Assume the turning stiffness between input shaft and driving gear is k12, the equation of input shaft as follow: ) ( 12 11 12 11 11 θ θ θ − − = k T J                                                                                                                     (1) In it, T is the input torque. When ignore friction while gear meshing, the dynamical model of the three involute gear was shown as figure 4. Assume the support bearing stiffness is kb1, kb2, kb3, apply the generalized coordinate of the input shaft is X1=﹛θ11, θ12,y12﹜, mass matrix is M1=diag(J11, J12, m12), then the vibration equation as follow: 12 1 31 12 12 31 21 12 12 21 12 12 ) ( ) ( y k r y r y rk r y r y rk y m b s s ss s s ss− + + − + − + − = θ θ θ θ                     (2)  ) ( ) ( ) ( 12 31221 12212 11 12 12 12 θ θ θ θ θ θ θ + + − − + − = ss ssrk rk k J                                                      (3) In it the force between driving and driven gear is  ) ( 21 21 12 12 12 θ θ s s ss sr y r y rk F + − + =  ;    ) ( 31 31 12 12 13 θ θ s s ss sr y r y rk F − − − =   Dynamical Model of Crankshaft. Four ring-plates connected the crankshaft with support bearing, the acting force of crankshaft was shown in figure 5. The crankshaft has two vibration directions; one is the torsional vibration in z direction, the other is the bending vibration on the direction of acting force, the model was shown in figure 6. Define the generalized coordinate of crankshaft as flow: Ty y y y y X } , , , , , , , , , { 25 24 23 22 21 25 24 23 22 21 2 θ θ θ θ θ =  Mass matrix as: ) , , , , , , , , , ( 25 24 3 22 22 25 24 23 22 21 2 m m m m m J J J J J diag M m =      Calculate the bending stiffness of crankshaft by compliance coefficient method, as shown in figure 6, can obtain its compliance. the coefficient as follow. Then the dynamical equation of the crankshaft is:  21 2 21 12 21 21 ) ( y k rk y m b ss+ + = θ θ                                                                                           (4)  [ ]2/]       ][I[ cos ]       ][ [ ] y  y  y  [           )][ ( ]       ][ [ ]   y  y  ][ m  m  m  ['7 6 5 4'7 6 5 4'25 24 23 22'25 24 23 22'25 24 23 22 25 24 23 22ϕ ϕ ϕ ϕ θ θ θ θ θθ θ θ θ− − − +⋅ + − − =l k Ia k yI k KW Ia k y y mbh bhbh bh                   (5) ) ( ) ( 21 22 21 21 21 12 12 21 21 θ θ θ θ θ − + + − + = k r y r y rk J s s ss                                                                  (6) ]2/ cos ) ( [ ) ( ) ( 2 6 2 24 2 2 2 2 2 2 2 θ ϕ θ θ θ θ θ θ θ m i i zb k i i j j iil a y a k k k J − − + − + − + − =         (i=2, 3, 4 j=1, 2, 3, k=3, 4, 5,m=4, 5, 6)           (7) ]2/ cos ) ( [ ) ( 7 25 7 25 25 24 24 25 25 θ ϕ θ θ θ θ θ l a y a k k J zb + − + − + − =                                                (8) 31 3 31 12 31 31 ) ( y k rk y m b ss+ − − = θ θ                                                                                                        (9)[ ]2/]       ][I[ cos ]       ][ [ ] y  y  y  [           )][ ( ]       ][ [ ]   y  y  ][ m  m  m  ['7 6 5 4'7 6 5 4'35 34 33 32'35 34 33 32'35 34 33 32 35 34 33 32ϕ ϕ ϕ ϕ θ θ θ θ θθ θ θ θ− − + +⋅ + − − =l k Ia k yI k KW Ia k y y mbh bhbh bh                   (10)          ]2/ cos ) ( [ ) ( ) ( 3 3 3 3 3 3 3 3 33 θ ϕ θ θ θ θ θ θ θ m i m i zb i k i i j j iil a y a k k k J − − + − + − + − =             (i=2, 3, 4 j=1, 2, 3, k=3, 4, 5,m=4, 5, 6)  (11) ]2/ cos ) ( [ ) ( 7 35 5 35 35 34 34 35 35 θ ϕ θ θ θ θ θ l a y a k k J zb − − + − + − =                                               (12) In it:   ]2/ cos ) ( [ 2 2 2 θ ϕ θ θ i j i j zb il a y k F + − + − =   ]2/ cos ) ( [ 3 3 3 θ ϕ θ θ i j i j zb il a y k F + − + − =  (i=4,5,6,7,j=2,3,4,5) Are the acting force between crankshaft 1,2 and the inner gear of ring-plate 4,5,6 and 7.   Dynamical Model of Ring-plate. Because the ring-plate has enough stiffness, taken it as a rigid body, assume it has a vibration angle θ in plane motion, has a swinging angle φ rotating on its center of mass, the dynamical model was shown in figure 7. The generalized coordinate and mass matrix was defined as follow: { }Ti iX ϕ θ , =         ) , ('i iJ J diag M =   ( i=4,5,6 7 )    Dynamical equation served as: ) ( ) ( ) ( 881 3222i ip ng i j zb i j zb iia r r k a k a k J θ ϕ θ θ θ θ θ θ − + + − + − =    (i=4,5,6,7  j=2,3,4,5 )      (13) ) ( 2/ cos 2/ cos 81 8 3 2 θ θ ϕ θ θ θ r a r r k l F l F J i ip p ng i i ii+ − + − − = ′         (i=4,5,6,7)                        (14)  Dynamical Model of Cycloid Wheel and Output Shaft.   Being the cycloid wheel has relative big diameter, taken it as a lumped mass, ignore its bending vibration and supporting bearing’s stiffness, taken into account its torsinoal rigidity.  The generalized coordinate and mass matrix of cycloid wheel and output shaft was defined as follow:                                      { }TX 82 81 8 ,ϕ θ =              ) , ( 82 81 8 J J diag M =  Dynamical equation as:   ] 4 ) ( ) [( ) ( 881 8 7 6 5 4 7 6 5 4 8 81 82 81 81 81 r r a r k k J ng θ ϕ ϕ ϕ ϕ θ θ θ θ θ θ θ − + + + − + + + + − =       (15) ) ( 82 81 81 82 82 θ θ θ − = k J                                                                                                       (16) Global Dynamical Equation. Connecting all above-mentioned equations we can obtain the common dynamical equation: [ ]{ } [ ]{ } [ ] { } { } F X K X C X M = + +                                                                                                 (17) In the equation, 双曲柄环板式针摆线齿轮行星传动英文文献和中文翻译(2):http://www.751com.cn/fanyi/lunwen_56501.html