Analytic conditions and frequency for targeted energy transfer are investigated in a 2-dof system comprising of a linear oscillator coupled to a nonlinear energy sink. The vibration equations are formulated as a dimensionless form, then the dynamic flow and Hamiltonian function for the underlined conservative system are derived by complex-averaging method. Based on the Hamiltonian and discrete breather theory in phase dynamics, the mass ratio and initial value condition linked to a completely energy transfer in the conservative system are determined, and the frequency of the targeted transfer is calculated by Jacobian elliptic integral. As a result of numerical simulation, the irreversibility of targeted energy transfer in damped system is demonstrated.