# Vibration Reduction of a Timoshenko Beam with Multiple Parallel Nonlinear Energy Sinks
## 2 Methods
### Modeling
\begin{array}{l}EI\frac{\partial^{4}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}-\rho I \Big{(}1+\frac{F}{\gamma G}\Big{)}\frac{\partial^{4}w(x,t)}{\partial x^{2} \partial t^{2}}\\ +\frac{\rho I\partial^{4}w(x,t)}{\gamma G\partial t^{4}}=q(x,t)+\frac{\rho I }{\gamma AG}\frac{\partial^{2}q(x,t)}{\partial t^{2}}-\frac{EI}{\gamma AG}\frac {\partial^{2}q(x,t)}{\partial x^{2}}+F(x^{*},t)\\ +\frac{\rho I}{\gamma AG}\frac{\partial^{2}F(x^{*},t)}{\partial t^{2}}+\sum \limits_{i=1}^{N}\Big{(}k_{i}(w(d_{i},t)-z_{i}(t))^{3}+c_{i}\Big{(}\frac{ \partial w(d_{i},t)}{\partial t}-\frac{\partial z_{i}(t)}{\partial t}\Big{)} \Big{)}\\ \sum\limits_{i=1}^{N}\Big{(}m_{i}\frac{\mathrm{d}z_{i}(t)}{\mathrm{d}t^{2}}+k_{ i}(z_{i}(t)-w(d_{i},t))^{3}+c_{i}\Big{(}\frac{\mathrm{d}z_{i}(t)}{\mathrm{d}t}- \frac{\partial w(d_{i},t)}{\partial t}\Big{)}\Big{)}=0\end{array}  \tag{6}\]  
where \(m_{i}\), \(k_{i}\), and \(c_{i}\) are the parameters of the \(i\)th NES on the beam. In order to express equations of motion in dimensionless form, the following variables are defined:  
\[ \begin{array}{l}\overline{w}_{i}(x,t)=\frac{w_{i}(x,t)}{L},\overline{x}= \frac{x}{L},\overline{z}_{i}=\frac{z}{L},\overline{t}=\frac{t}{L^{2}}\sqrt{ \frac{EI}{\rho A}},\\ \overline{C}=\frac{\sqrt{\rho AEI}}{\sqrt{\rho AEI}},\overline{c_{i}}=\frac{c_{ i}}{\sqrt{\rho AEI}},p=\Big{(}1+\frac{F}{\gamma G}\Big{)}\frac{I}{\Lambda L^{2}},\\ q=\frac{EI^{2}}{\gamma\Lambda G\Lambda^{2}},\overline{K_{i}}=\frac{k_{i}L^{0}}{EI },r_{i}=\frac{l\overline{K}_{i}L^{2}}{\gamma A^{2}G\Gamma^{2}},\\ \mu=\frac{L^{3}}{EI},\eta=\frac{I}{\gamma A^{2}G\Gamma^{2}},\varepsilon_{i}= \frac{m_{i}}{\rho A},\overline{f}=\frac{f}{\omega_{0}}\end{array}  \tag{7}\]  
and thus, the equations of motion for the [PERSON] theory in dimensionless form are obtained as:  
\[ \begin{array}{l}\frac{\partial^{2}\overline{w}(\overline{x}_{i}\overline{I })}{\partial t^{2}}+\frac{\partial^{4}\overline{w}(\overline{x}_{i}\overline{I })}{\partial\overline{x}^{4}}+\overline{C}\frac{\partial\overline{w}(\overline {x}_{i}\overline{I})}{\partial\overline{t}}-\\ p\frac{\partial^{4}\overline{w}(\overline{x}_{i}\overline{I})}{\partial x^{2} \partial t^{2}}+\frac{q}{\gamma G}\frac{\partial^{4}\overline{w}(\overline{x}_ {i}\overline{I})}{\partial\overline{t}^{4}}+\sum\limits_{i=1}^{N}\left(\begin{array} []{c}\overline{K_{i}}\big{(}\overline{w}\big{(}\overline{x}_{i},\overline{I }\big{)}-\overline{z}_{i}\big{(}\overline{I}\big{)}\big{)}^{3}+\\ \left(\frac{\partial^{2}}{\partial t^{2}}r_{i}\big{(}\overline{w}\big{(} \overline{x}_{i},\overline{I}\big{)}-\overline{z}_{i}\big{(}\overline{I}\big{)} \big{)}^{3}\right)\\ +\varepsilon_{i}\Big{(}\frac{\partial\overline{w}(\overline{x}_{i}\overline{I })}{\partial\overline{t}}-\frac{\mathrm{d}\overline{z}(\overline{I})}{ \mathrm{d}t}\Big{)}\end{array}\right)\\ +\varepsilon\Big{(}\frac{\partial^{4}\overline{w}(\overline{x}_{i}\overline{I })}{\partial\overline{t}^{3}}-\frac{\mathrm{d}^{2}\overline{w}(\overline{t})}{ \mathrm{d}t^{3}}\Big{)}=\mu\overline{F}\big{(}\overline{x}^{*},\overline{t} \big{)}+\eta\frac{\partial^{2}}{\partial\overline{t}^{2}}\overline{F}\big{(} \overline{x}^{*},\overline{t}\big{)}\\ \sum\limits_{i=1}^{N}\varepsilon_{i}\frac{\mathrm{d}^{2}\overline{w}(\overline{x}_{i} \overline{I})}{\mathrm{d}t^{2}}+\sum\limits_{i=1}^{N}\left(\overline{K}_{i} \big{(}\overline{z}_{i}\big{(}\overline{I}\big{)}-\overline{w}\big{(}\overline{x}_ {i}\overline{I}\big{)}\big{)}+\overline{c}_{i}\Big{(}\frac{\mathrm{d}\overline{z}_{i} ^{*}\overline{I}}{\mathrm{d}t}-\frac{\partial\overline{w}(\overline{x}_{i}\overline{I })}{\partial\overline{t}}\Big{)}\right)=0\end{array}  \tag{8}\]By eliminating the effects of shear deformation and rotary inertial from Equation (8), the equations of motion according to Euler-Bernoulli theory are obtained as follows:  
\[