We deal with dynamics of the \(\beta \) -Fermi–Pasta–Ulam–Tsingou chain with one free end, subjected to the sudden sinusoidal force. We examine the evolution of the total energy supplied into the chain at large times. In the harmonic case ( \(\beta =0\) ), the energy grows in time linearly at the driving frequencies, corresponding to nonzero group velocities. The rate of energy supply is shown to decrease with increasing excitation frequency. Loading with the cut-off frequency, corresponding to zero group velocity, results in the energy growth proportionally to \(\sqrt{t}\) . Explanation of behavior in time of the energy is proposed by analysis of the obtained approximate closed-form expressions for the particle velocity. In the weak anharmonic case, large-time asymptotic approximation for the total energy is obtained by using the renormalized dispersion relation. The approximation allows one to estimate rate of the energy supply at the driving frequencies, which belong both in the pass-band and in the stop-band of the harmonic chain. Consistency of the asymptotic estimates with the results of numerical simulations is discussed.
