Dynamic and energy analysis of frictional contact instabilities on a lumped system

Abstract

When dealing with complex mechanical systems, the frictional contact is at the origin of significant changes in their dynamic behavior. The presence of frictional contact can give rise to mode-coupling instabilities that produce harmonic friction induced vibrations. Unstable vibrations can reach large amplitude that could compromise the structural and surface integrity of the system and are often associated with annoying noise emission. The study of this kind of dynamic instability has been the subject of many studies ranging from both theoretical and numerical analysis of simple lumped models to numerical and experimental investigation on real mechanical systems, such as automotive brakes, typically affected by such issue. In this paper the numerical analysis of a lumped system constituted by several degrees of freedom in frictional contact with a slider is presented, where the introduction of friction can give rise to an unstable dynamic behavior. Two different approaches are used to investigate the effects of friction forces. The first approach, the Complex Eigenvalues Analysis, allows for calculating the complex eigenvalues of the linear system that can be characterized by a positive real part (i.e. negative modal damping). The complex eigenvalues and eigenvectors of the system are investigated with respect to friction. In the second approach a non linear model has been developed accounting for the stick–slip–detachment behavior at the interface to solve the time history solution and analyze the unstable vibration. The effects of boundary conditions and of system parameters are investigated. Results comparison between the two different approaches highlights how nonlinearities affect the time history solution. The lumped model allows for a detailed analysis of the energy flows between the boundary and the system during self-excited vibrations, which are at the origin of the selection between the predicted unstable mode.

Appendix: System matrices in sliding condition

With reference to the system in Fig. 1 the system matrices used in (3) to solve the complex eigenvalue problem can be defined writing the Lagrangian equation of the system to find the equation of motion considering the normal and tangential forces at the contact.

Applying the sliding condition on both the contacting points:

$$ \left\{ \begin{array}{l} x_{4}=\dot{x}_{4}=\ddot{x}_{4}=0 \quad \hbox {and} \,T_{1}=\mu N_{1}\\ x_{8}=\dot{x}_8=\ddot{x}_{8}=0 \quad \hbox {and} \, T_{2}=\mu N_{2}\\ \end{array} \right. $$

(18)

the number of DoFs of the system can be reduced up to 6.

The coordinate vector of the system is in this case:

$${\varvec{x}}=[x_{1}\,x_{2}\,x_{3}\,x_{5}\,x_{6}\,x_{7}]^{T} $$

(19)

and the mass and stiffness matrices are:

$$ {\mathbf{M}}= \hbox{diag} ([m_{1}\,m_{1}\,m_{2}\,m_{3}\,m_{3}\,m_{4}]) $$

(20)

$$\begin{aligned} {\mathbf{K}} = \begin{array}{l} \left[ \begin{array}{cccccc} k_{1}+k_{3}+k_{8} & 0 & -k_{8} a_{1} & -k_{3}& 0 & 0\\ 0 & k_{2}+k_{5}+k_{6} & -k_{2} b_{1} & 0 & -k_{6} & 0\\ -k_{8}(a_{1}+\mu b_{1}) & k_{2}(\mu a_{1}-b_{1}) & k_{8} a_{1}^{2}+k_{2} b_{1}^{2}+\mu (k_{8}-k_{2})a_{1} b_{1} & 0 & 0 &\\ -k_{3}& 0 & 0 & k_{3}+k_{9} & 0 & -k_{9} a_2\\ 0 & -k_{6} & 0 & 0 & k_{4}+k_{6}+k_{7} & -k_{4} b_{2}\\ 0 & 0 & 0 & -k_{9}(a_{2}+\mu b_{2}) & k_4(\mu a_{2}-b_{2}) & k_{9} a_{2}^{2} + k_{4} b_{2}^{2} + \mu (k_{9}-k_{4})a_{2} b_{2}\\ \end{array}\right] \end{array} \end{aligned} $$

(21)

where:

$$ a_{i}=\cos \theta_{i} \quad \hbox {and} \quad b_{i}=\sin \theta _i $$

(22)

The damping matrix is defined as proportional to the stiffness matrix by means of the proportional coefficient β defined in Table 1:

$$ {\mathbf{C}}=\beta {\mathbf{K}} $$

(23)

Finally, the forces vector at the second member of (3) assumes the following value:

$$ {\varvec{F}}=[ 0-k_{5} \delta_{1}-F_{Ex,1}\,0\,0-k_{7}\delta_{2}-F_{Ex,2}0 ]^{T} $$

(24)

The same approach can be applied for each contact condition.