with transient phenomena, in steady-state forced vibration, failures generally occur because of high cycle fatigue effects. Therefore, control of forced vibrations is of primary importance in many fields of mechanical engineering. It will be demonstrated that the energy pumping phenomenon can achieve this control function.

Derivation of periodic solutions
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We consider an harmonic excitation, $f(t)=P\cos\omega t$ in the system (\[eq:mvtR\]). The derivation of periodic solution uses the averaging procedure of section \[sect:AsymptoticAnalysis\], in which the two displacements variables take the form of (\[eq:solapprox\]), [*ie*]{}: $$x(\tau,\eta)=a_x(\eta) \cos(\tau +\varphi_x(\eta)) \mbox{ and } v(\tau,\eta)=a_v(\eta) \cos(\tau +\varphi_v(\eta))
    \label{eq:SS-solapprox}$$ where $\tau=\omega t$.

The term $G(a,\varphi,\tau)$ in equations (\[eq:SlowStand\]) and (\[eq:SlowMoy\]) is respectively for $x$ and $v$:

$$G_x(a_x,\varphi_x,\tau)=-\lambda_0\omega a_x\sin(\tau +\varphi_x) +  \omega_0^2a_x\cos(\tau +\varphi_x) - \epsilon_0 a_v\cos(\tau +\varphi_v) - P\cos \tau$$

$$G_v(a_v,\varphi_v,\tau)=-\lambda_1\omega a_v\sin(\tau +\varphi_v) +  \omega_1^2a_v\cos(\tau +\varphi_v) - \epsilon_1 a_x\sin(\tau +\varphi_x) + a_r(a_v,\varphi_v) \cos(\tau+\varphi_r(a_v,\varphi_v))$$

Then applying the relations (\[eq:SlowMoy\]) to the present example, one obtains:

\[eq:SlowMoyApp\] $$a_x^{\prime} = -\dfrac{\epsilon_0}{2\omega}a_v\sin(\varphi_x-\varphi_v) -\frac{\lambda_0}{2}a_x -\frac{P}{2\omega}\sin\varphi_x$$ $$a_v^{\prime} = \dfrac{\epsilon_1}{2\omega}a_x\sin(\varphi_x-\varphi_v) -\frac{\lambda_1}{2}a_v +\dfrac{1}{2\omega} a_r\sin(\varphi_v-\varphi_r)$$ $$a_x \varphi_x^{\prime} = \frac{\omega_0^2-\omega^2}{2\omega} -\dfrac{\epsilon_0}{2\omega}a_v\cos(\varphi_x-\varphi_v)-\frac{P}{2\omega}\cos\varphi_x$$ $$a_v \varphi_v^{\prime} = \frac{\omega_1^2-\omega^2}{2\omega} -\dfrac{\epsilon_1}{2\omega}a_x\cos(\varphi_x-\varphi_v) +\dfrac{1}{2\omega} a_r\cos(\varphi_v-\varphi_r)$$

As for the free response, the fixed points of system (\[eq:SlowMoyApp\]) are to the main approximation to the forced response. Next, these forced response will be investigated 