Eigen-proper vibration of continuous plates (Q2730549)

The paper deals with investigation of the problem of the linear and nonlinear vibrations of continuous rectangular plates. In the case of linear vibration of rectangular \(M\)-spans continuous isotropic plate of constant thickness \(h\) the problem of determination of frequency and form of such plate is reduced to eigenvalue and eigenfunctions problem for equation NEWLINE\[NEWLINE{\partial^4 W_{m}(x,y)\over \partial x^4}+ {\partial^4 W_{m}(x,y)\over \partial x^2\partial y^2}+ {\partial^4 W_{m}(x,y)\over \partial y^4}=\lambda^2 W_{m}(x,y),NEWLINE\]NEWLINE with the boundary condition NEWLINE\[NEWLINEW_{m}|_{\Gamma}=0, \xi_1{\partial W_{1}(0,y)\over \partial x}+\eta_1 {\partial^2 W_{1}(0,y)\over \partial x^2}= \xi_2{\partial W_{M}(l_{M},y)\over \partial x}+ \eta_2{\partial^2 W_{M}(l_{M},y)\over \partial x^2}=0,NEWLINE\]NEWLINE NEWLINE\[NEWLINE\sigma_1{\partial W_{m}(x,0)\over \partial x}+\tau_1 {\partial^2 W_{m}(x,0)\over \partial x^2}= \sigma_2{\partial W_{m}(x,b)\over \partial x}+ \tau_2{\partial^2 W_{m}(x,b)\over \partial x^2}=0,NEWLINE\]NEWLINE \(m=1,2,\ldots,M,\) and the conjunction conditions NEWLINE\[NEWLINEW_{m}(l_{m},y)=0, {\partial W_{m}(l_{m},y)\over \partial x}= {\partial W_{m+1}(l_{m},y)\over \partial x}, {\partial^2 W_{m}(l_{m},y)\over \partial x^2}= {\partial^2 W_{m+1}(l_{m},y)\over \partial x^2},NEWLINE\]NEWLINE \(m=1,2,\ldots,M-1.\) Here \(l_{m}\) is the length of span \(m\); \(b\) is the width of plate; \(\lambda^2=\rho h\omega^2/D\); \(\rho\) is the material density; \(D\) is the cylindrical stiffness; \(\omega\) is the eigen-proper vibration frequency; \(W_{m}\) is the normal deflection in span \(m\). In some special case the author finds the explicit solution of the considered problem. In general case the approximate solution is obtained. Also the nonlinear vibration is investigated.