30 Nonlinear Time-Varying Dynamic Analysis of a Multi-Mesh Spur Gear Train 313 where k1m and k2m are the mean components of k1 .t/ and k2 .t/, respectively, !ij .i, j D1, 2/ is the characteristic frequency, and ij .i, j D1, 2/ defines the damping of the system. Moreover, a dimensionless time parameter is obtained by setting t D t!c, where !c is one of the characteristic frequencies. Also, dimensionless displacements are defined as pi .t/ Dpi .t/=bc, ei .t/ Dei .t/=bc, and bi Dbi=bc .i D1, 2/ by employing a characteristic length bc. Using these dimensionless parameters and letting !ij D!ij=!c .i, j D1, 2/, the following dimensionless equations of motion are obtained: p001.t/ p002.t/ C2 11 !11 12!12 21!21 22!22 p01.t/ p02.t/ C !2 11k1.t/ !2 12k2.t/ !2 21k1.t/ !2 22k2.t/ g1.t/ g2.t/ D( F.1/ m Ce001.t/ F.2/ m Ce002.t/) , (30.6a) where gi.t/ D8< : pi.t/ bi, pi.t/>bi 0, jpi.t/j bi pi.t/ Cbi, pi.t/< bi , i D1, 2, (30.6b) F.1/ m .t/ D 1 !2 cbc r1 I1 T1 C r2 I2 T2 , F.2/ m .t/ D 1 !2 cbc r2 I2 T2 C r3 I3 T3 . (30.6c) 30.2.2 Period-One Dynamics The multi-term harmonic balance method coupled with discrete Fourier Transform process and the numerical continuation method, which has been successfully applied in [3, 4, 15, 16], is used in this study to solve the dimensionless equations of motion for pi .i D1, 2/. The solution is periodic based on the assumption that both excitations and time-varying parameters are periodic [15]. This also implies that the nonlinear displacement functions gi.t/ .i D1, 2/ can also be described periodically. The harmonic expression for mesh stiffness and static transmission error can be written as ki.t/ D1C AX aD1 h .i/ 2a cos.a t/ C .i/ 2aC1 sin.a t/i, .i D1, 2/, (30.7a) ei.t/ D JX jD1 h " .i/ 2j cos.j t/ C" .i/ 2jC1 sin.j t/i, .i D1, 2/. (30.7b) The mean values of the static transmission errors are set to zero since only the second order derivatives of them are included in the equations of motion (30.6a) as parts of the excitation terms on the right hand side of the equation. Given the periodic excitations of Eqs. (30.7a, 30.7b), the steady-state solution is assumed to be of the form pi.t/ Du .i/ 1 C RX rD1 h u.i/ 2r cos.r t/ Cu .i/ 2rC1 sin.r t/i, .i D1, 2/, (30.8) which can be differentiated to yield p0i.t/ D RX rD1 h .r /u .i/ 2r sin.r t/ C.r /u .i/ 2rC1 cos.r t/i, .i D1, 2/, (30.9a) p00i .t/ D RX rD1 h .r /2u .i/ 2r cos.r t/ C.r / 2u.i/ 2rC1 sin.r t/i, .i D1, 2/. (30.9b) Then, the time series of nonlinear restoring forces can be obtained by sampling N points within one fundamental mesh period. Here, N must be larger than 2R where R is the highest harmonics of the solution in order to avoid aliasing errors. Hence, the time series of the nonlinear restoring forces are
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