r 8 /2). Still further, the center of gravity  531   g  of the mass body  531 , the rivet  553 , and the rivet  544  (fulcrum of the mass body  541 ) are located at positions spaced away from the rotation center RC by the distance (R 8 +r 8 ). 
     In the centrifugal pendulum vibration absorbing apparatus  520 , the torsional stiffness mechanism  530  functions similarly to the torsional stiffness mechanism  20  and the torsional stiffness mechanism  30  of the damper apparatus  10  of  FIG. 1 , and the torsional stiffness mechanism  540  functions similarly to the torsional stiffness mechanism  40  of the damper apparatus  10  of  FIG. 1 . Therefore, when the centrifugal pendulum vibration absorbing apparatus  520  is in the swinging state, that is, when the mass body  531  and the mass body  541  deviate from their positions in the stationary state of the centrifugal pendulum vibration absorbing apparatus  520 , a force proportional to the square of the angular velocity Ω of the engine EG is applied to the torsional stiffness mechanism  530  in a direction in which the swing amount of the spring member  230  (deviation from the stationary state) is reduced similarly to the torsional stiffness mechanism  20  and the torsional stiffness mechanism  30 , and a force proportional to the square of the angular velocity Ω of the engine EG is applied to the torsional stiffness mechanism  540  in a direction in which the swing amount of the torsional stiffness mechanism  540  is increased similarly to the torsional stiffness mechanism  40 . Thus, it can be considered that the torsional stiffness mechanism  530  has a positive torsional stiffness k 81  proportional to the square of the angular velocity Ω of the engine EG similarly to the torsional stiffness mechanism  20  and the torsional stiffness mechanism  30  and the torsional stiffness mechanism  540  has a negative torsional stiffness k 82  proportional to the square of the angular velocity Ω of the engine EG similarly to the torsional stiffness mechanism  40 . In the centrifugal pendulum vibration absorbing apparatus  520 , it can be considered that the torsional stiffness mechanism  530  and the torsional stiffness mechanism  540  act on the driving member  511  in parallel. Therefore, an overall torsional stiffness k (=k 81 −k 82 ) of the torsional stiffness mechanisms  530  and  540  can be reduced. Through the movement of the mass body  531  of the torsional stiffness mechanism  530  and the mass body  541  of the torsional stiffness mechanism  540 , the pivot  555  coupled to the mass bodies  531  and  541  via the guide links  551  and  552  moves along the guide hole  511   h . In this manner, a vibration having a phase opposite to that of the vibration transferred from the engine EG to the driving member  511  is applied to the driving member  511  from the centrifugal pendulum vibration absorbing apparatus  520 , whereby vibrations of the driving member  511  and the driven member  15  can be absorbed (damped). Further, the rotation speed range of the engine EG in which high vibration damping performance can be exerted for the driving member  511  and the driven member  15  can be extended by appropriately setting the torsional stiffnesses k 81  and k 82  of the fourth and fifth torsional stiffness mechanisms  530  and  540 . 
     The inventors have found that the equation of motion of the centrifugal pendulum vibration absorbing apparatus  520  can be represented by Expression (25). In Expression (25), “m 81 ” represents a mass of the mass body  531 , “m 82 ” represents a mass of the mass body  541 , “r 8 ” represents each of the distance between the rivet  534  (fulcrum of the mass body  531 ) and the center of gravity  531   g  of the mass body  531  and the distance between the rivet  544  (fulcrum of the mass body  541 ) and the center of gravity  541   g  of the mass body  541 , “R 8 ” represents the distance between the rotation center RC and the fulcrum of the mass body  531  (position of the rivet  534 ), “48” represents a swing angle of each of the mass bodies  531  and  541  (each of an angle between the reference line L 81  and the extending direction of the arm  533  and an angle between the reference line L 81  and the extending direction of the arm  543 ), and “θ 8 ” represents a rotation angle (rotational position) of the driving member  511  that is a vibration damping target. Assuming that the driving member  511  is rotating at a constant speed, a “second-order derivative of θ 8 ” is a value “0” and a “first-order derivative of θ 8 ” is the angular velocity of the engine EG in Expression (25). Assuming that the angle $8 is infinitesimal, that is, “sin ϕ 8 ≈ϕ 8  and cos ϕ 8 =1”, Expression (25) is transformed to obtain Expression (26). In Expression (26), it can be considered that a coefficient “{m 81 ·R 8 −m 82 . (R 8 +r 8 )}·Ω 2 ” of the swing angle ϕ 8  of each of the mass bodies  531  and  541  corresponds to the overall torsional stiffness k (=k 81 −k 82 ) of the torsional stiffness mechanisms  530  and  540 . By using Expression (26), a natural frequency fn can be represented by Expression (27). Thus, the order n of the centrifugal pendulum vibration absorbing apparatus  520  can be represented by Expression (28). In the centrifugal pendulum vibration absorbing apparatus  520 , it is necessary that the values in the radical signs of Expression (27) and Expression (28) be positive values. Therefore, it can be said that the function of a dynamic vibration absorber can be exerted when Expression (29) is satisfied. 
     
       
         
           
             
               
                 
                   
                       
                   
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     In the centrifugal pendulum vibration absorbing apparatus  520 , the following effect is attained as well. As a comparative example, there is considered a centrifugal pendulum vibration absorbing apparatus  520 B of  FIG. 23 , in which the torsional stiffness mechanism  530  of the centrifugal pendulum vibration absorbing apparatus  520  of  FIG. 20  to  FIG. 22  is provided but the guide hole  511   h , the torsional stiffness mechanism  540 , and the coupling mechanism  550  are omitted. In the centrifugal pendulum vibration absorbing apparatus  520  and the centrifugal pendulum vibration absorbing apparatus  520 B, satisfactory vibration damping performance can be exerted when the orders of those apparatuses are equal to the order of the vibration transferred from the engine EG to the driving member  511 . In the case of the centrifugal pendulum vibration absorbing apparatus  520 B of  FIG. 23 , an order n′ of the centrifugal pendulum vibration absorbing apparatus  520  can be represented by Expression (30). In Expression (30), “r 8 ” represents the distance between the rivet  534  and the center of gravity  531   g  of the mass body  531 , and “R 8 ” represents the distance between the rotation center RC and the fulcrum of the mass body  531  (position of the rivet  534 ). As described above, in the centrifugal pendulum vibration absorbing apparatus  520 , the function of the dynamic vibration absorber can be exerted when Expression (29) is satisfied. Therefore, it can be said that the distance R 8  can be increased by increasing the mass m 82  of the mass body  541  within that range. For example, there is considered a case where the engine EG has two cylinders. In this case, it is necessary that the distance R 8  and the distance r 8  be set equal to each other in order that the order n′ of the centrifugal pendulum vibration absorbing apparatus  520 B be equal to a value “1” (order of the vibration transferred from the engine EG to the driving member  511 ). Thus, the centrifugal pendulum vibration absorbing apparatus  520  may occupy the majority of the face (axial end face) of the driving member  511 . When the order n of the centrifugal pendulum vibration absorbing apparatus  520  is set equal to the value “1”, it is demonstrated from Expression (28) that the distance R 8  can be set larger than the distance r 8  by appropriately setting the masses m 81  and m 82  of the mass bodies  531  and  541  (for example, when the ratio between the masses m 81  and m 82  of the mass bodies  531  and  541  is set to 2:1, “R 8 = 4 ·r 8 ” can hold). Thus, in the centrifugal pendulum vibration absorbing apparatus  520 , the distance R 8  can be set larger than that of the centrifugal pendulum vibration absorbing apparatus  520 B of the comparative example. Accordingly, it is possible to secure a larger space on the face side of the inner peripheral portion of the driving member  511 . 
     
       
         
           
             
               
                 
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                     Math 
                     . 
                     
                         
                     
                      
                     16 
                   
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                   ( 
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     In the damper apparatus  510  described above, the torsional stiffness mechanism  530  is structured to have the positive torsional stiffness k 81  proportional to the square of the angular velocity Ω of the engine EG, but may be structured to have a constant positive torsional stiffness irrespective of the rotation speed of the engine EG. 
     In the damper apparatus  510  described above, the turbine runner  5  of the torque converter TC is fixed to the driven member  15 . As indicated by a long dashed double-short dashed line in  FIG. 19 , the turbine runner  5  may be fixed to the driving member  511 . 
     In the damper apparatus  510  described above, the centrifugal pendulum vibration absorbing apparatus  520  is coupled to the driving member  511 , but may be coupled to the driven member  15 . 
     As described above, a first vibration damping apparatus disclosed herein is summarized as follows. The first vibration damping apparatus is the vibration damping apparatus ( 10 ,  110 ) having the plurality of rotational elements including the input element ( 11 ) to which the torque from the engine (EG) is transferred, and the output element ( 15 ). The vibration damping apparatus ( 10 ,  110 ) includes the first torsional stiffness mechanism ( 20 ) arranged between the input element ( 11 ) and the output element ( 15 ) and having the positive torsional stiffness, and the second torsional stiffness mechanism ( 40 ,  140 ) configured to act in parallel to the first torsional stiffness mechanism ( 20 ) between the input element ( 11 ) and the output element ( 15 ) and having the negative torsional stiffness. The torsional stiffness of the second torsional stiffness mechanism ( 40 ,  140 ) increases on the negative side as the rotation speed of the engine (EG) increases. 
     In the first vibration damping apparatus disclosed herein, the first torsional stiffness mechanism having the positive torsional stiffness and the second torsional stiffness mechanism having the negative torsional stiffness act in parallel between the input element to which the torque from the engine is transferred and the output element. Thus, the overall torsional stiffness of the plurality of torsional stiffness mechanisms including the first torsional stiffness mechanism and the second torsional stiffness mechanism (corresponding to a combined spring rate in a case of springs) can be reduced. Further, the torsional stiffness of the second torsional stiffness mechanism increases on the negative side as the rotation speed of the engine increases. Thus, the overall torsional stiffness of the plurality of torsional stiffness mechanisms can appropriately change in response to the rotation speed of the engine. As a result, it is possible to extend the rotation speed range in which high vibration damping performance can be exerted for the input element to which the torque from the engine is transferred. 
     In the first vibration damping apparatus disclosed herein, the first torsional stiffness mechanisms ( 20 ,  240 ) and the second torsional stiffness mechanisms ( 40 ,  140 ,  250 ,  350 ,  450 ) may be arranged so as to be arrayed in the circumferential direction of the vibration damping apparatus ( 10 ,  110 ,  210 ,  310 ,  410 ). In this case, the first torsional stiffness mechanisms ( 20 ,  240 ) and the second torsional stiffness mechanisms ( 40 ,  140 ,  250 ,  350 ,  450 ) may be arranged so as to be alternately arrayed in the circumferential direction. 
     In the first vibration damping apparatus disclosed herein, the second torsional stiffness mechanism ( 40 ,  140 ) may include the negative coupling member ( 41 ) having the relationship of the revolute pair with one of the two rotational elements coupled via the second torsional stiffness mechanism ( 40 ,  140 ) and the sliding pair with the other of the two rotational elements. The center of gravity of the negative coupling member ( 41 ) may be shifted outward in the radial direction of the vibration damping apparatus ( 10 ,  110 ) as the relative torsion angle between the two rotational elements increases, and shifted inward in the radial direction as the relative torsion angle between the two rotational elements decreases. In this case, when the relative torsion angle is formed between the two rotational elements, the second torsional stiffness mechanism ( 40 ,  140 ) may operate so as to increase the relative torsion angle between the two rotational elements. Further, the two rotational elements may be formed annularly with different diameters and arranged concentrically. The negative coupling member ( 41 ) may be coupled to the one of the two rotational elements in a freely rotatable manner, and coupled to the other of the two rotational elements so as to freely rotate and to freely move in the extending direction of the negative coupling member ( 41 ). When the relative torsion angle between the two rotational elements is zero, the center of gravity of the negative coupling member ( 41 ) may be located on the inner side in the radial direction with respect to the positions where the negative coupling member ( 41 ) is coupled to the one of the two rotational elements and the other of the two rotational elements. 
     In the first vibration damping apparatus disclosed herein, the torsional stiffness of the first torsional stiffness mechanism ( 20 ) may increase on the positive side as the rotation speed of the engine (EG) increases. In this case, the first torsional stiffness mechanism ( 20 ) may include the positive coupling member ( 21 ) having the relationship of the revolute pair with one of the input element ( 11 ) and the output element ( 15 ) and the sliding pair with the other of the input element ( 11 ) and the output element ( 15 ). The center of gravity of the positive coupling member ( 21 ) may be shifted inward in the radial direction of the vibration damping apparatus ( 10 ,  110 ) as the relative torsion angle between the input element ( 11 ) and the output element ( 15 ) increases, and shifted outward in the radial direction as the relative torsion angle between the input element ( 11 ) and the output element ( 15 ) decreases. In this case, when the relative torsion angle is formed between the two rotational elements, the first torsional stiffness mechanism ( 20 ) may operate so as to reduce the relative torsion angle between the two rotational elements. Further, the input element ( 11 ) and the output element ( 15 ) may be formed annularly with different diameters and arranged concentrically. The positive coupling member ( 21 ) may be coupled to the one of the input element ( 11 ) and the output element ( 15 ) in a freely rotatable manner, and coupled to the other of the input element ( 11 ) and the output element ( 15 ) so as to freely rotate and to freely move in the extending direction of the positive coupling member ( 21 ). When the relative torsion angle between the input element ( 11 ) and the output element ( 15 ) is zero, the center of gravity of the positive coupling member ( 21 ) may be located on the outer side in the radial direction with respect to the positions where the positive coupling member ( 21 ) is supported by the one of the input element ( 11 ) and the output element ( 15 ) and the other of the input element ( 11 ) and the output element ( 15 ). 
     The first vibration damping apparatus disclosed herein may further include the third torsional stiffness mechanism ( 30 ) having the positive torsional stiffness. The plurality of rotational elements ( 11 ,  12 ,  15 ) may include the intermediate element ( 12 ) arranged between the input element ( 11 ) and the output element ( 15 ). The second torsional stiffness mechanism ( 40 ) may be arranged as one of intermediation between the input element ( 11 ) and the intermediate element ( 15 ) and intermediation between the intermediate element ( 12 ) and the output element ( 15 ). The third torsional stiffness mechanism ( 30 ) may be arranged as the other of the intermediation between the input element ( 11 ) and the intermediate element ( 12 ) and the intermediation between the intermediate element ( 12 ) and the output element ( 15 ). 
     In the first vibration damping apparatus disclosed herein, the second torsional stiffness mechanism ( 250 ,  350 ,  450 ) may be arranged so as to extend in the radial direction of the vibration damping apparatus ( 210 ,  310 ,  410 ). 
     In the first vibration damping apparatus disclosed herein in the aspect in which the second torsional stiffness mechanism is arranged so as to extend in the radial direction of the vibration damping apparatus, the inner rotational element ( 215 ), which is a rotational element located on the inner side in the radial direction out of the two rotational elements ( 212 ,  215 ) coupled via the second torsional stiffness mechanism ( 250 ), may have the guide hole ( 215   h ) formed so as to extend along the radial direction. The second torsional stiffness mechanism ( 250 ) may include the mass body ( 255 ) movable along the guide hole ( 215   h ), and the spring ( 251 ) coupled to the mass body ( 255 ) and the outer rotational element ( 212 ), which is a rotational element located on the outer side in the radial direction out of the two rotational elements ( 212 ,  215 ), and compressed shorter than its equilibrium length when the relative torsion angle between the two rotational elements ( 212 ,  215 ) is zero. In this case, the torsional stiffness of the second torsional stiffness mechanism ( 250 ) is the overall torsional stiffness of the second torsional mechanism ( 250 ) that is defined by including the spring rate of the spring ( 251 ). 
     In the first vibration damping apparatus disclosed herein in the aspect in which the second torsional stiffness mechanism is arranged so as to extend in the radial direction of the vibration damping apparatus, the inner rotational element ( 215 ), which is a rotational element located on the inner side in the radial direction out of the two rotational elements ( 212 ,  215 ) coupled via the second torsional stiffness mechanism ( 250 ), may have the guide hole ( 215   h ) formed so as to extend along the radial direction. The second torsional stiffness mechanism ( 350 ) may include the movement member ( 255 ) movable along the guide hole ( 215   h ), the spring ( 251 ) coupled to the movement member ( 255 ) and the outer rotational element ( 212 ), which is a rotational element located on the outer side in the radial direction out of the two rotational elements ( 212 ,  215 ), and compressed shorter than its equilibrium length when the relative torsion angle between the two rotational elements ( 212 ,  215 ) is zero, and the positional adjuster ( 360 ) configured to adjust the position of the movement member ( 255 ) in the radial direction. In this case, it is appropriate to adjust the position of the movement member so that the position of the movement member is shifted more radially outward as the rotation speed of the engine increases. In this case, the torsional stiffness of the second torsional stiffness mechanism ( 350 ) is the overall torsional stiffness of the second torsional mechanism ( 350 ) that is defined by including the spring rate of the spring ( 251 ). 
     In the first vibration damping apparatus disclosed herein in the aspect in which the second torsional stiffness mechanism is arranged so as to extend in the radial direction of the vibration damping apparatus, the second torsional stiffness mechanism ( 450 ) may include the variable pitch coil spring ( 451 ) in which the pitch of the active coil portion is variable. The variable pitch coil spring ( 451 ) may be compressed shorter than its equilibrium length when the relative torsion angle between the two rotational elements is zero. In this case, the pitch of the active coil portion of the variable pitch coil spring ( 451 ) may be smaller on the outer side in the radial direction than the inner side in the radial direction. 
     In the first vibration damping apparatus disclosed herein in the aspect in which the second torsional stiffness mechanism is arranged so as to extend in the radial direction of the vibration damping apparatus, the first torsional stiffness mechanism ( 240 ) may be arranged so as to extend in the circumferential direction of the vibration damping apparatus ( 210 ,  310 ,  410 ). In this case, the first torsional stiffness mechanism ( 240 ) may be the variable pitch coil spring in which the pitch of the active coil portion is variable. The pitch of the active coil portion of the variable pitch coil spring of the first torsional stiffness mechanism ( 240 ) may be smaller at the center of the first torsional stiffness mechanism ( 240 ) in its extending direction than both ends of the first torsional stiffness mechanism ( 240 ) in its extending direction. 
     The first vibration damping apparatus disclosed herein in the aspect in which the second torsional stiffness mechanism is arranged so as to extend in the radial direction of the vibration damping apparatus may further include the third torsional stiffness mechanism ( 220 ) and the fourth torsional stiffness mechanism ( 230 ) having the positive torsional stiffnesses. The plurality of rotational elements may include the intermediate element ( 213 ) arranged between the input element ( 211 ) and the output element ( 215 ). The third torsional stiffness mechanism ( 220 ) may be arranged between the input element ( 211 ) and the intermediate element ( 213 ). The fourth torsional stiffness mechanism ( 230 ) may be arranged between the intermediate element ( 213 ) and the output element ( 215 ). In this case, the first torsional stiffness mechanism ( 240 ) may operate when the relative torsion angle between the input element ( 211 ) and the output element ( 215 ) is equal to or larger than the predetermined torsion angle. 
     A second vibration damping apparatus ( 520 ) disclosed herein is summarized as follows. The second vibration damping apparatus ( 520 ) is the vibration damping apparatus ( 520 ) configured to damp the vibration of the rotational element ( 511 ) to which the torque from the engine (EG) is transferred. The vibration damping apparatus ( 520 ) includes the first torsional stiffness mechanism ( 530 ) coupled to the rotational element ( 511 ) in a freely rotatable manner and having the positive torsional stiffness, the second torsional stiffness mechanism ( 540 ) coupled to the rotational element ( 511 ) in a freely rotatable manner and having the negative torsional stiffness, and the coupling mechanism ( 550 ) that couples the first torsional stiffness mechanism ( 530 ) and the second torsional stiffness mechanism ( 540 ) to each other. The torsional stiffness of the second torsional stiffness mechanism ( 540 ) increases on the negative side as the rotation speed of the engine (EG) increases. 
     In the second vibration damping apparatus disclosed herein, the first torsional stiffness mechanism coupled in a freely rotatable manner to the rotational element to which the torque from the engine is transferred and having the positive torsional stiffness and the second torsional stiffness mechanism coupled to the rotational element in a freely rotatable manner and having the negative torsional stiffness are coupled to each other via the coupling mechanism. In this structure, it can be considered that the first torsional stiffness mechanism and the second torsional stiffness mechanism act on the rotational element in parallel. Therefore, the overall torsional stiffness of the plurality of torsional stiffness mechanisms including the first torsional stiffness mechanism and the second torsional stiffness mechanism can be reduced. Further, in this structure, when the first torsional stiffness mechanism and the second torsional stiffness mechanism deviate from their positions in the stationary state due to the occurrence of fluctuation in the rotation of the rotational element, a vibration having a phase opposite to that of the vibration transferred from the engine to the rotational element is applied to the rotational element from the vibration damping apparatus so that the first torsional stiffness mechanism may return to its position in the stationary state and the second torsional stiffness mechanism may increase the amount of the deviation. Thus, the vibration of the rotational element can be absorbed (damped). Further, the torsional stiffness of the second torsional stiffness mechanism increases on the negative side as the rotation speed of the engine increases. Thus, the overall torsional stiffness of the plurality of torsional stiffness mechanisms including the first torsional stiffness mechanism and the second torsional stiffness mechanism can appropriately change in response to the rotation speed of the engine. As a result, it is possible to extend the rotation speed range in which high vibration damping performance can be exerted for the rotational element to which the torque from the engine is transferred. 
     In the second vibration damping apparatus disclosed herein, the torsional stiffness of the first torsional stiffness mechanism ( 530 ) may increase on the positive side as the rotation speed of the engine (EG) increases. 
     In this case, the first torsional stiffness mechanism ( 530 ) may include the first mass body ( 531 ) having the relationship of the revolute pair with the rotational element ( 511 ) at a first position on the rotational element ( 511 ) and having the center of gravity that is located on the outer side in the radial direction of the vibration damping apparatus ( 520 ) with respect to the first position in the stationary state. The center of gravity of the first mass body ( 531 ) may be shifted inward in the radial direction as the swing amount of the rotational element ( 511 ) increases, and shifted outward in the radial direction as the swing amount of the rotational element ( 511 ) decreases. The second torsional stiffness mechanism ( 540 ) may include the second mass body ( 541 ) having the relationship of the revolute pair with the rotational element ( 511 ) at a second position on the rotational element ( 511 ) that is located on the outer side in the radial direction with respect to the first position and having the center of gravity that is located on the inner side in the radial direction with respect to the second position in the stationary state. The center of gravity of the second mass body ( 541 ) may be shifted outward in the radial direction as the swing amount of the rotational element ( 511 ) increases, and shifted inward in the radial direction as the swing amount of the rotational element ( 511 ) decreases. 
     In this case, the rotational element ( 511 ) may have the guide hole ( 511   h ) formed so as to extend in a predetermined direction. The first position may be located on the inner side in the radial direction with respect to the guide hole ( 511   h ). The second position may be located on the outer side in the radial direction with respect to the guide hole ( 511   h ). The coupling mechanism ( 550 ) may include the first link ( 551 ) having the relationship of the revolute pair with the first mass body ( 531 ) at one end, the second link ( 552 ) having the relationship of the revolute pair with the second mass body ( 541 ) at one end, and the pivot ( 555 ) configured to move along the guide hole ( 511   h ) and having the relationship of the revolute pair with the other end of the first link ( 551 ) and the other end of the second link ( 552 ). 
     In this case, the first link ( 551 ) may have the relationship of the revolute pair with the first mass body ( 531 ) at the position of the center of gravity of the first mass body ( 531 ). The second link ( 552 ) may have the relationship of the revolute pair with the second mass body ( 541 ) at the position of the center of gravity of the second mass body ( 541 ). Each of the distance between the first position and the center of gravity of the first mass body ( 531 ) and the distance between the second position and the center of gravity of the second mass body ( 541 ) may be a first distance. Each of the distance between the center of gravity of the first mass body ( 531 ) and the pivot ( 555 ) and the distance between the center of gravity of the second mass body ( 541 ) and the pivot ( 555 ) may be a second distance that is a half of the first distance. 
     Although the embodiment of the present disclosure has been described above, the present disclosure is not limited to the embodiment. It is understood that various embodiments may be adopted without departing from the spirit of the present disclosure. 
     INDUSTRIAL APPLICABILITY 
     The present disclosure is applicable to, for example, industry for manufacturing the vibration damping apparatus.