**4. Equations governing dynamics of flexible elastic shells**

In a frame of non-linear classical theory we study a shell with constant stiffness and density and harmonic load *q* =*q*0sin(*ωpt*), where *q*0 - amplitude of excitation, *ωp*- frequency of excitation. In the rectangular co-ordinates the 3D space is:

$$\mathcal{Q} = \{ \mathbf{x}\_1 \; \; \; \mathbf{x}\_2 \; \; \; \mathbf{x}\_3 \mid (\mathbf{x}\_1 \; \; \; \mathbf{x}\_2) \in \mathsf{I}[0; a] \times \mathsf{I}[0; b] \; \; \; \mathbf{x}\_3 \in \mathsf{I} - \mathsf{h} \; \; \; / 2; \mathsf{h} \; \; / 2 \}, \; 0 \le t < \infty \; \; \; \; \}$$

Application of the Lyapunov Exponents and Wavelets to Study and Control of Plates and Shells http://dx.doi.org/10.5772/57452 9

**Table 3.** Wavelets spectra

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

> Fourier spectrum (130<*t*<225) of chaotic vibrations

> > 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

In a frame of non-linear classical theory we study a shell with constant stiffness and density and harmonic load *q* =*q*0sin(*ωpt*), where *q*0 - amplitude of excitation, *ωp*- frequency of excitation.

frequency increases with an increase of the number of the wavelet zero moments.

**4. Equations governing dynamics of flexible elastic shells**

*Ω* ={*x*1, *x*2, *x*<sup>3</sup> |(*x*1, *x*2)∈ 0; *a* × 0; *b* , *x*3∈ −*h* / 2;*h* / 2 }, 0≤*t* <*∞*.

In the rectangular co-ordinates the 3D space is:

Fourier spectrum (230<*t*<286) of periodic vibrations

implies an increase of the frequency resolution.

Fourier spectrum (50<*t*<56) of quasiperiodic vibrations

8 Computational and Numerical Simulations

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

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

$$\begin{split} &\frac{1}{12(1-\mu^{2})} \left(\nabla\_{\lambda}{}^{4}w\right) - k\_{x\_{2}}\frac{\partial^{2}F}{\partial x\_{1}^{2}} - k\_{x\_{1}}\frac{\partial^{2}F}{\partial x\_{2}^{2}} - L(w,F) + \frac{\partial^{2}w}{\partial t^{2}} + \varepsilon \frac{\partial w}{\partial t} - q(\mathbf{x}\_{1}, \mathbf{x}\_{2}, t) = 0, \\ &\nabla\_{\lambda}{}^{4}F + k\_{x\_{2}}\frac{\partial^{2}w}{\partial \mathbf{x}\_{1}^{2}} + k\_{x\_{1}}\frac{\partial^{2}w}{\partial \mathbf{x}\_{2}^{2}} + \frac{1}{2}L(w,w) = 0, \end{split} \tag{8}$$

whereas equations regarding displacements follow

( ) ( ) 1 2 1 2 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> 1 1 1 1 0, 2 2 2 2 1 1 1 1 2 2 2 2 *x x x x u u v ww w w w ww u k k x x xx x <sup>x</sup> x x xx x x t uv v w ww w w ww k k x x x x x x <sup>x</sup> x xx x <sup>x</sup>* m m m m m m m m m m ¶ -¶ + ¶ ¶¶ ¶ + ¶ ¶ - ¶¶ ¶ + + + -+ + + -= ¶ ¶ ¶¶ ¶ ¶ ¶ ¶ ¶¶ ¶ ¶ ¶ + ¶ ¶ -¶ ¶ ¶¶ +¶ ¶ - ¶¶ ++ - + + + <sup>+</sup> ¶ ¶ ¶ ¶ ¶ ¶ ¶ ¶ ¶¶ ¶ ¶ ( ) ( ) ( ) 1 2 2 1 2 1 1 2 2 1 2 2 2 1 2 4 2 2 <sup>2</sup> 1 2 <sup>1</sup> 2 2 111 2 2 2 1 222 0, <sup>1</sup> <sup>2</sup> 2 1 2 2 *x x x x x x x y xx x x v t u v k k <sup>w</sup> w k k k k k k kk w x x <sup>x</sup> k k w wu v w u v x xxx x x x x w v xxx* m m m m l m m m ¶ + = ¶ <sup>é</sup> ¶ ¶ <sup>+</sup> æ ö ¶ êÑ- + - + + ++ - ç ÷ - ç ÷ <sup>ê</sup> ¶ ¶ ¶ <sup>ë</sup> è ø <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 ¶¶ ¶ - ¶¶¶ 2 <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 e 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 (9)

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

1 2 1 2

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

<sup>=</sup> åå <sup>=</sup> åå (14)

p

 p

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

11

( )sin( )sin( ), ( )sin( )sin( ).

In what follows we investigate vibrations of flexible shells via the BGM versus the partition number *N* in periodic (Fig. 1) and chaotic (Fig. 2) zones. We consider the point *A*(*q*0, *ωp*)= *A*(5;25)∈{*q*0, *ωp*} marked on the type vibration chart (Fig. 3), which belongs to harmonic vibrations zone. For all *N* harmonic (one frequency) vibrations are observed. Increase of the approximations number *N* =11, 13, 15 implies the full coincidence regarding both frequency and amplitude. Further, we study the BGM convergence in a chaotic zone (Fig. 2). We analyze the point *B*(*q*0, *ωp*)= *B*(180;38)∈{*q*0, *ωp*} on the chart (Fig. 3). Although we cannot achieve the signal convergence as it occurred in the previous case, but we get the

**Figure 1.** Time histories (*w*(0.5, 0.5, *t*)) and power spectrum for different approximations of the Bubnov-Galerkin

We investigate further the convergence of the FDM versus a number of partitions of the mesh *m*×*n* for a flexible rectangular shell taking into account the same parameters. We consider first the shell center motion in an harmonic zone (Fig. 4). Let us fix the point

method for *t* ∈ 40;40.4 .

1 1 1 1

*i j i j*

= = = =

**5. Numerical analysis of shells non-linear dynamics**

p

*ij ij*

 p

convergence in average sense through Fourier integrals of power spectra.

*w A t ix jx F B t ix jx*

*N N N N*

*<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> – known non-linear operator, *w*(*x*1, *x*2, *t*)– element normal; *u*(*x*1, *x*2, *t*), *v*(*x*1, *x*2, *t*)– element displacement regarding *x*1 and *x*2, respec‐ tively; *F* (*x*1, *x*2, *t*)- stress function; *ε* – dissipation coefficient; *E*– Young modulus; *h* – height of the transversal panel cross section; *γ*– specific unit gravity of the shell material; *g*– Earth acceleration; *t*– time; *q* =*q*0sin(*ωt*) – external load.

The following non-dimensional parameters are introduced:

$$\begin{aligned} \lambda &= \frac{a}{h}, \quad \lambda\_1 = \frac{a}{b}, \quad \overline{w} = \frac{w}{h}, \quad \overline{u} = \frac{ua}{h^2}, \quad \overline{v} = \frac{va}{h^2}, \quad \overline{x\_1} = \frac{x\_1}{a}, \quad \overline{x\_2} = \frac{x\_2}{b}, \quad \overline{t} = \frac{t}{\tau}, \quad \tau = \frac{a}{c}, \\\ c &= \sqrt{\frac{E\_\mathcal{S}}{\mathcal{Y}}}, \quad \overline{c} = \frac{a}{c}, \quad \overline{q} = \frac{qa^4}{h^4E}, \quad \overline{k\_{x\_1}} = \frac{k\_{x\_1}a}{\lambda}, \quad \overline{k\_{x\_2}} = \frac{k\_{x\_2}b}{\lambda}, \quad \overline{F} = \frac{F}{Eh^3}. \end{aligned} \tag{10}$$

System of differential equations (4.1)-(4.2) should be supplemented by boundary and initial conditions (see [11]. Since we cannot solve the stated problems analytically, we reduce the problem to ODEs and solve it numerically by the fourth-order Runge-Kutta method (see [15] for more details).

As the first example, we consider a spherical rectangular shell governed by equations (4.1) and with the following homogeneous conditions:

$$\|\mathbf{w}\|\_{\sf T} = \mathbf{0}; \quad M\_{\sf n}\,\vert\_{\sf T} = \mathbf{0}; \quad N\_{\sf n}\,\vert\_{\sf T} = \mathbf{0}; \quad \varepsilon\_{\sf n}\,\vert\_{\sf T} = \mathbf{0} \quad \text{for} \quad \mathbf{x}\_1 = \mathbf{0}; \mathbf{1}, \quad \mathbf{x}\_2 = \mathbf{0}; \mathbf{1}, \tag{11}$$

which can be recast to the form

$$
\delta w = 0; \quad \frac{\partial^2 w}{\partial \mathbf{x}\_1^2} = 0; \quad F = 0; \quad \frac{\partial^2 F}{\partial \mathbf{x}\_1^2} = 0; \quad w = 0; \quad \frac{\partial^2 w}{\partial \mathbf{x}\_2^2} = 0; \quad F = 0; \quad \frac{\partial^2 F}{\partial \mathbf{x}\_2^2} = 0 \tag{12}
$$

and the following initial conditions

$$\left.w(\mathbf{x}\_1, \mathbf{x}\_2)\right|\_{t=0} = 0, \qquad \frac{\partial w}{\partial t} = 0. \tag{13}$$

Geometric parameters of the shell curvature *kx*<sup>1</sup> = *kx*<sup>2</sup> =24, *λ* =1, and the damping coefficient *ε* =1. We apply the BGM in Vlasov's form, and the we are looping for the functions w and F, satisfying (4.5), in the following form

$$\text{for } w = \sum\_{i=1}^{N} \sum\_{j=1}^{N} A\_{ij}(t) \sin(i\pi x\_1) \sin(j\pi x\_2), \qquad F = \sum\_{i=1}^{N} \sum\_{j=1}^{N} B\_{ij}(t) \sin(i\pi x\_1) \sin(j\pi x\_2). \tag{14}$$

### **5. Numerical analysis of shells non-linear dynamics**

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

∂2*w* ∂ *x*1∂ *x*<sup>2</sup>

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

element normal; *u*(*x*1, *x*2, *t*), *v*(*x*1, *x*2, *t*)– element displacement regarding *x*1 and *x*2, respec‐ tively; *F* (*x*1, *x*2, *t*)- stress function; *ε* – dissipation coefficient; *E*– Young modulus; *h* – height of the transversal panel cross section; *γ*– specific unit gravity of the shell material; *g*– Earth

1 2

*x x* G G GG = = == = = (11)

¶¶¶¶ = == = = == = ¶¶ ¶¶ (12)

*w*

= *kx*<sup>2</sup>

¶ = = ¶ (13)

=24, *λ* =1, and the damping coefficient

4 3

1 2

 l

1 2

l

*c h E Eh*

System of differential equations (4.1)-(4.2) should be supplemented by boundary and initial conditions (see [11]. Since we cannot solve the stated problems analytically, we reduce the problem to ODEs and solve it numerically by the fourth-order Runge-Kutta method (see [15]

As the first example, we consider a spherical rectangular shell governed by equations (4.1) and

1 2 | 0; | 0; | 0; | 0 for 0;1, 0;1, *wM N n nn* e

2222 22 22 11 22 0; 0; 0; 0; 0; 0; 0; 0 *wF wF wFwF xx xx*

12 0 ( , )| 0, 0. *<sup>t</sup>*

*t* <sup>=</sup>

*ε* =1. We apply the BGM in Vlasov's form, and the we are looping for the functions w and F,

*wx x*

, , , , , , ,,,

*a a w ua va x x t a wu v x x t hbh h h ab c*

= = = = = = = ==

,, , , , . *x x x x*

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

acceleration; *t*– time; *q* =*q*0sin(*ωt*) – external load.

e

with the following homogeneous conditions:

The following non-dimensional parameters are introduced:

1 2 2 1 2 4

*Eg ka kb a F qa c qk k F*

= == = = =

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

l

for more details).

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

10 Computational and Numerical Simulations

 l

> g

which can be recast to the form

and the following initial conditions

satisfying (4.5), in the following form

Geometric parameters of the shell curvature *kx*<sup>1</sup>

*λ* 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

– known non-linear operator, *w*(*x*1, *x*2, *t*)–

t

t

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

(10)

In what follows we investigate vibrations of flexible shells via the BGM versus the partition number *N* in periodic (Fig. 1) and chaotic (Fig. 2) zones. We consider the point *A*(*q*0, *ωp*)= *A*(5;25)∈{*q*0, *ωp*} marked on the type vibration chart (Fig. 3), which belongs to harmonic vibrations zone. For all *N* harmonic (one frequency) vibrations are observed. Increase of the approximations number *N* =11, 13, 15 implies the full coincidence regarding both frequency and amplitude. Further, we study the BGM convergence in a chaotic zone (Fig. 2). We analyze the point *B*(*q*0, *ωp*)= *B*(180;38)∈{*q*0, *ωp*} on the chart (Fig. 3). Although we cannot achieve the signal convergence as it occurred in the previous case, but we get the convergence in average sense through Fourier integrals of power spectra.

**Figure 1.** Time histories (*w*(0.5, 0.5, *t*)) and power spectrum for different approximations of the Bubnov-Galerkin method for *t* ∈ 40;40.4 .

We investigate further the convergence of the FDM versus a number of partitions of the mesh *m*×*n* for a flexible rectangular shell taking into account the same parameters. We consider first the shell center motion in an harmonic zone (Fig. 4). Let us fix the point power spectra.

and the following initial conditions

Geometric parameters of the shell curvature 24 <sup>1</sup> <sup>2</sup> *kx kx* , 1

**5. Numerical analysis of shells non-linear dynamics** 

in a chaotic zone (Fig. 2). We analyze the point ( , ) (180;38) { , } <sup>0</sup> *<sup>p</sup>* <sup>0</sup> *<sup>p</sup> B q*

chaotic (Fig. 2) zones. We consider the point *A*(*q*<sup>0</sup> ,

Vlasov's form, and the we are looping for the functions w and F, satisfying (4.5), in the following form

1 1 1 1

 

*N N N N ij ij i j i j*

**Figure 1.** Time histories ( (0.5,0.5, )) *w t* and power spectrum for different approximations of the Bubnov-Galerkin method for [40;40.4] *t* .

12 0 ( , ) | 0, 0. *<sup>t</sup> <sup>w</sup> wx x <sup>t</sup>*

*w At ix jx F Bt ix jx*

*<sup>p</sup>* ) *A*(5;25)*q*<sup>0</sup> ,

belongs to harmonic vibrations zone. For all *N* harmonic (one frequency) vibrations are observed. Increase of the approximations number 15 *N* 11,13, implies the full coincidence regarding both frequency and amplitude. Further, we study the BGM convergence

*B q*

the signal convergence as it occurred in the previous case, but we get the convergence in average sense through Fourier integrals of

In what follows we investigate vibrations of flexible shells via the BGM versus the partition number *N* in periodic (Fig. 1) and

1 2 1 2

(4.7)

( )sin( )sin( ), ( )sin( )sin( ).

(4.6)

  *<sup>p</sup>* marked on the type vibration chart (Fig. 3), which

on the chart (Fig. 3). Although we cannot achieve

. We apply the BGM in

, and the damping coefficient 1

*<sup>t</sup>* [40;41] . **Figure 2.** Time histories (*w*(0.5, 0.5, *<sup>t</sup>*)) and power spectrum for different approximations of the Bubnov-Galerkin method for *t* ∈ 40;41 .

**Figure 2.** Time histories ( (0.5,0.5, )) *w t* and power spectrum for different approximations of the Bubnov-Galerkin method for

**Figure 3.** Vibration charts in the control parameters plane *q*<sup>0</sup> ,**Figure 3.** Vibration charts in the control parameters plane {*q*0, ω*p*}.

*A*(*q*0, *ωp*)= *A*(5;25)∈{*q*0, *ωp*} on the vibration chart. Three curves are reported in Fig. 4 for *n* =*m*=8; 16; 32. For all partition points *n* =*m*=8; 16; 32 harmonic vibrations are observed. On the other hand for given *n* =*m* time histories differ from each other (see Fig. 5), but their convergence regarding the Fourier power spectra are evident (*n* =*m*= 16; 32). We investigate further the convergence of the FDM versus a number of partitions of the mesh *m n* for a flexible rectangular shell taking into account the same parameters. We consider first the shell center motion in an harmonic zone (Fig. 4). Let us fix the point *A*(*q*<sup>0</sup> ,*<sup>p</sup>* ) *A*(5;25)*q*<sup>0</sup> , *<sup>p</sup>* on the vibration chart. Three curves are reported in Fig. 4 for *n m* 8; 16; 32 . For all partition points *n m* 8; 16; 32 harmonic vibrations are observed. On the other hand for given *n m* time histories differ from each other (see Fig. 5), but their convergence regarding the Fourier power spectra are evident ( *n m* 16; 32 ).

*<sup>p</sup>* .

> <sup>0</sup> 180, 25 *<sup>p</sup> q*

> > *m n* 8

Increase of the resolution implies the results improvement. Results obtained for the cases 1d and 1e coincide in full. In Figure 3 vibration charts for the rectangular spherical shell, which have been obtained using the BGM for N = 11, as well as the FDM for partition number 8х8

**Figure 5.** Dependence ( (0.5,0.5, )) *w t* and power spectra for different partition *m n* 8;16;32 in a chaotic zone for *t* [40;45] . **Figure 5.** Dependence (*w*(0.5, 0.5, *t*)) and power spectra for different partition *m*=*n* =8;16;32 in a chaotic zone for

**Figure 4.** Deflections (*w*(0.5, 0.5, *t*)) and power spectra for different partitions *m*=*n* =8;16;32 in a periodic zone for

**Figure 4.** Deflections ( (0.5,0.5, )) *w t* and power spectra for different partitions 8;16;32 *m n* in a periodic zone for [40;40.5] *t* .

Bubnov-Galerkin method (*N*=11) Finite Difference Method (*m*=*n*=8) **Figure 3.** Vibration charts in the control parameters plane *q*<sup>0</sup> ,

We investigate further the convergence of the FDM versus a number of partitions of the mesh *nm* for a flexible rectangular shell taking into account the same parameters. We consider first the shell center motion in an harmonic zone (Fig. 4). Let us fix the

points *mn* 32;16;8 harmonic vibrations are observed. On the other hand for given *n m* time histories differ from each other

(see Fig. 5), but their convergence regarding the Fourier power spectra are evident ( *mn* 32;16 ).

*<sup>p</sup>* on the vibration chart. Three curves are reported in Fig. 4 for *mn* 32;16;8 . For all partition

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

*<sup>p</sup>* .

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

13

<sup>0</sup> 180, 25 *<sup>p</sup> q* 

*m n* 8

*m n* 16

*m n* 32

) regarding the spatial coordinate are shown. In both cases the initial

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

*t* ∈ 40;45 .

*t* ∈ 40;40.5 .

point 0

*<sup>p</sup> qAqA* <sup>0</sup> ,)25;5(),(

*m n* 16

*m n* 32

We now compare the results obtained through two qualitatively different approaches, i.e. FDM and BGM. The convergence of those two methods is numerically confirmed with respect to time histories and Fourier power spectra for small amplitude of excitation. Although in the case of chaotic vibrations the convergence regarding time series is not achieved, but it is achieved with respect to integral Fourier characteristics.

In what follows we address the problem of vibration chart identification versus a step of variation of the control parameters *q*0 and *ωp*. We consider the vibration charts for five applied resolutions given in Table 4: а – resolution 50×50, b -100×100, c –200×200, d –300×300, e - 400×400.

> **Figure 4.** Deflections ( (0.5,0.5, )) *w t* and power spectra for different partitions 8;16;32 *m n* in a periodic zone for [40;40.5] *t* .

**Figure 5.** Dependence ( (0.5,0.5, )) *w t* and power spectra for different partition *m n* 8;16;32 in a chaotic zone for *t* [40;45] .

Application of the Lyapunov Exponents and Wavelets to Study and Control of Plates and Shells http://dx.doi.org/10.5772/57452 13

Bubnov-Galerkin method (*N*=11) Finite Difference Method (*m*=*n*=8)

**Figure 4.** Deflections (*w*(0.5, 0.5, *t*)) and power spectra for different partitions *m*=*n* =8;16;32 in a periodic zone for *t* ∈ 40;40.5 . **Figure 4.** Deflections ( (0.5,0.5, )) *w t* and power spectra for different

partitions 8;16;32 *m n* in a periodic zone for [40;40.5] *t* .

*A*(*q*0, *ωp*)= *A*(5;25)∈{*q*0, *ωp*} on the vibration chart. Three curves are reported in Fig. 4 for *n* =*m*=8; 16; 32. For all partition points *n* =*m*=8; 16; 32 harmonic vibrations are observed. On the other hand for given *n* =*m* time histories differ from each other (see Fig. 5), but their

points *n m* 8; 16; 32 harmonic vibrations are observed. On the other hand for given *n m* time histories differ from each other

We investigate further the convergence of the FDM versus a number of partitions of the mesh *m n* for a flexible rectangular shell taking into account the same parameters. We consider first the shell center motion in an harmonic zone (Fig. 4). Let us fix the

*<sup>p</sup>* on the vibration chart. Three curves are reported in Fig. 4 for *n m* 8; 16; 32 . For all partition

Bubnov-Galerkin method (*N*=11) Finite Difference Method (*m*=*n*=8) **Figure 3.** Vibration charts in the control parameters plane *q*<sup>0</sup> ,

*<sup>p</sup>* .

> <sup>0</sup> 180, 25 *<sup>p</sup> q*

> > *m n* 8

*m n* 16

*m n* 32

12 0 ( , ) | 0, 0. *<sup>t</sup> <sup>w</sup> wx x <sup>t</sup>*

*w At ix jx F Bt ix jx*

*<sup>p</sup>* ) *A*(5;25)*q*<sup>0</sup> ,

belongs to harmonic vibrations zone. For all *N* harmonic (one frequency) vibrations are observed. Increase of the approximations number 15 *N* 11,13, implies the full coincidence regarding both frequency and amplitude. Further, we study the BGM convergence

*B q*

the signal convergence as it occurred in the previous case, but we get the convergence in average sense through Fourier integrals of

**Figure 1.** Time histories ( (0.5,0.5, )) *w t* and power spectrum for different approximations of the Bubnov-Galerkin method for [40;40.4] *t* .

**Figure 2.** Time histories ( (0.5,0.5, )) *w t* and power spectrum for different approximations of the Bubnov-Galerkin method for *<sup>t</sup>* [40;41] . **Figure 2.** Time histories (*w*(0.5, 0.5, *<sup>t</sup>*)) and power spectrum for different approximations of the Bubnov-Galerkin

In what follows we investigate vibrations of flexible shells via the BGM versus the partition number *N* in periodic (Fig. 1) and

1 2 1 2

(4.7)

( )sin( )sin( ), ( )sin( )sin( ).

(4.6)

  *<sup>p</sup>* marked on the type vibration chart (Fig. 3), which

on the chart (Fig. 3). Although we cannot achieve

<sup>0</sup> 180, 25 *<sup>p</sup> q* 

*N* 5

point 0

*N* 9

*N* 15

. We apply the BGM in

, and the damping coefficient 1

We now compare the results obtained through two qualitatively different approaches, i.e. FDM and BGM. The convergence of those two methods is numerically confirmed with respect to time histories and Fourier power spectra for small amplitude of excitation. Although in the case of chaotic vibrations the convergence regarding time series is not achieved, but it is

In what follows we address the problem of vibration chart identification versus a step of variation of the control parameters *q*0 and *ωp*. We consider the vibration charts for five applied resolutions given in Table 4: а – resolution 50×50, b -100×100, c –200×200, d –300×300, e -

> **Figure 4.** Deflections ( (0.5,0.5, )) *w t* and power spectra for different partitions 8;16;32 *m n* in a periodic zone for [40;40.5] *t* .

**Figure 5.** Dependence ( (0.5,0.5, )) *w t* and power spectra for different partition *m n* 8;16;32 in a chaotic zone for *t* [40;45] .

convergence regarding the Fourier power spectra are evident (*n* =*m*= 16; 32).

(see Fig. 5), but their convergence regarding the Fourier power spectra are evident ( *n m* 16; 32 ).

achieved with respect to integral Fourier characteristics.

400×400.

point *A*(*q*<sup>0</sup> ,

*<sup>p</sup>* ) *A*(5;25)*q*<sup>0</sup> ,

**Figure 3.** Vibration charts in the control parameters plane {*q*0, ω*p*}.

and the following initial conditions

power spectra.

method for *t* ∈ 40;41 .

Geometric parameters of the shell curvature 24 <sup>1</sup> <sup>2</sup> *kx kx* , 1

**5. Numerical analysis of shells non-linear dynamics** 

in a chaotic zone (Fig. 2). We analyze the point ( , ) (180;38) { , } <sup>0</sup> *<sup>p</sup>* <sup>0</sup> *<sup>p</sup> B q*

chaotic (Fig. 2) zones. We consider the point *A*(*q*<sup>0</sup> ,

12 Computational and Numerical Simulations

Vlasov's form, and the we are looping for the functions w and F, satisfying (4.5), in the following form

1 1 1 1

 

*N N N N ij ij i j i j*

**Figure 5.** Dependence (*w*(0.5, 0.5, *t*)) and power spectra for different partition *m*=*n* =8;16;32 in a chaotic zone for *t* ∈ 40;45 .

**Figure 5.** Dependence ( (0.5,0.5, )) *w t* and power spectra for different partition *m n* 8;16;32 in a chaotic zone for *t* [40;45] .

Increase of the resolution implies the results improvement. Results obtained for the cases 1d and 1e coincide in full. In Figure 3 vibration charts for the rectangular spherical shell, which have been obtained using the BGM for N = 11, as well as the FDM for partition number 8х8 and accuracy *O*(*h* <sup>2</sup> ) regarding the spatial coordinate are shown. In both cases the initial

Therefore, in the case of our 2D mechanical object we have achieved only the integral conver‐

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

The second computational example deals with the infinite cylindrical panel harmonically and

( , ) 0,

1 1 1 1 (0, ) (1, ) (0, ) (1, ) '' (0, ) '' (1, ) 0, *w t wt u t ut w t w t x x x x* = = = = = = (16)

1 111 *wx wx ux ux* ( ,0) ( ,0) ( ,0) ( ,0) 0. = = = = & & (17)

50×50 100×100 200×200 300×300

e

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

(15)

15

gence regarding the Fourier spectrum, and hence one my study a 1D problem.

2 2 2 2 3

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

*uw u k L ww x t x*

<sup>2</sup> 4 22 2 4 2 1 2 2

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

*wu w w w w k kw w L uw L ww q x xt x x <sup>t</sup>*

1

*x*

1 1

transversally loaded. In this case equations (4.2) yield

1 1 1 1

**Table 5.** Vibration charts {*q*0, ω*p*} for different *n* and different resolutions

1 1

*x x*

12 2

with the following boundary

and initial conditions

l

*n* =16

*n* =32

*n* =40

**Table 4.** Charts of shell vibrations in the {*q*0, ω*p*} plane.

problem is solved via the fourth order Runge-Kutta method. It should be emphasized that in order to remove potential errors in the obtained results and in order to confirm its reliability and validity, one needs to apply different numerical approaches while studying non-linear dynamics of 2D mechanical objects. The obtained vibration charts allow controlling the vibration regimes and a transition form dangerous zones to those of the required/engineering acceptable zones. One may observe that both charts obtained via the qualitatively different approaches are close to each other. For small values of the load (*q*<sup>0</sup> ≤30) the system exhibits periodic vibrations. Increase of the excitation amplitude implies occurrence of chaotic dynam‐ ics. There are also zones of the Hopf bifurcations and two-frequency quasi-periodic orbits. However, both charts exhibit a qualitative difference in the resonance zone (*ω<sup>p</sup>* =*ω*0), i.e. errors introduced by numerical techniques increase in the resonance conditions. However, increase of of the partition number of the applied mesh as well as increase of the applied terms of the series for the case of BGM again yields the reliable and validated results, although with a higher costs of time computation.

Approximation of *N* =15 for the BGM and of *n* =*m*=16for the FDM are most suitable to keep the validated results as well as economically reasonable computation time. Computational time of the BGM is lower than that of the FDM, since in the latter case we need to solve a system of algebraic equations (for *n* =*m*=8 we have 64 equations, for *n* =*m*=16-256 equations, *n* =*m*=32-1024 equations), which requires additional computational time.

Therefore, in the case of our 2D mechanical object we have achieved only the integral conver‐ gence regarding the Fourier spectrum, and hence one my study a 1D problem.

The second computational example deals with the infinite cylindrical panel harmonically and transversally loaded. In this case equations (4.2) yield

$$\begin{aligned} \frac{\hat{\sigma}^2 u}{\hat{\sigma} \mathbf{x}\_1^2} - k\_{x\_1} \frac{\partial w}{\partial \mathbf{x}\_1} + L\_3(w, w) - \frac{\partial u^2}{\partial t} &= 0, \\ \frac{1}{\lambda^2} \left\{ -\frac{1}{12} \frac{\hat{\sigma}^4 w}{\hat{\sigma} \mathbf{x}\_1^4} + k\_{x\_1} \left[ \frac{\partial u}{\partial \mathbf{x}\_1} - k\_{x\_1} w - \frac{1}{2} \left( \frac{\partial w}{\partial \mathbf{x}\_1} \right)^2 - w \frac{\hat{\sigma}^2 w}{\hat{\sigma} \mathbf{x}\_1^2} \right] + L\_1(u, w) + L\_2(w, w) \right\} + q - \frac{\hat{\sigma}^2 w}{\hat{\sigma} t} - \mathcal{s} \frac{\hat{\sigma} w}{\hat{\sigma} t} &= 0. \end{aligned} \tag{15}$$

with the following boundary

$$w(0,t) = w(1,t) = \mu(0,t) = \mu(1,t) = w\prescript{\circ}{}{\mathop{\mathbf{w}}}\_{\mathbf{x}\_1\mathbf{x}\_1}(0,t) = w\prescript{\circ}{}{\mathop{\mathbf{w}}}\_{\mathbf{x}\_1\mathbf{x}\_1}(1,t) = 0,\tag{16}$$

and initial conditions

problem is solved via the fourth order Runge-Kutta method. It should be emphasized that in order to remove potential errors in the obtained results and in order to confirm its reliability and validity, one needs to apply different numerical approaches while studying non-linear dynamics of 2D mechanical objects. The obtained vibration charts allow controlling the vibration regimes and a transition form dangerous zones to those of the required/engineering acceptable zones. One may observe that both charts obtained via the qualitatively different approaches are close to each other. For small values of the load (*q*<sup>0</sup> ≤30) the system exhibits periodic vibrations. Increase of the excitation amplitude implies occurrence of chaotic dynam‐ ics. There are also zones of the Hopf bifurcations and two-frequency quasi-periodic orbits. However, both charts exhibit a qualitative difference in the resonance zone (*ω<sup>p</sup>* =*ω*0), i.e. errors introduced by numerical techniques increase in the resonance conditions. However, increase of of the partition number of the applied mesh as well as increase of the applied terms of the series for the case of BGM again yields the reliable and validated results, although with a higher

50×50 100×100 200×200

300×300 400×400

**Table 4.** Charts of shell vibrations in the {*q*0, ω*p*} plane.

14 Computational and Numerical Simulations

Approximation of *N* =15 for the BGM and of *n* =*m*=16for the FDM are most suitable to keep the validated results as well as economically reasonable computation time. Computational time of the BGM is lower than that of the FDM, since in the latter case we need to solve a system of algebraic equations (for *n* =*m*=8 we have 64 equations, for *n* =*m*=16-256 equations,

*n* =*m*=32-1024 equations), which requires additional computational time.

costs of time computation.

$$
\hbar \mathbf{w}(\mathbf{x}\_1, 0) = \dot{\mathbf{w}}(\mathbf{x}\_1, 0) = \boldsymbol{\mu}(\mathbf{x}\_1, 0) = \dot{\boldsymbol{\mu}}(\mathbf{x}\_1, 0) = 0. \tag{17}
$$

**Table 5.** Vibration charts {*q*0, ω*p*} for different *n* and different resolutions

*<sup>ε</sup>* =1;*<sup>λ</sup>* <sup>=</sup> *<sup>a</sup>*

(Table 6) for *n*=16, 32, 40.

convergence is not achieved.

applied through computations:

**•** PDEs are solved via FEM;

**•** Charts resolution –200×200;

**6. Concluding remarks**

**Acknowledgements**

keeping the reliable and validated results.

**•** Number of partitions regarding the spatial coordinate –*n* =80.

than 16) wavelets while studying plates/shells dynamics.

*<sup>h</sup>* =50;*ω<sup>p</sup>* =8, 625;*q*<sup>0</sup> =59000;*kx*<sup>1</sup>

=0.

For the given parameters the system is in a chaotic regime, what is approved by the reported charts (Table 5, point А). In what follows we analyze the obtained signal versus number of partition of the spatial coordinate. Namely, we construct the power spectra and Poincaré maps

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

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

17

Comparison of the amplitudes of the analyzed time histories allows to conclude that the convergence is achieved for. Now, taking into account the Poincaré maps we that for *n* ≥60 a chaotic strange attractor appears, which has not been detected for a smaller number of applied partitions. The Fourier power spectrum exhibits a remarkable localization of dominating frequencies including the excitation frequency for different introduced partitions, but the

Therefore, taking into account the mentioned remarks, the following parameters have been

In the chapter part devoted to study non-linear vibrations of shells, we have chosen suitable charts resolutions and partition numbers to follow various scenarios of transitions form regular to chaotic dynamics and to optimal reduction of the computational time simultaneously

In the chapter part devoted to plate analysis we have shown that the obtained validated spectra of the Lyapunov exponents allow for the estimation of Kaplan-Yorke dimension, Sinai-Kolmogorov entropy, and velocity of the phase space compression. Furthermore, we have illustrated that the complex Morlet and Gauss wavelets have better localization with respect to frequency, in comparison to their real analogues, but time localization is better for the real wavelets. Therefore, one may either apply real or complex Morlet and Gauss (order bigger

This work has been supported by the grant RFFI No 12-01-31204, and the National Science Centre of Poland under the grant MAESTRO 2, No. 2012/04/A/ST8/00738, for years 2013-2016.

**Table 6.** Time histories (*w*(0.5, *t*)), Fourier spectra, and Poincaré maps for different panel partitions.

Observe that neither the input governing equations (5.1) nor more complex system of equations (4.2) cannot be solved analytically. The results obtained through FDM have been compared with the results obtained FEM (Finite Element Method) in the Bubnov-Galerkin form. The following characteristics are applied: time history (*w*(0.5, *t*)), power spectrum and the Morlet wavelets. Charts of the vibration regimes present all peculiarities and a general picture of the studied non-linear process within the investigated intervals of the control parameters. However, one would expect an optimal choice of the applied nodes (*n*) of the mesh, as well a number of partition of the excitation frequency (*ωp*) and the excitation amplitude (*q*0). In Table 5 charts of the vibration regimes on the control parameters plane {*ωp*, *q*0} and the interval partition regarding *x*1∈ 0;1 , including the applied color notation is presented for {*ωp*, *q*0} =50×50; 100×100; 200×200; 300×300, and the interval *x*1∈ 0;1 is divided into 16, 32 and 40 parts.

We have already mentioned that a crucial role in analysis plays the time of computation to get time histories and carry out their analysis. Taking into account the reported results we conclude that the most optimal chart resolution is that of 200×200; 300×300.

It should be noted that the constructed charts on a basis of only power spectra are not sufficient to decide about the results convergence versus the partition number (*n*) regarding the spatial coordinate. In order to get the validated results we apply the following fixed parameters (boundary conditions (5.2) and initial conditions (5.3)):

$$
\omega\_{\varepsilon} = 1; \lambda = \frac{a}{h} = 50; \omega\_p = 8, \ 625; q\_0 = 59000; k\_{x\_1} = 0.1
$$

*n*=16 *n*=32 *n*=60 *n*=80

**Table 6.** Time histories (*w*(0.5, *t*)), Fourier spectra, and Poincaré maps for different panel partitions.

and 40 parts.

16 Computational and Numerical Simulations

Observe that neither the input governing equations (5.1) nor more complex system of equations (4.2) cannot be solved analytically. The results obtained through FDM have been compared with the results obtained FEM (Finite Element Method) in the Bubnov-Galerkin form. The following characteristics are applied: time history (*w*(0.5, *t*)), power spectrum and the Morlet wavelets. Charts of the vibration regimes present all peculiarities and a general picture of the studied non-linear process within the investigated intervals of the control parameters. However, one would expect an optimal choice of the applied nodes (*n*) of the mesh, as well a number of partition of the excitation frequency (*ωp*) and the excitation amplitude (*q*0). In Table 5 charts of the vibration regimes on the control parameters plane {*ωp*, *q*0} and the interval partition regarding *x*1∈ 0;1 , including the applied color notation is presented for {*ωp*, *q*0} =50×50; 100×100; 200×200; 300×300, and the interval *x*1∈ 0;1 is divided into 16, 32

We have already mentioned that a crucial role in analysis plays the time of computation to get time histories and carry out their analysis. Taking into account the reported results we

It should be noted that the constructed charts on a basis of only power spectra are not sufficient to decide about the results convergence versus the partition number (*n*) regarding the spatial coordinate. In order to get the validated results we apply the following fixed parameters

conclude that the most optimal chart resolution is that of 200×200; 300×300.

(boundary conditions (5.2) and initial conditions (5.3)):

For the given parameters the system is in a chaotic regime, what is approved by the reported charts (Table 5, point А). In what follows we analyze the obtained signal versus number of partition of the spatial coordinate. Namely, we construct the power spectra and Poincaré maps (Table 6) for *n*=16, 32, 40.

Comparison of the amplitudes of the analyzed time histories allows to conclude that the convergence is achieved for. Now, taking into account the Poincaré maps we that for *n* ≥60 a chaotic strange attractor appears, which has not been detected for a smaller number of applied partitions. The Fourier power spectrum exhibits a remarkable localization of dominating frequencies including the excitation frequency for different introduced partitions, but the convergence is not achieved.

Therefore, taking into account the mentioned remarks, the following parameters have been applied through computations:


In the chapter part devoted to study non-linear vibrations of shells, we have chosen suitable charts resolutions and partition numbers to follow various scenarios of transitions form regular to chaotic dynamics and to optimal reduction of the computational time simultaneously keeping the reliable and validated results.
