4 Nonlinear Dynamic Analysis of a Spiral Bevel Geared System 35 4.2.2 Solution Method 4.2.2.1 Multi-Term Harmonic Balance Method with DFT In order to solve the nonlinear equations of motion for x, multi-term Harmonic Balance Method coupled with discrete Fourier Transform, which has been successfully applied in [3, 6–10], is used. Since static transmission error excitation is periodic, the solution and the nonlinear displacement function fn(ıd(t) em(t)) can also be described periodically [8]. The harmonic expressions for the static transmission error and its derivative with respect to time can be expressed as em.t/ D J X jD1 "cj cos.j!t/ C"sj sin.j!t/ ; (4.10a) Pem.t/ D J X jD1 .j!/"cj sin.j!t/ C.j!/"sj cos.j!t/ : (4.10b) Given the periodic excitation defined by Eq. (4.10a), the steady-state solution can as well be assumed periodic as x.t/ Du0 C R X rD1 Œucr cos.r!t/ Cusr sin.r!t/ ; (4.11) which can be differentiated to yield Px.t/ D R X rD1 Œ .r!/ucr sin.r!t/ C.r!/usr cos.r!t/ : (4.12) Sampling Npoints within one fundamental mesh period, the time series of the dynamic mesh force can be obtained as Fm.tn/ Dkmfn hpxp .tn/ hgxg .tn/ em.tn/ Ccm hpPxp .tn/ hgPxg .tn/ Pem.tn/ ; (4.13) where tn Dn (nD0,1,2, , N 1). Here, D2 /(N!) and N must be larger than 2R where R is the highest harmonic number used in the solution in order to avoid aliasing errors. Similarly, the dynamic mesh force can be represented by Fourier series as follows Fm.t/ DFm0 C R X rD1 Fm.cr/ cos.r!t/ CFm.sr/ sin.r!t/ ; (4.14) where, utilizing discrete Fourier Transform, Fm0 D 1 N N 1 X nD0 Fm; (4.15a) Fm.cr/ D 2 N N 1 X nD0 Fmcos 2 rn N ; (4.15b) Fm.sr/ D 2 N N 1 X nD0 Fmsin 2 rn N : (4.15c) Fourier coefficients of the nonlinear forcing vector FN(t) can be obtained by using the extended form of the coordinate transformation vectors, p, given in Eq. (4.9) as FN.t/ DFN.0/ C R X rD1 FN.cr/ cos.r!t/ CFN.cr/ sin.r!t/ ; (4.16a)
RkJQdWJsaXNoZXIy MTMzNzEzMQ==