## 2 The Equations of Motion and Boundary Conditions  
The free-free beam with unequal end masses considered in this study is shown in Figure 1a. The masses on the left side and right side of the beam are \(M_{1}\) and \(M_{2}\), respectively. \(J_{i}\) represents the rotary inertia of the tip masses about the beam end. \(G_{i}\) represents the center of the masses and \(d_{i}\) represents the distances between the end of the beam and the center of the masses. The point B is the connection point between the end mass and the beam. The beam is homogeneous, prismatic, and has a length of \(L\). The equation of free transverse vibration of an undamped Euler-Bernoulli beam [36] with a uniform cross-section is described by the following differential equation for small deflection:  
\[EI\frac{\partial^{4}w(x,t)}{\partial x^{4}}+\rho A\frac{\partial^{2}w(x,t)}{ \partial t^{2}}=0,\ 0<x<L \tag{1}\]  
where \(w(x,t)\) is the transverse deflection of the beam, \(EI\) is the bending stiffness of the beam, and \(\rho A\) is the mass density per unit length of the beam. In Equation (1), \(A\) and \(I\) represent the cross-sectional area and the moment of inertia of the beam cross-section; respectively; \(x\) denotes the axial coordinate along the beam axis, where \(x=0\) is at the left end of the beam; and \(t\) represents time. Two unequal end masses, \(M_{1}\) and \(M_{2}\) are attached to the beam at both ends. We can therefore obtain the following boundary conditions of the beam by the variational method:  
at \(x=0\),  
\[EI\frac{\partial^{2}w(0,t)}{\partial x^{2}}=\left(J_{1}+M_{1}d_{1}^{2}\right) \frac{\partial^{3}w(0,t)}{\partial x\partial t^{2}}+M_{1}d_{1}\frac{\partial^ {2}w(0,t)}{\partial t^{2}}, \tag{2}\]  
Figure 1: (**a**) The free–free beam with unequal end masses; (**b**) an alternative attachment of the end mass that has its gravity center point at \(\xi<1\). \[EI\frac{\partial^{3}w(0,t)}{\partial x^{3}}=-M_{1}\frac{\partial^{2}w(0,t)}{ \partial t^{2}}-M_{1}d_{1}\frac{\partial^{3}w(0,t)}{\partial x\partial t^{2}} \tag{3}\]  
similarly at \(x=L\),  
\[EI\frac{\partial^{2}w(L,t)}{\partial x^{2}}=-\Big{(}J_{2}+M_{2}d_{2}^{2}\Big{)} \frac{\partial^{3}w(L,t)}{\partial x\partial t^{2}}-M_{2}d_{2}\frac{\partial^ {2}w(L,t)}{\partial t^{2}}, \tag{4}\]  
\[EI\frac{\partial^{3}w(L,t)}{\partial x^{3}}=M_{2}\frac{\partial^{2}w(L,t)}{ \partial t^{2}}+M_{2}d_{2}\frac{\partial^{3}w(L,t)}{\partial x\partial t^{2}} \tag{5}\]  
For a solution of Equation (1), we can take a harmonic function as \(w=\bar{w}(x)\sin(\omega t)\), and Equation (1) turns into:  
\[EI\frac{\partial^{4}\bar{w}(x,t)}{\partial x^{4}}-\rho A\omega^{2}\bar{w}(x,t )=0,\;0<x<L \tag{6}\]  
The following non-dimensional variables are defined for simplicity in calculations:  
\[ \begin{array}{c}\tilde{w}(x)=W(\xi),\;\beta^{4}=L^{4}\omega^{2}\frac{\rho A }{EI},\;\xi=\frac{\chi}{L},\;\Omega_{1}=\frac{M_{1}}{\rho A\lambda},\;\Omega_ {2}=\frac{M_{2}}{\rho A\lambda},\\ \alpha_{1}=\frac{(J_{1}+M_{1}d_{1}^{2})}{\rho A\lambda^{3}},\;\alpha_{2}=\frac {(J_{2}+M_{2}d_{2}^{2})}{\rho A\lambda^{3}},\;\varepsilon_{1}=\frac{d_{1}}{L},\;\varepsilon_{2}=\frac{d_{2}}{L}\end{array}  \tag{7}\]  
Considering the above terms, Equation (6) is rewritten as:  
\[W^{IV}-\beta^{4}W=0,\;0<\xi<1, \tag{8}\]  
with corresponding boundary conditions  
at \(\xi=0\),  
\[W^{\prime\prime}(0)=-\alpha_{1}\beta^{4}W^{\prime}(0)-\Omega_{1}\varepsilon_{1 }\beta^{4}W(0) \tag{9}\]  
\[W^{\prime\prime\prime}(0)=\Omega_{1}\beta^{4}W(0)+\Omega_{1}\varepsilon_{1} \beta^{4}W^{\prime}(0) \tag{10}\]  
similarly at \(\xi=1\),  
\[W^{\prime\prime}(1)=\alpha_{2}\beta^{4}W^{\prime}(1)+\Omega_{2}\varepsilon_{2 }\beta^{4}W(1) \tag{11}\]  
\[W^{\prime\prime\prime}(1)=-\Omega_{2}\beta^{4}W(1)-\Omega_{2}\varepsilon_{2} \beta^{4}W^{\prime}(1), \tag{12}\]  
where the prime denotes differentiation with respect to \(\omega^{\prime}\).