**Application of the Lyapunov Exponents and Wavelets to Study and Control of Plates and Shells**

J. Awrejcewicz, V.А. Krysko, I.V. Papkova, Т.V. Yakovleva, N.A. Zagniboroda, М.V. Zhigalov, A.V. Krysko, V. Dobriyan, E.Yu. Krylovа and S.A. Mitskevich

Additional information is available at the end of the chapter

http://dx.doi.org/10.5772/57452

**1. Introduction**

We study regular and chaotic vibrations of continuous mechanical systems using the following structural members: plates, flexible shells and cylindrical panels. They are often used in various measurement devices, and numerous ship and planes constructions. Owing to a high devel‐ opment of technology and due to industrial requirements, in many cases the mentioned structural members are subjected to action of high intensity loads being spatially and time dependent. The structural member exhibit either regular or chaotic dynamics, and hence an important question arises: How to predict safe and dangerous regimes of behavior of the studied mechanical objects?

In order to solve the mentioned problems, the investigations are carried out using the ach‐ ievements of the qualitative analysis of differential equations and non-linear dynamics. Namely, we analyze time histories (signals), phase and modal portraits, Poincaré sections, autocorrelation functions, Lyapunov exponents, Fourier and wavelet spectra. Charts of vibration regimes versus the load excitation amplitude and frequency are constructed, which allow to control the vibration character of plates and shells.

Usually the data provided by numerical experiments are presented in time domain. In other words, we take time as an independent co-ordinate, and amplitude as a dependent co-ordinate, and the studied signal as analyzed through its amplitude-time representation. However, in order to understand deeply non-linear continuous systems subjected to various types of load actions and in order to fully understand the occurring dynamics, we have to apply the

information hidden in the spectral signal characteristics. The Fourier transformation has been applied for a long time. However, it has been demonstrated recently that the Fourier analysis (FFT) is reliable only for the study of frequency components of stationary processes, i.e. the processes which through the whole period of investigation keep constant frequency compo‐ nents in time. It happens that in particular the dynamics of continuous mechanical systems may exhibit quite complicated output, and their frequency characteristics may change strongly in time. This is why in spite of the standard Fourier approach the wavelet analysis is applied allowing us to detect and understand many interesting non-linear phenomena of the men‐ tioned mechanical systems.

2

*a w ua x t w u xt*

= = = ==

, ,, ,

 e

*hh a h a a Eg qa p q p p h E*

g

== ==

l

t

and the following initial conditions

*x* ∈ 0, 1 into 120 parts.

with the help of the neural networks approach.

neural network self-teaching process ( *dW*

conditions:

, , ,,,

and bars over the non-dimensional quantities have been already omitted in equations (1).

We demonstrate how to determine four first Lyapunov exponents applying pinned boundary

The boundary value problem (1), (3), (4) is reduced to the Cauchy problem via FDM (Finite Difference Method) of the second order accuracy. The obtained ODEs are solved by the Runge-Kutta method of the fourth and sixth orders. Validity and reliability of the obtained numerical results are confirmed by the FEM results (Finite Element Method). The initial problem of infinite dimension is substituted by that of finite dimension via partition of the interval

One of the ways to compute the spectrum of Lyapunov exponents is the neural network approach based on the generalized Benettin algorithm. It includes the following successive steps: 1. Choice of the appropriate time delay via tests; 2. Computation of an embedding space dimension; 3. Reconstruction of pseudo-phase trajectories using the method of time delays; 4. Neural network approximation; 5. Teaching of neural networks to compute successive iteration vectors; 6. Computation of the spectrum through the generalized Benettin algorithm

We apply the neural network with the following properties: It is an analog network regarding the input data (information is delivered through real numbers); It is self-organized with respect to its teaching aspects (output space of solutions is defined only through the input data); It belongs to the neural networks of straight signal distributions (all neural network couplings come from the input neurons and go to the output neurons); the neural network possesses dynamic couplings (control and improvement of synaptic couplings is carried out during the

Let point *x*<sup>0</sup> belong to attractor *A* of a dynamical system. The trajectory of evolution of point is called the unperturbed trajectory. We choose the positive quantity *ε* essentially less than the attractor dimension. Next, we choose an arbitrary (perturbed) point in a way to satisfy the

4 4

Application of the Lyapunov Exponents and Wavelets to Study and Control of Plates and Shells

// // (0, ) (1, ) (0, ) (1, ) (0, ) (1, ) 0, *w t wt u t ut w t w t xx xx* = === = = (3)

*wx wx ux ux* ( ,0) ( ,0) ( ,0) ( ,0). = == & & (4)

*dt* ≠0), where *W* stands for the net weight coefficients).

t

(2)

3

http://dx.doi.org/10.5772/57452

Chaotic dynamics of structural members has been investigated by many researchers [1-10]. In this work we propose a novel approach to study non-linear vibrations of a plate based on the neural network approach and we analyze dynamics of flexible shells with constant stiffness and density subjected to harmonic load action. In the latter case mathematical model is built on the Kirchhoff-Love hypothesis and taking into account non-linear relation between deformation and displacement in the von Kármán form. This approach yields a system of nonlinear PDEs regarding the deflection function and stresses (Airy's function) as well as the system of equations regarding displacements [11]. We use further FDM with approximation *O*(*h* <sup>2</sup> ) and BGM in higher approximations, which allows to study the system with infinite numbers of degrees of freedom without any truncation of the obtained system of ODEs, which is solved via the fourth-order Runge-Kutta method.
