Alzuwayer B* and Haque I
International Center for Automotive Research, Clemson University, USA
Received Date: December 13, 2016; Accepted Date: January 04, 2017; Published Date: January 08, 2017
Citation: Alzuwayer B, Haque I (2017) Influence of Friction Models on the Dynamics of 2-DOF Friction Oscillator. J Appl Mech Eng 6:246. doi: 10.4172/2168- 9873.1000246
Copyright: © 2017 Alzuwayer B, et al. This is an open-access article distributed under the terms of the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original author and source are credited.
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The dynamics of two Degrees of Freedom (DOF) mass-spring-damper system, resting on a moving belt are investigated. Due to friction force generated between the moving belt and the masses, the friction model plays a significant role on the energy transfer between the contacting surfaces. Therefore, in this study a steady-state and a dynamic friction models are implemented. The influence of these friction models on the system response is investigated. Additionally, the both systems where subjected to a number of initial conditions, to address the system dependency on the initial conditions. Finally, using the provided governing equations and the systems jacobain matrices Lyapunov exponent spectra were calculated.
Dynamics; Friction oscillator
Friction characteristics are important factor in many engineering applications, especially when the friction force contributes in the energy transfer between different components. For example, conveyor belts, engine accessories drive system, continuously variable transmission etc., in these systems the energy transfer is typically limited by the maximum friction force that develops between any two contacting surfaces, when the friction force limit is reached slip is always expected. However, due to the non-smooth mechanics of the friction force it introduces a nonlinearity into the system and possible irregular dynamics that might lead to a chaotic behavior. For instance, Popp and Stekter , investigated the self-sustained vibrations in different mechanical systems with friction. The authors observed the influence of the stick-slip on the system dynamics that led to a rich bifurcation and chaotic behavior. Srivastava and Haque [2,3], examined the friction characteristics of a 2-DOF mechanical oscillator, in effort to understand a possible and similar dynamics in the push-belt type cvt. In their study, the authors implemented two different friction models, namely Coulomb and Stribeck Models. The reported results indicated a mild chaotic behavior in the Stribeck model. However, using such steady-state friction models raises the question about the possibility and existence of the chaotic behavior in dynamic or state-dependent friction models. Gdaniec et al. , analyzed the dynamics of 1-DOF oscillator with LuGre dynamic friction model, the authors predicted a period doubling indicating a chaotic behavior, however, the 1-DOF oscillator is cannot serve the objective to answer the question in this research. Therefore, the dynamics of 2-Dof mechanical oscillator is going to be investigated under two different friction models, and the concept of Lyapunov exponents is going to be adopted in order to address any possible chaotic behavior.
The system under investigation consist of: two blocks resting on a moving belt and coupled by springs and dampers, as depicted in Figure 1. The friction force is generated between the blocks and the moving belt, this frictional force is going to be investigated using both Stribeck steady state friction model (case 1) and LuGre state dependent friction model (case 2). Without loss of generality, the equations of motion are given as:
Here, m,c, k,μ and g are mass, damping, stiffness, coefficient of friction and the gravity, respectively. It is to be noted that the coefficients of friction μ1 and μ2 can be represented by the appropriate friction model in each case. Assuming, that the friction model is in a steady-state or static form, the system represented by, has 4-dimesions in the phase space. However, if the friction model is dynamic and statedependent, then the system dimension increases according to friction model representation, this is going to be introduced in the subsequent section.
Case 1: Stribeck-steady-state model
In many applications, the contacting surfaces are exposed to lubricant; therefore, the chances of different lubrication regimes existence might be possible. From Tribological point of view, there are different friction regimes between the contacting surfaces where the coefficient of friction is continuously changing based on the relative speed, lubricant viscosity and load. For very low relative speeds, the asperities of contacting surfaces are close to each other; this regime is defined as the boundary lubrication. In the boundary lubrication regime, the coefficient of friction is high, due to resistance and interaction of contact asperities and the minimal influence of the lubricant. Furthermore, since this regime provides high coefficient of friction, the operation in this regime can increase the amount of energy transmitted through friction, while on the other hand it might increase the wear rate. As the relative speed between the surfaces increases, the distance between the contacting surfaces increases proportionally, allowing the admission of lubricant or increased lubricant layer height and reduced coefficient of friction, this regime is called the mixed or partial lubrication regime on the Stribeck curve (Figure 2).
The transition between the regimes (boundary to partial lubrication), involves several physical and Tribological properties such as surface texture, surface roughness, lubricant viscosity, fluid layer height etc. However, it is possible to adopt a friction model that represents the Stribeck effect, this type of models is steady state in nature which had been implemented in the literature widely [5,6]. Therefore, to simplify the frictional behavior and the transition from friction to slip with the influence of Stribeck incorporated, a steady state friction model for the coefficient of friction is going to be adopted, and this model is given as:
Where, is the relative velocity between the belt speed v and the block velocity , α is the Coulomb friction coefficient, β is the stiction to Coulomb ratio, β is the coefficient of friction growth rate and δ is the Stribeck curve decay rate. Substituting the friction model given by in , yields:
In order to further solve and analyze the governing equations, the system can be reduced to a set of first order differential equations, by letting, as a result can be written as:
Given the system representation as above, it is possible to show that the system have equilibrium points are influenced by the friction model
Additionally, the system jacobian:
Case 2: LuGre friction model
LuGre friction model  is a detailed dynamic model that is statedependent and is able to capture several frictional behaviors  as well as the influence of the contacting surfaces properties. In this model, the asperities between the contacting surfaces are represented by elastic bristles that reassemble several randomly spaced spring-damper couplings between the sliding bodies (Figure 3). These couplings can be lumped into one spring-damper and a viscous damper to represent the viscous frictional force resisting the bodies sliding motion.
The LuGre friction model can be represented as a first order Ordinary Differential Equation as follows:
In this form, σ0 ,σ1 ,σ2 represent the bristle stiffness, the damping coefficient and the viscous damping, respectively, while, the bristle average deflection rate is given as and is the relative velocity as defined previously. Additionally, a slip speed dependent function that describes the transition from stiction to Coulomb friction . Substituting and into again, the system of governing equations would be as following:
It is evident from the governing equations the system has 6-dimesnions, due to dynamics of the friction model. Following the same procedure, letting can be represented as the following first order differential equations:
The increased system dimension will also increase size of the Jacobian matrix; therefore, it is going to be included in the appendix.
The systems of equations given by , for the Stribeck friction, and for the LuGre friction, were implemented in Matlab® package. These systems were solved for a number of different sets of initial conditions. The results for both systems dynamics are reported and Lyapunov exponent spectra were calculated to address any possible chaotic behavior.
For the Stribeck friction model, Figures 4 and 5 depict the displacements and velocities responses, for both masses. It is apparent that the system dynamics is characterized by an oscillatory behavior, where both masses oscillations are in-phase. However, due to the different initial conditions, it was observed that some of the initial conditions sets have an impact on the on the phase portraits too, where the systems undergoes a limit cycle as can be noticed from Figures 6 and 7. Concerning the phase trajectories and their intersecting with each other, it is commonly indicates chaos, however, the portrait shown does not include the whole dimensions of the system. Thus, the chaotic behavior cannot be confirmed from the 2-D phase portraits only.
Regarding the dynamic LuGre friction model, since this model takes into account several friction characteristics, such as the pre-sliding, hysteresis, viscous friction and the contact asperities properties, it is expected that the system behavior would be richer in terms of its dynamics and in comparison with the steady-state friction model.
Figures 8 and 9 show the displacement and velocity response, from the responses this system depends heavily on the initial conditions when compared with the Stribeck friction model. Here, the oscillatory motion still exists, however, there are some responses that are driven towered the equilibrium points, which is confirmed by the phase portraits for the two mass, shown in Figures 10 and 11. In these phasespace plots, the trajectories show the stick region is obvious, where the mass velocity is equal to the belt speed. Furthermore, some limit cycles are also appearing for some of the initial conditions as reported in the Stribeck results. Again, these phase portrait cannot fully indicate the chaotic behavior, since in this case; the system has 6-dimesions in the phase plane. Thus, alternative methods should be used to investigate the chaotic behavior such as Lyapunov exponent spectrum. Lyapunov exponents can be determined either from the series or by solving the system equations; the former approach is usually used to determine the largest Lyapunov exponent from experimental data or if it is difficult to model the system mathematically. However, in this study, the governing equations and the system’s jacobian matrices are already derived and provided in order to estimate Lyapunov spectrum. Following Wolf’s algorithm , the estimated Lyapunov spectra for Stribeck and LuGre systems are depicted in Figures 12 and 13. In the case of 2-DOF oscillator with Stribeck friction model, the largest Lyapunov exponent was positive with a value of 0.007409, although this value is close to zero; however, it means that any two close trajectories would eventually diverge from each other, which indicates a mild chaotic behavior as reported by Srivastava and Haque . On the other hand, the largest Lyapunov in the case of 2-DOF oscillator with LuGre friction was 0.07269. In this case, Lyapunov exponent is greater than the Stribeck case, meaning that the any two close trajectories for the LuGre case would diverge faster than it would be in the Stribeck case, as a result a chaotic behavior can be expected.
The influence of two different friction model on the dynamics of 2-DOF oscillator is investigated. The oscillator consists of two mass blocks resting on a moving belt and coupled by springs-dampers. The first friction model is a steady-state model that depends solely on the relative speed between the blocks-belt interface and it aims to capture the Stribeck curve during stiction and sliding. Whereas, the second friction model (LuGre friction model) is a dynamic state-dependent model represented by a 1st order ODE. This dynamic friction model represent richer friction characteristics such as stick, slip, pre-sliding and viscous friction. The two systems equations were derived and solved in effort to understand the influence of the dynamic friction models on the system response and any possible chaotic behavior. However, depending on the phase portraits only did not help in confirming the existence of chaotic behavior due to the larger phase-space dimension. Therefore, the Lyapunov exponent spectra approach was implemented to address any positive exponents. It was observed that the system is dependent on the initial conditions and their positive .
Lyapunov exponents in both friction cases. Thus, the initial conditions for such systems and friction models should investigated properly along with the system parameters to avoid unstable and chaotic dynamics.