This study presents a new analytical model for nonlinear dynamics of a discrete rectangular membrane that is subjected to external harmonic force. It has recently been shown that the corresponding autonomous system admits a series of nonlinear normal modes. In this paper, we describe stationary and non-stationary dynamics on a single mode manifold. We suggest a simple formula for the amplitude-frequency response in both conservative and non-conservative cases and present an analytical expression (in parametric space) for thresholds for all possible bifurcations. Theoretical results obtained through asymptotic approach are confirmed by the experimental data. Experiments on the shaking table show that amplitude-frequency response to external force in a real system matches our theory. Substantial hysteresis is observed in the regimes with increasing and decreasing frequency of external force. The obtained results may be used in designing nonlinear energy sinks.