In this paper, the dynamic response of a harmonically forced Linear Oscillator (LO) strongly coupled to a Nonlinear Energy Sink (NES) is investigated both theoretically and experimentally. The system studied comprises a LO with an embedded, purely cubic NES. The behavior of the system is analyzed in the vicinity of 1 : 1 resonance. The complexification-averaging technique is used to obtain modulation equations and the associated fixed points. These modulation equations are analyzed using asymptotic expansion to study the regimes related to relaxation oscillation of the slow flow called Strongly Modulated Response (SMR). The zones where SMR occurs are computed using a mapping procedure. The Slow Invariant Manifolds (SIM) is used to derive a proper optimization procedure. It is shown that there is an optimal zone in the forcing amplitude-nonlinear stiffness parameter plane, where SMR occurs without having a high amplitude detached resonance tongue. Two experimental setups are presented. One is not optimized and has a relatively high mass ratio (≈ 13%) and the other one is optimized and exhibits strong mass asymmetry (mass ratio ≈ 1%). Different frequency response curves and associated zones of SMR are obtained for various forcing amplitudes. The reported experimental results confirm the design procedure, and the possible application of NES for vibration mitigation under periodic forcing.