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.

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