The considered system is composed of a linear oscillator, termed as the primary system, coupled to an essentially nonlinear oscillator, termed as the NES: \[ \begin{aligned} M\ddot y&+\varepsilon\lambda_1\dot y+\varepsilon\lambda(\dot y -\dot v)+\varepsilon(u-v)+ ky=0,\\ m\ddot v&+\varepsilon\lambda_2 \dot v+\varepsilon\lambda(\dot v-\dot y)+ \varepsilon(v-y)+Cv^3=0. \end{aligned} \] Variables \(y\) and \(v\) refer to the displacement of the primary system and of the NES, respectively. Weak coupling and damping is assured by requiring that \(\varepsilon\ll 1\), all other variables are treated as \(O(1)\) quantities; provided the input energy is high enough, a strongly nonlinear system can therefore be investigated. The periodic orbits of the undamped system \(\lambda=\lambda_1=\lambda_2=0\) are firstly studied. Then the authors examine the basic machanism for nonlinear energy pumping. In summary, the underlying dynamical phenomenon is a \(1\:1\) resonance capture. Numerical simulations are performed in a particular case, and many experimental results are given.