tailieunhanh - Nonlinear oscillators under delay control

In this paper, oscillations and stability of nonlinear oscillators with time delay are studied by means of the asymptotic method of nonlinear mechanics. Harmonic, superharmonic, subharmonic and parametric resonances of a Duffings oscillator are analyzed. Analytical method in combination with a computer is used. | Vietnam Journal of Mechanics, NCST of Vietnam Vol. ~1, 1999, No 2, (75 - 88) NONLINEAR OSCILLATORS UNDER DELAY CONTROL NGUYEN VAN DAO Vietnam National University, Hanoi 19 Le Thanh Tong, Hanoi, Vietnam ABSTRACT. In this paper, oscillations and stability of nonlinear oscillators with time delay are studied by means of the asymptotic method of nonlinear mechanics. Harmonic, superharmonic, subharmonic and parametric resonances of a 's oscillator are analyzed. Analytical method in combination with a computer is used. 1. Introduction The harmonically forced Duffing's oscillator with time delay state feedback has been investigated in .[1] by using the method of multiple scales [2]. Both primary and 1/3 subharmonic resonances of the Duffing's oscillator with weak nonlinearity and weak delay feedback have been e~amined. As shown in [1] the simplest model for various controlled nonlinear systems, ., active vehicle suspension systems when the nonlinearity in tires is taken into account, is described by a second order differential equation with time delay in the form d 2 x(t) dt 2 dx(t) + x(t) = -2€--;u---- JLX 3(t) + 2ux(t- ~) + 2v dx(t- ~) dt + 2pcos .Xt, () where €, JL, u, v and ~ are constants. To study all possible simple resonances in the dynamic system governed by equation (}, in the present paper it is supposed that between the external frequency A and the natural frequency 1 there exists a relationship of the form A= n + cu, () where n = !?. is a rational number, p and q are integers. We suppose that paq rameters €, JL, u, v are small. The smallness of these parameters is insured by introducing small positive parameter c. 75 2. Harmonic Resonance f Assuming that n = 1 and equation () in the form is a small quantity of e - order, we can rewrite () where F dx(t) = -2€---;u--- 3 (t) + 2ux(t- ~) + 2v dx(tdt ~) + 2pcos >t. () The solution of equation () is found in the form x(t) a cos .