tailieunhanh - Interaction between linear and cubic parametric excitations

In the present paper, the same system will be examined in the case general, assuming that there exists certain phase shift between two parametric excitations. As it will be shown, the form of the resonance curve is diversity and can be classified by using critical representative points. | Vietnam Journal of Mechanics, NCNST of Vietnam T. XX, 1998, No 4 (20- 29) INTERACTION BETWEEN LINEAR AND CUBIC PARAMETRIC EXCITATIONS NGUYEN VAN DINH - TRAN KIM CHI Institute of Mechanics In [1], a quasi-linear oscillating system with non linear restoring element harmonically depending on time has been studied. The mentioned element is represented by two terms-the linear and the cubic. We can consider the system examined as the one subjected to two parametric excitations. The oscillations thus result from the interaction between theiie two excitations. In the present paper, the same system will be examined in the case general, assuming that there exists certain phase shift between two parametric excitations. As it will be shown, the form of the resonance curve is diversity and can be classified by using critical representative points [2]. §l System under consideration and equations of stationary osdllations Let us consider a quasilinear system described by the differential equation: x+w 2 x = e:{ h± -,x3 + 2pxcos2wt+2qx3 cos2(wt +a)}. () where: x-oscillatory variable; h ~ 0-damping coefficient; 2p > 0, 2q > 0-intensities of linear and cubic parametric excitations, respectively; 2w-common frequency; 2a (0 :::; 2a 0-small parameter; 0 (system with damping), the two compatibility conditions are only satisfied at I the compatible ensemble is reduced to I. At this unique compatible point I, () is always satisfied, but (} requires: () 23 So, I is only critical if the damping is weak enough. In Fig. 1, the resonance curves have been plotted for fixed values u = 11' /2; p = , q = , "/ = and for various values of h. The point I, the segment J 1 J 2 and two straight-lines--(-1-)---ferm-theresenance curve of the system with aut damping (h = 0). The resonance curves (2), (3) correspond to h = ; h = respectively. We see that, for h > 0 (system with damping), the critical segment J 1 h disappears and the resonance curve .