tailieunhanh - On the interaction between self-excited and forced oscillations

The influence of the exciting force to resonant characteristics in a model of the small oscillation is investigated. Averaging method is used to determine resonant stationary solutions of all possible resonant oscillations and their stability. The numerical simulations used for oscillations in time domain and the analytic approaches give consistent results. | Vietnam Journal of Mechanics, NCST of Vietnam Vol. 24, 2002, No 2 (73 - 83) ON THE INTERACTION BETWEEN SELF-EXCITED AND FORCED OSCILLATIONS THAI MANH CAU, NGUYEN VAN KHANG Hanoi University of Technology ABSTRACT. The influence of the exciting force to resonant characteristics in a model of the small oscillation is invest igated . Averaging method is used to determine resonant stationary solutions of all possible resonant oscillations and their stability. The numerical simulations used for oscillations in time domain and the analytic approaches give consistent results. 1. Introduction Investigation of non-linear problems in dynamics of machines is actual in engineering practice. Interaction of various kinds of non-linear oscillations is one of significant technical problems. Some researches are shown in [1, 2, 3]. The differential equation of some oscillatory models in machines is written in the form X - biX. + b2X. 2 + b3X. 3 + /2X 2 2 + / 3X 3 ·+ WoX - BO = E COS "t H . () This paper presents an examination on the influence of the exciting force to resonant characteristics in a model of the small oscillation. Averaging method and numerical simulation are used to determine stationary solutions and their stability. Numerical simulations by using the MATLAB program are applied. 2. Averaging Method Assume the oscillation to be small, let x 0 be a root of the equation /3X 3 2 + / 2X 2 + WoX - BO = o· By using the new variable U =X - Xo , the equation ( ) is transformed into the following form ii,+ w2 u = cf(u , u) + E cos fit , where w2 = 313x6 + 2/2Xo + w5, c is a formal parameter indicating the smallness of the right side, 73 () cf (U, U.) = -/2*U 2 - /3U 3 + b1U. and 12 = 'Y2 + 3/3Xo . - b2U. 2 - b3U. 3 , () . Case of small exciting force Assume the exciting force to be small. Let transform the time t into a nondimensional quantity r, 7= n -t, m where mis some positive integer. The equation () becomes u" + .