UDC 518.61:534.121.1

ON THE SOLUTION OF THE DIFFERENTIAL EQUATION OF FORCED VIBRATIONS OF RECTANGULAR ORTHOTROPIC PLATES ON AN ELASTIC FOUNDATION WITH TWO CHARACTERISTICS

F. Badalov, T. Shirinkulov

The paper gives an approximate solution of the differential equation of forced vibrations of orthotropic plates on an elastic foundation with two bed coefficients of P. L. Pasternak [4], or on a foundation in the form of a linearly deformable layer of finite thickness of V. Z. Vlasov [1]. The solution is obtained by the method of direct methods, developed in [2—4, 6], when the values of the deflection and of the angle of rotation are prescribed on the edges of the plate. Setting, in the obtained solutions, \(C_3 = 0\), or \(C_2 = 0\), or \(C_1 = 0\), or combinations of them, we obtain various cases generalized in the present investigation. From the formulas obtained it is easy to compile computer programs for calculating the design quantities.

The differential equation of bending of the plate is

\[ D_1 \frac{\partial^4 W}{\partial x^4} + 2D_3 \frac{\partial^4 W}{\partial x^2 \partial y^2} + D_2 \frac{\partial^4 W}{\partial y^4} + m \frac{\partial^2 W}{\partial t^2} + C_1 W - \]
\[ - C_2 \left( \frac{\partial^2 W}{\partial x^2} + \frac{\partial^2 W}{\partial y^2} \right) + C_3 \frac{\partial W}{\partial t} = q(x,y,t), \tag{1} \]

where

\[ D_1 = \frac{E_1 h^3}{12(1-\mu_1\mu_2)}; \qquad D_2 = \frac{E_2 h^3}{12(1-\mu_1\mu_2)}; \]

\[ D_3 = D_1 \mu_2 + 2D_k; \qquad D_k = \frac{G h^3}{12}; \]

\(E_1\) and \(E_2\), \(\mu_1\) and \(\mu_2\) are, respectively, the moduli of elasticity and Poisson’s ratios for the principal directions; \(G\) is the shear modulus; \(h\) is the height (thickness) of the plate; \(W\) is the vertical displacement of the plate; \(q(x,y,t)\) is the acting load; \(m \dfrac{\partial^2 W}{\partial t^2}\) are the inertial forces; \(C_2\left(\dfrac{\partial^2 W}{\partial x^2}+\dfrac{\partial^2 W}{\partial y^2}\right)\) are the resistance forces of the elastic foundation; \(m\) is the mass of the plate and the attached foundation; \(C_3 \dfrac{\partial W}{\partial t}\) are the other resistance forces; \(C_1\) and \(C_2\) are the characteristics of the elastic foundation.

Let us solve differential equation (1) by the method of direct methods for a rectangular domain under the initial ...

\[ W=U_1(x,y), \qquad \frac{\partial W}{\partial t}=U_2(x,y) \quad \text{for } t=0 \tag{2} \]

and boundary

\[ W\big|_s=\bar{\varphi}(s,t), \qquad \frac{\partial W}{\partial n}\bigg|_s=\bar{\psi}(s,t) \tag{3} \]

conditions.

Let \(s\) be the boundary of the rectangular domain \(\bar{G}\), and let it consist of four pieces, which we denote by \(x_0\) and \(x_{N+1}\), and \(y_0\), \(y_{n+1}\). We divide the domain \(\bar{G}\) into \(n+1\) and \(N+1\) parts, respectively, in the directions of the coordinate axes \(y\) and \(x\). We denote the subdivision steps in \(y\) by \(l\), and in \(x\) by \(d\).

First consider the solution \(W(x,y,t)\) of equation (1) for \(y=y_0+kl=y_k\) \((k=1,2,\ldots,n)\), i.e.

\[ W(x,y_k,t)=W_k(x,t)=W_k. \]

Then equation (1) is rewritten in the form

\[ D_1\frac{\partial^4 W_k}{\partial x^4} +2D_3\frac{\partial^4 W}{\partial x^2\partial y^2}\bigg|_{y=y_k} +D_2\frac{\partial^4 W}{\partial y^4}\bigg|_{y=y_k} +m\frac{\partial^2 W_k}{\partial t^2} + \]

\[ +C_1W_k -C_2\left( \frac{\partial^2 W_k}{\partial x^2} + \frac{\partial^2 W}{\partial y^2}\bigg|_{y=y_k} \right) +C_3\frac{\partial W_k}{\partial t} =q_k(x,t). \tag{4} \]

Transform the fourth derivatives with respect to \(y\) of the unknown function
\(\left.\dfrac{\partial^4 W}{\partial y^4}\right|_{y=y_k}\) and
\(\left.\dfrac{\partial^4 W}{\partial x^2\partial y^2}\right|_{y=y_k}\)
as follows.

Expanding the functions \(W(x,y_k+l,t)\), \(W(x,y_k-l,t)\) in a Taylor series and adding the resulting expressions, we obtain

\[ \delta^2 W_k = l^2\frac{\partial^2 W}{\partial y^2}\bigg|_{y=y_k} + \frac{l^4}{12}\frac{\partial^4 W}{\partial y^4}\bigg|_{y=y_k} + O(l^6), \tag{5} \]

where

\[ \delta^2 W_k=W_{k+1}-2W_k+W_{k-1}. \]

Similarly,

\[ \delta^2 \frac{\partial^2 W}{\partial y^2}\bigg|_{y=y_k} = l^2\frac{\partial^4 W}{\partial y^4}\bigg|_{y=y_k} + \frac{l^4}{12}\frac{\partial^6 W}{\partial y^6}\bigg|_{y=y_k} + O(l^6). \tag{6} \]

According to equalities (5) and (6), we shall have

\[ \delta^4 W_k = l^4\frac{\partial^4 W}{\partial y^4}\bigg|_{y=y_k} + \frac{l^4}{12}\delta^2 \frac{\partial^4 W}{\partial y^4}\bigg|_{y=y_k} + \]

\[ + \frac{l^6}{12}\frac{\partial^6 W}{\partial y^6}\bigg|_{y=y_k} + O(l^6), \tag{7} \]

where

\[ \delta^4 W_k = W_{k+2}-4W_{k+1}+6W_k-4W_{k-1}+W_{k-2}. \]

In formula (6), retaining terms of order \(l^2\), we have

\[ \delta^2 \frac{\partial^4 W}{\partial y^4}\bigg|_{y=y_k} = l^2\frac{\partial^6 W}{\partial y^6}\bigg|_{y=y_k} + O(l^4). \tag{8} \]

Excluding from equalities (7) and (8) \(\left. \dfrac{\partial^6 W}{\partial y^6}\right|_{y=y_k}\) and discarding terms of order \(O(l^6)\), we obtain the following approximate formula:

\[ \delta^4 W_k = l^4 \left. \frac{\partial^4 W}{\partial y^4}\right|_{y=y_k} + \frac{l^4}{6}\,\delta^2 \left. \frac{\partial^4 W}{\partial y^4}\right|_{y=y_k}. \tag{9} \]

Now transform the term \(\left. \dfrac{\partial^4 W}{\partial x^2 \partial y^2}\right|_{y=y_k}\):

\[ \left. \frac{\partial^4 W}{\partial x^2 \partial y^2}\right|_{y=y_k} = \frac{1}{l^2}\,\delta^2 \frac{\partial^2 W_k}{\partial x^2}. \tag{10} \]

Formula (9), having accuracy \(O(l^6)\), can be used to solve equation (4). Indeed, using (10), from equation (4) we have

\[ \begin{aligned} D_2 \left. \frac{\partial^4 W}{\partial y^4}\right|_{y=y_k} ={}& - D_1 \frac{\partial^4 W_k}{\partial x^4} -\frac{2D_3}{l^2}\,\delta^2 \frac{\partial^2 W_k}{\partial x^2} - m \frac{\partial^2 W_k}{\partial t^2} \\ &- C_1 W_k + C_2 \left( \frac{\partial^2 W_k}{\partial x^2} + \frac{\delta^2 W_k}{l^2} \right) - C_3 \frac{\partial W_k}{\partial t} + q_k(x,t). \end{aligned} \tag{11} \]

Hence, replacing in formula (9) the fourth partial derivatives with respect to \(y\) by their values from formula (11), in order to determine the function \(W_k(x,t)\), we obtain the following system of partial differential equations with respect to \(x,t\):

\[ \begin{aligned} &D_1 \frac{\partial^4 W_k}{\partial x^4} + \frac{D_1}{6}\,\delta^2 \frac{\partial^4 W_k}{\partial x^4} - C_2 \frac{\partial^2 W_k}{\partial x^2} + \left( \frac{2D_3}{l^2} - \frac{C_2}{6} \right) \delta^2 \frac{\partial^2 W_k}{\partial x^2} + \frac{D_3}{3l^2}\,\delta^4 \frac{\partial^2 W_k}{\partial x^2} + m \frac{\partial^2 W_k}{\partial t^2} + \frac{m}{6}\,\delta^2 \frac{\partial^2 W_k}{\partial t^2} \\ &\quad + C_3 \frac{\partial W_k}{\partial t} + \frac{C_3}{6}\,\delta^2 \frac{\partial W_k}{\partial t} + C_1 W_k + \left( \frac{C_1}{6} - \frac{C_2}{l^2} \right) \delta^2 W_k + \frac{1}{l^2} \left( \frac{D_2}{l^2} - \frac{C_2}{6} \right) \delta^4 W_k = q_k + \frac{1}{6}\,\delta^2 q_k. \end{aligned} \tag{12} \]

Here we have \(n\) equations with \(n+4\) unknown functions; the four unknown functions are determined from the boundary conditions (3). These conditions, approximately, with accuracy up to \(l^2\), are written in the form

\[ \begin{aligned} W_{-1}(x,t) &= \overline{\varphi}_0(x,t) - l\,\overline{\psi}_0(x,t),\\ W_0(x,t) &= \overline{\varphi}_0(x,t),\\ W_{n+1}(x,t) &= \overline{\varphi}_{n+1}(x,t),\\ W_{n+2}(x,t) &= \overline{\varphi}_{n+1}(x,t) + l\,\overline{\psi}_{n+1}(x,t). \end{aligned} \tag{13} \]

We transform system (12) into a system each equation of which contains only one unknown function. For this, consider the vectors:

\[ U = [W_1(x,t),\quad W_2(x,t),\ldots, W_n(x,t)], \]

\[ F = [F_1(x,t),\quad F_2(x,t),\ldots, F_n(x,t)]. \]

and introduce the matrix

\[ M= \left| \begin{array}{rrrrrrrrr} -2&1&0&0&\cdots&0&0&0\\ 1&-2&1&0&\cdots&0&0&0\\ 0&1&-2&1&\cdots&0&0&0\\ \cdots&\cdots&\cdots&\cdots&\cdots&\cdots&\cdots&\cdots&\cdots\\ \cdots&\cdots&\cdots&\cdots&\cdots&\cdots&\cdots&\cdots&\cdots\\ 0&0&0&0&\cdots&1&-2&1\\ 0&0&0&0&\cdots&0&1&-2 \end{array} \right|. \]

Then system (12), in matrix form, is written as follows:

\[ D_1\left(E+\frac{M}{6}\right)\frac{\partial^4 U}{\partial x^4} + \left[ -C_2E+\left(\frac{2D_3}{l^2}-\frac{C_2}{6}\right)M +\frac{D_3}{3l^2}M^2 \right]\frac{\partial^2 U}{\partial x^2} + \]

\[ +\,m\left(E+\frac{M}{6}\right)\frac{\partial^2 U}{\partial t^2} + C_3\left(E+\frac{M}{6}\right)\frac{\partial U}{\partial t} + \]

\[ + \left[ C_1E+\left(\frac{C_1}{6}-\frac{C_2}{l^2}\right)M +\frac{1}{l^2}\left(\frac{D_2}{l^2}-\frac{C_2}{6}\right)M^2 \right]U=F, \tag{14} \]

where \(E\) is the identity matrix;

\[ F_1=q_1^*(x,t)-\frac{D_1}{6}\frac{\partial^4 W_0}{\partial x^4} -\left(\frac{2D_3}{l^2}-\frac{C_2}{6}\right)\frac{\partial^2 W_0}{\partial x^2} - \]

\[ -\frac{D_3}{3l^2} \left( \frac{\partial^2 W_{-1}}{\partial x^2} -4\frac{\partial^2 W_0}{\partial x^2} +\frac{\partial^2 W_1}{\partial x^2} \right) -\frac{m}{6}\frac{\partial^2 W_0}{\partial t^2} -\frac{C_3}{6}\frac{\partial W_0}{\partial t} + \]

\[ +\left(\frac{C_1}{6}-\frac{C_2}{l^2}\right)W_0 -\frac{1}{l^2}\left(\frac{D_2}{l^2}-\frac{C_2}{6}\right)(W_{-1}-4W_0+W_1), \]

\[ F_2=q_2^*(x,t)-\frac{D_3}{3l^2}\frac{\partial^2 W_0}{\partial x^2} -\frac{1}{l^2}\left(\frac{D_2}{l^2}-\frac{C_2}{6}\right)W_0, \]

\[ F_k=q_k^*(x,t);\qquad k=3,4,\ldots,n-2. \]

\[ F_{n-1}=q_{n-1}^*(x,t)-\frac{D_3}{3l^2}\frac{\partial^2 W_{n+1}}{\partial x^2} -\frac{1}{l^2}\left(\frac{D_2}{l^2}-\frac{C_2}{6}\right)W_{n+1}, \]

\[ F_n=q_n^*(x,t)-\frac{D_1}{6}\frac{\partial^4 W_{n+1}}{\partial x^4} -\left(\frac{2D_3}{l^2}-\frac{C_2}{6}\right)\frac{\partial^2 W_{n+1}}{\partial x^2} - \]

\[ -\frac{D_3}{3l^2} \left( \frac{\partial^2 W_{n+2}}{\partial x^2} -4\frac{\partial^2 W_{n+1}}{\partial x^2} +\frac{\partial^2 W_n}{\partial x^2} \right) -\frac{m}{6}\frac{\partial^2 W_{n+1}}{\partial t^2} - \]

\[ -\frac{C_3}{6}\frac{\partial W_{n+1}}{\partial t} +\left(\frac{C_1}{6}-\frac{C_2}{l^2}\right)W_{n+1} -\frac{1}{l^2}\left(\frac{D_2}{l^2}-\frac{C_2}{6}\right)\times \]

\[ \times (W_{n+2}-4W_{n+1}+W_n), \]

\[ q_k^*(x,t)=q_k(x,t)+\frac{1}{6}\delta^2 q_k(x,t). \]

For the unknown functions \(W_1(x,t)\) and \(W_n(x,t)\), which enter into \(F_1\) and \(F_n\), in the first approximation we take

\[ W_1(x,t)=\overline{\varphi}_0(x,t)+l\overline{\psi}_0(x,t), \]

\[ W_n(x,t)=\overline{\varphi}_{n+1}(x,t)-l\overline{\psi}_{n+1}(x,t). \]

As shown in the work of V. N. Faddeeva [5], the matrix \(M\), by means of orthogonal transformations, is always reduced to diagonal form:

\[ B^{-1}MB=(\lambda_1,\lambda_2,\ldots,\lambda_n)= \begin{Vmatrix} \lambda_1 & 0 & 0 & \ldots & 0\\ 0 & \lambda_2 & 0 & \ldots & 0\\ \ldots & \ldots & \ldots & \ldots & \ldots\\ \ldots & \ldots & \ldots & \ldots & \ldots\\ 0 & 0 & \ldots & \ldots & \lambda_n \end{Vmatrix}, \]

where

\[ B=B^{-1}=\|b_{s,k}\|= \left\|(-1)^{k+s}\sqrt{\frac{2}{n+1}}\sin\frac{sk\pi}{n+1}\right\|, \qquad \lambda_s=-2\left(1+\cos\frac{s\pi}{n+1}\right). \]

Introduce the notation:

\[ BU=[\varphi_1(x,t),\varphi_2(x,t),\ldots,\varphi_n(x,t)]=A, \]

\[ BF=[g_k(x,t),g_2(x,t),\ldots,g_n(x,t)]=C. \tag{15} \]

Then we shall have

\[ \begin{aligned} &D_1\left(1+\frac{\lambda_1}{6},\;1+\frac{\lambda_2}{6},\ldots,\;1+\frac{\lambda_n}{6}\right) \frac{\partial^4 A}{\partial x^4} +\Bigg[-C_2+ \\ &\quad+\left(\frac{2D_3}{l^2}-\frac{C_2}{6}\right)\lambda_1 +\frac{D_3}{3l^2}\lambda_1^2,\ldots, -C_2+\left(\frac{2D_3}{l^2}-\frac{C_2}{6}\right)\lambda_n+ \\ &\quad+\frac{D_3}{3l^2}\lambda_n^2\Bigg]\frac{\partial^2 A}{\partial x^2} +m\left(1+\frac{\lambda_1}{6},\ldots,1+\frac{\lambda_n}{6}\right) \frac{\partial^2 A}{\partial t^2}+ \\ &\quad+C_3\left(1+\frac{\lambda_1}{6},\ldots,1+\frac{\lambda_n}{6}\right) \frac{\partial A}{\partial t} +\Bigg[C_1+\left(\frac{C_1}{6}-\frac{C_2}{l^2}\right)\lambda_1+ \\ &\quad+\frac{1}{l^2}\left(\frac{D_2}{l^2}-\frac{C_2}{6}\right)\lambda_1^2,\ldots, C_1+\left(\frac{C_1}{6}-\frac{C_2}{l^2}\right)\lambda_n+\frac{1}{l^2}\left(\frac{D_2}{l^2}-\right. \\ &\quad\left.\left.-\frac{C_2}{6}\right)\lambda_n^2\right]A=C. \end{aligned} \tag{16} \]

Equation (16) is equivalent to \(n\) independent equations

\[ \frac{\partial^4\varphi_k}{\partial x^4} +\alpha_k\frac{\partial^2\varphi_k}{\partial x^2} +\beta_k\varphi_k +\overline{m}\frac{\partial^2\varphi_k}{\partial t^2} + \overline{C}_3\frac{\partial\varphi_k}{\partial t} = \overline{g}_k \qquad (k=1,2,\ldots,n), \tag{17} \]

where

\[ \alpha_k = \frac{ -C_2 + \left(\frac{2D_3}{l^2} - \frac{C_2}{6}\right)\lambda_k + \frac{D_3}{3l^2}\lambda_k^2 }{ D_1\left(1+\frac{\lambda_k}{6}\right) }; \]

\[ \beta_k = \frac{ C_1 + \left(\frac{C_1}{6} - \frac{C_2}{l^2}\right)\lambda_k + \frac{1}{l^2}\left(\frac{D_2}{l^2} - \frac{C_2}{6}\right)\lambda_k^2 }{ D_1\left(1+\frac{\lambda_k}{6}\right) }; \]

\[ \bar m=\frac{m}{D_1}; \qquad \bar C_3=\frac{C_3}{D_1}; \qquad \bar g_k=\frac{g_k(x,t)}{D_1\left(1+\frac{\lambda_k}{6}\right)} . \]

Let us solve the differential equation (17), again applying the method of lines under the boundary conditions (3), and transform the fourth derivatives with respect to \(x\) of the unknown function, as was done with respect to \(y\):

\[ \delta^4\varphi_{k,i} = d^4\left.\frac{\partial^4\varphi_k}{\partial x^4}\right|_{x=x_i} + \frac{d^4}{6}\delta^2 \left.\frac{\partial^4\varphi_k}{\partial x^4}\right|_{x=x_i} \tag{18} \]

(here and below the operator \(\delta\) acts with respect to the index \(i\)).

Formula (18), having accuracy \(O(d^6)\), can be used to solve equation (17). Indeed, for \(x=x_0+id=x_i\) \((i=1, 2, \ldots, N)\), from equation (17) we have

\[ \left.\frac{\partial^4\varphi_k}{\partial x^4}\right|_{x=x_i} = -\frac{\alpha_k}{d^2}\delta^2\varphi_{k,i}(t) -\beta_k\varphi_{k,i}(t) -\bar m\,\varphi_{k,i}''(t) - \]

\[ -\bar C_3\varphi_{k,i}'(t)+\bar g_{k,i}(t). \tag{19} \]

Hence, replacing in formula (18) the fourth partial derivatives with respect to \(x\) by their values (19), to determine the function \(\varphi_{k,i}(t)\) we obtain the following systems of ordinary differential equations of the type of M. G. Slobodyansky:

\[ \varphi_{k,i}''(t) + \frac{1}{6}\delta^2\varphi_{k,i}''(t) + \tilde C_3\varphi_{k,i}'(t) + \frac{\tilde C_3}{6}\delta^2\varphi_{k,i}'(t) + \]

\[ +\gamma_k\delta^2\varphi_{k,i}(t) +\sigma_k\delta^4\varphi_{k,i}(t) +\bar\beta_k\varphi_{k,i}(t) = \tilde g_{k,i}(t) + \frac{1}{6}\delta^2\tilde g_{k,i}(t), \tag{20} \]

where

\[ \tilde C_3=\frac{C_3}{m}; \qquad \gamma_k= \frac{ D_1\left(\frac{\beta_k}{6}+\frac{\alpha_k}{d^2}\right) }{m}; \]

\[ \sigma_k= \frac{ D_1\left(\frac{1}{d^2}+\frac{\alpha_k}{6}\right) }{md^2}; \qquad \bar\beta_k=\frac{D_1\beta_k}{m}; \qquad \tilde g_{k,i}(t)=\frac{D_1\bar g_{k,i}(t)}{m}. \]

System (20) in matrix form is written as follows:

\[ \left( E+\frac{M}{6}\right)V''(t)+\widetilde C_3\left(E+\frac{M}{6}\right)V'(t)+(\gamma_k M+\sigma_k M^2+ \]

\[ +\overline{\beta}_k E)V(t)=R(t), \tag{21} \]

where

\[ V=[\varphi_{k,1}(t),\ \varphi_{k,2}(t),\ldots,\varphi_{k,N}(t)]; \]

\[ R(t)=[R_{k,1}(t),\ R_{k,2}(t),\ldots,\ R_{k,N}(t)]; \]

\[ R_{k,1}(t)=\widetilde g^{*}_{k,1}(t)-\frac{1}{6}\varphi''_{k,0}(t)-\frac{\widetilde C_3}{6}\varphi'_{k,0}(t)-\gamma_k\varphi_{k,0}(t)- \]

\[ -\sigma_k[\varphi_{k,1}(t)-4\varphi_{k,0}(t)+\varphi_{k,-1}(t)]; \]

\[ R_{k,2}(t)=\widetilde g^{*}_{k,2}(t)-\sigma_k\varphi_{k,0}(t); \]

\[ R_{k,i}(t)=\widetilde g^{*}_{k,i}(t)\qquad (i=3,4,\ldots,N-2); \]

\[ R_{k,N-1}(t)=\widetilde g^{*}_{k,N-1}(t)-\sigma_k\varphi_{k,N+1}(t); \]

\[ R_{k,N}(t)=\widetilde g^{*}_{k,N}(t)-\frac{1}{6}\varphi''_{k,N+1}(t)-\frac{\widetilde C_3}{6}\varphi'_{k,N+1}(t)- \]

\[ -\gamma_k\varphi_{k,N+1}(t)-\sigma_k[\varphi_{k,N+2}(t)-4\varphi_{k,N+1}(t)+\varphi_{k,N}(t)]; \]

\[ \widetilde g^{*}_{k}(t)=\widetilde g_{k,i}(t)+\frac{1}{6}\delta^2\widetilde g_{k,i}(t). \]

For \(\varphi_{k,1}(t)\), \(\varphi_{k,N}(t)\)—unknown functions entering \(R_1(t)\) and \(R_N(t)\)—in the first approximation we take

\[ \varphi_{k,1}(t)=\widetilde\varphi_{k,0}(t)+d\widetilde\psi_{k,0}(t) =\sum_{i=1}^{N} b_{k,i}\,[\overline\varphi_{i,0}(t)+d\,\overline\psi_{i,0}(t)]; \]

\[ \varphi_{k,N}(t)=\widetilde\varphi_{k,N+1}(t)-d\widetilde\psi_{k,N+1}(t) =\sum_{i=1}^{N} b_{k,i}\,[\overline\varphi_{i,N+1}(t)-d\,\overline\psi_{i,N+1}(t)]. \]

Denoting in equation (21)

\[ BV=[V_{k,1}(t),\ldots,V_{k,N}(t)]=\Phi(t), \tag{22} \]

\[ BR=[C_{k,1}(t),\ldots,C_{k,N}(t)]=T(t), \]

we obtain \(N\) independent equations:

\[ V''_{k,\nu}(t)+\widetilde C_3 V'_{k,\nu}(t)+\omega^2_{k,\nu}V_{k,\nu}(t)=\overline C_{k,\nu}(t) \qquad (\nu=1,2,\ldots,N), \tag{23} \]

where

\[ \omega^2_{k,\nu}=\frac{\gamma_k\lambda_\nu+\sigma_k\lambda_\nu^2+\overline\beta_k}{1+\dfrac{\lambda_\nu}{6}}; \qquad \overline C_{k,\nu}(t)=\frac{C_{k,\nu}(t)}{1+\dfrac{\lambda_\nu}{6}}. \]

The integral of equation (23) for \(4\omega^2_{k,\nu}>\widetilde C_3^{\,2}\) will be

\[ V_{k,\nu}(t)=e^{-\frac{\widetilde C_3}{2}t}\left(A_{k,\nu}\sin \overline\omega_{k,\nu}t+B_{k,\nu}\cos \overline\omega_{k,\nu}t\right)+ \]
\[ +\frac{1}{\overline\omega_{k,\nu}}\int_0^t e^{-\frac{\widetilde C_3}{2}(t-\tau)}\overline C_{k,\nu}(\tau)\sin \overline\omega_{k,\nu}(t-\tau)\,d\tau, \tag{24} \]

where

\[ \overline\omega_{k,\nu}=\frac{1}{2}\sqrt{4\omega_{k,\nu}^{2}-\widetilde C_3^{\,2}}, \]

and for \(4\omega_{k,\nu}^{2}<\widetilde C_3^{\,2}\)

\[ V_{k,\nu}(t)=e^{-\frac{\widetilde C_3}{2}t}\left(\widetilde A_{k,\nu}\operatorname{sh}\widetilde\omega_{k,\nu}t+\widetilde B_{k,\nu}\operatorname{ch}\widetilde\omega_{k,\nu}t+\right. \]
\[ \left.+\frac{1}{\widetilde\omega_{k,\nu}}\int_0^t e^{-\frac{\widetilde C_3}{2}(t-\tau)}\overline C_{k,\nu}(\tau)\operatorname{sh}\widetilde\omega_{k,\nu}(t-\tau)\,d\tau\right), \tag{25} \]

where

\[ \widetilde\omega_{k,\nu}=\frac{1}{2}\sqrt{\widetilde C_3^{\,2}-4\omega_{k,\nu}^{2}}. \]

In the case \(4\omega_{k,\nu}^{2}=\widetilde C_3^{\,2}\), the solution of equation (23) will be

\[ V_{k,\nu}(t)=\left(\widetilde A_{k,\nu}+\widetilde B_{k,\nu}t\right)e^{-\frac{\widetilde C_3}{2}t} +\int_0^t (t-\tau)e^{-\frac{\widetilde C_3}{2}(t-\tau)}\overline C_{k,\nu}(\tau)\,d\tau . \tag{26} \]

Now, passing back to the functions \(\varphi_{k,i}(t)\), on the basis of (22),

\[ V(t)=B^{-1}\Phi(t)=B\Phi \]

or

\[ \varphi_{k,i}(t)=\sum_{\nu=1}^{N} b_{k,\nu}V_{i,\nu}(t). \]

According to the notation (15), we have

\[ U(t)=B^{-1}A=BA \]

or, for \(4\omega_{k,\nu}^{2}>\widetilde C_3^{\,2}\),

\[ W_{k,i}(t)=\frac{2}{\sqrt{(n+1)(N+1)}}\sum_{j=1}^{n}\sum_{\nu=1}^{N}(-1)^{i+\nu+k+j} \sin\frac{kj\pi}{n+1}\sin\frac{i\nu\pi}{N+1}\times \]
\[ \times\left[ e^{-\frac{\widetilde C_3}{2}t}\left(A_{j,\nu}\sin\overline\omega_{j,\nu}t+B_{j,\nu}\cos\overline\omega_{j,\nu}t\right)+ \right. \]
\[ \left. +\frac{1}{\omega_{j,\nu}}\int_0^t e^{-\frac{\widetilde C_3}{2}(t-\tau)} \overline C_{j,\nu}(\tau)\sin\overline\omega_{j,\nu}(t-\tau)\,d\tau \right]. \tag{27} \]

For the remaining cases the solution of equation (1) can be written analogously.

The arbitrary constants \(A_{j,\nu},\ \overline{A}_{j,\nu},\ \widetilde{A}_{j,\nu},\ B_{j,\nu},\ \overline{B}_{j,\nu},\ \widetilde{B}_{j,\nu}\) are found from the initial conditions.

For natural vibrations of plates rigidly clamped along the edges, the boundary conditions will be

\[ W\big|_s = 0;\qquad \frac{\partial W}{\partial n}\bigg|_s = 0. \]

In this case, in formulas (24)—(26) all \(\overline{C}_{j,\nu}(t)\) become zero, as a result of which the last integral terms in these formulas are absent.

If the resistance forces are not taken into account in the solutions obtained, one must put \(C_3 = 0\).

In conclusion, the authors thank the reviewer for a number of valuable comments made during the review of the present work.

References

1. Vlasov V. Z. Beams, plates, and shells on an elastic foundation. Fizmatgiz, 1960.
2. Langenbakh A. Approximate solutions of the biharmonic equation. Vestnik LGU, No. 3, series of mathematical sciences, 1956.
3. Slobodyanskii M. G. PMM, 3, issue 1, 1939.
4. Pasternak P. L. Fundamentals of a new method for calculating foundations on an elastic foundation with the aid of two bedding coefficients. Moscow, Gosstroiizdat, 1954.
5. Faddeeva V. N. Methods of lines as applied to certain boundary-value problems. Proceedings of the V. A. Steklov Mathematical Institute of the Academy of Sciences of the USSR, 28, 1949.
6. Yunusov K. S. Application of the method of lines to the solution of boundary-value problems for partial differential equations. Scientific Notes of the Kazan State Pedagogical Institute, issue 10. Kazan, 1955.

Received by the editors
April 5, 1965

Institute of Mechanics and Computing Center
Academy of Sciences of the Uzbek SSR