Strongly nonlinear vibrations of damped oscillators with two nonsmooth limits

A family of strongly nonlinear oscillators with a generalized power form elastic force and viscous damping is considered. An explicit analytical solution is obtained as a combination of smooth and nonsmooth functions. Two different nonsmooth functions involved into the solution are associated with two different nonsmooth limits of the oscillator as the exponent becomes either zero or infinity. As a result, the solution is drastically simplified to give the best match with numerical tests if approaching any of the two limits.