Investigation of nonlinear vibrations of thin plates with consideration of damping

N. F. Morozov

Submitted 1967-01-01 | SovietRxiv: ru-196701.56157 | Translated from Russian

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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