# Friction and Frequency Response Curves Essay

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First published in:

Archive of Applied Mechanics 71 (2001) 463±472 Ó Springer-Verlag 2001

Passive vibration absorber with dry friction
A. Hartung, H. Schmieg, P. Vielsack

Summary The properties of a passive vibration absorber with dry friction signi®cantly differ from those of the classical linear absorber. The exceptional phenomenon is the possibility of suppressing all excited modes. This effect is in¯uenced to a small extent by a special shape of the friction characteristic, but mainly by an appropriately adjusted threshold of the static friction. The theoretical predictions are con®rmed by experimental investigations. Keywords Nonlinear Absorber, Dry Friction, Experiment

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1 Introduction A passive vibration absorber is a mass-spring subsystem coupled to a superstructure to control its oscillations under the action of periodic excitation. A simple form of this arrangement is shown in Fig. 1 where M1 is a mass emulating the superstructure and K1 is its mounting spring. The second mass, M2 , the coupling spring K2 and a viscous damper d constitute the absorber system. The superstructure is driven by a harmonic base motion with amplitude A and angular frequency X. Let x1 be the displacement of M1 and x2 the displacement of M2 , respectively. So far, the problem is well known from elementary textbooks on linear vibration theory. Now, a friction device is added to the substructure which turns the problem into a strongly nonlinear mechanical system. The law for the friction force R must be de®ned in a way that R is an active force if the device slides, and a passive one if the device sticks. This gives strict separation between stick and slip phases during motion. Classical investigations on motions of mechanical systems with dry friction are mostly based on deterministic laws which are de®ned by the product of a dynamic friction coef®cient, depending on the relative velocity at the contact area, and the normal pressure, generally depending on time, [1]. In the following, the normal force is assumed to be constant during motion. Then, the dynamic friction force can be reduced to a simple expression _ _ R  Rd sgn x2  ax2 : Introducing the threshold value Rs for the static friction force, three possibilities will be investigated as plotted in Fig. 2. The simplest possibility is Coulomb's law (Fig. 2a). Here, Rs is equal to Rd , and the dynamic _ force R depends only on the direction of sliding and not on the value of the relative velocity x2 . In the case of a decreasing characteristic (Fig. 2b), the equality Rs  Rd still holds, but the _ friction force depends linearly on the relative velocity x2 with a negative slope a < 0. In the _ third case (Fig. 2c), the value Rd of dynamic friction remains constant for x T 0, but the static friction coef®cient Rs is larger than Rd . The ®rst question is whether or not different laws lead to signi®cantly different responses and phenomena of the vibration absorber. Secondly, the total behaviour of the mechanical system is of interest, compared with the well-known ef®ciency of the classical linear vibration absorber. And ®nally, experimental investigations should con®rm the theoretical results.

Received 10 January 2000; accepted for publication 26 September 2000 A. Hartung, H. Schmieg (&), P. Vielsack È È Institut fur Mechanik, Universitat Karlsruhe, D-76128 Karlsruhe, Germany Fax: (0721) 608 7990 E-mail: Mechanik@bau-verm.uni-karlsruhe.de
EVA-STAR (Elektronisches Volltextarchiv – Scientific Articles Repository) http://digbib.ubka.uni-karlsruhe.de/volltexte/8382001

Fig. 1. Mechanical model

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Fig. 2a±c. Friction laws. a Coulomb's law, b decreasing characteristic, c static friction larger than dynamic friction

2 Equation of motion and integration procedure Comprehensive literature on the subject of nonsmooth dynamical systems has been made available in the last decade, [2]. The motion of the nonsmooth dynamical system considered _ can consist of three…