## 2 Description of the Vibrating System  
A vibrational system with 3 DOF is described in Figure 1, in which a damped elastic pendulum with normal length \(l_{0}\), non-linear stiffness \(K_{1}\) and \(K_{2}\), and pendulum mass \(m_{1}\) is considered. A pivot point \(O_{1}\) of the pendulum is limited to move in an elliptic route with stationary angular velocity \(\Omega\), while the pendulum's other end is attached to a linear absorber of mass \(m_{2}\), a normal length \(l_{10}\), and linear stiffness \(K_{3}\). According to the sketch of Figure 1, we may write the corresponding coordinates of \(O_{1}\) to the axes \(OX\) and \(OY\) in the forms \(a\cos(\Omega t)\) and \(b\sin(\Omega t)\), respectively. Here, the ellipse's minor and major axes are represented by \(a\) and \(b\), respectively. On the supplementary circle \(b\), the equivalent point of \(O_{1}\) will be assigned by \(Q\). The system's motion is addressed to be under the influence of applied harmonic force \(F(t)=F_{1}\cos(\Omega_{1}t)\) at the spring's radial direction, as well as a harmonic moment \(M(t)=M_{0}\cos(\Omega_{2}t)\) at \(O_{1}\) in the anticlockwise direction. Here, \(\Omega_{1},\Omega_{2}\) and \(F_{1},M_{0}\) are the frequencies and amplitudes of \(F(t)\) and \(M(t)\). The extensions of the spring and absorber are supposed to be \(Z(t)\) and \(\xi(t)\), respectively. Furthermore, \(C_{1},C_{2}\), and \(C_{3}\) are thought to indicate the coefficients of viscous damping for the spring's longitudinal, swing oscillations, and the absorber's elongation, respectively. The kinetic and potential energies \(T\) and \(V\) of the system can be expressed in the following ways  
Figure 1: The examined model. \[ \begin{array}{ll}T&=\frac{1}{2}(m_{1}+m_{2})\Big{\{}\Omega^{2}[a^{2}\sin^{2}( \Omega t)+b^{2}\cos^{2}(\Omega t)]+Z^{\prime 2}+(l+Z)^{2}{\Phi^{\prime}}^{2}\\ &+2\Omega Z^{\prime}[b\cos(\Omega t)\sin\Phi-a\sin(\Omega t)\cos\Phi]+2\Omega( l+Z){\Phi^{\prime}}\\ &\times[b\cos(\Omega t)\cos\Phi+a\sin(\Omega t)\sin\Phi]\Big{\}}+\frac{1}{2}m_{ 2}\Big{\{}\tilde{\varsigma}^{\prime 2}+(l_{1}+\tilde{\varsigma})^{2}{\Phi^{ \prime}}^{2}\\ &-2\Omega(l_{1}+\tilde{\varsigma}){\Phi^{\prime}}[b\cos(\Omega t)\sin\Phi+a \sin(\Omega t)\cos\Phi]+2\Omega\tilde{\varsigma}^{\prime}\\ &\times[b\cos(\Omega t)\cos\Phi-a\sin(\Omega t)\sin\Phi]+2Z^{\prime}\tilde{ \varsigma}^{\prime}\sin(2\[PERSON])+2(l_{1}+\tilde{\varsigma})\\ &\times Z^{\prime}{\Phi^{\prime}}\cos(2\[PERSON])+2(l+Z){\Phi^{\prime}}\tilde{ \varsigma}^{\prime}\cos(2\Phi)-2(l+Z)(l_{1}+\tilde{\varsigma}){\Phi^{\prime}} ^{2}\sin(2\Phi)\Big{\}},\\ V&=\frac{1}{2}K_{1}(Z+Z_{r})^{2}+\frac{1}{3}K_{2}(Z+Z_{r})^{3}+\frac{1}{2}K_{3 }(\tilde{\varsigma}+\tilde{\varsigma}_{r})^{2}+m_{1}g(l+Z)(1-\cos\Phi)\\ &+m_{2}g[(l+Z)(1-\cos\Phi)+(l_{1}+\tilde{\varsigma})\sin\Phi];\\ l&=l_{0}+Z_{r},\quad l_{1}=l_{0}+\tilde{\varsigma}_{r}.\end{array}  \tag{1}\]  
Here, \(Z_{r}\) and \(\tilde{\varsigma}_{r}\) are the spring's and the absorber's static elongations, respectively; the angle formed by the vertical at \(O_{1}\) and the line passing across \(O_{1}m_{1}\) are denoted by \(\Phi;g\) is the acceleration of gravity, and the primes are the differentiation in relation to time \(t\). According to Equation (1), [PERSON]'s function \(L=T-V\) can be determined, and then, the controlling system of motion can be obtained using the next equations of Lagrange [30]  
\[ \begin{array}{l}\frac{d}{dt}(\frac{\partial L}{\partial T})-\frac{\partial L }{\partial Z}=Q_{Z},\\ \frac{d}{dt}(\frac{\partial L}{\partial\Phi})-\frac{\partial L}{\partial\Phi} =Q_{\Phi},\\ \frac{d}{dt}(\frac{\partial L}{\partial\Phi})-\frac{\partial L}{\partial\Phi} =Q_{\xi}.\end{array}  \tag{2}\]  
Here, \((Q_{Z},Q_{\Phi},Q_{\xi})\) are the system's non-conservative generalized forces, while \((Z,\Phi,\tilde{\varsigma})\) and \((Z^{\prime},{\Phi^{\prime}},\tilde{\varsigma}^{\prime})\) are the generalized coordinates and velocities, respectively. The forms of \(Q_{Z},Q_{\Phi}\), and \(Q_{\xi}\) are  
\[