The systems considered in this work are composed of weakly coupled, linear and essentially nonlinear (nonlinearizable) components. In Part I of this work we present numerical evidence of energy pumping in coupled nonlinear mechanical oscillators, i.e., of one-way (irreversible) “channeling” of externally imparted energy from the linear to the nonlinear part of the system, provided that the energy is above a critical level. Clearly, no such phenomenon is possible in the linear system. To obtain a better understanding of the energy pumping phenomenon we first analyze the dynamics of the underlying Hamiltonian system (corresponding to zero damping). First we reduce the equations of motion on an isoenergetic manifold of the dynamical flow, and then compute subharmonic orbits by employing nonsmooth transformation of coordinates which lead to nonlinear boundary value problems. It is conjectured that a 1:1 stable subharmonic orbit of the underlying Hamiltonian system is mainly responsible for the energy pumping phenomenon. This orbit cannot be excited at sufficiently low energies. In Part II of this work the energy pumping phenomenon is further analyzed, and it is shown that it is caused by transient resonance capture on a 1:1 resonance manifold of the system.