Localized oscillation of an Euler--Bernoulli beam with time-varying parameters on a visco-elastic foundation: asymptotics, adiabatic invariant, and equivalent Hamiltonian system

We consider localized oscillation of an Euler–Bernoulli beam on a visco-elastic foundation coupled to a damped discrete oscillator. All parameters of the system independently vary in time in a slow manner. For the conservative case, we use three various analytic approaches. Namely, these are asymptotics, the method based on the adiabatic invariance of the action of a trapped wave, and the consideration of the equivalent Hamiltonian system. All approaches result in the same formula for the amplitude of oscillation. In the dissipative case, we obtain the amplitude of oscillation only utilizing the asymptotic approach.