Analytical investigation of periodic solutions for a coupled oscillator with dry friction

In this paper, we present an analytical method to investigate the behavior of a two degrees of freedom oscillator excited by dry friction. The system consists of two masses connected by linear springs. These two masses are in contact with a driving belt moving at a constant velocity. The contact forces between the masses and the belt are obtained from Coulomb’s friction laws. A set of periodic solutions involving a global sticking phase followed by several other phases where one or both masses are slipping, are found in close form. Stability conditions related to these solutions are obtained. INTRODUCTION Vibrating systems excited by dry friction are frequently encountered in technical applications. These systems are strongly non-linear and they are usually modeled as spring-mass oscillators. This paper is the continuation of the work [1], where we present an analytical method to investigate the behavior of a two degrees of freedom oscillator with dry friction. In this former paper, two kinds of periodic solutions including stick slip phases were found. In the following, a new kind of periodic motion is investigated, involving for each period a global stick phase [2] for each masses, followed by several other phases where one or both masses are slipping. NOMENCLATURE C,D,H,Γ = 4by 4 matrices depending on time. 2 1 , F F = dry friction forces. f c b a J J J J J , , , , = Jacobian matrices. 2 1 , , K K K = stiffness matrices. V = belt velocity. 2 1 , k k = springs stiffness. 2 1 ,m m = blocks mass. t’ = time. t = non dimensional time. d c b a t t t t , , , = non dimensional times of switches. 2 1 ,u u = non dimensional friction forces. 2 1 , r r u u = non dimensional static friction forces. 2 1 , s s u u = non dimensional sliding friction forces. ) 4 ,.. 1 ( , = Σ i i = switching surfaces. χ = non dimensional stiffness. η = non dimensional mass. 3 2 1 , , , ω ω ω ω = eigenfrequencies. DESCRIPTION OF THE MODEL Figure 1: Coupled oscillator with dry friction. The system consists of two masses 2 1 ,m m connected by linear springs of stiffness 2 1 , k k (Fig.1). These two masses are in contact with a driving belt moving at a constant velocity. The contact forces 2 1 ,F F between the masses and the belt are obtained assuming Coulomb’s friction laws. 1 m 2 m 1 k 2 k