Stability analysis of state dependent dynamic frictional sliding
Two models are investigated; 1) a single-degree-of-freedom system and 2) a two-degree-of-freedom one. By 1), a spring-mass slider has coordinate \(x\), and the free end of the spring has a constant velocity. Tangential force exerted on the mass is \(t_T\), the normal one is \(t_N\). By 2), the mass has a (variable) coordinate \(y>0\) in the ``normal'' direction; in this case \(t_N\) is variable, too. The model of the frictional force is described by a state variable \(\alpha\): \(\dot t_T =\varepsilon_T \left[v-{1 \over\eta} \left\langle {\varphi\over t_N} \right \rangle \text{sign} (t_T) \right]\), \(\dot\alpha =-{1\over D_c \eta} \left \langle {\varphi \over t_N} \right \rangle\left[ \alpha+C_2 {\langle \varphi \rangle \over t_N\eta} \right]\), where the dot means the differentiation according to the time, \(\langle\;\rangle\) means positive part of the operand, \(\varepsilon_T\) is a tangential stiffness, \(v= \dot x\), \(\eta\) is a frictional viscosity, \(D_c\) is a ``critical'' sliding distance, \(\varphi \doteq | t_T| -t_N (\mu+K_1 (\alpha))\), \(\mu\) is the static friction coefficient, and \(K_1\) is a user-prescribed function \(K_1(\alpha) = \mu (e^{\alpha/r}-1)\), where \(r\) is a constant. For model 2), an another force is supposed in the normal direction due to asperity collisions, together with a coupling between \(t_N\) and \(y\). By model 1), a linearization about the steady state gives a critical spring stiffness below which oscillations can occur. The authors perform a nonlinear stability analysis by numerical calculations. Further, for both models they give conditions for stable sliding and describe periodic and chaotic motions. Comparisons with experimental results are made, and numerous diagrams depict results for particular cases.
scientific article; zbMATH DE number 1081041