# Weight Effects on Vertical Transverse Vibration of a Beam with a Nonlinear Energy Sink
## 2 The Dynamic Modal Accounting for Weight  
Consider an elastic beam coupled with a nonlinear energy sink, as shown in Figure 1. The beam is assumed to be a non-extensible Euler-Bernoulli beam without shear and torsional deformation effects. Both ends of the beam are fixed. The beam has a Young's modulus \(E\), moment of inertia \(I\), density \(\rho\), cross-section area \(A\), and length \(L\). The beam is subjected to concentrated force \(F\cos\left(\Omega T\right)\), and its transverse motion is subjected to a viscous damping force with coefficient \(C\). The nonlinear energy sink consists of mass \(m\), a cubic nonlinear stiffness with coefficient \(K\), and a viscous damping with coefficient \(\Lambda\). The ratio of the nonlinear energy sink mass to the beam mass per length is \(\mu\), and \(m=\mu\rho A\). The beam bending motion can be described by transverse displacement \(W(X,\,T)\), where \(X\) and \(T\) are the axial and the time coordinates, respectively. The governing equation is as follows:  
\[ \begin{array}{l}EI\frac{\partial^{2}W(X,T)}{\partial X^{4}}+\rho A\frac{ \partial^{2}W(X,T)}{\partial T^{2}}+C\frac{\partial W(X,T)}{\partial T}\\ +\Big{\{}K[W(X_{c},T)-V(T)]^{3}+\Lambda[\frac{2W(X_{c},T)}{\partial T}-\dot{ V}(T)]\Big{\}}\delta(X-X_{c})=-F\cos(\Omega T)\delta(X-S)-\rho AgL\\ \mu(\rho A)\ddot{V}(T)+\Big{\{}K[V(T)-W(X_{c},T)]^{3}+\Lambda[\dot{V}(T)-\dot{ W}(X_{c},T)]\Big{\}}=-\mu\rho AgL\end{array}  \tag{1}\]  
where \(V\) is the displacement of nonlinear energy sink mass, and \(X_{c}\) and \(S\) are the locations of the nonlinear energy sink and the concentrated force. The boundary conditions at both ends are as follows:  
\[W(0,T)=0,\frac{\partial W(0,T)}{\partial X}=0;W(L,T)=0,\frac{\partial W(L,T)}{ \partial X}=0 \tag{2}\]  
The equilibrium configurations produced by the weights satisfy the static equations derived from Equation (1), as follows:  
\[EI\frac{\mathrm{d}^{4}W_{0}(X)}{\mathrm{d}X^{4}}+K[W_{0}(X_{c})-V_{0}(T)]^{3} \delta(X-X_{c})=-\rho AgLK[V_{0}(T)-W_{0}(X_{c})]^{3}=-\mu\rho AgL \tag{3}\]  
Under boundary condition (2), the static equilibrium configurations of the beam and the nonlinear energy sink can be solved from Equation (3), as follows:  
\[ \begin{array}{l}W_{0}(X)=\left\{\begin{array}{l}\frac{\mu\rho Ag(-X_{c})^ {2}X^{2}}{6 EI}\left[3\frac{X_{c}}{L}-\left(1+2\frac{X_{c}}{L}\right)\frac{X}{ L}\right]+\frac{\rho AgX^{2}}{2 EI}\left(X^{2}-2 LX+L^{2}\right)\,\left(0\leq X \leq X_{c}\right)\\ \frac{\mu\rho AgX^{2}(1-X)^{2}}{4 EI}\left[\frac{X_{c}}{L}-\left(1+2\frac{L-X_ {c}}{L}\right)\frac{X}{L}\right]+\frac{\rho AgX^{2}}{2 EI}\left(X^{2}-2 LX+L^{2} \right)\,\left(X_{c}\leq X\leq L\right)\\ V_{0}=\sqrt[3]{\frac{-\mu\rho SI}{EK}}+\frac{\rho AgX_{c}^{2}}{3 EI}\left[\frac {\mu(L-X_{c})^{2}X_{c}}{L}\left(2-\frac{X_{c}}{L}\right)-\frac{1}{8}\left({X_ {c}}^{2}-2 LX_{c}+L^{2}\right)\right]\end{array}\right.\end{array}  \tag{4}\]  
Taking the static equilibrium configurations as the coordinate origin, consequently, \(W_{1}(X,T)=W(X,\,T)-W_{0}(X)\) and \(V_{1}(T)=V(T)-V_{0}\), namely, as follows:  
Figure 1: A mechanical model of an elastic beam coupled with a nonlinear energy sink. \[ \begin{array}{l}W_{1}(X,T)=\left\{\begin{array}{l}W(X,T)-\frac{\mu\rho Ag(L-X_{c} )^{2}X^{2}}{6 EIL}\Big{[}3\frac{X_{c}}{L}-\left(1+2\frac{X_{c}}{L}\right)\frac{ X}{L}\Big{]}-\frac{\rho AgX^{2}}{24 ET}\left(X^{2}-2 LX+L^{2}\right)\,(0\leq X\leq X _{c})\\ W(X,T)-\frac{\mu\rho AgX^{2}(L-X)^{2}}{4 EIL}\Big{[}\frac{X_{c}}{L}-\left(1+2 \frac{L-X_{c}}{L}\right)\frac{X}{L}\Big{]}-\frac{\rho AgX^{2}}{24 ET}\left(X^{2} -2 LX+L^{2}\right)\,(X_{c}\leq X\leq L)\\ V_{1}(T)=V(T)-\sqrt[3]{\frac{-\mu\rho GL}{EK}}-\frac{\rho AgX^{2}}{3 EI}\bigg{[} \frac{u(L-X_{c})^{2}X_{c}}{L}\Big{(}2-\frac{X_{c}}{L}\Big{)}-\frac{1}{8}\Big{(} X_{c}^{2}-2 LX_{c}+L^{2}\Big{)}\bigg{]}\end{array}\right.\end{array}  \tag{5}\]  
The substitution of Equation (5) into Equation (1) leads to the following governing equation, measured from the equilibrium configurations:  
\[