Full Text
UDC 534.121.1
MATHEMATICS
N. F. MOROZOV
ON NONLINEAR OSCILLATIONS OF THIN PLATES WITH ACCOUNT OF ROTATIONAL INERTIA
(Presented by Academician I. N. Vekua, 6 XII 1966)
The problem of nonlinear oscillations of a plate was studied in works \((^{1-4})\), where the system of equations
\[
\rho h w_{tt}+D\Delta^2 w=Z(x,y,t)+F_{xx}w_{yy}+F_{yy}w_{xx}-2F_{xy}w_{xy},
\]
\[
\Delta^2 F=hE\left[w_{xy}^{2}-w_{xx}w_{yy}\right]^* .
\tag{1}
\]
The contour \(\Gamma\) bounds the domain \(\Omega\) of the variables \(x\) and \(y\). The differential equations (1) are non-wave equations: their integration leads to instantaneous propagation of disturbances, which contradicts the general dynamical equations of the theory of elasticity. It should also be noted that the study of equations (1) is connected with substantial mathematical difficulties, for example, in the question of uniqueness of the solution.
In the present work the same problem will be studied with account of the rotational inertia of the plate elements. Then we obtain the system of equations (see \((^{5,6})\))
\[
D\Delta^2 w-\gamma^2\Delta w_{tt}+w_{tt}=Z(x,y,t)+F_{xx}w_{yy}+F_{yy}w_{xx}-2F_{xy}w_{xy},
\]
\[
\Delta^2F=hE\left[w_{xy}^{2}-w_{xx}w_{yy}\right].
\tag{2}
\]
Consider equations (2) with boundary conditions (3) and (4)
\[ w|_{\Gamma}=\partial w/\partial \nu|_{\Gamma}=0; \tag{3} \]
\[ F|_{\Gamma}=\partial F/\partial \nu|_{\Gamma}=0 \tag{4} \]
and initial conditions
\[ w|_{t=0}=w_t|_{t=0}=0. \tag{5} \]
In the proposed work it will be proved: the existence of a generalized solution of problem (2)—(5) in a certain energy space; the uniqueness of the solution of problem (2)—(5) in the same space; with account of dissipative terms, a qualitative study of the solution for “large” time will be carried out.
Let \(H\) be the space obtained by completing, in the norm (6), the set of smooth functions defined in the cylinder \(Q=[0,T]\times\Omega\) and satisfying conditions (3):
\[ \|w\|_{H}^{2}=\int_{0}^{T}\int_{\Omega}\left[w_t^{2}+\gamma^{2}\operatorname{grad}^{2}w_t+D(\Delta w)^2\right]\,d\Omega\,dt. \tag{6} \]
Analogously to \((^{1,7})\), we introduce the concept of a generalized solution of problem (2)—(5) in the space \(H\).
Consider the auxiliary problem
\[ D\Delta^2w-\gamma^2\Delta w_{tt}+w_{tt}=f(x,y,t); \tag{7} \]
\[ w|_{\Gamma}=\partial w/\partial \nu|_{\Gamma}=0; \tag{8} \]
\[ w|_{t=0}=w_t|_{t=0}=0. \tag{9} \]
\[ \text{* In what follows we shall assume } \rho h=1. \]
It is easy to show, following (1), that for the generalized solution of problem (7)—(9) the equality
\[ \int_{\Omega} D(\Delta w)^2\,d\Omega+\gamma^2\int_{\Omega}\operatorname{grad}^2 w_t\,d\Omega+\int_{\Omega}w_t^2\,d\Omega = 2\int_0^T\int_{\Omega} w_t f\,d\Omega\,dt . \tag{10} \]
is valid.
We now seek Bubnov–Galerkin approximations to the generalized solution of problem (2)—(5) in the space \(H\). We choose \(w^{(n)}\) in the form
\[ w^{(n)}=\sum_{k=1}^{n}q_k^{(n)}(t)\psi_k(x,y). \tag{11} \]
Here \(\psi_k\) are the eigenfunctions of the biharmonic operator under conditions (3), and \(F^{(n)}\) is determined from the equation
\[ \Delta^2 F^{(n)}=hE\left[(w_{xy}^{(n)})^2-w_{xx}^{(n)}w_{yy}^{(n)}\right] \tag{12} \]
and the boundary conditions (4)
\[ F^{(n)}\big|_{\Gamma}=\partial F^{(n)}/\partial \nu\big|_{\Gamma}=0. \]
Then for \(q_k^{(n)}\) we obtain the system of ordinary differential equations
\[ \ddot q_k^{(n)}+\sum_{i=1}^{n}c_{ik}\ddot q_i^{(n)} + \frac{\partial \Phi_n\left(q_1^{(n)},q_2^{(n)},\ldots,q_n^{(n)}\right)} {\partial q_k^{(n)}} = Z_k(t). \tag{13} \]
Here
\[ Z_k=\int_{\Omega} Z(x,y,t)\psi_k\,d\Omega;\qquad c_{ik}=-\int_{\Omega}\Delta\psi_i\cdot\psi_k\,d\Omega; \]
\(\Phi_n\) is a positive definite functional
\[ \Phi_n=\frac12\int_{\Omega}(\Delta w^{(n)})^2\,d\Omega+ \frac12\int_{\Omega}(\Delta F^{(n)})^2\,d\Omega . \tag{14} \]
By virtue of (13) and (14) we obtain the a priori equality
\[ D\int_{\Omega}[\Delta w^{(n)}]^2\,d\Omega +\gamma^2\int_{\Omega}\left[(w_{xt}^{(n)})^2+(w_{yt}^{(n)})^2\right]\,d\Omega +\int_{\Omega}(w_t^{(n)})^2\,d\Omega+ \]
\[ +\frac{1}{Eh}\int_{\Omega}[\Delta F^{(n)}]^2\,d\Omega = 2\int_0^T\int_{\Omega}Z w_t^{(n)}\,d\Omega\,dt \tag{15} \]
and, as a consequence of (15), the a priori estimate
\[ \|w^{(n)}\|_H<C. \]
It follows from this that the set of approximate solutions \(w^{(n)}\) is weakly compact in \(H\), and every weak limit of \(w^{(n)}\) in \(H\) is a generalized solution of problem (2)—(5).
We now show that the generalized solution of problem (2)—(5) is unique in the space \(H\). Let \(w_1\) and \(w_2\) be two distinct solutions from the space \(H\). Then \(v=w_1-w_2\) solves problem (7)—(9) with right-hand side \(f\)
\[ f\equiv F_{1xx}w_{1yy}+F_{1yy}w_{1xx}-2F_{1xy}w_{1xy} -F_{2xx}w_{2yy}-F_{2yy}w_{2xx}+2F_{2xy}w_{2xy}. \]
By virtue of (10) we obtain the inequality
\[ D\int_{\Omega}(\Delta v)^2\,d\Omega +\gamma^2\int_{\Omega}(v_{xt}^2+v_{yt}^2)\,d\Omega +\int_{\Omega}v_t^2\,d\Omega = 2\int_0^T\int_{\Omega}v_t f\,d\Omega\,dt = \]
\[ = 2\int_0^T\int_{\Omega} \left\{ v_{tx}\left[ w_{1y}F_{1xy}-w_{2y}F_{2xy}-w_{1x}F_{1yy}+w_{2x}F_{2yy} \right] +\right. \]
\[ \left. +\,v_{ty}\left[ w_{1x}F_{1xy}-w_{2x}F_{2xy}-w_{1y}F_{1xx}+w_{2y}F_{2xx} \right] \right\}\,d\Omega\,dt \leq \]
\[
\ll \frac{2}{\gamma}\|v\|_{H}\|v_y\|_{L_4Q}\|F_{1xy}\|_{L_4Q}+
\]
\[
+\frac{2}{\gamma}\|v\|_{H}\|w_{2y}\|_{L_4Q}\|F_{1xy}-F_{2xy}\|_{L_4Q}+\cdots \ll C\|v\|_{H}^{2}.
\tag{16}
\]
In deriving inequality (16), the embedding theorems, the properties of equation (2), and also the fact that the solutions \(w_1\) and \(w_2\) belong to the space \(H\) were used essentially.
Integrating inequality (16) from \(0\) to \(T\), we obtain
\[ \|v\|_{H}^{2}\ll CT\|v\|_{H}^{2}. \]
Hence it follows that \(v\equiv 0\), and the generalized solution in the space \(H\) is unique.
If the effect of damping is taken into account, then system (2) is transformed into the form
\[ D\Delta^2 w-\gamma^2\Delta w_{tt}+w_{tt}+\varepsilon_1 w_t-\varepsilon_2\Delta w_t =Z+F_{xx}w_{yy}+F_{yy}w_{xx}-2F_{xy}w_{xy}; \tag{17a} \]
\[ \Delta^2F=hE[w_{xy}^2-w_{xx}w_{yy}]; \tag{17b} \]
\[ w|_{\Gamma}=\partial w/\partial \nu|_{\Gamma}=F|_{\Gamma}=\partial F/\partial \nu|_{\Gamma}=0. \tag{17c} \]
Introduce the notation
\[ \|w\|_{H\Omega}^{2}=\int_{\Omega}D(\Delta w)^2\,d\Omega +\int_{\Omega}\gamma^2\operatorname{grad}^2 w_t\,d\Omega +\int_{\Omega}w_t^2\,d\Omega; \tag{18} \]
\[ \|w\|_{HQ_\tau}^{2}=\int_0^\tau \|w\|_{H\Omega}^{2}\,dt; \tag{19} \]
\[ \|w\|_{\hat H Q_\tau}=\max_{0\le t\le \tau}\|w\|_{H\Omega}. \tag{20} \]
The existence and uniqueness theorems in the space \(H_{Q_\tau}\) are established for system (17) analogously to what was set forth for system (2). Equalities of the type (10) and (15) ensure that the solution belongs to the space \(\hat H_{Q_\tau}\).
Let us note that the Bubnov—Galerkin system for problem (17) has the form
\[ \ddot q_k^{(n)}+\varepsilon_1\dot q_k^{(n)} +\sum_{i=1}^{n}c_{ik}\ddot q_i^{(n)} +\varepsilon_2\sum_{i=1}^{n}c_{ik}\dot q_i^{(n)} +\frac{\partial\Phi_n\bigl(q_1^{(n)},\ldots q_m^{(n)}\bigr)} {\partial q_k^{(n)}}=Z_k(t) \tag{21} \]
(see formulas (13) and (14)).
We now apply methods of ordinary differential equations (see \(({}^8)\)) for a further investigation of system (17). Consider the function \(V_n\)
\[
V_n=\frac12\sum_{k=1}^{n}\dot q_k^{(n)2}+\Phi_n
=\frac12\sum_{i,k=1}^{n}c_{ki}\dot q_k^{(n)}\dot q_i^{(n)}+
\]
\[
+c_0^2\left[\frac12\sum_{k=1}^{n}q_k^{(n)2}
+\frac{1}{\varepsilon_1}\sum_{i,k=1}^{n}c_{ik}\dot q_k^{(n)}q_i^{(n)}
+\right.
\]
\[
\left.
+\frac{1}{\varepsilon_1}\sum_{k=1}^{n}\dot q_k^{(n)}q_k^{(n)}
+\frac{\varepsilon_2}{2\varepsilon_1}\sum_{i,k=1}^{n}c_{ik}q_i^{(n)}q_k^{(n)}
\right].
\tag{22}
\]
Here \(c_0\) is a suitably chosen constant, \(c_0=c_0(\varepsilon_1,\varepsilon_2)\). For \(V_n\) the estimates
\[ c_1\|w^{(n)}\|_{H\Omega}^{2}\geq V_n \geq c_2\|w^{(n)}\|_{H\Omega}^{2}. \]
Differentiate \(V_n\) with respect to \(t\); by virtue of equations (21)
\[ \begin{aligned} \frac{dV_n}{dt} &= -\varepsilon_1 \sum_{k=1}^{n} \dot q_k^{(n)2} -\varepsilon_2 \sum_{ik=1}^{n} c_{ik} q_i^{(n)} \dot q_k^{(n)} +{}\\ &\quad + c_0^2 \left[ -\sum_{k=1}^{n} q_k^{(n)} \frac{\partial \Phi_n}{\partial q_k^{(n)}} +\frac{1}{\varepsilon_1}\sum_{k=1}^{n}\dot q_k^{(n)2} +\frac{1}{\varepsilon_1}\sum_{i,k=1}^{n} c_{ik} q_i^{(n)} \dot q_k^{(n)} \right] +{}\\ &\quad +\sum_{k=1}^{n}\dot q_k^{(n)} Z_k +c_0^2\sum_{k=1}^{n}q_k^{(n)} Z_k \le -a_0^2 V_n + b_0^2 Z_0^2 . \end{aligned} \]
Here
\[ Z_0^2=\max_{0\le t\le +\infty}\int_{\Omega} Z^2(x,y,t)\,d\Omega . \]
By virtue of (23) and (24), the function \(V_n\) is a Lyapunov-type function (see (9)) for the system (21). With the aid of \(V_n\) it is established that the system (21), and consequently (17), is generalized dissipative, namely: let \(\varphi_1\) and \(\varphi_2\) be admissible initial conditions for the system (17),
\[ w\big|_{t=0}=\varphi_1,\qquad w_t\big|_{t=0}=\varphi_2; \]
let \(w\) be a solution of system (17) under the indicated initial conditions. Then for all admissible \(\varphi_1\) and \(\varphi_2\) one can indicate an \(R>0\) such that to each admissible pair \(\varphi_1,\varphi_2\) there corresponds a \(t_0\), beginning with which the inequality \(\|w\|_{H_\Omega}^{2}\le R^2\) holds for almost all \(t\in[t_0,+\infty)\). We shall call the initial conditions admissible if \(\operatorname{grad}\varphi_2\in L_{2\Omega}\), \(\Delta\varphi_1\in L_{2\Omega}\). From the generalized dissipativity of system (17) it follows:
Theorem. Under a periodic load \(Z(x,y,t+T)=Z(x,y,t)\), problem (17) has in the space \(\hat H_{\Omega T}\) at least one periodic solution, and this solution can be found by the Bubnov–Galerkin method.
Remark. In the case of a circular symmetrically loaded plate under a periodic load \(Z(x,y,t+T)=Z(x,y,t)\), the system (17) is convergent for sufficiently large \(\varepsilon\) \((\varepsilon=\min(\varepsilon_1,\varepsilon_2))\), namely:
-
For all admissible initial conditions, in the space \(\hat H_{Q\infty}\) there exists a generalized solution of problem (17).
-
Equations (17) have a unique \(T\)-periodic generalized solution \(w_0(t)\in\hat H_{QT}\).
-
This solution is stable in the sense of Lyapunov in the metric \(\hat H_{Q\infty}\).
-
For any solution of problem (17) \(w(t)\in\hat H_{Q\infty}\), the limiting relation
\[ \lim_{t\to+\infty}\|w(t)-w_0(t)\|_{H_\Omega}=0 \]
holds.
Leningrad Technological Institute
of the Pulp and Paper Industry
Received
1 XII 1966
CITED LITERATURE
- I. I. Vorovich, Izv. AN SSSR, Ser. Mat., 21, No. 6, 747 (1957).
- N. F. Morozov, Izv. Vyssh. Uchebn. Zaved., Matematika, No. 3 (1965).
- N. F. Morozov, M. E. Yudovin, Tr. Leningradsk. Tekhnologich. Inst. Tsellyulozno-Bumazhn. Prom., issue 17 (1965).
- N. F. Morozov, Zhurn. Vychislit. Matem. i Matem. Fiz., 6, issue 4, 773 (1966).
- S. P. Timoshenko, Theory of Oscillations in Engineering, 1932.
- Ya. S. Uflyand, PMM, 12, issue 3 (1948).
- O. A. Ladyzhenskaya, Matem. Sbornik, 39, issue 4 (1956).
- V. A. Pliss, Nonlocal Problems in the Theory of Oscillations, “Nauka,” 1964.
- I. G. Malkin, Theory of Stability of Motion, Moscow–Leningrad, 1952.