Spring pendulum with dry and viscous damping

Free and forced oscillations of a torsion spring pendulum damped by viscous and dry friction are investigated analytically and with the help of numerical simulations. A simplified mathematical model is assumed (Coulomb law) which nevertheless can explain many peculiarities in behavior of various oscillatory systems with dry friction. The amplitude of free oscillations diminishes under dry friction linearly, and the motion stops after a final number of cycles. The amplitude of sinusoidally driven pendulum with dry friction grows at resonance without limit if the threshold is exceeded. At strong enough non-resonant sinusoidal forcing dry friction causes transients that typically lead to definite limit cycles — periodic steady-state regimes of symmetric non-sticking forced oscillations which are independent of initial conditions. However, at the subharmonic sinusoidal forcing interesting peculiarities of the steady-state response are revealed such as multiple coexisting regimes of asymmetric oscillations that depend on initial conditions. Under certain conditions simple dry friction pendulum shows complicated stick-slip motions and chaos.