Analytical vibration modelling and solution of bars with frictional clamps

The present study introduces a novel analytical solution to predict the nonlinear dynamic behaviour of bars under frictional clamping in axial and torsional motions. It investigates the vibration characteristics of straight bars with imperfect supports, which introduce dry friction at their contact interfaces. The bars are tightly clamped between rigid fixtures, with the tightening load acting as a normal load that induces friction, thus adding nonlinearity to the system. The model simplifies contact forces to point loads and utilises both the Jenkins and velocity-dependent friction models for simulating contact friction. These frictional forces are represented as solution-dependent external forces in the governing differential equation for bar vibration, which also includes appropriate boundary conditions. The equation is solved both analytically and through the numerical method of alternating frequency–time harmonic balance, to explore the influence of contact parameters on the bar's support system behaviour. Comparisons between the numerical and analytical results demonstrate strong agreement, confirming the model's accuracy and validity.