# Weight Effects on Vertical Transverse Vibration of a Beam with a Nonlinear Energy Sink
## 2 The Dynamic Modal Accounting for Weight
\begin{array}{l}EI\frac{\partial^{2}W_{1}}{\partial X^{4}}+\rho A\frac{ \partial^{2}W_{1}}{\partial X^{2}}+c\frac{\partial W_{1}}{\partial T}+\Bigg{[} K\bigg{(}W_{1}-V_{1}-\sqrt[3]{\frac{-\mu\rho AgL}{K}}\bigg{)}^{3}+\Lambda \Big{(}\frac{\partial W_{1}}{\partial T}-\dot{V}_{1}\Big{)}\Bigg{]}\delta(X-X _{c})=-F\cos(\Omega T)\delta(X-S)\\ \mu(\rho A)\ddot{V}_{1}+\Bigg{[}K\bigg{(}V_{1}+\sqrt[3]{\frac{-\mu\rho AgL}{K }}-W_{1}\bigg{)}^{3}+\Lambda\Big{(}\dot{V}_{1}-\frac{\partial W_{1}}{\partial T }\bigg{)}\Bigg{]}=-\mu\rho AgL\end{array}  \tag{6}\]  
Equation (6) can be transformed into the dimensionless form, as follows:  
\[ \begin{array}{l}\frac{\partial^{2}w_{1}}{\partial t^{2}}+c\frac{\partial w_ {1}}{\partial t}+\kappa^{2}\frac{\partial^{2}w_{1}}{\partial x^{4}}+\Bigg{\{} k\bigg{[}w_{1}(x,t)-v_{1}(t)-\sqrt[3]{\frac{-\mu\rho\rho L}{EK}}\bigg{]}^{3}+ \lambda\Big{[}\frac{\partial w_{1}(x,t)}{\partial t}-\dot{v}_{1}(t)\Big{]} \Bigg{\}}\delta(x-x_{c})=-f\cos(\omega t)\delta(x-s)\\ \mu\ddot{v}_{1}+k\bigg{[}v_{1}(t)-w_{1}(x_{c},t)+\sqrt[3]{\frac{-\mu\rho\rho L}{ E}}\bigg{]}^{3}+\lambda\Big{[}\dot{v}_{1}(t)-\frac{\partial w_{1}}{\partial t }(x_{c},t)\Big{]}=-\frac{\mu\rho\rho L}{E}\end{array}  \tag{7}\]  
where  
\[ \begin{array}{l}w_{1}=\frac{W_{1}}{L},v_{1}=\frac{V_{1}}{L},x=\frac{X}{L},x_ {c}=\frac{X_{c}}{L},s=\frac{S}{L},t=\frac{T}{L}\sqrt{\frac{E}{\rho}},c=\frac{C} {\Lambda}\sqrt{\frac{1}{\rho E}},\\ \kappa=\frac{1}{L}\sqrt{\frac{T}{\Lambda}},k=K\frac{L^{3}}{L\Lambda},\lambda= \frac{\Lambda}{\Lambda}\sqrt{\frac{1}{\rho E}},f=\frac{F}{E\Lambda},\omega= \Omega L\sqrt{\frac{\rho}{E}}\end{array}  \tag{8}\]  
It should be remarked that the beam weight does not appear in the governing equation, the first equation in Equation (7). It is balanced by the static deformation effect. The corresponding boundary conditions are as follows:  
\[w(0,t)=0,\frac{\partial w(0,t)}{\partial x}=0;w(1,t)=0,\frac{\partial w(1,t)}{ \partial x}=0 \tag{9}\]  
In all available works, the weights are assumed to be balanced by the static deflection. Thus, the corresponding governing equations measured from the equilibrium configurations are as follows:  
\[ \begin{array}{l}EI\frac{\partial^{2}W_{2}(X,T)}{\partial X^{4}}+\rho A\frac{ \partial^{2}W_{2}(X,T)}{\partial T^{2}}+C\frac{\partial W_{2}(X,T)}{\partial T }\\ +\Big{\{}K[W_{2}(X_{c},T)-V_{2}(T)]^{3}+\Lambda\Big{[}\frac{\partial W_{2}(X_{c},T)}{\partial T}-\dot{V}_{2}(T)]\Big{\}}\delta(X-X_{c})=-F\cos(\Omega T)\delta(X -S)\\ \mu(\rho A)\ddot{V}_{2}(T)+\Big{\{}K[V_{2}(T)-W_{2}(X_{c},T)]^{3}+\Lambda[\dot{V }_{2}(T)-\dot{W}_{2}(X_{c},T)]\Big{\}}=0\end{array}  \tag{10}\]  
Equation (10) can be transformed into the dimensionless form, as follows:  
\[ \begin{array}{l}\frac{\partial^{2}w_{2}}{\partial t^{2}}+c\frac{\partial w_ {2}}{\partial t}+\kappa^{2}\frac{\partial^{2}w_{2}}{\partial x^{4}}+\Big{\{} k[w_{2}(x,t)-v_{2}(t)]^{3}+\lambda\Big{[}\frac{\partial w_{2}(x,t)}{ \partial t}-\dot{v}_{2}(t)\Big{]}\Big{\}}\delta(x-x_{c})=-f\cos(\omega t)\delta(x -s)\\ \mu\ddot{v}_{2}+k[v_{2}(t)-w_{2}(x_{c},t)]^{3}+\lambda\Big{[}\dot{v}_{2}(t)- \frac{\partial w_{2}(x_{c},t)}{\partial t}\Big{]}=0\end{array}  \tag{11}\]  
where  
\[\begin{array}{l}w_{2}=\frac{W_{2}}{L},v_{2}=\frac{V_{2}}{L},x=\frac{X}{L},x_ {c}=\frac{X_{c}}{L},s=\frac{S}{L},t=\frac{T}{L}\sqrt{\frac{E}{\rho}},c=\frac{C} {\Lambda}\sqrt{\frac{1}{\rho E}},\\ \kappa=\frac{1}{L}\sqrt{\frac{T}{\Lambda}},k=K\frac{L^{3}}{E\Lambda},\lambda=\frac {\Lambda}{\Lambda}