tailieunhanh - Parametric vibration of mechanical system with several degrees of freedom under the action of electromagnetic force

Let us consider a vibrating system with n degrees of freedom which consists of a weightless cantilever beam carrying n concentrated masses m1, m2, . , mn (Fig. 1). The elastic elements of the vibrating system have stiffness ki, k2, . , kn. | Vietnam Journal of Mechanics, VAST, Vol. 29, No. 3 (2007), pp. 167 - 175 Special Issue Dedicated to the Memory of Prof. Nguyen Van Dao PARAMETRIC VIBRATION OF MECHANICAL SYSTEM WITH SEVERAL DEGREES OF FREEDOM UNDER THE ACTION OF ELECTROMAGNETIC FORCE NGUYEN VAN DAO Department of Methemathics and Physics Polytechnic Institute, Hanoi (This paper has been published in: Proceedings of Vibration Problems, 14, 1, , 1973 Institute of Fundamental Technological Research, Polish Academy of Sciences) 1. SYSTEMS WITH n DEGREES OF FREEDOM Let us consider a vibrating system with n degrees of freedom which consists of a weightless cantilever beam carrying n concentrated masses m1, m2, . , mn (Fig. 1) . The elastic elements of the vibrating system have stiffness ki, k2, . , kn. Fig. 1 Supposing that some sth mass is subjected to electromagnetic force, the differential equations of motion of the system considered can be written, in accordance with [1] in the form: :t vt, + Rq + ~q = E sin m1:h + k1(x1 - x2) = -h1±1 - ,61(x1 - x2) 3, m2x2 + ki (x2 - x1) + k2(x2 - x3) = -h2±2 (Lq) ,61 (x2 - xi) 3 - ,62(x2 - x3) 3, Nguyen Van Dao 168 msXs + ks-1(Xs - Xs-1) + ks(Xs - Xs+I) = -hsXs - f3s-1(Xs - Xs-1) 3 1 .2 DL 3 -f3s(Xs-Xs+l) +'i,q Dxs' mnXn + kn-1(Xn - Xn-l) + knXn = -hnXn - f3n-1(Xn - Xn-1) 3 - f3nx~. () Vve assume that L = L(xs) = Lo(l - a1Xs + a2x;), and that the friction forces and the non-linear terms in ( ) are small with respect to the remaining terms. Then, Eqs. () can be rewritten as: 1 Loq + Cq = Esinvt - µ[Loq(-a1x2 + k1(x1 m2x2 + ki (x2 - m1x1 x2) = µF1, x1) + k2(x2 - + a2x;) + qLo(-a1±s + 2a2xs±x)], x3) = µF2, () where µFi= -h1±1 - f31(x1 - x2) 3, µF2 = -h2±2 - f31(x2 - x1) 3 - f32(x2 - x3) 3, () µFn = -hn±n - f3n-1(Xn - Xn-1) 3 - f3nx~. We suppose that the characteristic equation of the homogeneous system + ki(x1 m2i2 + k1(x2 m1i1 x2) = 0, x1) + k2(x2 - x3) = 0, () mnXn + kn-I(Xn - Xn-l) + .