The objective of this paper is to examine the influence of non-linear parametric excitation to resonant characteristics of one degree of freedom system in which non-linear function contains derivatives of second order. | Vietnam Journal of Mechanics, NCNST of Vietnam T. XX, 1998, No 2 (27- 36) INFLUENCE OF NON-LINEAR PARAMETRIC EXCITATION ON RESONANT CHARACTERISTICS OF OSCILLATING SYSTEMS NGUYEN VAN KHANG - TRAN DINH SON Hanoi University of Technology 1. Introduction One of the most important tasks of oscillation investigation in engineering is to determine resonant regimes. Several problems related to parametric excitation have been quite thoroughly examined in [1, 2]. The objective of this paper is to examine the influence of non-linear parametric excitation to resonant characteristics of one degree of freedom system in which non-linear function contains derivatives of second order [3, 4]. 2. Averaging method for a system with second order derivative in the right side Consider an oscillating system described by the differential equation x + n 2 x + f(r) = c:F(r,x,x,x) () where f(r), F(r,x,x,x) are periodic functions of r with the period 211', n is an integer and c: is a small parameter. Using the variable transformation: x = r(r) cosnr + s(r) sinnr + x*(r)' x = -r(r)n sin nr + s(r)n cos nr + ±* (r) () where r and s are new variables, 00 f(r) = ~ + L(ai cosir + bi sinir), 0 ~=1 x*(r) = _ ao _.:;:. ai cosy'r + bi sinJr 2n 2 Lr=l :j¢n 27 n2 -P () Equations for new variables will be dr dr e: F( r,x * +rcosnr+ssmnr,x ·· . .* . ·· -rnsmnr+sncosnr, n - rn 2 cos nr- sn 2 sin nr- rn sinnr + sncos nr) sin nr, -=-- x* ds= e: F( r,x * +rcosnr+ssmnr,x . ·• -rnsmnr+sncosnr, . dr n x* -rn 2 cosnr- sn 2 sinnr- rnsinnr +sncosnr)cosnr. () The averaging method can be applied to the equations () to find out its approximate solutions [4]. First we write the equations () in the form: dr = - e: F( r,x * +rcosnr+ssmnr,x . . * -rnsmnr+sncosnr, . -d t n . x* - rn 2 cos nr - sn 2 sin nr) sin nr, {) e:F( r,x * + rcosnr + ssmnr,x . . * - rnsmnr+ . ds = -d sncosnr, t n x* - rn 2 cos nr - sn 2 sin nr) cos nr. Averaging the right hand sides of these equations
