Numerical frictional algorithm with implementation of closed form analytical solutions

Numerical difficulties, including underflow, drift, jump and numerical noise are encountered near transitions between slip and stick when conventional numerical integration methods are employed to yield the response for a frictional system (Barber and Wang, 2014). We present a new algorithm for two-dimensional frictional problems by making use of a series of closed form analytical solutions (Wang, 2014), which have been derived for a mass in sliding contact loaded by a constant external force. This algorithm can be used to yield the response of the system during dynamic slip periods including an accurate determination of the time and position of slip/stick transitions. We also develop an analytical patch for the recommencement of slip after stick, since zero initial velocity at the boundary of friction circle results in difficulties for both analytical (Wang, 2014) and numerical solutions (jump phenomenon) (Barber and Wang, 2014). A two-state boolean variable B is introduced to toggle between states of stick (B=0) and slip (B=1). The proposed new algorithm is explained using a flow chart. Accuracy and efficiency of the new algorithm are assessed by comparison with traditional algorithms in several examples.