In the biomass-fired boilers intensive deposit layer on the heating surface is formed. For deposit elimination various cleaning procedures are applied. The mechanical effect of cleaning is based on vibration of the system. The efficiency of the procedure depends on the parameters of the tube-deposit vibrating system. In the paper the system is assumed to be a truly nonlinear oscillator with linear damping which is excited with a periodical cleaning force. The exact nonlinear resonance for the system is determined and exact steady states of the oscillator are computed. Special attention is given to the influence of damping properties of the deposit on the system vibration and on the energy dissipation due to mismatch between the excitation and damping. For determination of the energy change an approximate analytic solving procedure is developed. The method represents the adopted version of the time variable amplitude and phase procedure. As an example, the tube-deposit system described with damped Duffing oscillator excited with Jacobi elliptic function is considered. Analytically obtained solution is compared with numerical one. The results are in good agreement and prove the correctness of the suggested model. Finally, the generalized mathematical model of the tube-slag oscillatory system gives prediction of nonlinear resonant vibration caused by periodical cleaning force.