Free vibration of the system of two viscoelastic beams coupled by viscoelastic interlayer (Q2719339)

The author considers the system of connected equations NEWLINE\[NEWLINE \begin{gathered} E_1I_1\bigg(1+c_1\,\frac{\partial}{\partial t}\bigg) \frac{\partial^4w_1}{\partial x^4}+ \mu_1\,\frac{\partial^2w_1}{\partial t^2}+ c\,\frac{\partial}{\partial t}(w_1-w_2) +k(w_1-w_2)=0,\\ E_2I_2\bigg(1+c_2\,\frac{\partial}{\partial t}\bigg) \frac{\partial^4w_2}{\partial x^4}+ \mu_2\,\frac{\partial^2w_2}{\partial t^2}+ c\,\frac{\partial}{\partial t}(w_1-w_2) +k(w_1-w_2)=0, \end{gathered}\tag{1} NEWLINE\]NEWLINE where \(\,w_1=w_1(x,t)\;,\) and \(\,w_2=w_2(x,t)\,\) are the deflections of the beams (hereafter the subscripts ``1'' and ``2'' mean the beam I and the beam II respectively); \(E_1\) and \(E_2\) are the Young's modules of the beams materials; \(I_1\) and \(I_2\) are the moments of inertia of the beams cross-sections; \(\mu_1\) and \(\mu_2\) are the masses of the beams per unit of length; \(c_1\) and \(c_2\) are the relative viscosity coefficients of the beams; \(c\) is the coefficient of viscosity of the interlayer; \(k\) is the coefficient of elasticit y of the interlayer; \(l\) is the length of the beams. After the separation of variables in the differential equations the boundary value problem is solved and two complex sequences are obtained: the sequence of frequencies and the sequence of modes of free vibration.