


ARTICLE 

Year : 2012  Volume
: 2
 Issue : 1  Page : 712 

Stability Analysis of Mode Coupling Vibration Systems with Coulomb Friction
Lilan Liu, Ziying Wu
School of Mechanical Engineering, Xi'an University of Technology, Xi'an, China
Date of Web Publication  24Mar2012 
Correspondence Address: Lilan Liu School of Mechanical Engineering, Xi'an University of Technology, Xi'an China
Source of Support: None, Conflict of Interest: None  Check 
DOI: 10.4103/09768580.93230
Abstract   
In the article, the stability analysis of mode coupling systems with Coulomb friction is performed. A twodegreeoffreedom (2DOF) vibration system with Coulomb friction is constructed. Taking the friction coefficient as a bifurcation parameter, a Hopf bifurcation graph is obtained using the Routh criterion. It is illustrated that Coulomb friction can induce selfexcited oscillation in the mode coupling system. The analytic expressions of the limit cycle amplitudes of the oscillation system are achieved based on the extended harmonic balance method. Moreover, the effects of some parameters on the size of limit cycles are investigated. All the research method and results presented can provide a valuable reference for the structure optimal design of mechanical systems and vibration reduction. Keywords: Friction, limit cycle, mode coupling vibration, stability
How to cite this article: Liu L, Wu Z. Stability Analysis of Mode Coupling Vibration Systems with Coulomb Friction. J Eng Technol 2012;2:712 
1. Introduction   
The selfexcited oscillation behavior induced by friction exists widely in mechanical systems, which has been a main cause of system instability, such as vibration and squeal noise. For the generation mechanism, many researchers intend to clarify it from different aspects ^{[1],[2]} . As a whole, there may be four types of mechanisms to describe frictioninduced vibration in brake systems ^{[3]} . The important one of them is the mode coupling instability. In the case of mode coupling, the system concerned will be instable and limit cycle will appear even with Coulomb friction. This negative effect is unfavorable and should be avoided in brake systems. So, to reduce or eliminate the instability caused by mode coupling, a better solution is to fall back on the optimum design of the system structure and physical parameters. However, it depends on full prior understanding of the influence of these parameters on the system stability. Due to complexity of the problem, one common method for stability analysis is modeling a linear or linearized model of brake systems ^{[4]} . In Ref. ^{[5]} , the instability mechanism of the steady sliding state has been understood well. Although vibration induced by friction in mode coupling systems has received considerable attention in recent years, there still exist some important aspects such as the effects of different physical parameters on the system stability and the strong nonlinearity induced by Coulomb friction has not been fully studied and are worth further discussion.
The object of this article is to investigate the stability and limit cycle behavior of mode coupling systems with Coulomb friction. The next is organized as follows: In Section 2, the mode coupling system subjected to Coulomb friction to be studied is introduced and modeled. In Section 3, the stability analysis of the system is carried out by using the Routh criterion. And, an analytical calculation method for the limit cycle amplitude based on the extended Harmonic Balance technique is presented in Section 4. The influence of physical parameters on the limit cycles is investigated in Section 5. Conclusion is made at last.
2. System Description and Modeling   
The mode coupling model used in this work is a 2DOF massconveyor belt system as shown in [Figure 1]. The conveyor belt moves with a constant velocity v_{o} . The mass is held by two springs (k_{1} and k_{2} ) and two dampers (C_{1} and C_{2} ) at two directions (x_{1} and x_{2} ). Because of the external force F _{e} , the mass is always contacts with the belt. And Coulomb friction force exists at the contact surfaces.
The origin form of this model is first presented by Hulten ^{[6],[7]} to study the problem of drum brake squeal, in which only the stiffness is included. The system may be instable because of mode coupling. After that, a damping part was added in the model by Sinou and Je'ze'quel ^{[8]} to understand the role of damping. In their work, however, to simplify the analysis, the direction of friction force is not considered in the course of vibration, as described by
where, F _{fi} is the friction force, F _{ni} is the normal force, and μ is the friction coefficient.
But as we know, the Coulomb friction force has its own direction which depends on the relative sliding velocity between two contact surfaces. So such an assumption may be not sufficient for the real dynamic behavior of mechanical systems.
In this article, therefore, to make up the shortage of above assumption, a complete Coulomb model considering direction is adopted
where, denotes the absolute velocity of the mass, and sgn(•) is the sign function.
Therefore, the motion equation of the nonlinear system shown in [Figure 1] is
where, x_{1} and x_{2} present the absolute displacements of the mass in the directions of x_{1} and_x_{2} , respectively.
In such a case, the external force F _{e} will cause a static displacement of the mass. So when there are the static equilibrium points of the system are:
Taking the static equilibrium points as a new origin of coordinate and inserting into Equation (3), the following can be obtained as
where, represent the displacements of the mass in the new coordinate, respectively. It can be seen in Equation (5) that the stiffness matrix is unsymmetrical because of the friction coefficient μ.
To generalize the system equation, the damping ratio ξ_{i} = c_{i} / (2w_{i}m)(i =1,2), and the natural frequency ω_{i} = √k_{i} /m(i =1,2)are introduced into Equation (5) and the next relation can be obtained as
It should be pointed out that the aim of the next work is to find out an appropriate analysis method. So the values of parameters used are prior known. In the next computation, the values of the three constant parameters are m=1 kg, v _{o} = ms ^{1} , and F _{e} = 4.9_N, respectively.
3. Stability Analysis   
For a nonlinear system like Equation (6), there is an explicit parameter μ in the stiffness matrix, and so the eigenvalues not only relate to the damping and frequency of the system but also relate to the friction coefficient. Therefore, its stability about the system static solution can be estimated by the eigenvalues of the linearized equation by using the Routh criterion ^{[9]} .
The linearized equation of Equation (6) is
And the characteristic equation of Equation (7) can be obtained as
with λ the eigenvalues of the linearized system.
So the eigenvalues of the linear system can be computed by solving the following characteristic polynomial
For Equation (9), if the real parts of the eigenvalues λ are all negative, the static solution of the system is stable. Otherwise it is unstable even if only one root has a positive real part. But solving Equation (9) is very complex. Fortunately, according to the Routh criterion, we do not have to solve the characteristic polynomial actually, and only the Routh coefficients are needed. If all the Routh coefficients are positive, the static solution of the system is stable. Therefore, using the Routh criterion and taking friction coefficient μ as a bifurcation parameter, the Hopf bifurcation point μ_{H} can be expressed by the other parameters as
with e= ω _{1} /ω _{2} and r = ξ _{1}/ξ _{2} .
Equation (10) indicates that the variation of the Hopf bifurcation point has a direct relation with the damping and frequency of the system, as shown in [Figure 2] with
ω_{1} = 1.1~2s ^{1}, ω _{2} = 1.508s ^{1}, ξ_{1} = 0.001~0.04 and ξ_{2} = 0.0102.
4. Amplitude Calculation of Limit Cycles   
From Section 3, it is illustrated that in the case of mode coupling, the vibration system will be instable and limit cycle will appear even with Coulomb friction. The conventional Harmonic Balance technique is a classical method to determine the amplitudes of limit cycles for nonlinear vibration systems. The approach is usually suitable to analyze onedimensional vibration problem, whereas the mode coupling phenomenon is involved in a 2DOF problem. To overcome this shortcoming, Hoffmann et al. ^{[10]} present a modified approach called extended harmonic balance with a main advantage of yielding multiDOF mode coupling system. So based on the procedure of the extended harmonic balance, the following calculation is put forward to calculate the limit cycle amplitudes of the oscillation system quantitatively.
The solutions of Equation (6) are supposed as
where denote the static displacements induced by the oscillation, represent the amplitudes of limit cycles, and ω is the harmonic oscillation frequency.
First, substitute Equation (11) into the nonlinear terms located in the right side of Equation (6), and there is
It can be seen that a constant and a first harmonic components are contained in the right side of Equation (12). But there are six parameters which are unknown. To obtain these parameters, first, the Fourier coefficients of the terms f_{1} (t) and f_{2} (t) are calculated by integration over one oscillation period as follows:
Similarly, the Fourier coefficients of the terms g _{1} (t) and g _{2} (t) can also be obtained as
When the above Fourier coefficients are all obtained, the right side of Equation (6) can therefore be represented as
Then, substitute both Equation (11) and Equation (13) into Equation (6) directly and compare the coefficients of the sine, cosine and constant terms. Consequently, six equations can be obtained as follows:
At last, according to these six equations, the six unknown parameters can solved by using the Newton's method. And so, the amplitudes of the limit cycles of the vibration system can be achieved.
5. Effects of Parameters on Limit Cycles   
To further understand the effects of some physical parameters on the size of limit cycles, the oscillation amplitudes varying with different parameter values are calculated using the analytical method presented in Section 4.
First, the variation of the limit cycles with the friction coefficient μ is investigated. The values of the used parameters are e = 1.325 with ω _{2} = 1.508s ^{1} , r = 0.98 with ξ _{2} = 0.0102 and μ=0.3~0.4, respectively. Under this condition, the value of the Hopf bifurcation point is μ_{H} =0.283 according to Equation (10). So when μ>μ_{H} , the system will be unstable. What's more, with the increasing of the friction coefficient_μ, the system will be more unstable and the oscillation amplitude becomes bigger. Consequently, as demonstrated in [Figure 3], it is shown that with the increasing of μ, the amplitudes of the limit cycles get greater in both x_{1} and x_{2} directions. Comparably, the variation in x _{1} direction is more obvious.
Next, the effect of the frequency ratio r = 0.98with ξ_{2} =0.0102, e=0.993~1.325 with ω_{2} = 1.508s ^{1} , and μ= 0.3, respectively. According to Equation (10), all the values of the Hopf bifurcation points with different frequency ratio e are listed in [Table 1]. It can be seen that the Hopf bifurcation point becomes higher with the increasing of the frequency ratio e. As we know, the higher value μ_{H} is beneficial for the system stability. So increasing e can diminish the oscillation amplitude. [Figure 4] gives the relationship between the oscillation amplitude and the frequency ratio e. It is shown that when the frequency ratio e increases, the amplitudes of the limit cycles decrease in both x_{1} and x_{2} directions.
In addition, the effect of the damping ratio r on limit cycles is also analyzed. The values of used parameters are e = 1.325 with ω_{2} = 1.508s ^{1} , r=0.490~2.941 with ξ_{2} =0.0102, and μ0.3, respectively. Similarly, the values of the Hopf bifurcation points with different damping ratio r are listed in [Table 2]. It can be seen that increasing the damping ratio r can increase or decrease the value of Hopf bifurcation point μ_{H}. So, increasing damping ratio may make a certain vibration mode of the system more unstable sometimes. The relationship between the oscillation amplitude and the frequency ratio r is illustrated in [Figure 5]. It is can be seen that when the damping ratio increases, the amplitudes of the limit cycles decrease in x_{1} direction while it increases in x_{2} direction. This means that adding more damping in the vibration system to reduce oscillation amplitudes is not feasible in some cases.
From the above figures and analysis, it can be known that in the considered region, the increasing of the friction coefficient and the frequency ratio can lead to regular variations of limit cycles. While for the damping ratio, because its effect pattern on the limit cycles is relatively complex, its variation tendency should be combined with detail stability analysis in prior.
6. Conclusion   
In this case study, a 2DOF mass conveyor belt system is built to investigate the problem of mode coupling vibration. The stability analysis of the system with Coulomb friction is carried out using the Routh criterion. The mathematical expression of the Hopf bifurcation point is given out using the Routh criterion. It is illustrated that both the frequency and the damping of the coupling modes have important effects on system stability and should be taken into account to avoid bad structure design. Moreover, the extended harmonic balance method is advanced, through which the analytic solutions of the limit cycle amplitudes of the nonlinear system are achieved. Furthermore, the effects of physical parameters on the size of limit cycles are discussed based on the analytic calculation. Among them, the values of the damping ratios should be considered carefully. All the work provides a better understanding of mode coupling mechanical system with Coulomb friction. And the methodology presented can be applied for mechanical structure design and parameter optimization.
References   
1.  G. P. Ostermeyer, "Dynamic friction laws and their impacton friction induced vibrations", Proceedings of the 28th SAEB rake Colloquium and Exhibition, Phoenix, 2010. 
2.  M. Graf, and G. P. Ostermeyer, "Instabilities in the sliding of continua with surface inertias: An initiation mechanism for brake noise", Journal of Sound and Vibration, Vol. 330, pp. 52695279, 2011. 
3.  N. Hoffmann, M. Fischer, R. Allgaier, and L. Gaul, "A minimal modal for studying properties of the modecoupling type instability in friction induced oscillations", Mechanics Research Communications, Vol. 29, pp.197205, 2002. 
4.  J. Kang, C. M. Krousgrill, and F. Sadeghi, "Comprehensive stability analysis of disc brake vibrations including gyroscopic, negative friction slope and modecoupling mechanisms", Journal of Sound and Vibration, Vol. 324, pp. 387407, 2009. 
5.  J. Behrendt, C. Weiss, and N. P. Hoffmann, "A numerical study on stickslip motion of a brake pad in steady sliding", Journal of Soundand Vibration, Vol. 330, pp. 636651, 2011. 
6.  J. Hultén, "Drum brake squeala selfexciting mechanism with constant friction", SAE Truck and Bus Conference, Detroit, pp. 695706, 1993. 
7.  J. Hultén, "Friction phenomena related to drum brake squeal instabilities", ASME Design Engineering Technical Conferences, Sacramento, pp. 588596, 1997. 
8.  J. J. Sinou, and L. Jézéquel, "Mode coupling instability in frictioninduced vibrations and its dependency on system parameters including damping", European Journal of Mechanics A/Solids, Vol. 26, pp. 106122, 2007. 
9.  X. Y. Wu, and S. X. Zhang, "Automatic Control Theory", Xi'an, China: Xidian University Publications; 2008. 
10.  N. Hoffmann, S. Bieser, and L. Gaul, "Harmonic balance and averaging techniques for stickslip limitcycle determination in modecoupling friction selfexcited systems", Technische Mechanik, Vol. 24, pp. 185197, 2004. 
Authors   
Lilan Liu is teaching in Mechanical Engineering Department of Xi'an University of Technology, China. she graduated in 2002; post graduated in 2005 and got PhD in 2010 in area of Mechanical Engineering. Her areas of interest include mechanical system friction identification and compensation.
Ziying Wu is an Associate Professor in Mechanical Engineering Department at Xi'an University of Technology, China. He graduated in 1998; post graduated in 2002 and got PhD in 2007 in area of Mechanical Engineering. His areas of interest include mechanical system identification and control.
[Figure 1], [Figure 2], [Figure 3], [Figure 4], [Figure 5]
[Table 1], [Table 2]
