tailieunhanh - Interaction of the elements characterizing the quadratic nonlinearity and forced excitation with the other excitations

In nonlinear systems, the first order of smallness terms of quadratic nonlinearity and the forced excitation with nonresonance frequency and the second order of smallness terms of linear friction, cubic nonlinearity, forced and parametric excitation with resonance frequencies have no effect on the oscillation in the first approximation. | T~p chi CO' hQc Journal of Mechanics, NCNST of Vietnam T. XIX, 1997, No 4 (11- 20) . INTERACTION OF THE ELEMENTS CHARACTERIZING THE QUADRATIC NONLINEARITY AND FORCED EXCITATION WITH THE OTHER EXCITATIONS NGUYEN VAN DAO Vietnam National Un~·versity, Hanoi Introduction In nonlinear systems, the first order of smallness terms of quadratic nonlinearity and the forced excitation with nonresonance frequency and the second order of smallness terms of linear friction, cubic nonlinearity, forced and parametric excitation with resonance frequencies have no effect on the oscillation in the first approximation. However, they do· interact one with another in the second approximation new nonlinear phenomena occur. The study of these phenomena, using the asymptotic method of nonlinear mechanics [1] with a digital computer, is our aim. 1. Interaction between the elements of quadratic nonlinearity and forced excitation themselves Let us consider a nonlinear system governed by the differential equation i + x = e[ax2 + q cos 2, 0) + e2u2 (a, t/>, 0) + e3 . , 11 0 =I"+ .p, {} ~; 2 = eA,(a)¢) +e A 2(a,,P) + . , ~~ =eBt(a,¢)+e2B2(a,,P) + . , where u;(a, ,P, 0) are periodic functions with period 21r with respect to both variables ,P and 0 and do not contain the first harmonics sinO, cosO. The functions A;(a,,P), B;(a,,P) are periodic with respect to the variable .p. These functions will be determined in the process of approximation calculations. Substituting the expressions () into equation {) and comparing the coefficients of e 1 we obtain 1 -2v( r)A, sin 0- 2av(r)B, cos 0 + v2 (r) ( aa:~ + "') = cos2 0 + q cos 21"{r). () Comparing the harmonics in {) gives: A1 = B 1 =0, {) 1 ( 2 ) q · u 1 = ----,--() - ----,--() + 2q cos 21/J cos 20 - ----,--() sm 21/> cos 20. b T ~ T 3v T {) 2 Comparing the coefficients of e2 in {) we get 2 - 2v(r)A2sinO- 2av(r)B2cosO + v 2 (rJ(a: ~ 8 + "•) = = 2aau 1 cosO+ Aacos 8 + .