Non-stationary friction-induced vibration with multiple contact points

Abstract

Modelling and simulating of the friction-induced vibration of a multi-point contact system are widely encountered and challenging problems. Non-smooth transitions between stick and slip and between contact and separation (during vibration) at each contact point can happen simultaneously in practice, which should be considered together in a theoretical model. This work is innovative in that it addresses the comprehensive dynamic analysis of a multi-point contact system considering the two types of complex non-smooth behaviour at the interface as well as mode-coupling instability, which has not been studied in previous research on multi-point contact dynamics, to the authors' best knowledge. To deal with this complex situation, a new mix-level time iteration scheme for the simulations of the non-smooth/discontinuous system with elastic contact and friction is formulated. This is an essential step as it provides a generic and effective approach that can be used for different systems with the same contact features regardless of the internal structural configurations of the systems. Interesting results and discoveries through a detailed dynamic analysis of a 10-DoF system with two sliders are reported: (1) the mass and mass ratio between the components linked with the contact interface are the essential factors of mode-coupling instability and mode-veering phenomenon through the stability analysis. These findings serve to guide the subsequent transient analysis, which is much more time-consuming and would otherwise be costly to use for revealing the roles of these masses; (2) the individual contributions of non-smoothness and mode-coupling instability, and the critical influences of the contact states, the normal compression force, and the belt speed on the vibration frequency and non-stationary vibration range of the components in the system are clarified from the complex dynamic behaviour.

Appendix

For the 10-DoF model, the kinetic energy of the system is:

$$ \begin{aligned} T = & \frac{1}{2}m_{1} \dot{x}_{1}^{2} + \frac{1}{2}m_{4} \dot{x}_{4}^{2} + \frac{1}{2}m_{2} \dot{x}_{2}^{2} + \frac{1}{2}m_{3} \dot{x}_{3}^{2} + \frac{1}{2}m_{2} \dot{y}_{2}^{2} + \frac{1}{2}m_{3} \dot{y}_{3}^{2} + \frac{1}{2}m_{4} \dot{y}_{4}^{2} \\ & + \frac{1}{2}m\dot{x}_{5}^{2} + \frac{1}{2}m\left( {\frac{{l_{c2} }}{{l_{c1} + l_{c2} }}\dot{y}_{5} + \frac{{l_{c1} }}{{l_{c1} + l_{c2} }}\dot{y}_{6} } \right)^{2} + \frac{1}{2}J\left( {\frac{{\dot{y}_{5} - \dot{y}_{6} }}{{l_{c1} + l_{c2} }}} \right)^{2} \\ \end{aligned} $$

The potential energy of the system is:

$$ \begin{aligned} V = & \frac{1}{2}\left[ {k_{1} x_{1}^{2} + k_{2} \left( {x_{2} - x_{1} } \right)^{2} + k_{3} \left( {x_{3} - x_{2} } \right)^{2} + k_{4} x_{3}^{2} + k_{6} \left( {y_{4} - y_{2} } \right)^{2} } \right. \\ & + k_{5} x_{4}^{2} + k_{7} \left[ {\left( {x_{3} - x_{4} } \right)\sin \theta - \left( {y_{3} - y_{4} } \right)\cos \theta } \right]^{2} \\ \left. \begin{aligned} & + k_{c1} \left( {y_{2} - y_{5} } \right)^{2} + k_{c2} \left( {y_{3} - y_{6} } \right)^{2} + k_{8} x_{5}^{2} \\ & + k_{9} \left( {\frac{{l_{k1} + l_{c2} }}{{l_{c1} + l_{c2} }}y_{5} - \frac{{l_{k1} - l_{c1} }}{{l_{c1} + l_{c2} }}y_{6} } \right)^{2} + k_{10} \left( {\frac{{l_{c2} - l_{k2} }}{{l_{c1} + l_{c2} }}y_{5} + \frac{{l_{k2} + l_{c1} }}{{l_{c1} + l_{c2} }}y_{6} } \right)^{2} \\ \end{aligned} \right] \\ \end{aligned} $$

in which $J$ is the moment of inertia $J = \frac{{m\left( {l_{c1} + l_{c2} } \right)^{2} }}{12}$.

The Lagrange’s equation is expressed as:

$$ \frac{\partial }{\partial t}\frac{\partial L}{{\partial \dot{q}_{i} }} + \frac{\partial L}{{\partial q_{i} }} = Q_{i} ,\;i = 1,2,...,9 $$

in which $L = T - V,q_{i}$ is the i^th element of generalised coordinate vector.

${\mathbf{q}} = \left[ {\begin{array}{*{20}c} {x_{1} } & {\begin{array}{*{20}c} {x_{4} } & {y_{4} } \\ \end{array} } & {x_{2} } & {x_{3} } & {y_{2} } & {y_{3} } & {x_{5} } & {y_{5} } & {y_{6} } \\ \end{array} } \right]^{T} ,$,

and $Q_{i}$ is the i^th element of the generalised force vector

$$ \left[ {\begin{array}{*{20}c} 0 & {\begin{array}{*{20}c} 0 & 0 \\ \end{array} } & {\mu k_{c1} \left( {y_{5} - y_{2} } \right)} & {\mu k_{c2} \left( {y_{6} - y_{3} } \right)} & 0 & 0 & { - \mu k_{c1} \left( {y_{5} - y_{2} } \right) - \mu k_{c2} \left( {y_{6} - y_{3} } \right)} & 0 & 0 \\ \end{array} } \right]^{T} , $$

The equation of motion of the 10-DoF model given in Eq. (17) can be obtained from the Lagrange’s equation. When the above mathematical expressions of the kinetic energy, potential energy, and the generalised force vector are substituted into the Lagrange’s equation, the mass matrix and the stiffness matrix can be derived as follows:

$$ {\mathbf{M}} = \left[ {\begin{array}{*{20}c} {m_{1} } & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & {m_{4} } & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & {m_{4} } & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & {m_{2} } & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & {m_{3} } & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & {m_{2} } & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & {m_{3} } & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & m & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & {m\left( {\frac{{l_{c2} }}{{l_{c1} + l_{c2} }}} \right)^{2} + \frac{J}{{\left( {l_{c1} + l_{c2} } \right)^{2} }}} & {m\frac{{l_{c1} l_{c2} }}{{\left( {l_{c1} + l_{c2} } \right)^{2} }} - \frac{J}{{\left( {l_{c1} + l_{c2} } \right)^{2} }}} \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & {m\frac{{l_{c1} l_{c2} }}{{\left( {l_{c1} + l_{c2} } \right)^{2} }} - \frac{J}{{\left( {l_{c1} + l_{c2} } \right)^{2} }}} & {m\left( {\frac{{l_{c1} }}{{l_{c1} + l_{c2} }}} \right)^{2} + \frac{J}{{\left( {l_{c1} + l_{c2} } \right)^{2} }}} \\ \end{array} } \right] $$

and

$$ {\mathbf{K}} = \left[ {\begin{array}{*{20}c} {k_{1} + k_{2} } & 0 & 0 & { - k_{2} } & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & {k_{5} + k_{7} \sin^{2} \theta } & { - k_{7} \sin \theta \cos \theta } & 0 & { - k_{7} \sin^{2} \theta } & 0 & {k_{7} \sin \theta \cos \theta } & 0 & 0 & 0 \\ 0 & { - k_{7} \sin \theta \cos \theta } & {k_{6} + k_{8} + k_{7} \cos \theta^{2} } & 0 & {k_{7} \sin \theta \cos \theta } & { - k_{6} } & { - k_{7} \cos \theta^{2} } & 0 & 0 & 0 \\ { - k_{1} } & 0 & 0 & {k_{2} + k_{3} } & { - k_{3} } & {\mu k_{c1} } & 0 & { - \mu k_{c1} } & 0 & 0 \\ 0 & { - k_{7} \sin^{2} \theta } & {k_{7} \sin \theta \cos \theta } & { - k_{3} } & {k_{3} + k_{4} + k_{7} \sin^{2} \theta } & 0 & { - k_{7} \sin \theta \cos \theta + \mu k_{c2} } & 0 & 0 & { - \mu k_{c2} } \\ 0 & 0 & { - k_{3} } & 0 & 0 & {k_{6} + k_{c1} } & 0 & 0 & { - k_{c1} } & 0 \\ 0 & {k_{7} \sin \theta \cos \theta } & { - k_{7} \cos^{2} \theta } & 0 & { - k_{7} \sin \theta \cos \theta } & 0 & {k_{7} \cos^{2} \theta + k_{c2} } & 0 & 0 & { - k_{c2} } \\ 0 & 0 & 0 & 0 & 0 & { - \mu k_{c1} } & { - \mu k_{c2} } & {k_{8} } & {\mu k_{c1} } & {\mu k_{c2} } \\ 0 & 0 & 0 & 0 & 0 & { - k_{c1} } & 0 & 0 & {k_{9,9} } & {k_{9,10} } \\ 0 & 0 & 0 & 0 & 0 & 0 & { - k_{c2} } & 0 & {k_{10,9} } & {k_{10,10} } \\ \end{array} } \right] $$

in which

$k_{9,9} = k_{9} \left( {\frac{{l_{k1} + l_{c2} }}{{l_{c1} + l_{c2} }}} \right)^{2} + k_{10} \left( {\frac{{l_{k2} - l_{c2} }}{{l_{c1} + l_{c2} }}} \right)^{2} + k_{c1}$,

$k_{9,10} = k_{10,9} = - \left[ {k_{9} \frac{{\left( {l_{k1} + l_{c2} } \right)\left( {l_{k1} - l_{c1} } \right)}}{{\left( {l_{c1} + l_{c2} } \right)^{2} }} + k_{10} \frac{{\left( {l_{k2} + l_{c1} } \right)\left( {l_{k2} - l_{c2} } \right)}}{{\left( {l_{c1} + l_{c2} } \right)^{2} }}} \right]$,

and $k_{10,10} = k_{9} \left( {\frac{{l_{k1} - l_{c1} }}{{l_{c1} + l_{c2} }}} \right)^{2} + k_{10} \left( {\frac{{l_{k2} + l_{c1} }}{{l_{c1} + l_{c2} }}} \right)^{2} + k_{c2}$.

When the resulting equation of motion is rewritten as Eq. (15), and then, the elements in K related to the unilateral contact are removed, h in Eq. (15) can be obtained:

$$ {\mathbf{h}} = - \left\{ {\begin{array}{*{20}c} {\left( {k_{1} + k_{2} } \right)x_{1} - k_{2} x_{2} } \\ {\left( {k_{5} + k_{7} \sin^{2} \theta } \right)x_{4}^{{}} - k_{7} \sin \theta \cos \theta y_{4} - k_{7} \sin^{2} \theta x_{3} + k_{7} \sin \theta \cos \theta y_{3} } \\ { - k_{7} \sin \theta \cos \theta x_{4} + \left( {k_{6} + k_{8} + k_{7} \cos \theta^{2} } \right)y_{4} - k_{6} y_{2} + k_{7} \sin \theta \cos \theta x_{3} - k_{7} \cos \theta^{2} y_{3} - F_{n} } \\ { - k_{1} x_{1} + \left( {k_{2} + k_{3} } \right)x_{2} - k_{3} x_{3} } \\ { - k_{7} \sin^{2} \theta x_{4} + k_{7} \sin \theta \cos \theta y_{4} - k_{3} x_{2} + \left( {k_{3} + k_{4} + k_{7} \sin^{2} \theta } \right)x_{3} - k_{7} \sin \theta \cos \theta y_{3} } \\ {k_{3} y_{2} - k_{6} y_{4} } \\ {k_{7} \sin \theta \cos \theta x_{4} - k_{7} \cos^{2} \theta y_{4} - k_{7} \sin \theta \cos \theta x_{3} + k_{7} \cos^{2} \theta y_{3} } \\ {k_{8} x_{5} } \\ {\left[ {k_{9} \left( {\frac{{l_{k1} + l_{c2} }}{{l_{c1} + l_{c2} }}} \right)^{2} + k_{10} \left( {\frac{{l_{k2} - l_{c2} }}{{l_{c1} + l_{c2} }}} \right)^{2} } \right]y_{5} - \left[ {k_{9} \frac{{\left( {l_{k1} + l_{c2} } \right)\left( {l_{k1} - l_{c1} } \right)}}{{\left( {l_{c1} + l_{c2} } \right)^{2} }} + k_{10} \frac{{\left( {l_{k2} + l_{c1} } \right)\left( {l_{k2} - l_{c2} } \right)}}{{\left( {l_{c1} + l_{c2} } \right)^{2} }}} \right]y_{6} } \\ { - \left[ {k_{9} \frac{{\left( {l_{k1} + l_{c2} } \right)\left( {l_{k1} - l_{c1} } \right)}}{{\left( {l_{c1} + l_{c2} } \right)^{2} }} + k_{10} \frac{{\left( {l_{k2} + l_{c1} } \right)\left( {l_{k2} - l_{c2} } \right)}}{{\left( {l_{c1} + l_{c2} } \right)^{2} }}} \right]y_{5} + \left[ {k_{9} \left( {\frac{{l_{k1} - l_{c1} }}{{l_{c1} + l_{c2} }}} \right)^{2} + k_{10} \left( {\frac{{l_{k2} + l_{c1} }}{{l_{c1} + l_{c2} }}} \right)^{2} } \right]y_{6} } \\ \end{array} } \right\} $$