In this paper, we study a Targeted Energy Transfer (TET) problem between a p degreesof-freedom (dof) linear master structure and several coupled parallel slave Nonlinear Energy Sink (NES) systems. In detail, each lth dof l = 1, 2, . . . , p contains n l parallel NES; so the linear structure has (n 1 + n 2 + · · · + n l + · · · + n p ) NES. We are interested to study analytically the TET phenomenon during the first mode of the compound system. To this end, complexification, averaging, and multiple scales methods are used.The system is studied under 1:1 resonance for the transient regime and under harmonic excitation. The influence of the system parameters is observed through dimensionless variables. An analytical criterion is defined to tune NES parameters which lead to an efficient TET for the transient and the forced regimes. It will be demonstrated that analytical results are in good agreement with numerical ones. This paper will be followed by a companion paper which mainly deals with the governing equations for compound nonlinear systems with trees of NES devices at each dof; then experimental results of a four storey structure with two parallel NES at the top floor which are tuned by the mentioned technique in the current paper will be demonstrated and commented upon.

International audienceIn this paper governing equations of general multi degrees of freedom (dof) systems with trees of parallel Nonlinear Energy Sink (NES) devices at each dof are derived. Then these equations are summarized for a 4 dof structure with two parallel NES at the 4th dof in order to control the first mode of the system. A prototype four storey structure with two parallel NES at the top floor is studied experimentally. The NES of the mentioned system is designed by endowing the suggested method in the Part I of this paper. A couple of experimental tests are carried out on the structure at the DGCB laboratory of the ENTPE, France. The aim is to control the first mode of the compound nonlinear system by demonstrating the efficiency of the parallel NES systems on the intended task