# Vibration Reduction of a Timoshenko Beam with Multiple Parallel Nonlinear Energy Sinks
## 2 Methods  
### Modeling  
A short beam coupled with multiple parallel NESs is shown in Figure 1. The length, thickness, and width of the beam are denoted by \(L\), \(H\), and \(B\), respectively. Generally, a short beam is defined as the ratio \(L/H\) within the range of 2.5-5 [36], and the shear deformation cannot be ignored for short beams. There are \(N\) nonlinear energy sinks on the beam. The mass, cubic stiffness, and damping of each NES are denoted by \(m\), \(k_{3}\), and \(c\), respectively. \(d_{i}\) is the distance of the \(i\)th nonlinear energy sink away from the left end of the beam. \(a\) is the place where a harmonic excitation \(F_{\mathrm{e}}=F\cos(2\pi\mathrm{f}t)\) is imposed, where \(F\) and \(f\) are the amplitude and frequency of the external harmonic excitation, respectively. In this section, the equations of motion are derived from the Timoshenko theory. It takes the effects of shear deformations and rotary inertias into account, and it is more applicable to short beams, deep beams, and laminated beams. According to the d'Alembert principle, the force and moment equilibrium equations can be written as:  
\[\left. \begin{array}{l}\rho A(x)\frac{\partial^{2}w(x,t)}{\partial x^{2}}=- \frac{\partial Q(x,t)}{\partial x}+q(x,t)+F(x^{*},t)\\ \rho I(x)\frac{\partial^{2}\rho(x,t)}{\partial t^{2}}=\frac{\partial M(x,t)}{ \partial x}-Q(x,t)\end{array} \right\} \tag{1}\]  
where \(\rho\) is the density of the beam material, \(A(x)\) is the area of the beam section, \(I(x)\) is the rotary inertial of the beam section, \(q\) (\(x\), \(t\)) is a continuous distributed force on the beam, \(F\) (\(x^{*}\), \(t\)) is a concentrated force, imposed at \(x=x^{*}\), \(M\) (\(x\), \(t\)) is the bending moment, and \(Q\) (\(x\), \(t\)) is the shearing force. The fundamental functions of the beam in material mechanics can be written as:  
\[\left. \begin{array}{l}M(x,t)=EI(x)\frac{\partial\theta(x,t)}{\partial x}\\ \theta(x,t)-\frac{\partial w(x,t)}{\partial x}=\frac{Q(x,t)}{\gamma A(x)G} \end{array} \right\} \tag{2}\]  
where \(\gamma\) is a constant, determined by the section shape of the beam, and \(G\) is the shearing modulus, determined by the beam material. Figure 1: A short beam coupled with multiple parallel nonlinear energy sinks. Analyzing the relationship between shear force and deformation, the total slope of the beam deflection curve can be written as:  
\[\partial w(x,t)/\partial x=\theta(x,t)-\psi(x,t) \tag{3}\]  
where \(\Psi\) (\(x,t\)) is the shearing angle, also called slope loss. Substituting Equations (2) and (3) into Equation (1), the vibration equation of a beam under external forces can be written as:  
\[ \begin{array}{l}EI\frac{\partial^{4}w(x,t)}{\partial x^{4}}+\rho A\frac{ \partial^{2}w(x,t)}{\partial t^{2}}+C\frac{\partial w(x,t)}{\partial t}-\rho I \Big{(}1+\frac{F}{\gamma G}\Big{)}\frac{\partial^{4}w(x,t)}{\partial x^{2} \partial t^{2}}\\ +\frac{\rho^{2}I\partial^{4}w(x,t)}{\gamma G\partial t^{4}}=q(x,t)+\frac{\rho I }{\gamma AG}\frac{\partial^{2}q(x,t)}{\partial t^{2}}-\frac{EI}{\gamma AG}\frac {\partial^{2}q(x,t)}{\partial x^{2}}\\ +F(x^{*},t)+\frac{\rho I}{\gamma AG}\frac{\partial^{2}F(x^{*},t)}{\partial t^{ 2}}\end{array}  \tag{4}\]  
where \(C\) is the structure damping coefficient of the beam. When the beam vibrates freely, the free vibration function can be written as:  
\[ \begin{array}{l}EI\frac{\partial^{4}w(x,t)}{\partial x^{4}}+\rho A\frac{ \partial^{2}w(x,t)}{\partial t^{2}}+C\frac{\partial w(x,t)}{\partial t}\\ -\rho I\Big{(}1+\frac{F}{\gamma G}\Big{)}\frac{\partial^{4}w(x,t)}{\partial x^ {2}\partial t^{2}}+e^{2}l\frac{\partial^{4}w(x,t)}{\gamma G\partial t^{4}}=0 \end{array}  \tag{5}\]  
where the 4 th and 5 th term in Equation (5) consider the shearing deformation and the effect of the moment of inertia, respectively. After attaching \(N\) NESs to the beam, the coupled vibration equations of the system can be written as:  
\[