Prediction and validation of the energy dissipation of a friction damper

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Abstract

Friction dampers can be a cheap and efficient way to reduce the vibration levels of a wide range of mechanical systems. In the present work it is shown that the maximum energy dissipation and corresponding optimum friction force of friction dampers with stiff localized contacts and large relative displacements within the contact, can be determined with sufficient accuracy using a dry (Coulomb) friction model. Both the numerical calculations with more complex friction models and the experimental results in a laboratory test set-up show that these two quantities are relatively robust properties of a system with friction. The numerical calculations are performed with several friction models currently used in the literature. For the stick phase smooth approximations like viscous damping or the arctan function are considered but also the non-smooth switch friction model is used. For the slip phase several models of the Stribeck effect are used. The test set-up for the laboratory experiments consists of a mass sliding on parallel ball-bearings, where additional friction is created by a sledge attached to the mass, which is pre-stressed against a friction plate. The measured energy dissipation is in good agreement with the theoretical results for Coulomb friction.

Introduction

Frictional forces arising from the relative motion of two contacting surfaces are a well-known source of energy dissipation. Sometimes this is an unwanted effect of the design, but it can also be intentionally used to increase the damping of a certain system in a simple and cost-effective way.

Friction dampers are used in very diverse applications: civil engineering, rotor machinery (mainly turbines), vehicles and ring dampers. The common goal in these applications is to find the optimum system parameters that lead to the highest amount of damping possible for a given friction damper design. This implies that a model of the system with friction is needed, where the main issue is finding the right friction model for that system. In general the friction models normally applied to friction damping can be divided into micro-slip models, where the contact interface is modeled with a number of friction elements in parallel (see for example [1], [2]) and macro-slip models where local contact is assumed and a single friction element is used. This paper focuses on systems with stiff localized contacts and where the relative displacements within the contact are relatively large. These systems do not require an accurate description of the pre-sliding behavior, and can be modeled using a single friction element. The most widespread model in the literature on friction damping is the classical Coulomb friction model (with equal static and dynamic friction forces). In [3] a procedure to design friction dampers for turbine blades is presented based on minimizing the blade stress, while an expression for the equivalent viscous damping coefficient for a single degree of freedom (SDOF) system with friction is found in [4] using the harmonic balance method. More recently, a study on friction damping applied to an isolation system for buildings has been presented in [5], where the optimum friction force is found by minimizing the response amplitude. In [6] approximate expressions for the energy dissipated per cycle and optimum friction force for a single degree of freedom (SDOF) system with friction are obtained applying the harmonic balance method. The classical Coulomb friction model has also been applied in models for structures connected by a friction damper [7], thin-walled friction dampers for turbine blades [8], damping of conical whirl in rotors [9], friction dampers for vehicle applications [10], [11] and friction dampers for railway wheels [12], [13].

There are very few works where more complex friction models are used to model friction damping [14], [15], [16], [17]. In his review on nonlinear passive vibration isolators [14], Ibrahim discusses friction-pendulum systems, where teflon sliding bearings are used and mentions that the friction coefficient strongly depends on the sliding velocity (Stribeck effect). A model for the Stribeck effect in teflon bearings was presented in [15] and compared to a Coulomb friction model. It was concluded that the use of Coulomb's constant friction model may result in a useful estimation of the peak response, provided that an appropriate value of the coefficient of friction is used. In [16] the friction force is modeled using a Coulomb friction model with different static and dynamic friction forces (FsFd). It is shown that for Fs>Fd0.64Fs (which is the range often found in practice [18], [19], [20], [21], [22]) the maximum energy dissipation and the corresponding optimum friction force are constant and equal to those obtained with the classical Coulomb friction model. This gives an indication that the choice of friction model might have a limited influence on the predicted maximum energy dissipation. A Stribeck model and numerical simulations have been used in [17] to determine the overall energy levels of two SDOF systems connected by a joint, but the results are not compared to the Coulomb model.

In the light of the literature study the following question arises: How close to reality should the friction model be in order to obtain good estimations of the energy dissipation and optimum system parameters? Or, in other words, is a simple model like the Coulomb model good enough to be applied to the design of friction dampers with stiff localized contacts and large relative displacements?

The goal of the present paper is to investigate whether an accurate friction model is needed to estimate the optimum friction force and maximum energy dissipation correctly and, therefore, to optimize the design of friction dampers. In order to answer these questions a SDOF model of a mass driven by a periodic force has been studied both numerically and experimentally. Several different friction models have been considered in the numerical study. The stick-phase is modeled using continuous approximations (viscous damping, arctan function) and implementations of the Stribeck effect and viscous damping are considered for the slip-phase. The stable periodic solutions have been found using a single-point shooting method. Once the periodic solution is known, the energy dissipated per cycle has been determined. It is shown that the use of continuous approximations to model the stick-phase (viscous damping, arctan function) has limited influence on the predicted energy dissipation and that the exact behavior during the stick-phase will not significantly influence the prediction of the maximum energy dissipation and the optimal friction force. Furthermore the optimum friction force and maximum energy dissipation are relatively insensitive to the choice of friction model in the sliding region, as long as realistic values of the parameters are used. The test set-up for the laboratory experiments consists of a mass sliding on parallel ball-bearings, where additional friction is created by a sledge attached to the mass, which is pre-stressed against a friction plate. No care has been taken to ensure pure dry (Coulomb) friction. Nevertheless, the measured energy dissipation is in good agreement with the theoretical results for Coulomb friction presented in [16], which supports the conclusions from the numerical analysis.

The paper is organized as follows. First the most important definitions and results from [16] are summarized in Section 2. In Section 3 the influence of the friction model is analyzed and the results are discussed. The experimental set-up and data processing are described in Section 4 and in Section 5 the experimental results are presented and compared to the theoretical results. Finally the conclusions are summarized in Section 6.

Section snippets

Background

In Fig. 1(a) a SDOF system with friction is shown excited by a periodic harmonic force f0(t)=F0sin(ω0t), where F0 is the force amplitude and ω0 the radian frequency. Such a SDOF system is often used to model friction dampers. Here the limiting case ωn/ω00 is considered, where ωn=k/m is the natural frequency of the SDOF system. In this case the SDOF system reduces to a free mass with friction excited by a periodic force as shown in Fig. 1(b). The free mass system with friction is chosen for its

Influence of stick phase

The stick phase has been modeled using three different friction laws: viscous damping, arctan function and switch model (see [24]). In all three cases a constant friction force is used in the slip phase and Fs=Fd. The corresponding equations are:

Viscous damping:fr=Fdsgn(v),|v|>η,Fdηv,|v|η.

Arctan function:Fr=Fd2πarctan(εv).

Switch model:Fr=Fdsgn(v),|v|>η,min{|Fex|,Fd}sgn(Fex)+αv,|v|η.The zero velocity interval η is normally chosen such that 1η>Tol, where Tol is the tolerance of the integration

Description of the experimental set-up and measurement procedure

The experimental investigation of energy dissipation of dry friction has been done at a set-up where a shaker forces a sledge back and forth. In Fig. 7 a top-view sketch of the set-up is depicted. As friction lip a steel part is used which is stiff in the translating direction and elastic in the direction perpendicular to the translation. A bearing ball is attached to the part and used as friction tip. The bearing balls slide along a polished silicium–carbonate plate which is fixed to the

Comparison of calculated and measured energy dissipation

It has already been mentioned that four measurement series have been completed and that, for each measurement series different excitation frequencies have been measured. The experimentally determined energy dissipation for the excitation frequencies of 13 and 16 Hz is compared to the analytical curve in Fig. 10, Fig. 11, respectively. The results for the four measurement series are shown. In every plot a distinction is made between decreasing and increasing normalized friction force and between

Discussion

The experimental results of the previous section show a very good agreement with the results from the analytical model presented in [16]. This is a remarkable result, since the theoretical results have been obtained for the most simple form of friction model, namely Coulomb friction, and the friction model that best describes the set-up is not known. This result validates the conclusions derived in Section 3, where numerical experiments for different friction models are reported and it is shown

Conclusions

In the present paper the influence of the friction model on the predicted energy dissipation has been investigated by means of numerical models and experiments on a free mass with friction. The free mass system with friction is chosen for its simplicity and ease of implementation, but it can be shown that for a wide range of parameter values the qualitative behavior of the free mass regarding the energy dissipated as a function of the friction force is similar to that of the mass–spring–damper

Acknowledgments

The authors would like to thank Dr. Pieter Nuij for his advice and assistance in setting up the experiments and Coen Blok for carrying them out.

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