ENOC-2005, Eindhoven, Netherlands, 7-12 August 2005 SYMMETRIC AND ASYMMETRIC MOTIONS OF A HARMONICALLY DRIVEN DRY-FRICTION OSCILLATOR G´ abor Csern´ ak HAS-BUTE Research Group on Dynamics of Machines and Vehicles Hungary csernak@mm.bme.hu G´ abor St´ ep´ an Department of Applied Mechanics Budapest University of Technology and Economics Hungary stepan@mm.bme.hu Abstract A harmonically excited dry friction oscillator is ex- amined analytically and numerically. We search for -periodic non-sticking solutions, where is the excitation frequency and . Using the assumption that there are only two turnarounds during each cycle, we find the exact expressions of the phase and the amplitude of the evolving periodic motions. We check the validity of our results in the case . The application of the condition of having strictly two turnaround points per cycle shows that the parameter domain of non-sticking solutions is smaller than it was published in earlier contributions. We prove that if the two phases of the motion are equal in length, the mo- tion is symmetric in the coordinate at almost all the values of the excitation frequency. However, we show that at excitation frequencies , innumerable asymmetric solutions exist. The analytical results are confirmed by numerical simulation. Key words Coulomb friction, stick-slip boundary, analytical for- mulae 1 Introduction The harmonically excited linear oscillator with vis- cous damping is a well-known system, its solutions can be found easily using conventional methods. However, if the oscillating body slips on a rough surface, the ef- fects of dry friction must also be taken into account, which throws difficulties in the way of finding the so- lutions. We are concerned with a harmonically excited, dry friction oscillator which is presented in Fig. 1. Dry friction force resists relative motion between con- tacting surfaces of the block and the ground. If the coefficient of dry friction is small, the mass slides and its velocity is zero only for the instant when it passes through zero [Shaw, 1986]. However, at great friction Rough surface Figure 1. The physical system coefficient, sticking may occur. It means that the mass remains at rest for a finite time after the velocity of the oscillator reaches zero. In this case the friction force adjusts itself to make equilibrium with other external forces acting on the body. The equation of motion of the analysed system is the following: (1) where denotes the time derivative and if [-1,1] if if (2) Thus, the sgn() function is meant to take (mathemat- ically) indetermined values between 1 and -1 at zero – this corresponds to the stick phase, when the actual value of the dry friction force is determined via further physical conditions (see Fig. 2). Rescaling time and displacement as and , respectively, one obtains (3) where denotes , , , and .