**2. Lyapunov exponents computation via neural networks an the generalized Benettin's algorithm**

We illustrate and demonstrate the efficiency of the neural network approach to study a flexible plate with an infinite length. The governing equations are within the Kirchhoff hypothesis and they have the following non-dimensional form [11]:

$$\begin{aligned} \frac{\hat{\boldsymbol{\sigma}}^2 \boldsymbol{u}}{\hat{\boldsymbol{\alpha}}^2} + L\_3(\boldsymbol{w}, \boldsymbol{w}) - \frac{\hat{\boldsymbol{\sigma}}^2 \boldsymbol{u}}{\hat{\boldsymbol{\sigma}}^2} &= 0, \\ \frac{1}{\hat{\boldsymbol{\lambda}}^2} \left\{ -\frac{1}{12} \frac{\hat{\boldsymbol{\sigma}}^4 \boldsymbol{w}}{\hat{\boldsymbol{\alpha}}^4} + L\_1(\boldsymbol{u}, \boldsymbol{w}) + L\_2(\boldsymbol{w}, \boldsymbol{w}) \right\} + q - \frac{\hat{\boldsymbol{\sigma}}^2 \boldsymbol{w}}{\hat{\boldsymbol{\sigma}}^2} - \boldsymbol{\varepsilon} \frac{\partial \boldsymbol{w}}{\hat{\boldsymbol{\sigma}} t} &= 0, \end{aligned} \tag{1}$$

where *L* 1(*u*, *w*), *L* 2(*w*, *w*), *L* <sup>3</sup>(*w*, *w*) – non-linear operators; *w*(*x*, *t*)– plate element bending in normal direction; *u*(*x*, *t*)– plate element longitudinal displacement; *ε*– dissipation coeffi‐ cient; *E*– Young modulus; *h* – height of the transversal panel cross section; *γ*– specific plate material gravity; *g* –Earth acceleration; *t* – time; *q* =*q*0sin(*ωpt*)– external load.

The non-dimensional parameters are as follows:

Application of the Lyapunov Exponents and Wavelets to Study and Control of Plates and Shells http://dx.doi.org/10.5772/57452 3

$$\begin{aligned} \lambda &= \frac{a}{h}, \quad \overline{w} = \frac{w}{h}, \quad \overline{u} = \frac{ua}{h^2}, \quad \overline{x} = \frac{x}{a}, \quad \overline{t} = \frac{t}{\tau}, \\\ \tau &= \frac{a}{p}, \quad p = \sqrt{\frac{\mathbb{E}g}{\gamma}}, \quad \overline{\varepsilon} = \frac{a}{p}, \quad \overline{q} = \frac{qa^4}{h^4 E} \end{aligned} \tag{2}$$

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 conditions:

$$\text{tr}(\mathbf{0},t) = \text{tr}(1,t) = \mathbf{u}(\mathbf{0},t) = \mathbf{u}(1,t) = \text{tr}\_{xx}^{\prime\prime}(\mathbf{0},t) = \text{tr}\_{xx}^{\prime\prime}(1,t) = \mathbf{0},\tag{3}$$

and the following initial conditions

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‐

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

) 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

We illustrate and demonstrate the efficiency of the neural network approach to study a flexible plate with an infinite length. The governing equations are within the Kirchhoff hypothesis and

( , ) 0,

**2. Lyapunov exponents computation via neural networks an the**

2 2 2 2 3 4 2

¶ ¶ + -= ¶ ¶

*u u L ww x t*

1 1 (, ) ( , ) 0, <sup>12</sup>

ì ü ¶ ¶ ¶ í ý - + + +- - = ï ï ¶ ¶ ¶ î þ

*<sup>w</sup> w w L uw L ww q x t <sup>t</sup>*

where *L* 1(*u*, *w*), *L* 2(*w*, *w*), *L* <sup>3</sup>(*w*, *w*) – non-linear operators; *w*(*x*, *t*)– plate element bending in normal direction; *u*(*x*, *t*)– plate element longitudinal displacement; *ε*– dissipation coeffi‐ cient; *E*– Young modulus; *h* – height of the transversal panel cross section; *γ*– specific plate

e

(1)

2 4 1 2 2

material gravity; *g* –Earth acceleration; *t* – time; *q* =*q*0sin(*ωpt*)– external load.

ï ï

tioned mechanical systems.

2 Computational and Numerical Simulations

is solved via the fourth-order Runge-Kutta method.

they have the following non-dimensional form [11]:

The non-dimensional parameters are as follows:

**generalized Benettin's algorithm**

l

*O*(*h* <sup>2</sup>

$$
\dot{w}(\mathbf{x},0) = \dot{w}(\mathbf{x},0) = \boldsymbol{u}(\mathbf{x},0) = \dot{\boldsymbol{u}}(\mathbf{x},0).\tag{4}
$$

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 *x* ∈ 0, 1 into 120 parts.

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 with the help of the neural networks approach.

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 neural network self-teaching process ( *dW dt* ≠0), where *W* stands for the net weight coefficients). 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 relation Then, we monitor the evolution of chosen and in time interval *T*, and the correspond‐ ing new points obtained after that time interval are denoted as and respectively. Vector is called the perturbation vector. We are ready to estimate *λ*:

$$
\tilde{\mathcal{L}}\_1 = \frac{1}{T} \ln \frac{\left\| \Delta \mathbf{x}\_1 \right\|}{\varepsilon}. \tag{5}
$$

d e

d e

**Table 1.** Plate output characteristics

/ <sup>=</sup> *<sup>x</sup>*<sup>1</sup> <sup>+</sup> *<sup>Δ</sup>x*<sup>1</sup>

/

map [12], the Lorenz system [13] and the logistic map.

average arithmetic quantity *λ*˜*<sup>i</sup>*

point *x*˜ <sup>1</sup>

f

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

5

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

f

*<sup>q</sup>*<sup>0</sup> =7×10<sup>3</sup>

Time interval *T* is chosen in a way to keep the perturbation amplitude less than the linear dimensions of the space non-homogeneity as well as less than the attractor dimension. We consider the unit normalized perturbation vector and the corresponding new perturbation

*x*0 and respectively. We repeat the described procedure *M* times, and we may estimate *λ* as an

approach has been tested using the standard classical examples including that of the Henon

. We extend the so far described approach by using points and *x*˜ 1 instead of

of those obtained on each computation step. The proposed

<sup>a</sup> <sup>b</sup> <sup>c</sup>

Application of the Lyapunov Exponents and Wavelets to Study and Control of Plates and Shells http://dx.doi.org/10.5772/57452 5

**Table 1.** Plate output characteristics

relation Then, we monitor the evolution of chosen and in time interval *T*, and the correspond‐ ing new points obtained after that time interval are denoted as and respectively. Vector is called

1

<sup>D</sup> <sup>=</sup> % (5)

f

e

*<sup>q</sup>*<sup>0</sup> =0.125×10<sup>3</sup>

*<sup>q</sup>*<sup>0</sup> =5×10<sup>3</sup>

<sup>1</sup> ln . *x*

1

<sup>a</sup> <sup>b</sup> <sup>c</sup>

<sup>a</sup> <sup>b</sup> <sup>c</sup>

d e

*T* l

the perturbation vector. We are ready to estimate *λ*:

4 Computational and Numerical Simulations

Time interval *T* is chosen in a way to keep the perturbation amplitude less than the linear dimensions of the space non-homogeneity as well as less than the attractor dimension. We consider the unit normalized perturbation vector and the corresponding new perturbation point *x*˜ <sup>1</sup> / <sup>=</sup> *<sup>x</sup>*<sup>1</sup> <sup>+</sup> *<sup>Δ</sup>x*<sup>1</sup> / . We extend the so far described approach by using points and *x*˜ 1 instead of *x*0 and respectively. We repeat the described procedure *M* times, and we may estimate *λ* as an average arithmetic quantity *λ*˜*<sup>i</sup>* of those obtained on each computation step. The proposed approach has been tested using the standard classical examples including that of the Henon map [12], the Lorenz system [13] and the logistic map.

We consider vibrations of our mechanical object with the following fixed parameters: *λ* =50, *ε* =1, *ω<sup>p</sup>* =7, *q* =*q*0sin(*ωpt*), and for the following amplitudes of the harmonic excitation: *<sup>q</sup>*<sup>0</sup> =0, <sup>125</sup>⋅10<sup>3</sup> ;5⋅10<sup>3</sup> ; 7⋅10<sup>3</sup> . In order to study chaotic dynamics of flexible plates we need to monitor and analyze the following output characteristics: time histories (a), phase (c) and modal portraits; phase portraits yielded by the neural networks approach (d); Fourier power spectra (b); wavelet spectra, Poincaré sections (e); spectra of Lyapunov exponents, where *d* stands for the fractional part of dimension and *h* is the Kolmogorov-Sinai entropy (f); auto‐ correlation functions (some of them are reported in Table 1). Analysis of the obtained results implies that for *q*<sup>0</sup> =0, <sup>125</sup>⋅10<sup>3</sup> ; periodic vibrations appear, whereas for 5⋅10<sup>3</sup> chaos is exhib‐ ited, and for 7⋅10<sup>3</sup> the hyper-chaotic vibrations occur.

### **3. Wavelets approach to study plate dynamics**

One of the first important tasks to be solved is associated with the choice of a suitable wavelet, which fits well with the stated problem. In order to solve this query we have studied the nonstationary signal (Table 2) obtained via the numerical experiment as an output of the mathe‐ matical model of the rectangular flexible isotropic plate subjected to the periodic shear load acting in the shell volume unit. The mathematical model is as follows [11]:

$$\begin{aligned} \frac{1}{12(1-\mu^2)} \left(\nabla\_\lambda^4 w\right) - L(w, F) + \frac{\hat{\sigma}^2 w}{\hat{\sigma} t^2} + \varepsilon \frac{\hat{\sigma} w}{\hat{\sigma} t} - q(\mathbf{x}\_1, \mathbf{x}\_2, t) + 2S \frac{\hat{\sigma}^2 w}{\hat{\sigma} \mathbf{x}\_1 \hat{\sigma} \mathbf{x}\_2} &= 0, \\ \nabla\_\lambda^4 F + \frac{1}{2} L(w, w) &= 0, \end{aligned} \tag{6}$$

where: 
$$\nabla\_{\lambda} \, ^4 \mathbf{E} = \frac{1}{\lambda^2} \frac{\partial^4}{\partial \mathbf{x}\_1^4} + \lambda^2 \frac{\partial^4}{\partial \mathbf{x}\_2^4} + 2 \frac{\partial^4}{\partial \mathbf{x}\_1 \partial \mathbf{x}\_2^2} \lambda$$

for

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

2 2

1 1 2 2

*w F wF x x x*

¶ ¶ = == = = ¶ ¶

¶ ¶ = == = = ¶ ¶

*w F wF x x x*

2 2

the number of partitions has been discussed by Awrejcewicz et al. [14].

approximation *О(h<sup>2</sup>*

The term <sup>2</sup>*<sup>S</sup>* <sup>∂</sup><sup>2</sup> *<sup>w</sup>*

Fourier power spectrum.

2 2 1

0, 0, 0, 0 0;1;

2 2 2

The external harmonic shear load has the form *S* =*s*0sin*ωpt*. PDEs governing dynamics of our investigated plate are reduced to the ODEs via the FDM (Finite Difference Method) with the

order Runge-Kutta method, and additionally on each of the iterations a large system of linear algebraic equations should be solved with respect to the stress (Airy's) function. Time integration step has been chosen using the Runge rule. The partition number of spatial coordinates is *n*=14 while applying FDM. Validity and reliability of the obtained results regarding

located in shell middle plane and it essentially influences non-linear dynamics of the investigated shell. The numerical simulation indicates that the output signal (shell vibra‐ tions) may change in time repeatedly. We apply this signal to choose a methodology suitable for the investigation of non-stationary processes, and in addition, we illustrate advantag‐ es and disadvantages of the standard Fourier approach versus the wavelet transform procedure. The studied signal has been obtained analyzing the system with the following fixed parameters: *s*<sup>0</sup> =8.4 and *ω<sup>p</sup>* =26. We show that frequency characteristics taken in different time intervals essentially differ from each other. It should be emphasized that the system stability loss occurs not only via the change of chosen control parameters but also keeps all of them fixed owing to the system time evolution. In the first time interval *t* ∈ [50;56] the shell exhibits two frequency quasi-periodic vibrations. Instead of the vanished excitation frequency, two independent frequencies have appeared. A further long time evolution of chaotic vibrations with the exhibition of a few dependent frequencies is observed. In the Fourier spectrum the excitation frequency is not visible. The last studied time interval corresponds to harmonic vibrations, which is also in agreement with the

We constructed wavelet spectra regarding the mentioned signal. We applied the follow‐ ing wavelets: Haar, Shannon-Kotelnikov, Meyer, Daubechies wavelets from db2 up to db 16, Coiflets and symlet wavelets, as well as the Morlet and Gauss (real and imaginary) wavelets, on the basis of the derivatives from 2 to 16. Haar and Shannon-Kotelnikov wavelets are not suitable for the analysis of shell structures. The first one is badly local‐ ized regarding frequency, whereas the second one, contrary to the previous wavelets, is badly localized in time. On the other hand, the analysis regarding the Doubechies wave‐ lets, as well as symlet and Coiflets wavelets implies an increase of the frequency resolu‐ tion assuming that the filter properties are increased. Neglecting differences regarding the

0, 0, 0, 0 0;1.

for

*)* regarding spatial co-ordinates. Next, ODEs are solved via the fourth-

<sup>∂</sup> *<sup>x</sup>*1<sup>∂</sup> *<sup>x</sup>*<sup>2</sup> introduced in the governing equations exhibits the action of shear stresses

(7)

7

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

*<sup>L</sup>* (*w*, *<sup>F</sup>* )= <sup>∂</sup>2*<sup>w</sup>* ∂ *x*<sup>1</sup> 2 ∂<sup>2</sup> *F* ∂ *x*<sup>2</sup> <sup>2</sup> + ∂2*w* ∂ *x*<sup>2</sup> 2 ∂<sup>2</sup> *F* ∂ *x*<sup>1</sup> <sup>2</sup> −2 ∂2*w* ∂ *x*1∂ *x*<sup>2</sup> ∂<sup>2</sup> *F* ∂ *x*1∂ *x*<sup>2</sup> – the known non-linear operator, whereas *w* and *F* stand for the plate deflection and Airy's function, respectively.

System (3.1) is reduced to the non-dimensional form using the following non-dimensional parameters: *<sup>λ</sup>* <sup>=</sup>*<sup>a</sup>* / *<sup>b</sup>*, *<sup>x</sup>*<sup>1</sup> <sup>=</sup>*ax*¯ 1, *<sup>x</sup>*<sup>2</sup> <sup>=</sup>*bx*¯ <sup>2</sup> – non-dimensional parameters regarding *x*1 and *x*2, respectively; *<sup>w</sup>* =2*<sup>h</sup> <sup>w</sup>*¯– deflection; *<sup>F</sup>* <sup>=</sup>*E*(2*<sup>h</sup>* ) 3 *<sup>F</sup>*¯– Airy's function; *<sup>t</sup>* <sup>=</sup>*t*0*<sup>t</sup>* ¯– time; *<sup>q</sup>* <sup>=</sup> *<sup>E</sup>*(2*<sup>h</sup>* ) 4 *a* 2 *<sup>b</sup>* <sup>2</sup> *q*¯– external load; *<sup>ε</sup>* =(2*<sup>h</sup>* )*ε*¯– damping coefficient of the surrounding medium, *<sup>S</sup>* <sup>=</sup> *<sup>E</sup>*(2*<sup>h</sup>* ) 3 *ab <sup>S</sup>*¯– external shear load. In the equations bars have already been omitted over the non-dimensional quantities. The following notation is introduced: *a*, *b*– plate dimensions regarding *x*1 and *x*2, respectively; *μ*– Poisson's coefficient. Zero value initial conditions and the following boundary value conditions are attached to system (2.6):

Application of the Lyapunov Exponents and Wavelets to Study and Control of Plates and Shells http://dx.doi.org/10.5772/57452 7

$$\begin{aligned} \varepsilon w &= 0, \quad \frac{\partial^2 w}{\partial \mathbf{x}\_1^2} = 0, \quad F = 0, \quad \frac{\partial^2 F}{\partial \mathbf{x}\_1^2} = 0 \quad \text{for} \quad \mathbf{x}\_1 = 0; 1;\\ \varepsilon w &= 0, \quad \frac{\partial^2 w}{\partial \mathbf{x}\_2^2} = 0, \quad F = 0, \quad \frac{\partial^2 F}{\partial \mathbf{x}\_2^2} = 0 \quad \text{for} \quad \mathbf{x}\_2 = 0; 1. \end{aligned} \tag{7}$$

The external harmonic shear load has the form *S* =*s*0sin*ωpt*. PDEs governing dynamics of our investigated plate are reduced to the ODEs via the FDM (Finite Difference Method) with the approximation *О(h<sup>2</sup> )* regarding spatial co-ordinates. Next, ODEs are solved via the fourthorder Runge-Kutta method, and additionally on each of the iterations a large system of linear algebraic equations should be solved with respect to the stress (Airy's) function. Time integration step has been chosen using the Runge rule. The partition number of spatial coordinates is *n*=14 while applying FDM. Validity and reliability of the obtained results regarding the number of partitions has been discussed by Awrejcewicz et al. [14].

We consider vibrations of our mechanical object with the following fixed parameters: *λ* =50, *ε* =1, *ω<sup>p</sup>* =7, *q* =*q*0sin(*ωpt*), and for the following amplitudes of the harmonic excitation:

monitor and analyze the following output characteristics: time histories (a), phase (c) and modal portraits; phase portraits yielded by the neural networks approach (d); Fourier power spectra (b); wavelet spectra, Poincaré sections (e); spectra of Lyapunov exponents, where *d* stands for the fractional part of dimension and *h* is the Kolmogorov-Sinai entropy (f); auto‐ correlation functions (some of them are reported in Table 1). Analysis of the obtained results

One of the first important tasks to be solved is associated with the choice of a suitable wavelet, which fits well with the stated problem. In order to solve this query we have studied the nonstationary signal (Table 2) obtained via the numerical experiment as an output of the mathe‐ matical model of the rectangular flexible isotropic plate subjected to the periodic shear load

( ) 2 2

e

*w w <sup>w</sup> w LwF qx x t S <sup>t</sup> <sup>t</sup> x x*

∂<sup>2</sup> *F* ∂ *x*1∂ *x*<sup>2</sup>

System (3.1) is reduced to the non-dimensional form using the following non-dimensional parameters: *<sup>λ</sup>* <sup>=</sup>*<sup>a</sup>* / *<sup>b</sup>*, *<sup>x</sup>*<sup>1</sup> <sup>=</sup>*ax*¯ 1, *<sup>x</sup>*<sup>2</sup> <sup>=</sup>*bx*¯ <sup>2</sup> – non-dimensional parameters regarding *x*1 and *x*2,

3

shear load. In the equations bars have already been omitted over the non-dimensional quantities. The following notation is introduced: *a*, *b*– plate dimensions regarding *x*1 and *x*2, respectively; *μ*– Poisson's coefficient. Zero value initial conditions and the following boundary

¶ ¶ ¶ Ñ- + + - + =

<sup>1</sup> (,) ( , , ) 2 0, 12(1 )


the hyper-chaotic vibrations occur.

acting in the shell volume unit. The mathematical model is as follows [11]:

2 2 1 2

∂2*w* ∂ *x*1∂ *x*<sup>2</sup>

external load; *<sup>ε</sup>* =(2*<sup>h</sup>* )*ε*¯– damping coefficient of the surrounding medium, *<sup>S</sup>* <sup>=</sup> *<sup>E</sup>*(2*<sup>h</sup>* )

*w* and *F* stand for the plate deflection and Airy's function, respectively.

where: ∇*<sup>λ</sup>* <sup>4</sup> <sup>=</sup> <sup>1</sup>

**3. Wavelets approach to study plate dynamics**

4

l

<sup>1</sup> ( , ) 0, <sup>2</sup>

∂2*w* ∂ *x*<sup>2</sup> 2 ∂<sup>2</sup> *F* ∂ *x*<sup>1</sup> <sup>2</sup> −2

respectively; *<sup>w</sup>* =2*<sup>h</sup> <sup>w</sup>*¯– deflection; *<sup>F</sup>* <sup>=</sup>*E*(2*<sup>h</sup>* )

value conditions are attached to system (2.6):

*F Lww*

Ñ+ =

m

4

l

∂ *x*<sup>1</sup> 2 ∂<sup>2</sup> *F* ∂ *x*<sup>2</sup> <sup>2</sup> +

*<sup>L</sup>* (*w*, *<sup>F</sup>* )= <sup>∂</sup>2*<sup>w</sup>*

. In order to study chaotic dynamics of flexible plates we need to

; periodic vibrations appear, whereas for 5⋅10<sup>3</sup> chaos is exhib‐

1 2

∂4 ∂ *x*<sup>1</sup>

<sup>4</sup> <sup>+</sup> *<sup>λ</sup>* <sup>2</sup> <sup>∂</sup><sup>4</sup> ∂ *x*<sup>2</sup> <sup>4</sup> + 2

– the known non-linear operator, whereas

*λ* 2

*<sup>F</sup>*¯– Airy's function; *<sup>t</sup>* <sup>=</sup>*t*0*<sup>t</sup>*

(6)

∂4 ∂ *x*<sup>1</sup> <sup>2</sup>∂ *x*<sup>2</sup> 2 ,

> 4 *a* 2 *<sup>b</sup>* <sup>2</sup> *q*¯–

¯– time; *<sup>q</sup>* <sup>=</sup> *<sup>E</sup>*(2*<sup>h</sup>* )

3 *ab <sup>S</sup>*¯– external

*<sup>q</sup>*<sup>0</sup> =0, <sup>125</sup>⋅10<sup>3</sup>

;5⋅10<sup>3</sup>

6 Computational and Numerical Simulations

implies that for *q*<sup>0</sup> =0, <sup>125</sup>⋅10<sup>3</sup>

ited, and for 7⋅10<sup>3</sup>

; 7⋅10<sup>3</sup>

The term <sup>2</sup>*<sup>S</sup>* <sup>∂</sup><sup>2</sup> *<sup>w</sup>* <sup>∂</sup> *<sup>x</sup>*1<sup>∂</sup> *<sup>x</sup>*<sup>2</sup> introduced in the governing equations exhibits the action of shear stresses located in shell middle plane and it essentially influences non-linear dynamics of the investigated shell. The numerical simulation indicates that the output signal (shell vibra‐ tions) may change in time repeatedly. We apply this signal to choose a methodology suitable for the investigation of non-stationary processes, and in addition, we illustrate advantag‐ es and disadvantages of the standard Fourier approach versus the wavelet transform procedure. The studied signal has been obtained analyzing the system with the following fixed parameters: *s*<sup>0</sup> =8.4 and *ω<sup>p</sup>* =26. We show that frequency characteristics taken in different time intervals essentially differ from each other. It should be emphasized that the system stability loss occurs not only via the change of chosen control parameters but also keeps all of them fixed owing to the system time evolution. In the first time interval *t* ∈ [50;56] the shell exhibits two frequency quasi-periodic vibrations. Instead of the vanished excitation frequency, two independent frequencies have appeared. A further long time evolution of chaotic vibrations with the exhibition of a few dependent frequencies is observed. In the Fourier spectrum the excitation frequency is not visible. The last studied time interval corresponds to harmonic vibrations, which is also in agreement with the Fourier power spectrum.

We constructed wavelet spectra regarding the mentioned signal. We applied the follow‐ ing wavelets: Haar, Shannon-Kotelnikov, Meyer, Daubechies wavelets from db2 up to db 16, Coiflets and symlet wavelets, as well as the Morlet and Gauss (real and imaginary) wavelets, on the basis of the derivatives from 2 to 16. Haar and Shannon-Kotelnikov wavelets are not suitable for the analysis of shell structures. The first one is badly local‐ ized regarding frequency, whereas the second one, contrary to the previous wavelets, is badly localized in time. On the other hand, the analysis regarding the Doubechies wave‐ lets, as well as symlet and Coiflets wavelets implies an increase of the frequency resolu‐ tion assuming that the filter properties are increased. Neglecting differences regarding the wavelet forms and the associated filters, the wavelet spectra obtained through the Dobe‐ chies wavelets as well as symlet and Coiflets wavelets are practically identical. However, their localization with respect to frequency is not suitable for the analysis of continuous systems dynamics. In the case of the Gauss functions, an increase of their derivative order implies an increase of the frequency resolution.

Wavelet spectrum: Меyеr Wavelet spectrum: Daubechies 16 Wavelet spectrum: complex Morlet

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

Wavelet spectrum: Morlet Wavelet spectrum: Gauss 16 Wavelet spectrum Gauss: 16 complex

The governing non-dimensional equations are given in the hybrid form:

1 2


*x x*

( )

m

2 2

4 2 2

<sup>1</sup> <sup>2</sup>

 m 1 2

*x x*

*x x x x x y xx*

m

1 1 1 1 2 2 2 2

<sup>1</sup> ( , ) 0, <sup>2</sup>

( )

*x x*

1 2

*u u v ww w w w ww u k k x x xx x <sup>x</sup> x x xx x x t*

¶ -¶ + ¶ ¶¶ ¶ + ¶ ¶ - ¶¶ ¶ + + + -+ + + -= ¶ ¶ ¶¶ ¶ ¶ ¶ ¶ ¶¶ ¶ ¶ ¶

m

*uv v w ww w w ww k k x x x x x x <sup>x</sup> x xx x <sup>x</sup>*

+ ¶ ¶ -¶ ¶ ¶¶ +¶ ¶ - ¶¶ ++ - + + + <sup>+</sup> ¶ ¶ ¶ ¶ ¶ ¶ ¶ ¶ ¶¶ ¶ ¶

( ) ( ) ( ) 1 2 2 1 2 1 1 2

m

ì ü é ù æö <sup>ù</sup> ï ï ¶ -¶ ¶ ¶ ¶ ¶ í ý ê ú + + ç ÷ + -+ + = <sup>ú</sup> ï ï ë û èø ¶ ¶¶ ¶ ¶ <sup>ú</sup> ¶ î þû

*u v k k <sup>w</sup> w k k k k k k kk w x x <sup>x</sup>*

<sup>é</sup> ¶ ¶ <sup>+</sup> æ ö ¶ êÑ- + - + + ++ - ç ÷ - ç ÷ <sup>ê</sup> ¶ ¶ ¶ <sup>ë</sup> è ø

2 111 2 2 2 1

*x xxx x x x x*

<sup>2</sup> 1 2 <sup>1</sup>

*k k w wu v w u v*

<sup>+</sup> æö é ù æ ö ¶ ¶¶ ¶ ¶ -¶ ¶ ¶ ì ü ï ï - - + + +- ç ÷ í ý ê ú ç ÷ ç ÷ ¶ ¶¶¶ ¶ ¶ ¶ ¶ èø ë û è ø ï ï î þ

1 1 1 1 0, 2 2 2 2

1

<sup>1</sup> <sup>2</sup> 1 12 1 <sup>1</sup> 0, <sup>2</sup> *u wu v w w <sup>q</sup> x xx x t <sup>t</sup>*

m

m m

m

 e

22 22 2 2 2 2 2 2 2 2 1 2 12 1 1 1 2 12 1 1 22 2 2 2 2 2 2 <sup>2</sup> 1 2 2 1 2 2 <sup>2</sup> 1 12 <sup>2</sup>

22 2

2 22 2 1 2

*F F ww w k k LwF qx x t xx t <sup>t</sup>*

¶ ¶ ¶¶ Ñ- - - ++ - =

e

 m

> m

2

2

*x x*

m

2 2 2 1 2

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

9

¶ + = ¶

*v t*

0,

(8)

(9)

<sup>1</sup> (,) ( , , ) 0, 12(1 )

( ) 2 1

2 1

*x x*

 m

> m

> > 2

m

2 2

2 2 1 2

¶ ¶ Ñ+ + + = ¶ ¶

*w w F k k Lww x x*

whereas equations regarding displacements follow

4

l

**Table 3.** Wavelets spectra

4

m

2 1

m m

*x x*

¶¶ ¶ - ¶¶¶

222

*w v xxx*

m

l

m

l

**Table 2.** Fourier spectra and a signal

Table 3 gives results associated with the application of different wavelets (Meyer, Morlet, complex Morlet, real and complex Gauss with 16 derivative order, Daubechies) to analyze nonlinear shell vibrations. One may conclude from Table 3 that the localization regarding frequency increases with an increase of the number of the wavelet zero moments.
