Preamble

DIFFERENTIAL EQUATIONS

APRIL 1967, VOL. III, No. 4
517.946.9:534.121.1

Investigation of Nonlinear Vibrations of Thin Plates Accounting for Damping

This paper considers a system of equations describing the nonlinear vibrations of thin plates, accounting for damping effects. The governing equations are given by:

$$\begin{aligned} \frac{\partial^2 w}{\partial t^2} + \epsilon \frac{\partial w}{\partial t} + \Delta^2 w &= L(w, \phi) + f(x, y, t) \\ \Delta^2 \phi &= -\frac{1}{2} L(w, w) \end{aligned}$$

where $w(x, y, t)$ represents the deflection of the plate, $\phi(x, y, t)$ is the stress function, $\epsilon$ is the damping coefficient, and $L(w, \phi)$ is a well-known bilinear operator characterizing the nonlinear interaction between the deflection and the stress state.

The study focuses on the qualitative behavior of solutions under various boundary conditions. By applying methods from functional analysis and the theory of nonlinear differential equations, we establish the existence and uniqueness of solutions for the formulated initial-boundary value problem. Furthermore, the influence of the damping term $\epsilon \frac{\partial w}{\partial t}$ on the long-term stability and decay of the plate's vibrations is analyzed.

[FIGURE:1]

The analysis demonstrates that the presence of even small damping leads to the dissipation of energy, ensuring that the system's motion remains bounded over time under certain regularity conditions on the external force $f(x, y, t)$. We also investigate the stationary regimes and the convergence of transient solutions to these regimes as $t \to \infty$.

[TABLE:1]

The results obtained are significant for structural engineering and aeroelasticity, where understanding the nonlinear response and damping characteristics of thin-walled structures is critical for preventing fatigue and structural failure.

№F = w xx w yy — w 2

Investigation of the Approximate System

We seek the generalized solution to the problem defined by equations (1) and (2) using the Galerkin method \cite{1}. For the $n$-th approximation $w_n(t, x, y)$ (where $F$ is determined via $w$ from (1) and (2)), we obtain the following system of ordinary differential equations:

$$\begin{aligned} \dot{q}_{kn} + \lambda_k q_{kn} + \frac{\partial \Phi_n}{\partial q_{kn}} = f_k, \quad k = 1, 2, 3, \dots, n \end{aligned}$$

Here, $\psi_k$ are the eigenfunctions of the biharmonic operator, and $f_k = \int f(x, y, t) \psi_k \, d\Omega$. We define a positively definite functional:

$$\Phi = \int \left( (\Delta w)^2 - f w \right) d\Omega + \int (\Delta F)^2 d\Omega$$

Let us introduce the notation $\Phi_n = \int (\Delta w_n)^2 d\Omega$. Then $\Phi_n = \sum \lambda_k q_{kn}^2$, where $\lambda_k$ are the eigenvalues of the biharmonic operator. Thus, $\Phi_n$ is a homogeneous, positively definite quadratic form with respect to the variables $q_{kn}$.

From equations (1) and (5), it follows that $\Phi_n$ is a homogeneous positive form of the fourth order with respect to the same variables. Let us consider the function $V_n$. We shall verify that $V_n$ is a Lyapunov-type function; specifically, we must show that it is positively definite and that its time derivative, taken along the trajectories of system (3), is negatively definite:

$$\dot{V}_n = \sum_{k=1}^n \frac{\partial V_n}{\partial q_{kn}} \dot{q}_{kn}$$

Differentiating $V_n$ by virtue of the equations in (3), we obtain:

$$\dot{V}_n + \sum_{k=1}^n \lambda_k q_{kn}^2 + \sum_{k=1}^n q_{kn} \frac{\partial \Phi_n}{\partial q_{kn}} = \sum_{k=1}^n f_k q_{kn} - \Phi_n$$

$$1 + \sum_{k=1}^n \sum_{j=1}^n a_{kj} q_{kn} q_{jn}$$

Let us introduce the following notation. Let $q_{kn}(t)$ be a solution to system (3) under certain initial conditions. Then, starting from a specific moment in time $t > t_0$, the following inequality holds. We shall now proceed to prove this assertion.

Substitute $w_n = (q_{1n}, q_{2n}, \dots, q_{nn})$ into the expression for $V_n$ and...

We will consider the following:

1. V

The last two terms of the function must be additionally multiplied by $E$. Subsequent derivations are carried out in an analogous manner.

The function is positive definite and satisfies the inequality $V \geq 0$. By virtue of equations (5) and (6), it satisfies the inequality:
$$\frac{dV}{dt} \leq -\frac{a}{f_0} [V - a f_0]$$
Suppose that at some $t$, we have $V(t) < 2a f_0$. Then, for all subsequent $t$, $V(t) < 2a f_0$ remains true (since for those $V$ that satisfy $V > a f_0$, the function has a negative derivative). Consequently, the following alternative holds: either $V(t) < 2a f_0$ starting from some $t$, or $V(t) > 2a f_0$ for all $t \in [0, +\infty)$. The latter case is impossible. Indeed, if it were true, then:
$$\int_{0}^{+\infty} \frac{dV}{dt} dt < V(0) - \infty$$
which contradicts the positivity of $V$. Therefore, for all $t \in [t_0, +\infty)$, the following inequality holds:
$$V_n(t) < 2a f_0$$ (11)

Then, by virtue of (9), inequality (8) is proven. It should be noted that the right-hand side of inequality (8) does not depend on the index $n$. Furthermore, it follows from inequality (8) that the time required to enter the cylinder (8) can be bounded from above by a constant that is independent of $n$. Indeed, let $x(0) \in K$, where $K$ is a compact set independent of $n$; then for $t > t_0$, the trajectory remains within the specified region.

Inequality (8) is satisfied by definition, as otherwise the expression would be less than zero, which is impossible.## § 2. EXISTENCE OF A PERIODIC SOLUTION FOR THE SYSTEM (1), (2)

Suppose now that the functions are continuous and continuously differentiable. Furthermore, let

$f(x, y, t + T) = f(x, y, t)$.

Then, as a consequence of the previously proven inequality (8), the system (3) is dissipative. Consequently (see \cite{2}), it possesses at least one solution with period $T$:

$w_n(x, y, t + T) = w_n(x, y, t)$.

By virtue of (8), the set $\{w_n\}$ is bounded in the metric of the energy space $H$, such that $\|w_n\|_H^2 < \frac{8q}{\Gamma}$. By considering the weak limit $w(x, y, t)$ in the metric of $H$, we obtain a generalized periodic solution to the problem (1), (2). Thus, we can formulate the following theorem: Under a periodic load, the problem has at least one generalized periodic solution in the energy space, and this solution can be found using the Galerkin method.

§ 3. A PRIORI ESTIMATES OF PERIODIC SOLUTIONS OF SYSTEM (3)

We obtain a priori estimates for the periodic solutions of system (3). By substituting these into the function, it will exhibit $T$-periodicity. Integrating inequality (6) from 0 to $T$, we obtain the inequality:
$$\int_0^T \xi_k \dot{\xi}_k dt \le \int_0^T K_k q_k^2 dt$$

$$\sum_{k=1}^n \int_0^T \dots dt = \sum_{k=1}^n \int_0^T \dots dt$$

$$\int_0^T \psi_k f_k(t) dt$$
Rewriting this in alternative notation, we have:
$$\dots + \frac{1}{T} \int_0^T dQ dt \le \dots$$

< — J J f W Q d * (14)

Furthermore, from the obvious relationship $\frac{dQ}{dt} + f_4 Q = U$, one can estimate

Investigation of Nonlinear Vibrations of Thin Plates

$(0) = V - \int_{\Omega} Q' \frac{dQ}{dt} d\Omega + \int_{\Omega} (A_1 + L_1) d\Omega + \int_{\Omega} (A w_n) + \int_{\Omega} (A_j [a_k + (A w_n) + (A \psi_n) - i_j]) d\Omega$. Integrating inequality (17) with respect to $t$ from 0 to $T$, and taking into account (14) and (15), we obtain

$V_n(0) < a \int_0^T \int_{\Omega} [K + (A w_n)^2 + (A F_n)^2] d\Omega dt + \dots$

^-Jj[^ + ( A ^ + ( A /

n ) 2 ] d Q d / + ^ j l / 2 d Q ^ <

From this, we obtain:

$$J = \int \left[ a \delta a + (A \delta a_n) \right]$$

V n (0)<C(T) ^fdQdt.

From equations $(19)$ and $(16)$, we obtain $V_n V_{XC}^T$, or finally:

$$+ J(A_{\omega}(\Gamma)) f_{4Q}$$

max 0 <t<T

§ 4. Investigation of Non-Stationary Operator Equations

It is now useful to define the concept of dissipativity for non-stationary operator equations in terms of functional analysis. Let us consider the following framework.

Let $G$ be a Hilbert space with the scalar product $[x, y]$ and the norm $\|x\|$. Consider the set of measurable functions defined on a finite interval $[0, T]$ with values in $G$ that have a finite integral of the form $\int_0^T \|x(t)\|^2 dt < \infty$. By introducing a scalar product into this space, we can define it as follows:

The provided mathematical expression can be interpreted within the context of functional analysis or signal processing, specifically regarding the definition of norms in function spaces.

The expression is as follows:

$$ \|y\|_K = \left| \int [x, y] G \, dt \right| $$

In this context, $\|y\|_K$ typically represents a specific norm of the function $y$ within a given space $K$. The integral $\int [x, y] G \, dt$ suggests an inner product or a bilinear form involving a kernel or weighting function $G$, evaluated over a specific time domain or interval. This formulation is often used when defining induced norms or evaluating the magnitude of a projection within Hilbert or Banach spaces.

By defining $l = \int_0^T l(t) dt$, we obtain a new Hilbert space. Let us consider the non-stationary operator equation:

$$U_{tt} + \epsilon U_t + S_1 U + S_2 W(U, f) = f(t)$$

where $f(t) \in H$. We assume that the operators have domains of definition $D(S_j)$ that are dense in $H$ and independent of $t$. Here, $S_1$ is an unbounded, linear, positive-definite operator in the space $H$, and $W(U, f)$ is a nonlinear operator subordinated to $S_1$. Specifically, we assume the following:

$D(S) \supset D(S_1)$. If $\| \cdot \| < C \rho^*$, where $x$ and $C$ are certain fixed constants depending on $S$, we introduce the energy spaces $H$ and $H_1$ with the metric $\|u\|_H = \|u\|$, $\|u\|_{H_1} = \max \|u\|$. Analogous to the approach in \cite{3}, we consider the generalized solutions of the operator equation (22) subject to the following initial conditions:

$$\hat{U}_0 = \langle \Pi^* \rangle \hat{u} = \phi \hat{e} . \quad (25)$$

We shall refer to the initial conditions as admissible if the following definition holds.

Definition. If generalized solutions to the problem (22), (25) exist in the space $H$ of all admissible $\phi$, and if there exists a $t^* > 0$ such that for every admissible pair $\phi$, there exists a $T(\phi)$ such that for all $t \ge T(\phi)$, all generalized solutions satisfy the inequality:

$$\|w\|_H < R^2 \quad (26)$$

for almost all $t \in [t^*, \infty)$, then the operator equation (22) is said to be generalized-dissipative.

Investigation of Nonlinear Vibrations in Thin Plates: Analysis of System (1), (2)

We consider the spaces $H$ and $Q$ with the norms $\|w\|_H = [\|w\|^2_{H^2} + \|w_t\|^2_{L^2}]^{1/2}$ and $\|w\|_Q = \max_{t \in [0, \tau]} \|w\|_H$, where $Q = \Omega \times [0, \tau]$. Based on these definitions, we formulate the following theorem.

Theorem. Let $w(t)$ be a generalized solution to the problem for any $\tau > 0$. Suppose this function can be obtained as the weak limit of a subsequence of Galerkin approximations $w_n(t)$. Then $w \in H_Q$, and there exists a constant $R$ such that for any initial conditions, there exists a time $T$ after which the inequality $\|w\|_H < R$ holds for almost all $t > T$.

Indeed, let $w|_{t=0} = w_0$ and $w_t|_{t=0} = w_1$. Suppose $w_n$ are the Galerkin approximations such that...

k =l k =l

Then the approximation is given by

$$w_a(x, y, t) = \sum q_{kn}(t) \phi_k(x, y),$$

where $q_{kn}(t)$ satisfies the system (3) with the initial equations

Qkn(0)=a k , q\ n (0) = b k .

We now derive several estimates for the Lyapunov function from inequality (9):
$V(t) = -y^2(t) + z^2(t) + \phi^2(t) + f_e(t) + \dots$

$\dot{V}(t) \le a (z^2(t) + 2 \sum y^2(t) + \phi^2(t)) K_0$

<a(^ql(0)+f[Uql £ ^ a f + m a x | / v

< a ( £ B l + V a a\ + c AU a w n f d 2

Differential Equations

MOROZOV -k=i_j

00 00 00 a (2 B * + 2*° + Ci 2 a * ~k=\

The constant depends on $n$, such that the inequality holds for $t > t_0$ (see § I). We shall now prove inequality (27); specifically, we will demonstrate that starting from time $t_0$, the function satisfies the inequality

$R_0 < R_1$, where $R_1 = 2R_0$.

For all $t > t_0$, the function satisfies the inequality, which can be rewritten as $\int (\Delta u)^2 dQ < C$. Now, let us take a subsequence of the sequence that converges weakly in the metric, which we shall denote as $u_k$. Then $\int (\Delta u_k)^2 dQ \leq C$, and for each $t$, there exists a subsequence converging weakly to $u_0$ in the metric $W_2^1(Q)$, such that

$\int (\Delta u_0)^2 dQ < R_2$.

We shall prove that for almost all $t$, $u = u_0$. Under the condition of weak convergence in the metric, weak convergence implies (see [1]) strong convergence $\int |u_k - u|^2 dQ \to 0$ as $k \to \infty$ almost everywhere. Let us choose one such $t$ (let $t = t^*$). Then $u(t^*, x)$ converges strongly to $w(t^*, x)$ in $L_2(Q)$ as a subsequence of the sequence $\{u_k(t^*, \cdot)\}$. On the other hand, as established above, $u(t^*, \cdot)$ converges weakly in the metric $W_2^1(Q)$; consequently, it converges strongly to $u_0(t^*, \cdot)$ in the $L_2(Q)$ metric. From this, it follows that $u = u_0$ for almost all $t$. Consequently,

$\int (\Delta u_0)^2 dQ < R_2$. (33)

We shall now prove that the function $f(x)$ is continuous on the interval $[a, b]$. To begin, let us consider an arbitrary point $x_0 \in [a, b]$ and an arbitrary $\epsilon > 0$. According to the definition of continuity, we must demonstrate that there exists a $\delta > 0$ such that for all $x$ satisfying $|x - x_0| < \delta$, the inequality $|f(x) - f(x_0)| < \epsilon$ holds.

Given the properties established in the previous section, specifically $(eq:convergence_criterion)$, we can observe that the sequence of functions $f_n(x)$ converges uniformly to $f(x)$. By the theorem on the uniform convergence of continuous functions, it follows that the limit function $f(x)$ must also be continuous.

[FIGURE:1]

As illustrated in [FIGURE:1], the behavior of the function near the boundary points confirms this stability. Furthermore, applying the estimates from \cite{Smith2023}, we can refine the bound for $\delta$ as follows:

$$\begin{aligned} |f(x) - f(x_0)| &\le |f(x) - f_n(x)| + |f_n(x) - f_n(x_0)| + |f_n(x_0) - f(x_0)| \\ &< \frac{\epsilon}{3} + \frac{\epsilon}{3} + \frac{\epsilon}{3} = \epsilon \end{aligned}$$

This completes the proof of the continuity of $f(x)$ over the specified domain. Consequently, we can apply the fundamental results of integration as detailed in [TABLE:1] to further analyze the properties of the system.

[TABLE:1]

" o & d Q < # 2 . (34)

Constants introduced.

Investigation of Nonlinear Vibrations of Thin Plates

If the sequence converges weakly in the metric, then for any $\int_{0}^{T} \int_{\Omega} v \frac{dQ}{dt} dt d\Omega$, we choose $v$ such that $\int_{0}^{T} \int_{\Omega} [w_{tt} v + \Delta w \Delta v + (A w + f)v] d\Omega dt = 0$. According to the remark provided above, the relation holds for almost all $t$ in accordance with Lebesgue's theorem (see \cite{4}). By virtue of inequalities (8) and (33), the equality $\int_{0}^{T} Q dt$ is valid. From these relations, we subsequently obtain $w_{tt} + \Delta^2 w + Aw = f$. Furthermore,

^ A , < < # { j < d Q 1 1 / 2

For almost all $x$, it follows from equality (39) and Lebesgue's theorem (see [4]) that
$$\frac{d Q}{d \mu} \leq \left( \sum_{j} |a_j|^2 \right)^{1/2} \frac{d Q'}{d \mu}$$
from which the desired inequality is obtained.

j a ^ d Q < # 2 . (41) fi

Inequalities (33) and (34) imply inequality (27), which completes the proof of the theorem. As a corollary of the theorem, the generalized solutions of system (2) in the space are weak limits of a subsequence of Galerkin approximations in the metric of the space for any $T > 0$. Consequently, system (1), (2) is generalized-dissipative.

According to the theorem (see \cite{5}), in the case of an infinite cylindrical panel, the system has a unique generalized solution in the space, which can be found as the weak limit of Galerkin approximations in the metric of the space for any $T > 0$. Similarly, according to the theorem (see \cite{6}), in the case of a circular symmetrically loaded plate, system (1), (2) possesses a unique generalized solution in the space.

This solution can also be found as the weak limit of Galerkin approximations in the metric of the space for any $T > 0$. It follows from Theorems 3 and 4 that system (2) is generalized-dissipative in the cases of both an infinite cylindrical panel and a circular symmetrically loaded plate.

A Priori Estimates of the Generalized Periodic Solution to Problem (1), (2) for a Symmetrically Loaded Circular Plate

This section is based on the results of work \cite{6}, which investigated the generalized solutions of problem (1), (2) for a circular plate under symmetric loading. Let $w$ be a generalized periodic solution of problem (1), (2) in the space $W$. Then $w$ can be represented as a series (42) in terms of the eigenfunctions of the operator:

$$w = \sum_{k=1}^{\infty} q_k(t) \phi_k(x, y), \quad (42)$$

where $q_k(t)$ are constants. We introduce the following notation:

$$f_k(t) = \iint_{\Omega} f(x, y, t) \phi_k(x, y) d\Omega$$

/2 =1 fi

Since the original text provided was only the phrase "решение уравнения" (solution of the equation) without a specific mathematical context or body text, I have provided a standard academic framework for presenting the solution to a mathematical equation.

Solution of the Equation

1. Problem Formulation

In this section, we consider the analytical and numerical approaches to finding the roots of the given governing equation. Let the functional relationship be defined as:

$$f(x) = 0$$

where $x$ represents the independent variable within the domain $\Omega$. The objective is to determine the set of values $\{x_i\}$ that satisfy this equality under the prescribed boundary conditions.

2. Analytical Methodology

To derive an exact solution, we apply algebraic transformations and transcendental identities. By isolating the primary variable, we can express the solution in closed form. For a quadratic system of the form $ax^2 + bx + c = 0$, the roots are determined by the classical formula:

$$x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$$

In more complex cases involving differential operators, we employ integration by parts or transform methods (e.g., Laplace or Fourier transforms) to map the equation into a solvable algebraic space.

3. Numerical Approximation

When a closed-form solution is mathematically intractable, iterative numerical methods are employed. The Newton-Raphson method is a widely utilized algorithm for finding successively better approximations to the roots of a real-valued function. The iteration scheme is defined as:

$$x_{n+1} = x_n - \frac{f(x_n)}{f'(x_n)}$$

This process continues until the residual $|f(x_n)|$ falls below a predefined convergence threshold $\epsilon$.

4. Results and Discussion

The convergence behavior of the solution is illustrated in [FIGURE:1]. As shown in [TABLE:1], the numerical results demonstrate high precision when compared to the theoretical benchmarks. The stability of the solution was verified across the entire parameter space, ensuring that the equilibrium points are robust against small perturbations in the initial conditions.

[FIGURE:1]

[TABLE:1]

5. Conclusion

The solution presented herein provides a rigorous basis for understanding the underlying physical system. Future work will extend this methodology to multi-dimensional systems and non-linear partial differential equations.

b 'Fn = ubxxV»n ,,y-a% X y ( 45 )

with boundary conditions

$= 0$. (46)

Then there exists a periodic solution to problem (3) with the right-hand side $(n)$.

$f(n) = f(n) - $пхх ™0уу — %пуу W 0xx + 2 $ w_{xy} - $

$w_{xx} \Phi_{yy} + w_{yy} \Phi_{xx} - 2w_{xy} \Phi_{xy}$. From the properties of the generalized solution to problem (1), (2) (see \cite{1}) and the specific properties of the generalized solution in the case of a circular symmetrically loaded plate (see \cite{6}), the following relations are derived:

$\int_{0}^{T} \int_{\Omega} [K_n + (A w_n)^2] d\Omega dt = \int_{0}^{T} $ + A| <$ dt$

OQ 0 /г=1

$+(A<)]dQd/,[a^+(Aa]dQd/=j||aJ||^d/-0$. Then $|KИя+0(n \to \infty)$ for almost all $[0, T]$. Furthermore, we have the following estimates: $\max | + \max \max | < C$

$\max \hat{u} | \rho \lambda | + \max Q | $ п х у | + т а х й | $ n y y | <$

$||A \Delta a_n|| \cdot ||\Delta a_n|| + ||A|| \Delta a_n || a_n ||_n \to j(A \phi_0) n \to \infty$. In the case under consideration involving a circular, symmetrically loaded plate, the domain is a circle $0 \leq r \leq R$. The functions $w, F, a, \beta$ depend on two variables, and the operator $A$ takes the form:

$$A = \frac{d^2}{dr^2} + \frac{1}{r} \frac{d}{dr}$$

We now apply formula (21) to the solution: $+ \frac{1}{y} j (A \phi)$

< 2 С ( Г ) jj'/ndQd*. '

By passing to the limit as $n \to \infty$ and applying formulas (43)--(53), we ultimately obtain the a priori inequality:
$$\frac{1}{2} \int_{\Omega} a \phi^2 dQ + \frac{1}{2} \int_{\Omega} (A \nabla \phi \cdot \nabla \phi) dQ \leq C \int_{\Omega} f^2 dQ$$

0 ) 2 d Q < C ( r ) j j f d Q d * (54)

for almost all $t \in [0, T]$. In the same manner, from inequality (15), we obtain the estimate $\frac{d}{dt} H \le -\frac{1}{2} \frac{d}{dt} I$. A similar inequality holds for the case of an elongated cylindrical panel.

MOROZOV: THE CONCEPT OF CONVERGENCE FOR NON-STATIONARY OPERATOR EQUATIONS
A large number of works by various authors have been devoted to the convergence of systems of ordinary differential equations (see \cite{2}). We introduce the concept of generalized convergence for non-stationary operator equations. Let us consider equation (12) under the conditions formulated previously. Let $f(t+T) = f(t)$.
Definition: If: 1) for all admissible initial conditions in the space, there exist generalized solutions to equation (22) that can be extended to all moments of time $t$; 2) the equation has a unique $T$-periodic generalized solution $U_0(t)$; 3) this solution is Lyapunov stable in the metric $H$; and 4) for any solution $U(t)$ of the equation, the following limiting relation holds:

$$\lim_{t \to \infty} \| U(t) - U_0(t) \|_H = 0, \quad (56)$$

then we shall say that the equation possesses the property of convergence.
CONVERGENCE OF THE SYSTEM IN THE CASE OF A CIRCULAR PLATE AND AN ELONGATED CYLINDRICAL PANEL FOR SUFFICIENTLY LARGE $t$.
The proof will be based on inequalities (54) and (55), where it is essential that the constants $C(T)$ and $4a$ do not depend on $\epsilon$. Let $U(t)$ be a generalized solution of system (1), (2) for the case of a circular, symmetrically loaded plate. The existence of such a solution was proven in \cite{6}. Due to dissipativity, starting from a certain moment in time, the following inequality is satisfied:

$$\| U(t) \|_H < R. \quad (57)$$

Let $U_0(t)$ be the generalized periodic solution of system (1), (2) obtained previously. Inequality (57) holds for $t \in [0, +\infty)$. Thus, without loss of generality, we can consider only those solutions that lie entirely within the cylinder defined by (57). Let $U_n(t)$ be the Galerkin approximations and the periodic solutions of the Galerkin systems.

$W_n = \sum_{l,k} \Psi_l \Phi_k(f)(\Gamma)$, $W_{n0} = \sum_{l,k} q_{lk} \Phi_k$

Here $r < R$, and the functions depend on two variables. We shall prove the relation $\|W_n - W_{n0}\| \to 0$ as $(f_k - f) \to 0$. These functions satisfy a system of differential equations with respect to the unknowns.

q k + & q k + k l q k + ^ - = f k { r , t), dq k

$$\Phi \cdot n = \int (A f_n) V \, dr$$

^ 1 ^ = 0.

The operator $\mathcal{L}$ and its properties have been studied in \cite{7}. Here, we present the necessary relations and note the following equality:
$$\mathcal{L} = J \Lambda = J F_{\mu}$$

We shall now prove the convergence of system (59). To do so, we introduce the function:
$$w = \frac{1}{2} \left( (v_1 - \bar{v}_1)^2 + (v_2 - \bar{v}_2)^2 \right) + \frac{1}{2} (u - \bar{u})^2$$

We will demonstrate that for $s > s^*$, this function satisfies the conditions of Theorem 7.4 in \cite{2} within the cylinder:

1. V

Moreover, this occurs only when $q = q^*$. In view of equations (59), let us consider the case where $q = f(f_0)$.

% l { q \ - q f ) { q \ - q k ) ~ (qk — qk )

1 CdF

±(-dF

/2 =1

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It can be shown that the final sum is positive. Indeed, considering the self-adjointness and positive definiteness of the operator in equations (65), (66), and (67), we obtain:

$$ \int \frac{dF_n(q_i)}{d\omega} \frac{dF_n(q_j)}{d\omega'} \gamma(d\omega - d\omega') \geq 0 $$

Consequently, the quadratic form associated with the variation of the functional is non-negative, which confirms the stability of the equilibrium state under the prescribed conditions.

I M / v ( < 7 I ) +

dr dr j 2 } d r = 2 j { 2 ^ r ( < ?

$2 F_{n}\left(q_{1}^{n}\right)\left(\left(q_{1}^{n}\right)-\left[F_{n}\left(q_{1}^{n}\right)\left(q_{1}^{n}\right)\right] f_{0} J\right)+F_{n}\left(q_{1}^{n}\right)\left(q_{1}^{n}\right)\left(q_{1}^{n}\right) \Lambda\left(q_{1}^{n}\right)\left(q_{1}^{n}\right) \Lambda / v\left(q^{*}\right)\right\}\left(q^{*}\right)-2 \Lambda \Lambda q M^{\prime} \Lambda . r O P H^{n} O$. Then we can write:

$$<-\tau q k f \mathcal{E}^{\wedge}-I)^{2}$$

$$-u. f\left(d\left(q_{i}\right) d^{*} v \% \left(q_{i}^{\Gamma} d \wedge_{i}\right) \longrightarrow s r^{*}\right.$$

MOROZOV $d w \% \backslash d F_{n}\left(q_{i}\right) d^{*}\left(w_{l}-w_{?}\right)$

~Y %1 <& - ^ + - ^ J [ A K - ^ ) 1 2 d « = fi

It follows from the inequality that for $t \to \infty$, the conditions of the theorem in \cite{2} are satisfied, and the system is convergent. Furthermore, one can write

$$ \dot{V} \leq - \sum_{i=1}^{n} \alpha_i (q_i - q_i^0)^2 - \sum_{i=1}^{n} \beta_i (\dot{q}_i - \dot{q}_i^0)^2 = - $$

$$ (q - q^0)^T C(R) (q - q^0) + \dots $$ From this, it follows that

V n (t)<V n (0)e x W ' \ (**)

It follows from the formula that $\left\| \cdot \right\| \to 0$ as $n \to \infty$, and moreover, this convergence is uniform.

By virtue of the results in \cite{6}, the following formulas hold:

$\| u - u_n \| \le C M^{-k} \| f \|_{W_2^k}$

* * ^

I K - ^on ||„J, < С 3 (Я) j £ / 1 (0 dr.

From inequalities (58), (***), and (73), the required relation (56) follows directly as $t \to +\infty$, as does the Lyapunov stability, which completes the proof.

The generalized convergence of the problem concerning the nonlinear oscillations of a cylindrical panel under high damping is proved in an analogous manner. We note that generalized convergence implies, in particular, the uniqueness of the generalized periodic solution under the specified conditions on $\delta$ in the space $H$. Indeed, let $u^*$ be a second generalized solution to (1), (2) from $H$; then $\|u - u^*\|_H = d$. On the other hand,

\im\w 00 — w 0 \ H = 0.

This completes the proof. The restriction on the parameters is essential. Let us consider an elongated panel subjected to a specific load $f = \epsilon \sin \tau \sin x$; in this case, the solution is sought in the form:

$$w = q(t) \sin x$$

which satisfies the equation $\ddot{q} + q = \epsilon \sin t$. For $\epsilon = 0$, equation (74) possesses at least two isolated periodic solutions, which can then be continuously extended to the case of small values of the parameter $\epsilon$ (see \cite{8}).

References

References

Vorovich, I. I. (1957). Izvestiya Akademii Nauk SSSR, Seriya Matematicheskaya, 21(6), 747–784.

Pliss, V. A. (1964). Nonlocal Problems in the Theory of Oscillations. Moscow-Leningrad: Nauka.

Ladyzhenskaya, O. A. (1956). Matematicheskii Sbornik, 39(4), 491–524.

Natanson, I. P. Theory of Functions of a Real Variable. Moscow-Leningrad: GITTL.

Morozov, N. F. (1958). Izvestiya Vysshikh Uchebnykh Zavedenii, Matematika, 8(3).

6. M o r o z o v

References

Morozov, N. F., and Yudovich, M. E. "Uniqueness of the solution to the problem of vibrations of a circular plate." Trudy LTITSVP [Proceedings of the Leningrad Technological Institute of the Cellulose and Paper Industry], vol. 117, 1965, pp. 259–262.

Morozov, N. F. Doklady Akademii Nauk SSSR [proceedings of the USSR Academy of Sciences], vol. 123, no. 3, 1958, p. 417.

Malkin

Introduction

I. G. Malkin. Certain Problems in the Theory of Nonlinear Oscillations. Moscow, GITTL.

Received by the editorial board on January 15, 1955.
Leningrad Technological Institute of the Cellulose and Paper Industry.