314 S.D. Yavuz et al. F.i/ k .tn/ D! 2 i1k1 .tn/g1 .tn/ C! 2 i2k2 .tn/g2 .tn/, .i D1, 2/, (30.10) where tn Dn .n D0, 1, 2, : : : N 1/. Here, D2 =.N /. In order to use multi-term HBM, the nonlinear restoring forces must be represented by Fourier series as F.i/ k .t/ DF .i/ k1 C RX rD1 h F.i/ k.2r/ cos.r t/ CF .i/ k.2rC1/ sin.r t/i, .i D1, 2/, (30.11) where the coefficients can be calculated using the discrete Fourier Transform as .r D1, 2, : : : R/ F.i/ k1 D 1 N N 1X nD0 F.i/ k , .i D1, 2/, (30.12a) F.i/ k.2r/ D 2 N N 1X nD0 F.i/ k cos 2 rn N , .i D1, 2/, (30.12b) F.i/ k.2rC1/ D 2 N N 1X nD0 F.i/ k sin 2 rn N , .i D1, 2/. (30.12c) Substituting Eqs. (30.7b), (30.9a, 30.9b), and (30.11) into Eq. (30.6a) and equating the coefficients of the like harmonic terms, a set of .4RC2/ nonlinear algebraic equations are obtained with.i D1, 2/ and .r D1, 2, : : : R/ S.i/ 1 DF .i/ k1 F.i/ m , (30.13a) S.i/ 2r D .r / 2u.i/ 2r C2 i1!i1 .r /u .1/ 2rC1 C2 i2!i2 .r /u .2/ 2rC1 C F.i/ k.2r/ C.r / 2" .i/ 2r , (30.13b) S.i/ 2rC1 D .r /2u .i/ 2rC1 2 i1!i1 .r /u .1/ 2r 2 i2!i2 .r /u .2/ 2r CF .i/ k.2rC1/ C.r /2" .i/ 2rC1 . (30.13c) Finally, the solution vector U D hu.1/ 1 , u .1/ 2 , : : : u .1/ 2R, u .1/ 2RC1 , u.2/ 1 , u .2/ 2 , : : : u .2/ 2R, u .2/ 2RC1i T is determined by using Newton’s Method with arc length continuation. Newton’s method can be applied as follows U.m/ DU.m 1/ J 1 .m 1/S.m 1/ , (30.14) where U(m) is the mth iterative solution based on the .m 1/ th solution and J 1 .m 1/ is the inverse of the Jacobian matrix of the vector function S estimated at the previous point .m 1/. The iteration procedure described by Eq. (30.14) is repeated until the vector norm of S(m) is below a predefined error limit for that excitation frequency. Furthermore, arclength continuation method is used in the solution and a new parameter, arc-length, which is the radius of a hypothetical sphere in which the next solution point will be searched, is chosen as the continuation parameter instead of the frequency in order to follow the solution path even at the turning points. Details of Newton’s method with arc-length continuation can be found in [22–24]. The Floquet theory is used to determine the stability of the steady state solutions pi, .i D1, 2/ obtained above. This is done by examining the stability of the perturbed solutionpi C pi, .i D1, 2/. The variational equation for the perturbation pi, .i D1, 2/ is p001.t/ p002.t/ C2 11 !11 12!12 21!21 22!22 p01.t/ p02.t/ C !2 11k1.t/ 1.t/ !2 12k2.t/ 2.t/ !2 21k1.t/ 1.t/ !2 22k2.t/ 2.t/ p1.t/ p2.t/ Df 0g, (30.15) where i(t) is a discontinuous separation function i.t/ D 1, jpi.t/j >1 0, jpi.t/j 1 , .i D1, 2/. (30.16)
RkJQdWJsaXNoZXIy MTMzNzEzMQ==