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. In this paper, the propagation of Rayleigh waves in a homogeneous isotropic elastic half-space coated with a thin weakly inhomogeneous isotropic elastic layer is investigated. The material parameters of the layer is assumed to depend arbitrarily continuously on the thickness variable. The contact between the layer and the half space is perfectly bonded. | Vietnam Journal of Mechanics, VAST, Vol. 38, No. 1 (2016), pp. 15 – 25 DOI:10.15625/0866-7136/38/1/5928 AN EFFICIENT NUMERICAL PROCEDURE FOR CALCULATING PERIODIC VIBRATIONS OF ELASTIC MECHANISMS Nguyen Van Khang, Nguyen Phong Dien∗ , Nguyen Sy Nam 1 Hanoi University of Science and Technology, Vietnam ∗ E-mail: dien.nguyenphong@hust.edu.vn Received February 23, 2015 Abstract. This paper proposes a numerical procedure based on the well-known Newmark integration method to determine initial conditions for the periodic solution of a system of linear differential equations with time-periodic coefficients. Based on this, steady-state periodic vibrations of mechanisms with elastic elements governed by linearized differential equations with time-periodic coefficients can be conveniently calculated. The proposed procedure is demonstrated by a dynamic model of a planar four-bar mechanism with the flexible coupler. Keywords: Steady-state vibration, elastic mechanism, Newmark integration method, mode superposition method, dynamic stability. 1. INTRODUCTION In high-speed machines, the motion of the transmission mechanisms is often composed of a combination of rigid body motion and elastic deformation [1, 2]. A review on the vibration and stability behavior of mechanisms with elastic links represents an update to earlier literatures surveys on this subject [3–5]. Many researchers have tried to represent the vibration of such mechanisms in a more and more realistic form. Up to now, the following models are used for modeling of flexible links of mechanisms: continuum models [6, 7], lumped parameter models [8, 9], finite element models [10–14]. In general, the mathematical formulation of this vibration problem is quite a complicated nonlinear differential equation, for which an exact solution is practically impossible. It is possible to calculate the transient solutions by the numerical methods. The linearized equations of motion of an elastic mechanism that performs the .
