∂l∂kQd�Xl

2 < ∂l∂kQd�Xl

field of a point on this fractal curve. Let us also consider a variable field Q Xi

, i ¼ ffiffiffiffiffiffi

dþXi � d�Xi

d�Xi � � � d�xi (7)

d�ζi � � ¼ 0 (8)

; t � � and the

d�Xk (9)

d�Xk > (10)

2

�1 p (5)

http://dx.doi.org/10.5772/intechopen.74435

51

Non-Linear Behaviours in the Dynamics of Some Biostructures

dt (6)

Vb i ¼ bdXi

Vi ¼ 1 2

scale resolution dependent (fractal velocity field).

of the bio-structure.

with

with

The "+" sign corresponds to forward processes of bio-structure's structural units, while the "�" sign correspond to the backwards ones;

(iv) The differential of the spatial coordinate field dXi ð Þ t; dt by means of which we can describe the bio-structure dynamics, is expressed as the sum of the two differentials, one of them being scale resolution independent (differential part d�xi ð Þt and the other one being scale resolution dependent (fractal part dþζi ð ÞÞ t i.e.,

$$d\_{\pm}X^{i}(t) = d\_{\pm}x^{i}(t) + d\_{\pm}\zeta^{i}(t) \tag{2}$$

(v) The non-differentiable part of the spatial coordinate field, by means of which we can describe the bio-structure dynamics, satisfies the fractal equation [14]:

$$d\_{\pm} \zeta^i(t, dt) = \lambda^i\_{\pm}(dt)^{1/D\_{\mathbb{F}}} \tag{3}$$

where λi � are constant coefficients that help to specify the fractalization type which describes the bio-structure dynamics. Also, DF defines the fractal dimension of the bio-structure nondifferentiable curve.

In this way, the bio-structure processes imply dynamics on geodesics having different fractal dimensions. This variety of fractal dimensions of the bio-structure geodesics is a result of the bio-structure's structure. For DF ¼ 2, quantum type processes are generated in bio-structure dynamics [15]. For DF < 2, correlative type processes are induced and for DF > 2 noncorrelative type processes are generated [6, 12, 13].

(vi) The differential time reflection invariance of any dynamical variable of the bio-structure is recovered by combining the derivatives dþ=dt and d�=dt in the non-differentiable operator

$$
\widehat{d}dt = \frac{1}{2} \left( \frac{d\_+ + d\_-}{dt} \right) - \frac{i}{2} \left( \frac{d\_+ - d\_-}{dt} \right) \tag{4}
$$

This is a natural result of the complex prolongation procedure applied to bio-structure dynamics [14, 16]. Applying now the non-differentiable operator to the spatial coordinate field, by means of which we can describe the bio-structure dynamics, yields the complex velocity field of the bio-structure.

$$
\widehat{V}^i = \frac{\widehat{d}X^i}{dt} = V^i - i\mathcal{U}^i,\\
\dot{\mathbf{i}} = \sqrt{-1} \tag{5}
$$

with

(1)

(4)

can be identified with dt, that is, δt � dt. Consequently, it will play the role of an independent variable. We shall use the notation dt for the usual time as in the Hamiltonian bio-

(iii) The dynamics of bio-structure's structural units are described by means of fractal variables. Since the differential time reflection invariance of any dynamical variable is broken, these functions depend on the space-time coordinates and also on the resolution scale. In consequence, one can define two derivatives of the variable field Q tð Þ ; dt at any point of the

> Q tð Þ� þ Δt;Δt Q tð Þ ;Δt Δt

> Q tð Þ� ;Δt Q tð Þ � Δt;Δt Δt

> > ð Þ t; dt by means of which we can describe

ð Þt and the other one being scale

ð Þt (2)

�ð Þ dt 1=DF (3)

structure dynamics;

bio-structure fractal curve:

50 Nonlinear Systems - Modeling, Estimation, and Stability

"�" sign correspond to the backwards ones;

resolution dependent (fractal part dþζi

correlative type processes are generated [6, 12, 13].

bddt ¼ 1 2

where λi

differentiable curve.

(iv) The differential of the spatial coordinate field dXi

dþQ tð Þ ; dt

d�Q tð Þ ; dt

being scale resolution independent (differential part d�xi

d�Xi

describe the bio-structure dynamics, satisfies the fractal equation [14]:

d�ζi

dt ¼ lim

dt ¼ lim

Δt!0þ

Δt!0�

The "+" sign corresponds to forward processes of bio-structure's structural units, while the

the bio-structure dynamics, is expressed as the sum of the two differentials, one of them

ð Þþ t d�ζi

� are constant coefficients that help to specify the fractalization type which describes

ð ÞÞ t i.e.,

(v) The non-differentiable part of the spatial coordinate field, by means of which we can

the bio-structure dynamics. Also, DF defines the fractal dimension of the bio-structure non-

In this way, the bio-structure processes imply dynamics on geodesics having different fractal dimensions. This variety of fractal dimensions of the bio-structure geodesics is a result of the bio-structure's structure. For DF ¼ 2, quantum type processes are generated in bio-structure dynamics [15]. For DF < 2, correlative type processes are induced and for DF > 2 non-

(vi) The differential time reflection invariance of any dynamical variable of the bio-structure is recovered by combining the derivatives dþ=dt and d�=dt in the non-differentiable operator

This is a natural result of the complex prolongation procedure applied to bio-structure dynamics [14, 16]. Applying now the non-differentiable operator to the spatial coordinate field, by

� i 2

dþ � d� dt � �

dþ þ d� dt � �

ð Þ¼ t; dt λi

ðÞ¼ t d�xi

$$V^i = \frac{1}{2} \frac{d\_+ X^i + d\_- X^i}{dt}, \\ \mathcal{U}^i = \frac{1}{2} \frac{d\_+ X^i - d\_- X^i}{dt} \tag{6}$$

The real part Vi of the bio-structure complex velocity field is differentiable and scale resolution independent (differentiable velocity field). The imaginary part Ui is non-differentiable and scale resolution dependent (fractal velocity field).

(vii) If we have no external constraint, one can find an infinite number of fractal curves (geodesics) relating any pair of points. This happens on all scales of bio-structure dynamics. Then, in the fractal space of the bio-structure, all the structural units are substituted with the geodesics themselves so that any external constraint can be interpreted as a selection of biostructure geodesics. The infinity of geodesics in the bundle, their non-differentiability and the two values of the derivative imply a generalized statistical fluid-like description of the bio-structure dynamics. Then, the average values of the bio-structure variables must be considered in the previously mentioned sense, so the average of d�Xi is

$$\left< d\_{\pm} X^{i} \right> \equiv d\_{\pm} x^{i} \tag{7}$$

with

$$
\langle d\_{\pm} \zeta^{i} \rangle = 0 \tag{8}
$$

The previous relation (8) implies that the average of the fractal fluctuations is null.

(viii) One can describe the bio-structure dynamics by means of a scale covariant derivative. Its explicit form can be obtained as follows. We assume that the bio-structure fractal curves are immersed in a 3-dimensional space. We also suppose that Xi is the spatial coordinate field of a point on this fractal curve. Let us also consider a variable field Q Xi ; t � � and the following Taylor expansion up to the second order

$$d\_{\pm}Q(X^{i},t) = \partial\_{t}Qdt + \partial\_{t}Qd\_{\pm}X^{i} + \frac{1}{2}\partial\_{t}\partial\_{k}Qd\_{\pm}X^{l}d\_{\pm}X^{k} \tag{9}$$

These relations are valid at any point and more for the points Xi on the bio-structure fractal curve which we have selected in Eq. (9). From here, forward and backward average values for bio-structure variables from Eq. (9) become

$$
\langle d\_{\pm}Q \rangle = \langle \partial\_{l}Qdt \rangle + \left\langle \partial\_{l}Qd\_{\pm}X^{l} \right\rangle + \frac{1}{2} < \partial\_{l}\partial\_{k}Qd\_{\pm}X^{l}d\_{\pm}X^{k} > \tag{10}
$$

We suppose that the average values of the all variable field Q and its derivatives coincide with themselves and the differentials d�Xi and dt are independent. Therefore, the average of their products coincides with the product of averages. Consequently, Eq. (10) becomes

$$d\_{\pm}Q = \partial\_{l}Qdt + \partial\_{l}Q < d\_{\pm}X^{i} > + \frac{1}{2}\partial\_{l}\partial\_{k}Q \langle d\_{\pm}X^{l}d\_{\pm}X^{k} \rangle \tag{11}$$

Even the average value of d�ζi is null, for the higher order of d�ζi the situation can still be different. Let us focus on the averages d�ζl d�ζk � �. Using Eq. (3) we can write

$$
\langle d\_{\pm} \zeta^{l} d\_{\pm} \zeta^{k} \rangle = \pm \lambda^{l}\_{\pm} \lambda^{k}\_{\pm} (dt)^{\binom{2}{\mathcal{D}\_{\mathbb{F}}} - 1} dt \tag{12}
$$

bd

bdVb i dt ¼ ∂tVb i

This means that the local acceleration ∂tVb i

complex coefficient of viscosity-type 4�1

where δil is the Kronecker's pseudo-tensor.

bdVi D dt ¼ ∂tVi

bdVi F dt ¼ ∂tVi

bdVb i dt ¼ ∂tVb i

form:

dt ¼ ∂t þ Vb i

þ Vb l ∂lVb i þ 1 4 ð Þ dt 2 DF � ��1

movements [9, 13, 17] of the bio-structure structural units, then:

λi þλl þ ¼ λi �λl

þ Vb l ∂lVb i

D þ Vl

F þ Vl

can be described by means of the following equations:

or more, by separating the motions on differential and fractal scale resolutions,

D∂lVi

D∂lVi

D � Vl

F þ Vl

Using the standard procedure from [18], the bio-structure dynamics at fractal scale resolution

∂i þ 1 4 ð Þ dt 2 DF � ��1

Let us now consider the principle of scale covariance (the physics laws – bio-structure dynamics specific – are invariant with respect to scale transformations) and postulate that the passage from the classical (differentiable) physics to the fractal (non-differentiable) physics can be implemented by replacing the standard time derivative d=dt by the non-differentiable operator bd=dt. In this way, this operator has the role of a scale covariant derivative. More precisely, it is used to write the bio-structure dynamics fundamental equations in the same form as in the classic (differentiable) case. In these conditions, applying the operator (18) to the complex velocity field (5), with no external constraint, the bio-structure geodesics take the

, the convection Vb l

make their balance at any point of the bio-structure fractal curve. Moreover, the presence of the

If the fractalization is achieved by Markov type stochastic processes, which involve Lévy type

� iλð Þ dt

2 DF � ��1

F þ λð Þ dt

F þ λð Þ dt

ð Þ dt 2 DF � ��1

that it is a rheological medium. So, it has memory, as a datum, by its own structure.

Under these conditions, the equation of bio-structure geodesics takes the simple form

Dlk∂l∂kVb i

∂lVb i

Dlk∂l∂kVb i

∂lVi F ¼ 0

∂lVi D ¼ 0

2 DF � ��1 ∂l

2 DF � ��1 ∂l

" #

" #

Dlk∂l∂k (18)

http://dx.doi.org/10.5772/intechopen.74435

Non-Linear Behaviours in the Dynamics of Some Biostructures

¼ 0 (19)

and the dissipation Dlk∂l∂kVb i

¼ 0 (21)

Dlk in the bio-structures dynamics specifies

� ¼ 2λδil (20)

,

53

(22)

where the sign + corresponds to dt > 0 and the sign – corresponds to dt < 0.

Then, Eq. (11) takes the form:

$$d\_{\pm}Q = \partial\_t Qdt + \partial\_t Q < d\_{\pm}X^i > + \frac{1}{2}\partial\_t \partial\_k Qd\_{\pm} \mathbf{x}^l d\_{\pm} \mathbf{x}^k \pm \frac{1}{2}\partial\_l \partial\_k Q \left[\lambda\_{\pm}^l \lambda\_{\pm}^k (dt)^{\left(\frac{2}{\Gamma\_l}\right)-1} dt\right] \tag{13}$$

If we divide by dt and neglect the terms that contain differential factors (for details, see the method from [13, 14]) we obtain:

$$\frac{d\pm Q}{dt} = \partial\_t Q + \nu^i\_{\pm} \partial\_i Q \pm \frac{1}{2} \lambda^l\_{\pm} \lambda^k\_{\pm} (dt)^{\binom{\frac{2}{l}}{l} - 1} \partial\_l \partial\_k Q \tag{14}$$

where νi þ ¼ dþxi dt , νi � ¼ d�xi dt .

These relations also allow us to define the operators

$$\frac{d\_{\pm}}{dt} = \partial\_t + \nu\_{\pm}^i \partial\_i \pm \frac{1}{2} \lambda\_{\pm}^l \lambda\_{\pm}^k (dt)^{\binom{\frac{2}{l}}{2} - 1} \partial\_l \partial\_k \tag{15}$$

Using Eqs. (4), (5), and (15), let us calculate the differentiable operator

$$\frac{\widehat{d}Q}{dt} = \partial\_t Q + \widehat{V}^i \partial\_i Q + \frac{1}{4} (dt)^{\binom{\frac{2}{D}}{D-i} - 1} D^k \partial\_l \partial\_k Q \tag{16}$$

where

$$\begin{aligned} D^{\mathbb{k}} &= d^{\mathbb{k}} - \overline{d}^{\mathbb{k}} \\ d^{\mathbb{k}} &= \lambda\_{+}^{\mathbb{k}} \lambda\_{+}^{\mathbb{k}} - \lambda\_{-}^{\mathbb{k}} \lambda\_{-}^{\mathbb{k}} \ \overline{d}^{\mathbb{k}} = \lambda\_{+}^{\mathbb{k}} \lambda\_{+}^{\mathbb{k}} + \lambda\_{-}^{\mathbb{k}} \lambda\_{-}^{\mathbb{k}} \end{aligned} \tag{17}$$

Eq. (16) also allows us to define the covariant derivative in the bio-structure dynamics

Non-Linear Behaviours in the Dynamics of Some Biostructures http://dx.doi.org/10.5772/intechopen.74435 53

$$\frac{d\widehat{d}}{dt} = \partial\_t + \widehat{V}^i \partial\_i + \frac{1}{4} (dt)^{\left(\frac{2}{D\_{\mathbb{F}}}\right) - 1} D^{\mathbb{K}} \partial\_{\mathbb{H}} \partial\_{\mathbb{k}} \tag{18}$$

Let us now consider the principle of scale covariance (the physics laws – bio-structure dynamics specific – are invariant with respect to scale transformations) and postulate that the passage from the classical (differentiable) physics to the fractal (non-differentiable) physics can be implemented by replacing the standard time derivative d=dt by the non-differentiable operator bd=dt. In this way, this operator has the role of a scale covariant derivative. More precisely, it is used to write the bio-structure dynamics fundamental equations in the same form as in the classic (differentiable) case. In these conditions, applying the operator (18) to the complex velocity field (5), with no external constraint, the bio-structure geodesics take the form:

$$\frac{d\widehat{V}^i}{dt} = \partial\_t \widehat{V}^i + \widehat{V}^l \partial\_l \widehat{V}^i + \frac{1}{4} (dt)^{\binom{2}{l} - 1} D^k \partial\_l \partial\_k \widehat{V}^i = 0 \tag{19}$$

This means that the local acceleration ∂tVb i , the convection Vb l ∂lVb i and the dissipation Dlk∂l∂kVb i , make their balance at any point of the bio-structure fractal curve. Moreover, the presence of the complex coefficient of viscosity-type 4�1 ð Þ dt 2 DF � ��1 Dlk in the bio-structures dynamics specifies that it is a rheological medium. So, it has memory, as a datum, by its own structure.

If the fractalization is achieved by Markov type stochastic processes, which involve Lévy type movements [9, 13, 17] of the bio-structure structural units, then:

$$
\lambda\_+^i \lambda\_+^l = \lambda\_-^i \lambda\_-^l = 2\lambda \delta^{il} \tag{20}
$$

where δil is the Kronecker's pseudo-tensor.

We suppose that the average values of the all variable field Q and its derivatives coincide with themselves and the differentials d�Xi and dt are independent. Therefore, the average of their

Even the average value of d�ζi is null, for the higher order of d�ζi the situation can still be

�λk �ð Þ dt

1 2

∂l∂kQ d�Xl

d�ζk � �. Using Eq. (3) we can write

2 DF � � �1

d�xk � 1 2

∂l∂kQ λl

2 DF � � �1

2 DF � � �1 �λk �ð Þ dt

d�Xk � � (11)

dt (12)

2 DF � � �1 dt

∂l∂kQ (14)

∂l∂k (15)

Dlk∂l∂kQ (16)

(13)

(17)

" #

products coincides with the product of averages. Consequently, Eq. (10) becomes

d�Q ¼ ∂tQdt þ ∂iQ < d�Xi > þ

d�ζl

d�ζk � � ¼ �λl

where the sign + corresponds to dt > 0 and the sign – corresponds to dt < 0.

1 2

�∂iQ �

�∂i � 1 2 λl �λk �ð Þ dt

> ∂iQ þ 1 4 ð Þ dt 2 DF � � �1

Dlk ¼ dlk � id

Eq. (16) also allows us to define the covariant derivative in the bio-structure dynamics

lk

∂l∂kQd�xl

If we divide by dt and neglect the terms that contain differential factors (for details, see the

1 2 λl �λk �ð Þ dt

different. Let us focus on the averages d�ζl

52 Nonlinear Systems - Modeling, Estimation, and Stability

d�Q ¼ ∂tQdt þ ∂iQ < d�Xi > þ

d�Q

d�

bdQ

dlk ¼ λl

dt ¼ ∂t þ νi

Using Eqs. (4), (5), and (15), let us calculate the differentiable operator

dt ¼ ∂tQ þ Vb i

þλk þ � λl �λk �, d lk ¼ λl þλk þ þ λl �λk �

� ¼ d�xi dt . These relations also allow us to define the operators

dt ¼ ∂tQ þ νi

Then, Eq. (11) takes the form:

method from [13, 14]) we obtain:

where νi

where

þ ¼ dþxi dt , νi

Under these conditions, the equation of bio-structure geodesics takes the simple form

$$\frac{d\widehat{V}^i}{dt} = \partial\_t \widehat{V}^i + \widehat{V}^l \partial\_l \widehat{V}^i - i\lambda (dt)^{\binom{2}{l}-1} D^{lk} \partial\_l \partial\_k \widehat{V}^i = 0 \tag{21}$$

or more, by separating the motions on differential and fractal scale resolutions,

$$\begin{aligned} \frac{\widehat{d}V\_D^i}{dt} &= \partial\_t V\_D^i + V\_D^l \partial\_l V\_D^i - \left[ V\_F^l + \lambda (dt)^{\binom{2}{\mathcal{D}\_V} - 1} \mathfrak{d}' \right] \mathfrak{d}\_l V\_F^i = 0 \\\\ \frac{\widehat{d}V\_F^i}{dt} &= \partial\_t V\_F^i + V\_D^l \partial\_l V\_F^i + \left[ V\_F^l + \lambda (dt)^{\binom{2}{\mathcal{D}\_V} - 1} \mathfrak{d}' \right] \mathfrak{d}\_l V\_D^i = 0 \end{aligned} \tag{22}$$

Using the standard procedure from [18], the bio-structure dynamics at fractal scale resolution can be described by means of the following equations:

$$
\partial\_t V\_F^i + V\_F^l \partial\_l V\_F^i = \lambda(dt)^{\binom{2}{\sum}-1} \partial^l \partial\_l V\_F^i \tag{23}
$$

$$
\partial\_i V\_F^i = 0 \tag{24}
$$

Relations (29) and (30) suggest that the bio-structure velocity field is highly non-linear through topological solitons of kink type (tanh), various non-topological solitons of breather type

For y ¼ 0, we obtain in relation (29) the flow critical velocity of the bio-structure in the form

and (30) will be the one in which we assign indexes for each component.

Vxð Þ¼ x; y ¼ 0 Vc ¼

while taking into account (31), relation (28) becomes

Θ ¼ r

dcð Þ¼ x; y ¼ 0 Θ

, v ¼ Vy w0 , Ω ¼

4:52=3 3 1 3 � Ω ζ 1 3

uð Þ¼ ζ; η

ωη ζ 2 3

þ ð∞

þ∞

V2

2rV2 c

xð Þ x; y dy ¼

so that the critical cross section of the strains lines tube of the bio-structure is given by:

), and through topological – non-topological soliton mixtures of kink-breather type

> 1:5 Θ 6ϱ � �2 3 � �

> > 2 DF � ��1 x

" #1=3 (31)

Non-Linear Behaviours in the Dynamics of Some Biostructures

http://dx.doi.org/10.5772/intechopen.74435

c ð Þ x; 0 dy, (32)

Θ 6ϱ � �2 3 y0

, (35)

λð Þ dt 2 DF � ��1 x0 " #2=3 (34)

(33)

55

(36)

λð Þ dt

þ ð dc

�dc

¼ 2:42 λð Þ dt

Θ 6ϱ � �2 3

DF � ��1 x0 " #1=3 , ω ¼

> sech2 0:5Ωωη ζ 2 3 !

" # !

Relations (29) and (30) can be strongly simplified if we introduce the normalized quantities

w0 λð Þ dt 2

where x0, y0, w0 are specific lengths and the specific velocity, respectively, of the laminar flow of

sech2 0:5Ωωη ζ 2 3 !

We present in Figures 1a, b and 2a, b the dependence of the normalized velocity field u on the normalized spatial coordinates ξ, η for various nonlinearity degrees (ω = 0.3; 6). The results

1:5Ω ζ 1 3

V2

2 DF � ��1 x

3 r Θ � �1=3

� tanh 0:5Ωωη ζ 2 3

" #2

(sech2

(sech2

ζ ¼ x x0 , η ¼ y y0

the bio-structure. It results that

, u ¼ Vx w0

vð Þ¼ ζ; η

Eq. (23) corresponds to the specific impulse conservation law at fractal scale resolution, while Eq. (24) corresponds to the states density conservation law at fractal scale resolution (we consider that the density of the bio-structure at fractal scale resolution is constant – incompressible bio-structure).

Since this equation system is non-linear, one could find relatively difficult finding the solutions for these equations [19, 20]. However, in the particular case of a stationary flow in a plane symmetry ð Þ x; y , there is an analytical solution of this system. Then, for VF ¼ Vx; Vy; 0 � �, Eqs. (23) and (24) take the form:

$$V\_x \frac{\partial V\_x}{\partial x} + V\_y \frac{\partial V\_x}{\partial x} = \lambda (dt)^{\left(\frac{2}{V\_y}\right) - 1} \frac{\partial^2 V\_x}{\partial y^2} \tag{25}$$

$$\frac{\partial V\_x}{\partial x} + \frac{\partial V\_y}{\partial y} = 0 \tag{26}$$

The boundary conditions of the flow are:

$$\lim\_{y \to 0} V\_y(x, y) = 0, \lim\_{y \to 0} \frac{\partial V\_x}{\partial y} = 0, \lim\_{y \to \infty} V\_x(x, y) = 0 \tag{27}$$

and the flux momentum per length unit is constant

$$\Theta = \rho \int\_{-\infty}^{+\infty} V\_x^2 dy = \text{const.}\tag{28}$$

Using the method from [18–20] for solving Eqs. (25) and (26), with the conditions (27) and (28), the following solutions result:

$$V\_x = \frac{\left[1.5\left(\frac{\oplus}{\text{sq}}\right)^{\frac{\pi}{3}}\right]}{\left[\lambda(dt)\left(\frac{\oplus}{\text{sq}}\right)^{-1}x\right]^{1/3}} \cdot \text{sech}^2 \cdot \frac{\left[\left(0.5\text{y}\right)\left(\frac{\oplus}{\text{sq}}\right)^{\frac{\pi}{3}}\right]}{\left[\lambda(dt)\left(\frac{\oplus}{\text{sq}}\right)^{-1}x\right]^{2/3}} \tag{29}$$

$$V\_y = \frac{\left[4.5\left(\frac{\oplus}{\text{sq}}\right)^{\frac{\pi}{3}}\right]}{\left[3\lambda(dt)\left(\frac{\oplus}{\text{sq}}\right)^{-1}x\right]^{1/3}} \cdot \left[\frac{\left[\text{y}\left(\frac{\oplus}{\text{sq}}\right)^{\frac{\pi}{3}}\right]}{\left[\lambda(dt)\left(\frac{\oplus}{\text{sq}}\right)^{-1}x\right]^{\frac{\pi}{3}}} \cdot \text{sech}^2 \cdot \frac{\left[\left(0.5\text{y}\right)\left(\frac{\oplus}{\text{sq}}\right)^{\frac{\pi}{3}}\right]}{\left[\lambda(dt)\left(\frac{\oplus}{\text{sq}}\right)^{-1}x\right]^{\frac{\pi}{3}}} \cdot \tanh\frac{\left[\left(0.5\text{y}\right)\left(\frac{\oplus}{\text{sq}}\right)^{\frac{\pi}{3}}\right]}{\left[\lambda(dt)\left(\frac{\oplus}{\text{sq}}\right)^{-1}x\right]^{\frac{\pi}{3}}} \tag{30}$$

Relations (29) and (30) suggest that the bio-structure velocity field is highly non-linear through topological solitons of kink type (tanh), various non-topological solitons of breather type (sech2 ), and through topological – non-topological soliton mixtures of kink-breather type (sech2 -tanh). Given the structural complexity of the bio-structure (which is given by its various structural units, that retains their own velocity field) an accurate way of writing relations (29) and (30) will be the one in which we assign indexes for each component.

For y ¼ 0, we obtain in relation (29) the flow critical velocity of the bio-structure in the form

$$V\_x(\mathbf{x}, y=0) = V\_c = \frac{\begin{bmatrix} 1.5 \left(\frac{\Theta}{\delta \mathbf{q}}\right)^{\frac{2}{3}} \\\\ \lambda (dt)^{\left(\frac{2}{D\_p}\right) - 1} \mathbf{x} \end{bmatrix}^{1/3}}{\begin{bmatrix} \lambda (dt)^{\left(\frac{2}{D\_p}\right) - 1} \mathbf{x} \end{bmatrix}^{1/3}} \tag{31}$$

while taking into account (31), relation (28) becomes

∂tVi F þ Vl

Vx ∂Vx ∂x

þ Vy

Vyð Þ¼ x; y 0, lim

Θ ¼ r

1:5 Θ 6ϱ � �2 3 � �

2 DF � � �1 x " #1=3 � sech2 �

> y Θ 6ϱ � �1 3 � �

" #2

3 sech2 �

λð Þ dt 2 DF � � �1 x

λð Þ dt

∂Vx

∂Vx ∂x þ ∂Vy

y!0

þ ð∞

�∞

∂Vx

V2

Using the method from [18–20] for solving Eqs. (25) and (26), with the conditions (27) and (28),

∂y ¼ 0, lim

pressible bio-structure).

Eqs. (23) and (24) take the form:

54 Nonlinear Systems - Modeling, Estimation, and Stability

The boundary conditions of the flow are:

the following solutions result:

4:5 Θ 6ϱ � �2 3 � �

" #1=3 �

3λð Þ dt 2 DF � � �1 x

Vy ¼

lim y!0

and the flux momentum per length unit is constant

Vx ¼

F∂lVi

∂iVi

Eq. (23) corresponds to the specific impulse conservation law at fractal scale resolution, while Eq. (24) corresponds to the states density conservation law at fractal scale resolution (we consider that the density of the bio-structure at fractal scale resolution is constant – incom-

Since this equation system is non-linear, one could find relatively difficult finding the solutions for these equations [19, 20]. However, in the particular case of a stationary flow in a plane symmetry ð Þ x; y , there is an analytical solution of this system. Then, for VF ¼ Vx; Vy; 0 � �,

∂x ¼ λð Þ dt

2 DF � � �1 ∂2 Vx

F ¼ λð Þ dt 2

DF � � �1 ∂l ∂lVi

F (23)

∂y2 (25)

∂y ¼ 0 (26)

xdy ¼ const: (28)

0:5y � � Θ 6ϱ � �1 3

> 2 DF � � �1 x

> > 3 � tanh

λð Þ dt

0:5y � � Θ 6ϱ � �1 3 � �

" #2

λð Þ dt 2 DF � � �1 x

� �

y!∞ Vxð Þ¼ x; y 0 (27)

" #2=3 (29)

0:5y � � Θ 6ϱ � �1 3 � �

" #2

3

(30)

λð Þ dt 2 DF � � �1 x

F ¼ 0 (24)

$$\Theta = \rho \int\_{+\circ}^{+\circ} V\_x^2(\mathbf{x}, y) dy = \int\_{-d\_c}^{+d\_c} V\_c^2(\mathbf{x}, 0) dy,\tag{32}$$

so that the critical cross section of the strains lines tube of the bio-structure is given by:

$$d\_c(\mathbf{x}, \mathbf{y} = \mathbf{0}) = \frac{\Theta}{2\rho V\_c^2} = 2.42 \left[ \lambda (dt)^{\binom{2}{\frac{2}{\|\mathbf{y}\|} - 1}} \mathbf{x} \right]^{\frac{2}{5}} \left( \frac{\rho}{\Theta} \right)^{1/3} \tag{33}$$

Relations (29) and (30) can be strongly simplified if we introduce the normalized quantities

$$\zeta = \frac{\mathbf{x}}{\mathbf{x}\_0}, \eta = \frac{\mathbf{y}}{y\_0}, u = \frac{V\_x}{w\_0}, v = \frac{V\_y}{w\_0}, \Omega = \frac{\left(\frac{\Theta}{\Theta}\right)^{\frac{2}{\gamma}}}{w\_0 \left[\lambda (dt)^{\left(\frac{2}{\Gamma\_0}\right) - 1} x\_0\right]^{1/3}}, \omega = \frac{\left(\frac{\Theta}{\Theta}\right)^{\frac{2}{\gamma}} y\_0}{\left[\lambda (dt)^{\left(\frac{2}{\Gamma\_0}\right) - 1} x\_0\right]^{2/3}}\tag{34}$$

where x0, y0, w0 are specific lengths and the specific velocity, respectively, of the laminar flow of the bio-structure. It results that

$$\ln(\zeta,\eta) = \frac{1.5\Omega}{\zeta^{\ddagger}}\text{sech}^2\left(\frac{0.5\Omega\omega\eta}{\zeta^{\ddagger}}\right),\tag{35}$$

$$w(\zeta,\eta) = \frac{4.5^{2/3}}{3^{\ddagger}} \cdot \frac{\Omega}{\zeta^{\ddagger}} \left[ \frac{\omega\eta}{\zeta^{\ddagger}} \text{sech}^2 \left( \frac{0.5\Omega\omega\eta}{\zeta^{\ddagger}} \right) - \tanh\left(\frac{0.5\Omega\omega\eta}{\zeta^{\ddagger}}\right) \right] \tag{36}$$

We present in Figures 1a, b and 2a, b the dependence of the normalized velocity field u on the normalized spatial coordinates ξ, η for various nonlinearity degrees (ω = 0.3; 6). The results

showcase that the velocity field on the bio-structure flow direction (ξ) is affected in a weak manner by the nonlinearity degree (the velocity always decreases on the flow axes regardless of the nonlinearity degree). Also, the bio-structure flow direction (η) is strongly affected. Biostructure flow starts from constant values on the η axes and with the increase of ω, preferential

Non-Linear Behaviours in the Dynamics of Some Biostructures

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57

In Figures 3a, b and 4a, b the dependences of the normalized velocity field v on the normalized spatial coordinates ξ, η for various non-linearity degrees (ω = 0.3; 6) are represented. For small non-linearity degrees, the variations (increase/decrease) of the velocity field have similar behaviors on both directions (ξ, η), while for higher values of the non-linearity degree these

Taking the above into account, the force that the bio-structure will exercise to the walls of the flow vessels is of great importance for the understanding of arterial occlusion and other

Figure 3. The 3D representation (a) and the contour plot (b) of the normalized velocity field v on the normalized spatial

bio-structure flow direction can be identified.

variations are only focused on a single direction (ξ).

circulatory system diseases.

coordinates ξ, η for the nonlinearity degree ω = 0.3.

Figure 1. The dependence of the normalized velocity field u on the normalized spatial coordinates ξ, η for the nonlinearity degree ω = 0.3: (a) 3D representation; contour plot (b).

Figure 2. The dependence of the normalized velocity field u on the normalized spatial coordinates ξ, η for the nonlinearity degree ω = 6: (a) 3D representation; contour plot (b).

showcase that the velocity field on the bio-structure flow direction (ξ) is affected in a weak manner by the nonlinearity degree (the velocity always decreases on the flow axes regardless of the nonlinearity degree). Also, the bio-structure flow direction (η) is strongly affected. Biostructure flow starts from constant values on the η axes and with the increase of ω, preferential bio-structure flow direction can be identified.

In Figures 3a, b and 4a, b the dependences of the normalized velocity field v on the normalized spatial coordinates ξ, η for various non-linearity degrees (ω = 0.3; 6) are represented. For small non-linearity degrees, the variations (increase/decrease) of the velocity field have similar behaviors on both directions (ξ, η), while for higher values of the non-linearity degree these variations are only focused on a single direction (ξ).

Taking the above into account, the force that the bio-structure will exercise to the walls of the flow vessels is of great importance for the understanding of arterial occlusion and other circulatory system diseases.

Figure 1. The dependence of the normalized velocity field u on the normalized spatial coordinates ξ, η for the non-

Figure 2. The dependence of the normalized velocity field u on the normalized spatial coordinates ξ, η for the non-

linearity degree ω = 0.3: (a) 3D representation; contour plot (b).

56 Nonlinear Systems - Modeling, Estimation, and Stability

linearity degree ω = 6: (a) 3D representation; contour plot (b).

Figure 3. The 3D representation (a) and the contour plot (b) of the normalized velocity field v on the normalized spatial coordinates ξ, η for the nonlinearity degree ω = 0.3.

Figure 4. The 3D representation (a) and the contour plot (b) of the normalized velocity field v on the normalized spatial coordinates ξ, η for the non-linearity degree ω = 6.

Figure 5. The dependence of the normalized force field F of a bio-structure flow on the vessels, on the normalized spatial coordinates ξ, η for two resolution scales: 3D representation (a); contour plot (b) for the non-linearity degree ω = 0.3.

Non-Linear Behaviours in the Dynamics of Some Biostructures

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59

Figure 6. The dependence of the normalized force field F of a bio-structure flow on the vessels, on the normalized spatial coordinates ξ, η for two resolution scales: 3D representation (a) and contour plot (b), for the non-linearity degree ω = 6.

In our case the normalized force is given by the relation:

$$\begin{aligned} f(\zeta,\eta) &= \partial\_{\zeta}\mu - \partial\_{\zeta}\upsilon \\ &= -1.5 \frac{\operatorname{Oscech}^{-2}\left(\frac{0.5\circ\eta}{\zeta^{\ddagger}}\right)\tanh\left(\frac{0.5\circ\eta}{\zeta^{\ddagger}}\right)\omega}{\zeta} - \frac{1}{\zeta^{\ddagger}} \\ &+ \left(0.9\Omega\left(-\frac{2\circ\eta\operatorname{sech}^{2}\left(\frac{0.5\circ\eta}{\zeta^{\ddagger}}\right)}{\zeta^{\ddagger}} + \frac{0.66\omega^{2}\eta^{2}\operatorname{sech}^{2}\left(\frac{0.5\circ\eta}{\zeta^{\ddagger}}\right)\tanh\left(\frac{0.5\circ\eta}{\zeta^{\ddagger}}\right)\omega}{\zeta^{\ddagger}}\right) \\ &+ \frac{0.33}{\left(1 - \tanh^{2}\left(\frac{0.5\circ\eta}{\zeta^{\ddagger}}\right)\right)\omega\eta}\right) \\ &0.32\left(\frac{\omega\eta\operatorname{sech}^{2}\left(\frac{0.5\circ\eta}{\zeta^{\ddagger}}\right)}{\zeta^{\ddagger}} - \tanh\left(\frac{0.5\circ\eta}{\zeta^{\ddagger}}\right)\right)3^{3/2} \\ &+ \frac{0.33\left(\frac{0.5\circ\eta}{\zeta^{\ddagger}}\right)}{\zeta^{\ddagger}}\end{aligned} (37)$$

Figure 5. The dependence of the normalized force field F of a bio-structure flow on the vessels, on the normalized spatial coordinates ξ, η for two resolution scales: 3D representation (a); contour plot (b) for the non-linearity degree ω = 0.3.

In our case the normalized force is given by the relation:

Ωsech 2 0:5ωη

ζ 2 3 !

ωηsech2 0:5ωη

ζ 5 3

! !

ζ 5 3

ωηsech2 0:5ωη

ζ 2 3

" #

ζ 2 3

ζ 2 3 !

> ζ 4 3

ζ 2 3 !

tanh 0:5ωη ζ 2 3 !

þ

ωη

1

CCCCA

� tanh 0:5ωη ζ 2 3 !

ζ � 1

Figure 4. The 3D representation (a) and the contour plot (b) of the normalized velocity field v on the normalized spatial

ω

ζ 1 3

> ζ 2 3 !

> > ζ 7 3

tanh 0:5ωη ζ 2 3 !

ω

(37)

0:66ω2η2sech2 0:5ωη

1

CCCCA 33=2

fð Þ¼ ζ; η ∂ηu � ∂ζv

coordinates ξ, η for the non-linearity degree ω = 6.

58 Nonlinear Systems - Modeling, Estimation, and Stability

¼ �1:5

0

BBBB@

þ

þ

0:3Ω

0

BBBB@

� 0:9Ω � 2

0

BBBB@

3

0:33 1 � tanh2 0:5ωη

Figure 6. The dependence of the normalized force field F of a bio-structure flow on the vessels, on the normalized spatial coordinates ξ, η for two resolution scales: 3D representation (a) and contour plot (b), for the non-linearity degree ω = 6.

In Figures 5a, b and 6a, b the normalized force field evolution on the two-flow direction (ξ, η) for various non-linear degrees is represented. It results that with the increase of the nonlinearity of the bio-structure the force toward the walls increases.

A second application was proposed for bio-structure flow, and of cholesterol deposition on the vessel walls. The results revealed the directional flow toward the walls. This could explain in our opinion the thickening effect which is one of the sources of arteriosclerosis. Moreover, our model imposes redefinition of "good" and "bad" cholesterol (which are traditionally associated with HDL and LDL respectively); instead they should be replaced by the following notions: specific cholesterol entities, associated with a certain non-differentiable curve, that have a major endothelial impact and specific cholesterol entities which have no or low endo-

Non-Linear Behaviours in the Dynamics of Some Biostructures

http://dx.doi.org/10.5772/intechopen.74435

61

There is currently a great number of works describing matter organization and behavior in all of its variations, from which we mention [27]. We consider that our bio-structure flow model could also be used to further development in the study of other complex systems dynamics (such as pulmonary and metabolic diseases or environmental systems). Moreover, possible

\*, Maricel Agop3 and Daniel Timofte4

1 Department Obstetrics Gynecology, Gr. T. Popa University of Medicine and Pharmacy, Iasi,

4 Department of Surgery, Gr. T. Popa University of Medicine and Pharmacy, Iasi, Romania

[1] Popescu DM. Hematologie Clinică (in Romanian). Bucharest: Medical Publishing House;

[3] Badii R, Politi A. Complexity: Hierarchical Structure and Scaling in Physics. Cambridge:

2 Faculty of Mathematics, Alexandru Ioan Cuza University from Iasi, Romania 3 Department of Physics, Gheorghe Asachi Technical University of Iasi, Romania

[2] Tesloianu ND. PAD and COPD in Smokers. Iași: Ars Longa; 2014

therapeutic treatments can be developed, e.g., new drug release mechanisms.

thelial impact.

Conflict of interest

Author details

Emil Anton1

Romania

References

1999

The authors declare no conflict of interest.

, Anna Gavrilut2

Cambridge University Press; 1997

\*Address all correspondence to<EMAIL_ADDRESS>
### 3. Discussions

The theory proposed in this chapter explains from a fractal viewpoint the atherogenesis process [21], basically "molding" to the classical anatomical and histopathological descriptions and completely respecting the process they postulate. In consequence, the fractal physics model sustains already accumulated morpho-pathological information and research. There are plenty of electronic and optical microscopy images that describe the spatial-temporal hologram of the phenomenon; we can thus discuss about the non-fractal – fractal and microscopic – macroscopic translation through holographically reproducible auto-similarity [21]. In this way, we affirm that fractality is the mathematical and semantic quintessence for defining atherogenesis, a process that can be physically characterized by fractal physics. This physics becomes in this situation more of a component rather than an explanation for the complex biological system represented by the atheroma plaque [22, 23].

In what concerns the recovery of such biological diseases, there are a huge number of techniques. We recall that external electrical stimulation can cause changes in the bio-structure vessels. Although atherosclerosis cause vasodilatation in the affected area and bio-structure flow remains unchanged for an extended period of time, the vascular wall stiffness will increase the pulse pressure. The purpose of the study developed in [24, 25] was to measure the effects of electrical stimulation (ES) on bio-structure flow and bio-structure pressure. All subjects received electrical stimulation at intensity sufficient to produce torque equal to 15% of the predetermined maximal voluntary contraction of their right quadriceps femor is muscle. The conclusions were that the increase in bio-structure flow occurred within 5 min after the onset of ES and dropped to resting levels within 1 min after a 10-min period of ES [25]. Kinesiotherapy or Kinesitherapy or kinesiatrics, is the therapeutic treatment of disease by passive and active muscular movements (as by massage) and of exercise [26]. From the physiotherapeutic viewpoint, an efficient treatment is directed toward improving bio-structure flow and also decreasing the disparity between the demand for bio-structure and its supply [22]. An effective vascular rehabilitation training program for improving walking efficiency and vascular remodeling in patients with diabetic atherosclerosis suffering from intermittent claudication could be a supervised treadmill walking exercise combined with Allen-Burger exercises [23].

### 4. Conclusions

The present chapter proposes a fractal model for the dynamics analysis of bio-structure flows. The fractal hydrodynamic equations were obtained and applied for the laminar flow of bio-structure.

A second application was proposed for bio-structure flow, and of cholesterol deposition on the vessel walls. The results revealed the directional flow toward the walls. This could explain in our opinion the thickening effect which is one of the sources of arteriosclerosis. Moreover, our model imposes redefinition of "good" and "bad" cholesterol (which are traditionally associated with HDL and LDL respectively); instead they should be replaced by the following notions: specific cholesterol entities, associated with a certain non-differentiable curve, that have a major endothelial impact and specific cholesterol entities which have no or low endothelial impact.

There is currently a great number of works describing matter organization and behavior in all of its variations, from which we mention [27]. We consider that our bio-structure flow model could also be used to further development in the study of other complex systems dynamics (such as pulmonary and metabolic diseases or environmental systems). Moreover, possible therapeutic treatments can be developed, e.g., new drug release mechanisms.

### Conflict of interest

In Figures 5a, b and 6a, b the normalized force field evolution on the two-flow direction (ξ, η) for various non-linear degrees is represented. It results that with the increase of the non-

The theory proposed in this chapter explains from a fractal viewpoint the atherogenesis process [21], basically "molding" to the classical anatomical and histopathological descriptions and completely respecting the process they postulate. In consequence, the fractal physics model sustains already accumulated morpho-pathological information and research. There are plenty of electronic and optical microscopy images that describe the spatial-temporal hologram of the phenomenon; we can thus discuss about the non-fractal – fractal and microscopic – macroscopic translation through holographically reproducible auto-similarity [21]. In this way, we affirm that fractality is the mathematical and semantic quintessence for defining atherogenesis, a process that can be physically characterized by fractal physics. This physics becomes in this situation more of a component rather than an explanation for the complex

In what concerns the recovery of such biological diseases, there are a huge number of techniques. We recall that external electrical stimulation can cause changes in the bio-structure vessels. Although atherosclerosis cause vasodilatation in the affected area and bio-structure flow remains unchanged for an extended period of time, the vascular wall stiffness will increase the pulse pressure. The purpose of the study developed in [24, 25] was to measure the effects of electrical stimulation (ES) on bio-structure flow and bio-structure pressure. All subjects received electrical stimulation at intensity sufficient to produce torque equal to 15% of the predetermined maximal voluntary contraction of their right quadriceps femor is muscle. The conclusions were that the increase in bio-structure flow occurred within 5 min after the onset of ES and dropped to resting levels within 1 min after a 10-min period of ES [25]. Kinesiotherapy or Kinesitherapy or kinesiatrics, is the therapeutic treatment of disease by passive and active muscular movements (as by massage) and of exercise [26]. From the physiotherapeutic viewpoint, an efficient treatment is directed toward improving bio-structure flow and also decreasing the disparity between the demand for bio-structure and its supply [22]. An effective vascular rehabilitation training program for improving walking efficiency and vascular remodeling in patients with diabetic atherosclerosis suffering from intermittent claudication could be a supervised treadmill walking

The present chapter proposes a fractal model for the dynamics analysis of bio-structure flows. The fractal hydrodynamic equations were obtained and applied for the laminar flow

linearity of the bio-structure the force toward the walls increases.

60 Nonlinear Systems - Modeling, Estimation, and Stability

biological system represented by the atheroma plaque [22, 23].

exercise combined with Allen-Burger exercises [23].

4. Conclusions

of bio-structure.

3. Discussions

The authors declare no conflict of interest.

### Author details

Emil Anton1 , Anna Gavrilut2 \*, Maricel Agop3 and Daniel Timofte4

\*Address all correspondence to<EMAIL_ADDRESS>
1 Department Obstetrics Gynecology, Gr. T. Popa University of Medicine and Pharmacy, Iasi, Romania

### References

[4] Michel OD, Thomas BG. Mathematical Modelling for Complex Fluids and Flows. NewYork: Springer; 2012

[22] Adelaide EC. Atherosclerosis Obliterans. The Australian Journal of Physiotherapy. 1958;

Non-Linear Behaviours in the Dynamics of Some Biostructures

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63

[23] Haas TL, Lloyd PG, Yang H-T, Terjung RL. Exercise training and peripheral arterial

[24] Singh RB, Mengi SA, Xu Y-J, Arneja AS, Dhalla NS. Pathogenesis of atherosclerosis: A

[25] Tracy JE, Currier DP, Threlkeld AJ. Comparison of selected pulse frequencies from two different electrical stimulators on bio-structure flow in healthy subjects. Physical Therapy.

[26] Vasileva D, Lubenova D, Mihova M, Dimitrova A, Grigorova-Petrova K. Influence of Kinesitherapy on Gaitin patients with ischemic stroke in the chronic period. Open Access

[27] Constantin B, Postolache P, Croitoru A, Nemes RM. Occupational bronchial asthmaclinical and epidemiological aspects. Journal of Environmental Protection and Ecology.

[28] Mitchel M. Complexity: A Guided Tour. Oxford: Oxford University Press; 2009

multifactorial process. Experimental and Clinical Cardiology. 2002;7:40-53

disease. Comprehensive Physiology. 2012;2:2933-3017

Macedonian Journal of Medical Sciences. 2015;3:619-623

4:19-22

1988;68:1526-1532

2015;16:517-520

[22] Adelaide EC. Atherosclerosis Obliterans. The Australian Journal of Physiotherapy. 1958; 4:19-22

[4] Michel OD, Thomas BG. Mathematical Modelling for Complex Fluids and Flows.

[5] Schwartz SM, Murry CE. Proliferation and the monoclonal origins of atherosclerotic

[6] Nottale L. Fractals and the quantum theory of space time. International Journal of Mod-

[7] Friedman MH, Deters OJ, Mark FF, et al. Arterial geometry affects hemodinamics. A

[8] Thomas YH. Multi-Scale Phenomena in Complex Fluids: Modelling, Analysis and Numerical Simulations. Singapore: World Scientific Publishing Company; 2009

[9] Mandelbrot BB. The Fractal Geometry of Nature (Updated and Augm. Ed.). New York:

[10] Cristescu CP. Non-linear Dynamics and Chaos. Theoretical Fundaments and Applications (in Romanian). Bucharest: Romanian Academy Publishing House; 2008

[12] Nottale L. Fractal Space-Time and Microphysics: Towards a Theory of Scale Relativity.

[13] Nottale L. Fractal Space-Time. A New Approach to Unifying Relativity and Quantum

[14] Merches I, Agop M. Differentiability and Fractality in Dynamics of Physical Systems.

[15] El Naschie MS, Rossler OE, Prigogine I. Quantum Mechanics, Diffusion and Chaotic

[18] Solovastru L-G, Ghizdovat V, Nedeff V, Lazar G, Eva L, Ochiuz L, Agop M, Popa RF. Non-linear effects at differentiable-non-differentiable scale transition in complex fluids.

[19] Batchelor GK. An Introduction to Fluid Dynamics. Cambridge: Cambridge University

[20] Landau LD, Lifshitz EM. Fluid Mechanics. 2nd ed. Oxford: Butterworth Heinemann; 1987 [21] Tesloianu ND, Ghizdovat V, Agop M. Flow Dynamics Via Non-differentiability and

. International Journal of Geometric

potential risk factor for atherosclerosis. Atherosclerosis. 1983;46:225-231

[11] Federer J, Aharoner A. Fractals in Physics. Amsterdam: North Holland; 1990

NewYork: Springer; 2012

62 Nonlinear Systems - Modeling, Estimation, and Stability

W.H. Freeman; 1983

ern Physics A. 1989;4:5047-5117

Singapore: World Scientific; 1995

Singapore: World Scientific; 2016

Fractals. Oxford: Elsevier; 1995

Press; 2000

Mechanics. London: Imperial College Press; 2011

[16] Cresson J. Non-differentiable deformations of ℝn

[17] Gouyet JF. Physique et Structures Fractales. Paris: Masson; 1992

Cardiovascular Diseases. Saarbrücken: Scholar's Press; 2015

Journal of Computational and Theoretical Nanoscience. 2016;13:1-6

Methods in Modern Physics. 2006;3:13-95

lesions. Annual Review of Medicine. 1998;49:437-460

**Chapter 4**

Provisional chapter

**Soliton-Like Solutions in the Problems of Vibrations of**

DOI: 10.5772/intechopen.74434

In the given chapter, free vibrations of different nonlinear mechanical systems with onedegree-of-freedom, two-degree-of-freedom, and multiple-degree-of-freedoms are reviewed with the emphasis on the vibratory regimes which could go over into the aperiodic motions under certain conditions. Such unfavorable and even dangerous regimes of vibrations resulting in the irreversible process of energy exchange from its one type to another type are discussed in detail. The solutions describing such processes are found analytically in

Keywords: soliton-like solution, nonlinear mechanical systems, free vibrations, method of

It is known [1] that the periodical transfer of energy from one type to another is made possible during vibrational processes occurring in nonlinear mechanical systems. This phenomenon is

Investigations on the energy exchange originate from the chapter [4], wherein the authors studied small nonlinear vibrations of a two-degree-of-freedom (2dof) system consisting of a load suspended on a linearly elastic spring and executing pendulum vibrations and vibrations along the spring's axis in the same vertical plane. In spite of the apparent simplicity of that system, it realistically explains some phenomena occurring during vibrations of more complex nonlinear systems and in particular describes all types of energy exchange from pendulum vibratory motions into oscillatory motions along the spring's axis, and vice versa: the periodic

> © 2016 The Author(s). Licensee InTech. This chapter is distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and eproduction in any medium, provided the original work is properly cited.

© 2018 The Author(s). Licensee IntechOpen. This chapter is distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use,

distribution, and reproduction in any medium, provided the original work is properly cited.

terms of functions, which are in frequent use in the theory of solitons.

Soliton-Like Solutions in the Problems of Vibrations of

**Nonlinear Mechanical Systems: Survey**

Nonlinear Mechanical Systems: Survey

Yury A. Rossikhin and Marina V. Shitikova

Yury A. Rossikhin and Marina V. Shitikova

multiple time scales, suspension bridge

http://dx.doi.org/10.5772/intechopen.74434

Abstract

1. Introduction

called energy exchange [2, 3].

Additional information is available at the end of the chapter

Additional information is available at the end of the chapter

#### **Soliton-Like Solutions in the Problems of Vibrations of Nonlinear Mechanical Systems: Survey** Soliton-Like Solutions in the Problems of Vibrations of Nonlinear Mechanical Systems: Survey

DOI: 10.5772/intechopen.74434

Yury A. Rossikhin and Marina V. Shitikova Yury A. Rossikhin and Marina V. Shitikova

Additional information is available at the end of the chapter Additional information is available at the end of the chapter

http://dx.doi.org/10.5772/intechopen.74434

#### Abstract

In the given chapter, free vibrations of different nonlinear mechanical systems with onedegree-of-freedom, two-degree-of-freedom, and multiple-degree-of-freedoms are reviewed with the emphasis on the vibratory regimes which could go over into the aperiodic motions under certain conditions. Such unfavorable and even dangerous regimes of vibrations resulting in the irreversible process of energy exchange from its one type to another type are discussed in detail. The solutions describing such processes are found analytically in terms of functions, which are in frequent use in the theory of solitons.

Keywords: soliton-like solution, nonlinear mechanical systems, free vibrations, method of multiple time scales, suspension bridge

### 1. Introduction

It is known [1] that the periodical transfer of energy from one type to another is made possible during vibrational processes occurring in nonlinear mechanical systems. This phenomenon is called energy exchange [2, 3].

Investigations on the energy exchange originate from the chapter [4], wherein the authors studied small nonlinear vibrations of a two-degree-of-freedom (2dof) system consisting of a load suspended on a linearly elastic spring and executing pendulum vibrations and vibrations along the spring's axis in the same vertical plane. In spite of the apparent simplicity of that system, it realistically explains some phenomena occurring during vibrations of more complex nonlinear systems and in particular describes all types of energy exchange from pendulum vibratory motions into oscillatory motions along the spring's axis, and vice versa: the periodic

© 2016 The Author(s). Licensee InTech. This chapter is distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and eproduction in any medium, provided the original work is properly cited. © 2018 The Author(s). Licensee IntechOpen. This chapter is distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

and aperiodic energy interchange, as well as stationary regimes during which the energy exchange is absent.

of the mechanical system, the second one may occur to be the most unfavorable. As for the behavior of a suspension bridge, then the most hazardous type is the irreversible transfer of the energy of vertical vibrations into the energy of its torsional vibrations in the case of a bisymmetrical stiffening girder or into the energy of flexural-torsional vibrations in the case of a mono-symmetrical girder. This is due to the fact that suspension bridges possess a rather higher flexural rigidity than torsional one, that is, they perceive better than those dynamic

Soliton-Like Solutions in the Problems of Vibrations of Nonlinear Mechanical Systems: Survey

http://dx.doi.org/10.5772/intechopen.74434

Solutions describing the one-sided energy transfer occurring in mechanical systems we shall call as soliton-like solutions, since the functions entering in such solutions are widely met in the

In this chapter, it is shown that solutions of such a type exist both in 1dof systems and in

The phenomenon of energy transfer, when one type of the energy completely and irreversibly goes into another type of the energy as time passes, can be observed on such a simple object as

In order to demonstrate this, let us consider the expression for the total mechanical energy of

ð1Þ

67

the mathematical pendulum which is combined from the kinetic energy

loads that result in vertical vibrations.

systems possessing two- and more degrees-of-freedom.

2. A one-degree-of-freedom system

a mathematical pendulum (Figure 1).

and the potential energy (Figure 1)

Figure 1. A mathematical pendulum.

theory of solitons [13, 14].

The energy-exchange mechanism in a similar nonlinear 2dof system has been studied in [5, 6]. The system was made up of two loads, one of which was suspended on a linearly elastic spring and executed vertical vibrations, and the other was suspended on an unstretched rod and executed pendulum vibrations in the same vertical plane. Reviews devoted to nonlinear vibrations of 2dof systems can be found in [2, 3].

However, the energy transfer is observed during free vibrations of different nonlinear mechanical systems: possessing one-degree-of-freedom (1dof), two- (2dof), and more degrees-of-freedom (multiple-dof), and as well as having infinite number of degrees-of-freedom (deformable solids). The internal resonance is realized when magnitudes of natural frequencies of two natural modes belonging to the different types of vibrations of the system (partial subsystems) are approximately equal to each other or one of them two to three times larger than the other. This phenomenon is particular evident in modern engineering structures which are very light and flexible due to the application of present-day materials, resulting in finite displacements of individual structural elements as well as of the structure as a whole. Among such constructions are suspension-combined systems: suspension and cable-stayed bridges, suspension roofs in large public and industrial buildings, and so on. Suspension-combined systems and suspension bridges, in particular, are distinguished by high esthetic merits, and many of them are referred to the most remarkable up-to-date engineering structures. For example, "Golden Gate" suspension bridge in San Francisco with the span of 1281 m, cable-stayed bridge in Cologne with the span of 690 m, suspension roofing of Olympic sport complex in Moscow, and many others.

The majority of papers devoted to the dynamic behavior of suspension-combined systems studies free nonlinear vibrations of suspension bridges with a thin-walled stiffening girder [7–11]. Different dynamic loads (wind, seismic excitation, moving loads, etc.) after the completion of acting on a suspended structure setup prolonged free nonlinear vibrations of this structure, in so doing both vertical and flexural-torsional vibrations could be excited. One of the most unfavorable nonlinear effects, which is observed in suspension systems during free vibrations, is just the "energy exchange" from one type of vibratory motions into the other under the conditions of the internal resonance.

The intensity and frequency of energy exchange between strongly coupled modes essentially depend on an absolute level of the initial amplitudes [7, 8, 11, 12] which is governed by the value of the initial mechanical energy of the system.

However, the qualitative character of the energy exchange is dependent on the relative level of initial amplitudes which is independent of the system's initial energy and is defined as the ratio of the initial amplitudes of the two interacting modes [9]. It has been found in [9] that in accordance with a value of that level, three types of an energy-exchange mechanism exist: twosided energy exchange (a periodic energy exchange from one subsystem to another), one-sided energy exchange (one subsystem completely or partially transfers the energy to another), and energy exchange does not occur (stationary vibrations). Among the three types of the behavior of the mechanical system, the second one may occur to be the most unfavorable. As for the behavior of a suspension bridge, then the most hazardous type is the irreversible transfer of the energy of vertical vibrations into the energy of its torsional vibrations in the case of a bisymmetrical stiffening girder or into the energy of flexural-torsional vibrations in the case of a mono-symmetrical girder. This is due to the fact that suspension bridges possess a rather higher flexural rigidity than torsional one, that is, they perceive better than those dynamic loads that result in vertical vibrations.

Solutions describing the one-sided energy transfer occurring in mechanical systems we shall call as soliton-like solutions, since the functions entering in such solutions are widely met in the theory of solitons [13, 14].

In this chapter, it is shown that solutions of such a type exist both in 1dof systems and in systems possessing two- and more degrees-of-freedom.

### 2. A one-degree-of-freedom system

and aperiodic energy interchange, as well as stationary regimes during which the energy

The energy-exchange mechanism in a similar nonlinear 2dof system has been studied in [5, 6]. The system was made up of two loads, one of which was suspended on a linearly elastic spring and executed vertical vibrations, and the other was suspended on an unstretched rod and executed pendulum vibrations in the same vertical plane. Reviews devoted to nonlinear

However, the energy transfer is observed during free vibrations of different nonlinear mechanical systems: possessing one-degree-of-freedom (1dof), two- (2dof), and more degrees-of-freedom (multiple-dof), and as well as having infinite number of degrees-of-freedom (deformable solids). The internal resonance is realized when magnitudes of natural frequencies of two natural modes belonging to the different types of vibrations of the system (partial subsystems) are approximately equal to each other or one of them two to three times larger than the other. This phenomenon is particular evident in modern engineering structures which are very light and flexible due to the application of present-day materials, resulting in finite displacements of individual structural elements as well as of the structure as a whole. Among such constructions are suspension-combined systems: suspension and cable-stayed bridges, suspension roofs in large public and industrial buildings, and so on. Suspension-combined systems and suspension bridges, in particular, are distinguished by high esthetic merits, and many of them are referred to the most remarkable up-to-date engineering structures. For example, "Golden Gate" suspension bridge in San Francisco with the span of 1281 m, cable-stayed bridge in Cologne with the span of 690 m, suspension roofing of Olympic sport complex in Moscow, and

The majority of papers devoted to the dynamic behavior of suspension-combined systems studies free nonlinear vibrations of suspension bridges with a thin-walled stiffening girder [7–11]. Different dynamic loads (wind, seismic excitation, moving loads, etc.) after the completion of acting on a suspended structure setup prolonged free nonlinear vibrations of this structure, in so doing both vertical and flexural-torsional vibrations could be excited. One of the most unfavorable nonlinear effects, which is observed in suspension systems during free vibrations, is just the "energy exchange" from one type of vibratory motions into the other

The intensity and frequency of energy exchange between strongly coupled modes essentially depend on an absolute level of the initial amplitudes [7, 8, 11, 12] which is governed by the

However, the qualitative character of the energy exchange is dependent on the relative level of initial amplitudes which is independent of the system's initial energy and is defined as the ratio of the initial amplitudes of the two interacting modes [9]. It has been found in [9] that in accordance with a value of that level, three types of an energy-exchange mechanism exist: twosided energy exchange (a periodic energy exchange from one subsystem to another), one-sided energy exchange (one subsystem completely or partially transfers the energy to another), and energy exchange does not occur (stationary vibrations). Among the three types of the behavior

exchange is absent.

many others.

vibrations of 2dof systems can be found in [2, 3].

66 Nonlinear Systems - Modeling, Estimation, and Stability

under the conditions of the internal resonance.

value of the initial mechanical energy of the system.

The phenomenon of energy transfer, when one type of the energy completely and irreversibly goes into another type of the energy as time passes, can be observed on such a simple object as a mathematical pendulum (Figure 1).

In order to demonstrate this, let us consider the expression for the total mechanical energy of the mathematical pendulum which is combined from the kinetic energy

$$T = \frac{1}{2} m \upsilon^2 = \frac{1}{2} m l^2 \left(\dot{\phi}\right)^2 \tag{1}$$

and the potential energy (Figure 1)

Figure 1. A mathematical pendulum.

$$
\prod = m \lg(OH) = 2mgl \sin^2 \left(\frac{\varphi}{2}\right) \tag{2}
$$

and has the form

$$E = T + \Pi = \frac{1}{2} \left. ml \right|^2 (\dot{\phi})^2 + 2mgl \sin^2 \left( \frac{\varphi}{2} \right), \tag{3}$$

where an overdot denotes a time derivative, l is the string length, g is the gravity acceleration, m is the load mass, is its velocity, and is the angle of the string's deflection from the vertical.

Rewrite Eq. (3) in the dimensionless form

$$\frac{\left(\dot{\varphi}\right)^{2}}{\left(\rho\_{0}^{2}\right)^{2}} + 4\sin^{2}\left(\frac{\varphi}{2}\right) = \frac{E}{E\_{0}}\,,\tag{4}$$

anywhere, and the angle , that is, the pendulum, tends to take the upper position of equilibrium which is an unstable one. As this takes place, the kinetic energy completely transforms into the potential energy. This solution is the soliton-like one, since the functions arctan

Soliton-Like Solutions in the Problems of Vibrations of Nonlinear Mechanical Systems: Survey

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69

at different magnitudes of the energy E, then solution (6) will correspond to the phase trajectory which is called as a separatrix. This line divides closed trajectories from nonclosed ones (Figure 2). Closed and nonclosed trajectories are consistent with the solutions for the periodic transfer of the potential and kinetic energies into each other, in doing so in the first case, the pendulum will vibrate, and in the second one, it will rotate around the point of suspension.

Now, consider a 2dof system presented in Figure 3. The kinetic T and potential Π energies of

where , k is the elastic spring rigidity, m1 and m2 are the masses of the first and second loads, respectively, y is the vertical displacement of the first load, and is the angle

ð8aÞ

ð8bÞ

If one represents the phase trajectories of the pendulum motion on the phase plane

and ch are frequently met in soliton solutions.

Figure 2. Phase portrait describing vibrations of mathematical pendulum.

3. A two-degree-of-freedom system

3.1. Governing equations

such a system have the form

of the pendulum's deflection.

where and

Consider the case of motion of the mathematical pendulum when its energy E is exactly equal to 4E0. Then, the law of conservation of energy Eq. (4) gives the simple relationship [15].

$$\frac{\left(\dot{\varphi}\right)^{2}}{\left.o\_{0}^{2}\right|} = 4\cos^{2}\left(\frac{\varphi}{2}\right)\tag{5a}$$

or

$$
\dot{\varphi} = 2\alpha\_0 \cos\left(\frac{\varphi}{2}\right). \tag{5b}
$$

Dividing the variables in Eq. (5b), integrating separately the right and left parts of the relationship obtained, and considering that at t = 0 yield

$$
\left\lfloor \ln \left( \frac{\pi - \varphi}{4} \right) \right\rfloor = -\alpha\_0 t \tag{6a}
$$

or

$$
\varphi = \pi - 4 \arctan\left(\mathbf{e}^{-\alpha\_{\Phi}}\right). \tag{6b}
$$

Differentiating Eq. (6b) over t, we find

$$\dot{\phi} = \frac{2a\_0}{\cosh(a\_0 t)}\tag{7}$$

Reference to Eqs. (6) and (7) shows that if the mathematical pendulum begins its motion from the extreme low position, then at t!∞ its velocity , in so doing does not vanish Soliton-Like Solutions in the Problems of Vibrations of Nonlinear Mechanical Systems: Survey http://dx.doi.org/10.5772/intechopen.74434 69

Figure 2. Phase portrait describing vibrations of mathematical pendulum.

anywhere, and the angle , that is, the pendulum, tends to take the upper position of equilibrium which is an unstable one. As this takes place, the kinetic energy completely transforms into the potential energy. This solution is the soliton-like one, since the functions arctan and ch are frequently met in soliton solutions.

If one represents the phase trajectories of the pendulum motion on the phase plane at different magnitudes of the energy E, then solution (6) will correspond to the phase trajectory which is called as a separatrix. This line divides closed trajectories from nonclosed ones (Figure 2). Closed and nonclosed trajectories are consistent with the solutions for the periodic transfer of the potential and kinetic energies into each other, in doing so in the first case, the pendulum will vibrate, and in the second one, it will rotate around the point of suspension.

#### 3. A two-degree-of-freedom system

#### 3.1. Governing equations

ð2Þ

ð3Þ

ð4Þ

ð5aÞ

ð5bÞ

ð6aÞ

ð6bÞ

ð7Þ

and has the form

Rewrite Eq. (3) in the dimensionless form

68 Nonlinear Systems - Modeling, Estimation, and Stability

ship obtained, and considering that at t = 0 yield

Differentiating Eq. (6b) over t, we find

where and

vertical.

or

or

where an overdot denotes a time derivative, l is the string length, g is the gravity acceleration, m is the load mass, is its velocity, and is the angle of the string's deflection from the

Consider the case of motion of the mathematical pendulum when its energy E is exactly equal to 4E0. Then, the law of conservation of energy Eq. (4) gives the simple relationship [15].

Dividing the variables in Eq. (5b), integrating separately the right and left parts of the relation-

Reference to Eqs. (6) and (7) shows that if the mathematical pendulum begins its motion from the extreme low position, then at t!∞ its velocity , in so doing does not vanish Now, consider a 2dof system presented in Figure 3. The kinetic T and potential Π energies of such a system have the form

$$T = \frac{1}{2} (m\_1 + m\_2) \dot{\mathbf{y}}^2 - m\_2 l \dot{\mathbf{y}} \dot{\boldsymbol{\varphi}} \sin \boldsymbol{\varphi} + \frac{1}{2} m\_2 l^2 \dot{\boldsymbol{\phi}}^2,\tag{8a}$$

$$
\Pi = \frac{1}{2}k(\mathbf{y} + \mathbf{y}\_{cm})^2 - m\_1\mathbf{g}\left(\mathbf{y} + \mathbf{y}\_{cm}\right) - m\_2\mathbf{g}\left(\mathbf{y} + \mathbf{y}\_{cm} + l\cos\varphi\right),
\tag{8b}
$$

where , k is the elastic spring rigidity, m1 and m2 are the masses of the first and second loads, respectively, y is the vertical displacement of the first load, and is the angle of the pendulum's deflection.

Figure 3. Scheme of a 2dof mechanical system.

Applying Lagrange equations of the second kind [15]

$$\frac{d}{dt}\left(\frac{\partial T}{\partial \dot{\mathcal{A}}}\right) - \frac{\partial T}{\partial \mathcal{A}} = -\frac{\partial \prod}{\partial \mathcal{A}} \qquad (\mathcal{A} = \mathcal{y}, \phi).$$

and considering Eq. (8), the system's equations of motion in the dimensionless form within an accuracy of the values of the second order of smallness with respect to y and can be written as follows:

$$
\ddot{\mathbf{y}} \stackrel{\ast}{\ast} + o \stackrel{\ast \ast^2}{\ast^2} \mathbf{y} \stackrel{\ast}{\ast} - a\rho \ddot{\boldsymbol{\varphi}} - a\dot{\boldsymbol{\varphi}}^2 = \mathbf{0}, \tag{9a}
$$

$$
\ddot{\boldsymbol{\varphi}} + \boldsymbol{\Omega} \stackrel{\ast \ast^2}{\ast} \boldsymbol{\varphi} - b\rho \dot{\boldsymbol{\varphi}} \dot{\mathbf{y}}^{\mathbb{N}} = \mathbf{0}, \tag{9b}
$$

ð10aÞ

71

http://dx.doi.org/10.5772/intechopen.74434

ð10bÞ

ð11aÞ

ð11bÞ

ð12Þ

ð13Þ

or the linear natural frequency and the linear natural frequency are equal to each other,

Soliton-Like Solutions in the Problems of Vibrations of Nonlinear Mechanical Systems: Survey

It is said that the system is being under the conditions of the two-to-one internal resonance or the one-to-one internal resonance if the condition Eq. (10a) or (10b) is fulfilled, respectively [2].

For analyzing nonlinear vibrations of the systems subjected to the internal resonance (10), assume that the amplitudes of vibrations are small but finite values and weakly vary with time. Then, perturbation technique could be used to construct the solution of the set of Eq. (9),

An approximate solution of Eq. (9) can be represented by an expansion in terms of different

and, particularly, the method of multiple time scales [16].

where Tn ¼ εnt (n = 0,1,2…), and is a small parameter.

Substituting Eq. (11) into Eq. (9), considering that

time scales limiting by the values of the third order of smallness in

and equating the coefficients of like powers of , one obtains, to order ,

that is,

3.2. Method of solution

to order ,

to order ,

where

$$\alpha \ast^{2} = \alpha^{2} \text{y}\_{0} \text{g}^{-1}, \quad \boldsymbol{\Omega} \ast^{2} = \boldsymbol{\Omega} \, \text{y}\_{0} \text{g}^{-1}, \quad \alpha^{2} = k \, (m\_{1} + m\_{2})^{-1}, \quad \boldsymbol{\Omega}^{2} = \boldsymbol{g} \, \boldsymbol{l}^{-1}$$

$$\mathbf{y}^{\*} = \mathbf{y} \, \mathbf{y}\_{0}^{-1}, \quad t^{\*} = t \sqrt{\mathbf{g} \, \mathbf{y}\_{0}^{-1}}, \quad \mathbf{y}\_{0} = m\_{1} \boldsymbol{k}^{-1} \mathbf{g}, \quad \boldsymbol{a} = \frac{m\_{2}}{m\_{1} + m\_{2}} \cdot \frac{l}{\mathbf{y}\_{0}}, \quad \boldsymbol{b} = \frac{\mathbf{y}\_{0}}{l}.$$

Suppose that the linear natural frequency is twice as large than the linear natural frequency , that is,

Soliton-Like Solutions in the Problems of Vibrations of Nonlinear Mechanical Systems: Survey http://dx.doi.org/10.5772/intechopen.74434 71

$$
\alpha \flat^\* = \mathfrak{D} \Omega^\* \tag{10a}
$$

or the linear natural frequency and the linear natural frequency are equal to each other, that is,

$$
\alpha \flat^\* = \Omega \upharpoonright. \tag{10b}
$$

It is said that the system is being under the conditions of the two-to-one internal resonance or the one-to-one internal resonance if the condition Eq. (10a) or (10b) is fulfilled, respectively [2].

For analyzing nonlinear vibrations of the systems subjected to the internal resonance (10), assume that the amplitudes of vibrations are small but finite values and weakly vary with time. Then, perturbation technique could be used to construct the solution of the set of Eq. (9), and, particularly, the method of multiple time scales [16].

#### 3.2. Method of solution

An approximate solution of Eq. (9) can be represented by an expansion in terms of different time scales limiting by the values of the third order of smallness in

$$\mathbf{y}^\*(t) = \varepsilon \mathbf{y}\_1(T\_0, T\_1, T\_{2,\dots}) + \varepsilon^2 \mathbf{y}\_2(T\_0, T\_1, T\_{2,\dots}) + \varepsilon^3 \mathbf{y}\_3(T\_0, T\_1, T\_{2,\dots}) + \dots \tag{11a}$$

$$
\varphi(t) = \varepsilon \wp\_1(T\_0, T\_1, T\_2, \dots) + \varepsilon^2 \wp\_2(T\_0, T\_1, T\_2, \dots) + \varepsilon^3 \wp\_3(T\_0, T\_1, T\_2, \dots) + \dots \quad , \tag{11b}
$$

where Tn ¼ εnt (n = 0,1,2…), and is a small parameter.

Substituting Eq. (11) into Eq. (9), considering that

$$\begin{aligned} \frac{d}{dt} &= D\_0 + \varepsilon \, D\_1 + \varepsilon^2 \, D\_2 + \dots, \qquad D\_n = \frac{\partial}{\partial T\_n} \, (n = 0, 1, 2, \dots) \\\\ \frac{d^2}{dt^2} &= \left( D\_0 + \varepsilon \, D\_1 + \varepsilon^2 \, D\_2 + \dots \right)^2 = D\_0^2 + 2\varepsilon \, D\_0 \, D\_1 + \varepsilon^2 \, (D\_0 D\_2 + D\_1^2) + \dots \end{aligned}$$

and equating the coefficients of like powers of , one obtains, to order ,

$$D\_0^2 \ y\_1 + \phi^{\*2} \ y\_1 = 0, \qquad D\_0^2 \wp\_1 + \mathfrak{Q}^{\*2} \ \wp\_1 = 0;\tag{12}$$

to order ,

ð9aÞ

ð9bÞ

Applying Lagrange equations of the second kind [15]

Figure 3. Scheme of a 2dof mechanical system.

70 Nonlinear Systems - Modeling, Estimation, and Stability

as follows:

where

, that is,

and considering Eq. (8), the system's equations of motion in the dimensionless form within an accuracy of the values of the second order of smallness with respect to y and can be written

Suppose that the linear natural frequency is twice as large than the linear natural frequency

$$\begin{aligned} D\_0^2 \text{ y}\_2 + a \ast^{\ast 2} \text{ y}\_2 &= -2D\_0 D\_1 \text{y}\_1 + a \varrho\_1 D\_0^2 \varrho\_1 + a (D\_0 \varrho\_1)^2, \\\\ D\_0^2 \varrho\_2 + \Omega \ast^{\ast 2} \varrho\_2 &= -2D\_0 D\_1 \varrho\_1 + b \varrho\_1 D\_0^2 \varrho\_1; \end{aligned} \tag{13}$$

to order ,

$$\begin{aligned} D\_0^2 \ y\_3 + a \ast^{\varphi\_2} y\_3 &= -2D\_0 D\_1 \mathbf{y}\_2 - (D\_1^2 + 2D\_0 D\_2) \mathbf{y}\_1 + a \varphi\_2 D\_0^2 \varphi\_1 \\ &+ 2a D\_0 \varphi\_1 (D\_1 \varphi\_1 + D\_0 \varphi\_2) + a \varphi\_1 (D\_0^2 \varphi\_2 + 2D\_0 D\_1 \varphi\_1), \\\\ D\_0^2 \varphi\_3 + \Omega^{\ast 2} \varphi\_3 &= -2D\_0 D\_1 \varphi\_2 - (D\_1^2 + 2D\_0 D\_2) \varphi\_1 \\ &+ b \varphi\_2 D\_0^2 \mathbf{y}\_1 + b \varphi\_1 (D\_0^2 \mathbf{y}\_2 + 2D\_0 D\_1 \mathbf{y}\_1) \end{aligned} \tag{14}$$

The solution of Eq. (12) could be sought in the form

$$\begin{aligned} \mathbf{y}\_1 &= A\_1(T\_1) \exp(i\alpha \ast^\* T\_0) + \overline{A}\_1(T\_1) \exp(-i\alpha \ast^\* T\_0), \\\\ \varrho\_1 &= A\_2(T\_1) \exp(i\Omega \ast^\* T\_0) + \overline{A}\_2(T\_1) \exp(-i\Omega \ast^\* T\_0), \end{aligned} \tag{15}$$

Representing the functions A1 and A2 in a polar form

we can rewrite the set of four differential equations as

ship with respect to T1 yield

first integral of the set of Eq. (19).

where .

account for Eq. (21) and

Introducing a new function such that

and substituting Eq. (21) in Eq. (19a), we have

where an overdot denotes differentiation with respect to T1, and .

Eliminating the value from Eqs. (19a) and (19b) and integrating the net relation-

Soliton-Like Solutions in the Problems of Vibrations of Nonlinear Mechanical Systems: Survey

http://dx.doi.org/10.5772/intechopen.74434

where E0 is the initial magnitude of the system's energy, which represents the law of conservation of the total mechanical energy of the system under consideration. Expression (20) is the

Doubling both sides of Eq. (19d) and subtracting from the net relationship Eq. (19c) with due

ð18Þ

73

ð19aÞ

ð19bÞ

ð19cÞ

ð19dÞ

ð20Þ

ð21Þ

ð22Þ

where A1 and A2 are unknown complex functions, while and are the complex conjugates of A1 and A2, respectively.

#### 3.2.1. The case of a two-to-one internal resonance

Substituting Eq. (15) into the right-hand sides of Eq. (13) yields

$$\begin{split} D\_0^2 \mathbf{y}\_2 + a \ast^{\ast 2} \mathbf{y}\_2 &= -2i\alpha \ast^{\ast} D\_1 A\_1 \exp(i\alpha \ast^{\ast} T\_0) - 2a A\_2^2 \Omega \ast^{\ast 2} \exp(2i\Omega \ast^{\ast} T\_0) + c\varepsilon, \\ D\_0^2 \varrho\_2 + \Omega \ast^{\ast 2} \varrho\_2 &= -2i\Omega \ast^{\ast} D\_1 A\_2 \exp(i\Omega \ast^{\ast} T\_0) - b A\_1 A\_2 \alpha \ast^{\ast 2} \exp[i(\alpha \ast^{\ast} + \Omega^{\ast}) T\_0] - \\ &- b A\_1 \overline{A}\_2 \alpha \ast^{\ast 2} \exp[i(\alpha \ast^{\ast} - \Omega^{\ast}) T\_0] + c\varepsilon, \end{split} (16)$$

where cc denotes complex conjugate parts of the preceding terms.

The functions , , entering into the right-hand sides of Eq. (16) produce secular terms in the expression for and , that is, the terms of the type of and . Since secular terms increase without any limits as time goes on, then there is a need to eliminate them by equating the coefficients standing at the enumerated functions to zero. As a result, we obtain

$$
tau^\* D\_1 A\_{\mathbb{I}} + a A\_2^2 \Omega^{\*\mathbb{I}^2} = 0,\tag{17a}$$

$$2i\Omega 2^\* D\_1 A\_2 + b A\_1 \overline{A}\_2 o^{\*\sharp^2} = 0.\tag{17b}$$

Multiply Eq. (17a) by and Eq. (17b) by and find the complex conjugate equations. Two mutually conjugated equations first add to each other and then subtract one from another. As a result of such a procedure, we obtain more convenient set of four equations:

Soliton-Like Solutions in the Problems of Vibrations of Nonlinear Mechanical Systems: Survey http://dx.doi.org/10.5772/intechopen.74434 73

$$\begin{split} i\alpha \ast^{\mathbb{A}} \left( \overline{A}\_{\textnormal{l}} D\_{\textnormal{l}} A\_{\textnormal{l}} - A\_{\textnormal{l}} D\_{\textnormal{l}} \overline{A}\_{\textnormal{l}} \right) + a \Omega \ast^{\mathbb{A}^{2}} \left( \overline{A}\_{\textnormal{l}} A\_{\textnormal{2}}^{2} + A\_{\textnormal{l}} \overline{A}\_{\textnormal{2}}^{2} \right) &= 0 \\ i\alpha \ast^{\mathbb{A}} \left( \overline{A}\_{\textnormal{l}} D\_{\textnormal{l}} A\_{\textnormal{l}} + A\_{\textnormal{l}} D\_{\textnormal{l}} \overline{A}\_{\textnormal{l}} \right) + a \Omega \ast^{\mathbb{A}^{2}} \left( \overline{A}\_{\textnormal{l}} A\_{\textnormal{2}}^{2} - A\_{\textnormal{l}} \overline{A}\_{\textnormal{2}}^{2} \right) &= 0 \\ 2i\Omega \ast^{\mathbb{A}} \left( \overline{A}\_{\textnormal{2}} D\_{\textnormal{l}} A\_{\textnormal{2}} - A\_{\textnormal{2}} D\_{\textnormal{l}} \overline{A}\_{\textnormal{2}} \right) + b \, bo \ast^{\mathbb{A}^{2}} \left( A\_{\textnormal{l}} \overline{A}\_{\textnormal{2}}^{2} + \overline{A}\_{\textnormal{l}} A\_{\textnormal{2}}^{2} \right) &= 0 \\ 2i\Omega \ast^{\mathbb{A}} \left( \overline{A}\_{\textnormal{2}} D\_{\textnormal{l}} A\_{\textnormal{2}} + A\_{\textnormal{2}} D\_{\textnormal{l}} \overline{A}\_{\textnormal{2}} \right) + b \, bo \ast^{\mathbb{A}^{2}} \left( A\_{\textnormal{1}} \overline{A}\_{\textnormal{2}}^{2} - \overline{A}\_{\textnormal{1}} A\_{\textnormal{2}}^{2} \right) &= 0. \end{split}$$

Representing the functions A1 and A2 in a polar form

ð14Þ

ð15Þ

ð16Þ

ð17aÞ

ð17bÞ

The solution of Eq. (12) could be sought in the form

72 Nonlinear Systems - Modeling, Estimation, and Stability

3.2.1. The case of a two-to-one internal resonance

Substituting Eq. (15) into the right-hand sides of Eq. (13) yields

where cc denotes complex conjugate parts of the preceding terms.

The functions , ,

of A1 and A2, respectively.

where A1 and A2 are unknown complex functions, while and are the complex conjugates

entering into the right-hand sides of Eq. (16) produce secular terms in the expression for and , that is, the terms of the type of and . Since secular terms increase without any limits as time goes on, then there is a need to eliminate them by equating the

Multiply Eq. (17a) by and Eq. (17b) by and find the complex conjugate equations. Two mutually conjugated equations first add to each other and then subtract one from another. As a

coefficients standing at the enumerated functions to zero. As a result, we obtain

result of such a procedure, we obtain more convenient set of four equations:

$$A\_1 = a\_1(T\_1) \exp[i\varphi\_1(T\_1)], \quad A\_2 = a\_2(T\_1) \exp[i\varphi\_2(T\_1)].\tag{18}$$

we can rewrite the set of four differential equations as

$$\left(a\_1^2\right)^\bullet = -\frac{1}{2}a a \omicron\prime^\* a\_1 a\_2^2 \sin\delta\,,\tag{19a}$$

$$\left(a\_2^2\right)^\bullet = 2b\,\alpha\prime\,a\_1a\_2^2\sin\delta\,,\tag{19b}$$

$$\dot{\phi}\_{\text{l}} = \frac{1}{4} a \,\phi \, \prescript{}{}{a}\_2^2 a\_{\text{l}}^{-1} \cos \delta \,\tag{19c}$$

$$
\dot{\varphi}\_2 = b\rho \, \prescript{\omega}{}{a}\_1 \cos \delta \,\, , \tag{19d}
$$

where an overdot denotes differentiation with respect to T1, and .

Eliminating the value from Eqs. (19a) and (19b) and integrating the net relationship with respect to T1 yield

$$a\_1^2 + \frac{1}{4}ab^{-1}a\_2^2 = E\_0 \,, \tag{20}$$

where E0 is the initial magnitude of the system's energy, which represents the law of conservation of the total mechanical energy of the system under consideration. Expression (20) is the first integral of the set of Eq. (19).

Introducing a new function such that

$$a\_1^2 = E\_0 \xi(T\_1), \qquad a\_2^2 = 4 \, b a^{-1} E\_0 [1 - \xi(T\_1)],\tag{21}$$

and substituting Eq. (21) in Eq. (19a), we have

$$\dot{\xi} = -B\sqrt{\xi} \,(1 - \xi) \sin \delta \,, \tag{22}$$

where .

Doubling both sides of Eq. (19d) and subtracting from the net relationship Eq. (19c) with due account for Eq. (21) and

$$\boldsymbol{\delta} = 2\dot{\boldsymbol{\phi}}\_2 - \dot{\boldsymbol{\phi}}\_1.$$

we obtain

$$\dot{\delta} = B \frac{3\xi - 1}{2\sqrt{\xi}} \cos \delta \,\,. \tag{23}$$

At G0 = 0, the integral in Eq. (28) can be calculated, in so doing, it possesses two magnitudes. Really, changing the variable in the integral in Eq. (28) at G0 = 0, we have the first

Soliton-Like Solutions in the Problems of Vibrations of Nonlinear Mechanical Systems: Survey

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Considering Eq. (29), the solutions of Eq. (28) may be written in the following form:

Solutions (30b) and (31b) at and 1 describe the motions corresponding to the one-sided energy exchange between pendulum's vibrations and vertical vibration of the load. As this takes place, and in the first and second solutions, respectively, with the increase in time . In other words, in the first solution, the energy of vibrations of the pendulum completely transforms into the energy of vertical vibrations of the load, but in the second solution, quite the reverse, the energy of vertical vibrations of the load completely goes into

ð29aÞ

75

ð29bÞ

ð30aÞ

ð30bÞ

ð31aÞ

ð31bÞ

magnitude

and the second magnitude

the first solution

and the second solution

the energy of vibrations of the pendulum.

or

or

Putting

$$
\dot{\delta} = \frac{d\delta}{d\xi} \dot{\xi} \tag{24}
$$

and substituting Eq. (24) into Eqs. (22) and (23), we are led to the equation

$$\frac{d\cos\delta}{d\xi} + \frac{1-3\xi}{2\xi(1-\xi)}\cos\delta = 0\,\,. \tag{25}$$

Separating the variables in Eq. (25) and integrating the equation obtained yield

$$\cos \delta = G\_0 \xi^{-1/2} (1 - \xi)^{-1} \tag{26a}$$

or

$$G(\xi, \delta) = \sqrt{\xi} \left( 1 - \xi \right) \cos \delta = G\_0(\xi\_0, \delta\_0) \, , \tag{26b}$$

where is an arbitrary constant determined from the initial conditions, and and are the initial magnitudes of the values ξ and δ, respectively. Note that relationship (26b) is the other first integral of the set of Eq. (19).

Finely, let us eliminate the value from Eqs. (26a) and (22), resulting in

$$\dot{\xi} = -B\,\xi^{1/2} \left( 1 - \xi \right) \sqrt{1 - \frac{G\_0^2}{\xi \left( 1 - \xi \right)^2}} \,\, \, \, \, \, \, \tag{27}$$

Separating the variables in Eq. (27) and integrating the net expression, we obtain implicitly the desired function

$$\int\_{\xi\_0}^{\xi} \frac{d\xi}{\left[\xi \left(1-\xi\right)^2 - G\_0^2\right]^{1/2}} = -BT\_{1,\epsilon} \tag{28}$$

where is the value defining the relative level in the initial amplitudes.

The integral in Eq. (28) can be transformed into an incomplete integral of the first kind, which is tabulated in [17].

At G0 = 0, the integral in Eq. (28) can be calculated, in so doing, it possesses two magnitudes. Really, changing the variable in the integral in Eq. (28) at G0 = 0, we have the first magnitude

$$\int\_{\frac{\tilde{\xi}}{\xi\_0}}^{\tilde{\xi}} \frac{d\tilde{\xi}}{\sqrt{\tilde{\xi}} \,(\tilde{\xi} - 1)} = 2 \int\_{\sqrt{\tilde{\xi}\_0}}^{\sqrt{\xi}} \frac{d\lambda}{\lambda^2 - 1} = \int\_{\sqrt{\xi\_0}}^{\sqrt{\xi}} \frac{d\lambda}{\lambda - 1} - \int\_{\sqrt{\xi\_0}}^{\sqrt{\xi}} \frac{d\lambda}{\lambda + 1} = \ln\left(\frac{\sqrt{\xi} - 1}{\sqrt{\xi} + 1}\right)\Big|\_{\xi\_0}^{\xi} \tag{29a}$$

and the second magnitude

$$\int\_{\frac{\tilde{\varphi}}{\varphi\_0}}^{\tilde{\varphi}} \frac{d\tilde{\varphi}}{\sqrt{\tilde{\varphi}}(1-\xi)} = 2 \int\_{\sqrt{\tilde{\varphi}\_0}}^{\sqrt{\tilde{\varphi}}} \frac{dx}{1-x^2} = \int\_{\sqrt{\tilde{\varphi}\_0}}^{\sqrt{\tilde{\varphi}}} \frac{dx}{1-x} - \int\_{\sqrt{\tilde{\varphi}\_0}}^{\sqrt{\tilde{\varphi}}} \frac{dx}{1+x} = \ln\left|\frac{1+\sqrt{\tilde{\varphi}}}{1-\sqrt{\tilde{\varphi}}}\right|\_{\frac{\tilde{\varphi}}{\tilde{\varphi}\_0}}^{\tilde{\varphi}}.\tag{29b}$$

Considering Eq. (29), the solutions of Eq. (28) may be written in the following form: the first solution

$$\ln \left| \frac{(\sqrt{\xi} - 1)(\sqrt{\xi\_0} + 1)}{(\sqrt{\xi} + 1)(\sqrt{\xi\_0} - 1)} \right| = -BT\_1 \tag{30a}$$

or

ð23Þ

ð24Þ

ð25Þ

ð26aÞ

ð26bÞ

ð27Þ

ð28Þ

we obtain

74 Nonlinear Systems - Modeling, Estimation, and Stability

Putting

or

desired function

is tabulated in [17].

and substituting Eq. (24) into Eqs. (22) and (23), we are led to the equation

Separating the variables in Eq. (25) and integrating the equation obtained yield

that relationship (26b) is the other first integral of the set of Eq. (19).

Finely, let us eliminate the value from Eqs. (26a) and (22), resulting in

where is the value defining the relative level in the initial amplitudes.

where is an arbitrary constant determined from the initial conditions, and and are the initial magnitudes of the values ξ and δ, respectively. Note

Separating the variables in Eq. (27) and integrating the net expression, we obtain implicitly the

The integral in Eq. (28) can be transformed into an incomplete integral of the first kind, which

$$\sqrt{\xi} = \frac{(1 + \sqrt{\xi\_0}) - (1 - \sqrt{\xi\_0})\mathbf{e}^{-BT\_1}}{(1 + \sqrt{\xi\_0}) + (1 - \sqrt{\xi\_0})\mathbf{e}^{-BT\_1}},\tag{30b}$$

and the second solution

$$\ln \left| \frac{(1 + \sqrt{\xi})(1 - \sqrt{\xi\_0})}{(1 - \sqrt{\xi})(1 + \sqrt{\xi\_0})} \right| = -BT\_1 \tag{31a}$$

or

$$\sqrt{\xi} = \frac{(1 + \sqrt{\xi\_0})\mathbf{e}^{-BT\_1} - (1 - \sqrt{\xi\_0})}{(1 + \sqrt{\xi\_0})\mathbf{e}^{-BT\_1} + (1 - \sqrt{\xi\_0})} \tag{31b}$$

Solutions (30b) and (31b) at and 1 describe the motions corresponding to the one-sided energy exchange between pendulum's vibrations and vertical vibration of the load. As this takes place, and in the first and second solutions, respectively, with the increase in time . In other words, in the first solution, the energy of vibrations of the pendulum completely transforms into the energy of vertical vibrations of the load, but in the second solution, quite the reverse, the energy of vertical vibrations of the load completely goes into the energy of vibrations of the pendulum.

In the first solution, the process of energy transfer occurs over an infinitely large time interval, which resembles the phenomenon of the transfer of the kinetic energy into potential one, which is described by the soliton-like solution (6b) for the mathematical pendulum.

In the second solution, the process of energy transfer occurs during a finite instant of the time from 0 till , where

$$T\_1 \* = -\frac{1}{B} \ln \left( \frac{1 - \sqrt{\xi\_0}}{1 + \sqrt{\xi\_0}} \right).$$

According to our classification, both of them are the soliton-like solutions. At from (30b), we obtain the known soliton-like solution in the form of a single kink [13].

$$
\sqrt{\xi} = \tanh\left(\frac{1}{2}\,\mathrm{BT}\_{\mathrm{l}}\right).
\tag{32}
$$

Physically speaking, this solution kink is responsible to the one-sided energy exchange when the energy of the pendulum vibration completely transforms with time into the energy of the vertical vibrations which energy was equal to zero at the initial moment of time, so the pendulum vibrations give way to the vertical vibrations.

In order to understand the physical meaning of the first integral (26b), let us introduce into consideration the phase plane and analyze on this plane the phase fluid flow that interprets the motion of the mechanical system in hand. The velocity vector V of the phase fluid particles motion has the components and . From Eqs. (22) and (23), it follows that

$$\nu\_{\tilde{\varphi}} = B \frac{\partial G}{\partial \mathcal{S}}, \qquad \nu\_{\mathcal{S}} = -B \frac{\partial G}{\partial \tilde{\varphi}} \tag{33}$$

bounded by the lines . The flow in each rectangle is isolated. On all four rectangle sides, and inside it the value preserves its sign. On the closed streamlines, a two-sided energy exchange takes place between the partial subsystems. Along the lines , a one-sided energy interchange occurs corresponding to pure amplitude-modulated aperiodic motions, in so doing on the lines with ascending flow of the phase fluid particles (an arrow is directed upwards), the aperiodic regime is described by Eq. (30b), and on the lines with descending flow (an arrow is directed downwards), the aperiodic regime is governed by Eq. (31b). On the line , there exists the boundary phasemodulated regime. The transition of fluid elements from the points with the coordinates

To construct the solution in the case of a one-to-one internal resonance (10b), it will suffice to restrict consideration to the terms of the order of and to consider the amplitudes and

The resonance (10b) is weaker than (10a), since in order to eliminate circular terms arising in the second approximation, it would suffice to consider the functions and dependent on only [18]. Under such an assumption, the set of equations providing the absence of circular

nates , correspond to the stable stationary regimes.

Figure 4. Phase portrait in the case of the two-to-one internal resonance .

3.2.2. The case of a one-to-one internal resonance

terms in the expressions for and has the form

as functions of and .

to the points proceeds instantly. The points with coordi-

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77

Writing the equation of a streamline of the phase fluid and substituting Eq. (33) in it,

we obtain that the function defined by the relationship (26b) is the stream function of the phase fluid. In other words, Eq. (26b) at different magnitudes of and governs a family of the streamlines of the phase fluid. Since the phase fluid is incompressible (div V = 0) and its flow is steady and solenoidal (rot V6¼0), then streamlines of the phase fluid will coincide with trajectories of the phase fluid particles motion.

Streamlines constructed according to the relationship

$$
\sqrt{\xi} \,(1 - \xi) \cos \delta = \sqrt{\xi\_0} \,(1 - \xi\_0) \cos \delta\_0 \tag{34}
$$

at different magnitudes of and are presented in Figure 4, where digits near the curves denote the magnitudes of the value . Reference to Figure 4 shows that all phase trajectories are closed lines located around the perimeter of the rectangle Soliton-Like Solutions in the Problems of Vibrations of Nonlinear Mechanical Systems: Survey http://dx.doi.org/10.5772/intechopen.74434 77

Figure 4. Phase portrait in the case of the two-to-one internal resonance .

bounded by the lines . The flow in each rectangle is isolated. On all four rectangle sides, and inside it the value preserves its sign. On the closed streamlines, a two-sided energy exchange takes place between the partial subsystems. Along the lines , a one-sided energy interchange occurs corresponding to pure amplitude-modulated aperiodic motions, in so doing on the lines with ascending flow of the phase fluid particles (an arrow is directed upwards), the aperiodic regime is described by Eq. (30b), and on the lines with descending flow (an arrow is directed downwards), the aperiodic regime is governed by Eq. (31b). On the line , there exists the boundary phasemodulated regime. The transition of fluid elements from the points with the coordinates to the points proceeds instantly. The points with coordinates , correspond to the stable stationary regimes.

#### 3.2.2. The case of a one-to-one internal resonance

In the first solution, the process of energy transfer occurs over an infinitely large time interval, which resembles the phenomenon of the transfer of the kinetic energy into potential one,

In the second solution, the process of energy transfer occurs during a finite instant of the time

According to our classification, both of them are the soliton-like solutions. At from

Physically speaking, this solution kink is responsible to the one-sided energy exchange when the energy of the pendulum vibration completely transforms with time into the energy of the vertical vibrations which energy was equal to zero at the initial moment of time, so the

In order to understand the physical meaning of the first integral (26b), let us introduce into consideration the phase plane and analyze on this plane the phase fluid flow that interprets the motion of the mechanical system in hand. The velocity vector V of the phase fluid particles motion has the components and . From Eqs. (22) and (23), it

Writing the equation of a streamline of the phase fluid and substituting Eq. (33) in it,

we obtain that the function defined by the relationship (26b) is the stream function of the phase fluid. In other words, Eq. (26b) at different magnitudes of and governs a family of the streamlines of the phase fluid. Since the phase fluid is incompressible (div V = 0) and its flow is steady and solenoidal (rot V6¼0), then streamlines of the phase fluid will coincide with

at different magnitudes of and are presented in Figure 4, where digits near the curves denote the magnitudes of the value . Reference to Figure 4 shows that all phase trajectories are closed lines located around the perimeter of the rectangle

ð32Þ

ð33Þ

ð34Þ

which is described by the soliton-like solution (6b) for the mathematical pendulum.

(30b), we obtain the known soliton-like solution in the form of a single kink [13].

pendulum vibrations give way to the vertical vibrations.

trajectories of the phase fluid particles motion.

Streamlines constructed according to the relationship

from 0 till , where

76 Nonlinear Systems - Modeling, Estimation, and Stability

follows that

To construct the solution in the case of a one-to-one internal resonance (10b), it will suffice to restrict consideration to the terms of the order of and to consider the amplitudes and as functions of and .

The resonance (10b) is weaker than (10a), since in order to eliminate circular terms arising in the second approximation, it would suffice to consider the functions and dependent on only [18]. Under such an assumption, the set of equations providing the absence of circular terms in the expressions for and has the form

$$\left(a\_{\rm l}^{2}\right)^{\bullet} = ab\,\alpha\prime\,a\_{\rm l}^{2}a\_{2}^{2}\sin\delta\,,\tag{35a}$$

$$\left(a\_2^2\right)^\bullet = -b^2 \alpha^\ast a\_1^2 a\_2^2 \sin \delta \; , \tag{35b}$$

$$\dot{\phi}\_1 = -aboo\,\ast\,a\_2^2 \left(\frac{1}{3} + \frac{1}{2}\cos\delta\right),\tag{35c}$$

$$\dot{\varphi}\_2 = -ab\,\phi^\* \left( \frac{1}{3}a\_1^2 - \frac{4}{3}a\_2^2 + \frac{1}{2}\frac{b}{a}a\_1^2 \cos\delta \right),\tag{35d}$$

where overdots denote differentiation with respect to , and . The two first integrals of the system (35) have the following form:

ð36Þ

ð38Þ

79

where the sign "+" fits to the initial magnitudes ,

Figure 5. Phase portrait in the case of the one-to-one internal resonance.

initial magnitudes , and

energy of the vertical vibrations.

The upper branch of the separatrix describes the partial irreversible energy transfer from the vertical vibrations to the pendulum vibrations, but the lower branch, on the contrary, is in compliance with partial irreversible transfer of the energy of the pendulum vibrations to the

The points with coordinates (points like a center)

Similar solutions corresponding to the one-sided energy interchange could be obtained for more complex nonlinear systems that describe dynamic behavior of real structures, as an

corresponding to the stable stationary regime are located inside closed streamlines.

4. System with an infinite number of degrees-of-freedom

and but the sign "–" conforms to the

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$$G(\xi, \delta) = \xi(1 - \xi)\cos\delta - \frac{a}{3b}\xi^2 - \frac{\mathsf{S}}{3}(1 - \xi)^2 = G\_0(\xi\_0, \delta\_0), \tag{37}$$

in so doing

$$\dot{\xi} = -\alpha \prime {}^{\mu} b^{2} E\_{0} \frac{\partial G}{\partial \delta}, \qquad \dot{\delta} = \alpha {}^{\mu} b^{2} E\_{0} \frac{\partial G}{\partial \xi}.$$

where , the function ξ (T2) is connected with and by the relationships

$$a\_1^2 = E\_0 \xi(T\_2), \qquad \qquad a\_2^2 = ba^{-1} \left[ 1 - \xi(T\_2) \right].$$

and the rest of the values have the same meaning as in the abovementioned case (10a).

Streamlines constructed according to Eq. (37) at different magnitudes of and are presented in Figure 5 when and . Magnitudes of the value that correspond to the streamlines are indicated by digits near the curves; the flow direction of the phase fluid elements is shown by arrows on the streamlines. Reference to Figure 5 shows that there exist two types of the streamlines, namely (1) nonclosed which correspond to the periodic change of amplitudes and the aperiodic change of phases and (2) closed ones which correspond to the periodic change of both amplitudes and phases. The alignment of the circulation zones resembles that of Von Karman vortex streets with a symmetric arrangement. The adjacent circulation zones osculate at the saddle points with the coordinates and , wherein the unstable stationary regime occurs.

On the boundary lines of these zones (separatrixes), the value , and the analytical solution corresponding to the soliton-like regime has the form

Soliton-Like Solutions in the Problems of Vibrations of Nonlinear Mechanical Systems: Survey http://dx.doi.org/10.5772/intechopen.74434 79

Figure 5. Phase portrait in the case of the one-to-one internal resonance.

ð35aÞ

ð35bÞ

ð35cÞ

ð35dÞ

ð36Þ

ð37Þ

where overdots denote differentiation with respect to , and .

where , the function ξ (T2) is connected

Streamlines constructed according to Eq. (37) at different magnitudes of and are presented in Figure 5 when and . Magnitudes of the value that correspond to the streamlines are indicated by digits near the curves; the flow direction of the phase fluid elements is shown by arrows on the streamlines. Reference to Figure 5 shows that there exist two types of the streamlines, namely (1) nonclosed which correspond to the periodic change of amplitudes and the aperiodic change of phases and (2) closed ones which correspond to the periodic change of both amplitudes and phases. The alignment of the circulation zones resembles that of Von Karman vortex streets with a symmetric arrangement.

and the rest of the values have the same meaning as in the abovementioned case (10a).

The adjacent circulation zones osculate at the saddle points with the coordinates

solution corresponding to the soliton-like regime has the form

On the boundary lines of these zones (separatrixes), the value , and the analytical

and , wherein the unstable stationary regime occurs.

The two first integrals of the system (35) have the following form:

in so doing

with and by the relationships

78 Nonlinear Systems - Modeling, Estimation, and Stability

$$\ln\left\|\frac{2\sqrt{-0.1154(\xi-0.5)^2+0.3397}+0.2308}{\xi-0.5}\right\|\_{\xi\_0}^{\xi}=\pm 0.083E\_0T\_2.\tag{38}$$

where the sign "+" fits to the initial magnitudes , and but the sign "–" conforms to the initial magnitudes , and

The upper branch of the separatrix describes the partial irreversible energy transfer from the vertical vibrations to the pendulum vibrations, but the lower branch, on the contrary, is in compliance with partial irreversible transfer of the energy of the pendulum vibrations to the energy of the vertical vibrations.

The points with coordinates (points like a center) corresponding to the stable stationary regime are located inside closed streamlines.

#### 4. System with an infinite number of degrees-of-freedom

Similar solutions corresponding to the one-sided energy interchange could be obtained for more complex nonlinear systems that describe dynamic behavior of real structures, as an

where the coefficients , , and are defined in [7]. Subsequently, for the

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An approximate solution of Eq. (40) for small but finite amplitudes could be written as an

The number of the independent time scales needed depends on the order to which the expansion is carried out. Here, is the first scale characterizing motions with the natural frequencies and , and are slow scales characterizing the modulations of the amplitudes

Substituting Eq. (41) into Eq. (40) and equating the coefficients of like powers of ε, we obtain on each step a set of two linear equations. On the first step, it is convenient to seek the solution

where and are unknown complex functions, and and are the complex conjugates of

Substituting Eq. (42) into the set of equations obtained on the first step and using the second step to eliminate secular terms, as well as representing the functions and in the polar form we are led to the following system of equations for

is the system's initial energy, , , and an overdot denotes differ-

ð41Þ

81

ð42Þ

ð43Þ

ð44Þ

ease of presentation, the indices n and m will be omitted.

and phases.

in the form:

and , respectively.

entiation with respect to .

The solution to Eq. (44) has the form

the case of the two-to-one internal resonance (10a):

Representing and considering Eq. (43) yield

where is an unknown function, , ,

expansion in terms of different time scales in the following form [16]:

Figure 6. Scheme of a suspension bridge.

example, for systems with an infinite number of degree-of-freedom. Among such systems are suspension bridges, the scheme of one of them is shown in Figure 6.

The suspension bridge scheme presents a bisymmetrical thin-walled stiffening girder, which is connected with two suspended cables by virtue of vertical suspensions. The cables are thrown over the pilons and are tensioned by anchor mechanisms. The suspensions are considered as inextensible and uniformly distributed along the stiffening girder. The cables are parabolic, and the contour of the girder's cross section is undeformable. The cross section l – l in Figure 6 illustrates the displacements of the girder's contour during vibratory motions of the suspension system. Reference to this scheme shows that the girder's contour translates as a rigid body vertically (in the y-axis direction) on the value of and rotates with respect to the girder's axis (the z-axis) through the angle of . The origin of the frame of references is in the center of gravity of the cross section.

It is known for suspension bridges [8] that some natural modes belonging to different types of vibrations could be coupled with each other, that is, the excitation of one natural mode gives rise to another one. Two modes interact more often than not, although the possibility for the interaction of a greater number of modes is not ruled out.

If only two modes predominate in the vibrational process, namely the vertical п-th mode with linear natural frequency and the torsional m-th mode with the natural frequency , such that the modes interaction is observed under the conditions (10a) or (10b), then the functions and can be approximately defined as

$$
\eta(z,t) \sim \mathbf{v}\_n(z) \, \mathbf{x}\_{\ln}(t), \quad \boldsymbol{\varrho}(z,t) \sim \Theta\_m(z) \, \mathbf{x}\_{2m}(t) \, . \tag{39}
$$

where x1n and x2m are the generalized displacements, and and are natural shapes of the two interacting modes of vibrations.

The resolving system of equations in a dimensionless form is written as [7, 8]

$$\begin{aligned} \ddot{\mathbf{x}}\_{1n} + a \mathbf{e}\_{0n}^2 \mathbf{x}\_{1n} + a\_{11}^n \mathbf{x}\_{1n}^2 + a\_{22}^{nm} \mathbf{x}\_{2m}^2 + \left[ b\_{11}^n \mathbf{x}\_{1n}^2 + b\_{22}^{nm} \mathbf{x}\_{2m}^2 \right] \mathbf{x}\_{1n} &= \mathbf{0}, \\ \ddot{\mathbf{x}}\_{2m} + \mathbf{\Omega}\_{0m}^2 \mathbf{x}\_{2m} + a\_{12}^{nm} \mathbf{x}\_{1n} \mathbf{x}\_{2m} + \left( c\_{11}^{nm} \mathbf{x}\_{1n}^2 + c\_{22}^m \mathbf{x}\_{2m}^2 \right) \mathbf{x}\_{2m} &= \mathbf{0}, \end{aligned} \tag{40}$$

where the coefficients , , and are defined in [7]. Subsequently, for the ease of presentation, the indices n and m will be omitted.

An approximate solution of Eq. (40) for small but finite amplitudes could be written as an expansion in terms of different time scales in the following form [16]:

$$\begin{aligned} \boldsymbol{\chi}\_{1}(t) &= \varepsilon \boldsymbol{\chi}\_{1}(T\_{0}, T\_{1}, T\_{2}, \ldots) + \varepsilon^{2} \boldsymbol{\chi}\_{12}(T\_{0}, T\_{1}, T\_{2}, \ldots) + \ldots, \\ \boldsymbol{\chi}\_{2}(t) &= \varepsilon \boldsymbol{\chi}\_{21}(T\_{0}, T\_{1}, T\_{2}, \ldots) + \varepsilon^{2} \boldsymbol{\chi}\_{22}(T\_{0}, T\_{1}, T\_{2}, \ldots) + \ldots \end{aligned} \tag{41}$$

The number of the independent time scales needed depends on the order to which the expansion is carried out. Here, is the first scale characterizing motions with the natural frequencies and , and are slow scales characterizing the modulations of the amplitudes and phases.

Substituting Eq. (41) into Eq. (40) and equating the coefficients of like powers of ε, we obtain on each step a set of two linear equations. On the first step, it is convenient to seek the solution in the form:

$$\begin{aligned} \chi\_{11} &= A\_1(T\_1, T\_2) \exp(i o\_0 T\_0) + A\_1(T\_1, T\_2) \exp(-i o\_0 T\_0), \\ \chi\_{21} &= A\_2(T\_1, T\_2) \exp(i o\_0 T\_0) + \overline{A}\_2(T\_1, T\_2) \exp(-i o\_0 T\_0), \end{aligned} \tag{42}$$

where and are unknown complex functions, and and are the complex conjugates of and , respectively.

Substituting Eq. (42) into the set of equations obtained on the first step and using the second step to eliminate secular terms, as well as representing the functions and in the polar form we are led to the following system of equations for the case of the two-to-one internal resonance (10a):

$$\begin{aligned} \dot{\tilde{\xi}} &= -b(1-\tilde{\xi})\sqrt{\tilde{\xi}}\sin\chi, \\ \dot{\gamma} &= -\frac{1}{2}b(1-3\tilde{\xi})\tilde{\xi}^{-1/2}\cos\chi, \end{aligned} \tag{43}$$

where is an unknown function, , , is the system's initial energy, , , and an overdot denotes differentiation with respect to .

Representing and considering Eq. (43) yield

$$\frac{d\cos\chi}{d\xi} + \frac{1}{2}\frac{1-3\xi}{\xi(1-\xi)}\cos\chi = 0.\tag{44}$$

The solution to Eq. (44) has the form

example, for systems with an infinite number of degree-of-freedom. Among such systems are

The suspension bridge scheme presents a bisymmetrical thin-walled stiffening girder, which is connected with two suspended cables by virtue of vertical suspensions. The cables are thrown over the pilons and are tensioned by anchor mechanisms. The suspensions are considered as inextensible and uniformly distributed along the stiffening girder. The cables are parabolic, and the contour of the girder's cross section is undeformable. The cross section l – l in Figure 6 illustrates the displacements of the girder's contour during vibratory motions of the suspension system. Reference to this scheme shows that the girder's contour translates as a rigid body vertically (in the y-axis direction) on the value of and rotates with respect to the girder's axis (the z-axis) through the angle of . The origin of the frame of references is in the center

It is known for suspension bridges [8] that some natural modes belonging to different types of vibrations could be coupled with each other, that is, the excitation of one natural mode gives rise to another one. Two modes interact more often than not, although the possibility for the

If only two modes predominate in the vibrational process, namely the vertical п-th mode with linear natural frequency and the torsional m-th mode with the natural frequency , such that the modes interaction is observed under the conditions (10a) or (10b), then the functions

where x1n and x2m are the generalized displacements, and and are natural shapes

The resolving system of equations in a dimensionless form is written as [7, 8]

ð39Þ

ð40Þ

suspension bridges, the scheme of one of them is shown in Figure 6.

interaction of a greater number of modes is not ruled out.

and can be approximately defined as

of the two interacting modes of vibrations.

of gravity of the cross section.

Figure 6. Scheme of a suspension bridge.

80 Nonlinear Systems - Modeling, Estimation, and Stability

$$G\_{\mathbb{I}}(\xi,\mathcal{Y}) = \xi^{\mathbb{I}/2} (1-\xi) \cos \mathcal{Y} = G\_{\mathbb{I}}^{0},\tag{45}$$

It can be shown that under the relationship among the natural frequencies rad/s (a two-to-one internal resonance between the sixth symmetrical mode of vertical vibrations and the first symmetrical mode of torsional vibrations), one can obtain the analytical solution in the form of a single kink (32), where B should be replaced by the coefficient b defined by the system's parameters according to Eq. (43). The physical sense of this solution kink is that it is responsible for the one-sided energy exchange when the energy of the torsional vibrations completely transforms into the energy of the vertical vibrations with time, so that the torsional vibrations initiate the

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rad/s (a one-to-one internal resonance), the analytical solutions may be found by solving Eq. (49),

In the first case of Eq. (50), the coefficients and in the integral (49) become zero, and the analytical solution corresponding to the separatrix describes a one-sided energy transfer from the vertical vibration to the torsional vibration (a low aperiodic regime), which leads in time to the conversion of the flexural-torsional vibrations to the predominantly torsional vibrations. This regime is the most unfavorable and dangerous for suspension bridges. In the second case of Eq. (50), the analytical solution corresponding to the separatrix

vibration (an upper aperiodic regime), so that the flexural-torsional vibrations evolve into the

The solutions obtained may be interpreted on the phase plane by virtue of streamlines of the phase fluid which is demonstrated in Figures 5 and 7 for solutions (32) and (50), respectively. Digits near curves indicate the magnitudes of the values and corresponding to the

The analysis of the phase portraits in terms of the variables ξ and γ for various oscillatory regimes demonstrates that they contain both closed and nonclosed streamlines which are separated by the curves separatrixes. Along the separatrixes, one succeeds in finding analytical solutions that are inherently soliton-like solutions in the theory of vibrations and describe the

Note that soliton-like solutions could be found also in an analytical form for the case of free damped vibrations of a suspension bridge, when damping features of the system are described

complete one-sided energy transfer from one subsystem to another.

describes a one-sided energy transfer from the torsional vibration to the vertical

ð50Þ

83

Under the relationships among the natural frequencies, rad/s and

where denotes the evaluation at the upper and lower limits of integration.

predominantly vertical vibrations with time.

vertical vibrations [20].

respectively, as [20]

streamlines.

where is an arbitrary constant determined from the initial conditions. Note that relationship (45) is similar to the first integral (26b) for a two-degree-of-freedom system.

In the case of the one-to-one internal resonance (10b), we seek the solution in the form of Eq. (42) also. Using the procedure for the elimination of secular terms, we obtain the following set of equations:

$$\begin{aligned} \dot{\tilde{\varphi}} &= \frac{1}{2} \Gamma\_2 E\_0 \tilde{\varphi} (1 - \tilde{\varphi}) \sin \gamma, \\ \frac{1}{2} \dot{\gamma} &= \frac{1}{4} \Gamma\_2 E\_0 (1 - 2\tilde{\varphi}) \cos \gamma - (\lambda\_1 - \lambda\_3) E\_0 \tilde{\varphi} - \frac{\Gamma\_2}{\Gamma\_1} (\lambda\_2 - \lambda\_4) E\_0 (1 - \tilde{\varphi}), \end{aligned} \tag{46}$$

where is an unknown function, , is the system's initial energy, , , an overdot denotes differentiation with respect to , and the coefficients and dependent upon the system parameters [8].

Representing and using Eq. (46) yield

$$\frac{d\cos\chi}{d\xi} + \frac{1 - 2\xi}{\xi(1 - \xi)}\cos\chi - \frac{4(\lambda\_1 - \lambda\_3)}{\Gamma\_2(1 - \xi)} - \frac{4(\lambda\_2 - \lambda\_4)}{\Gamma\_1\xi} = 0. \tag{47}$$

The solution to Eq. (47) has the form

$$G\_2(\xi, \gamma) = \tilde{\xi}(1 - \tilde{\xi}) \cos \chi - 2(\lambda\_1 - \lambda\_3) \Gamma\_2^{-1} \tilde{\xi}^2 + 2(\lambda\_2 - \lambda\_4) \Gamma\_1^{-1} (1 - \tilde{\xi})^2 = G\_2^0,\qquad(48)$$

where is an arbitrary constant determined from the initial conditions.

Eliminating the variable γ in Eq. (48) and in the second equation of (46) and integrating over yield

$$\frac{1}{\sqrt{m\_1 m\_2}} \int\_{\xi\_0}^{\xi} \frac{d\xi}{\sqrt{(\xi^2 + p\_1 \xi + q\_1)(\xi^2 + p\_2 \xi + q\_2)}} = \frac{E\_0}{2\Gamma\_1} T\_2,\tag{49}$$

where is a value determined by the relative level of the initial amplitudes, and the quantities , , and are the coefficients [11]. The integral in Eq. (49) can be transformed to an incomplete elliptic integral of the first kind [17].

#### 4.1. Soliton-like solutions

As examples, the nonlinear free vibrations of the Golden Gate Bridge in San Francisco are considered. All geometrical data, as well as natural frequency spectra and mode shapes for this one of the most beautiful suspension bridges, are available in [19].

It can be shown that under the relationship among the natural frequencies rad/s (a two-to-one internal resonance between the sixth symmetrical mode of vertical vibrations and the first symmetrical mode of torsional vibrations), one can obtain the analytical solution in the form of a single kink (32), where B should be replaced by the coefficient b defined by the system's parameters according to Eq. (43). The physical sense of this solution kink is that it is responsible for the one-sided energy exchange when the energy of the torsional vibrations completely transforms into the energy of the vertical vibrations with time, so that the torsional vibrations initiate the vertical vibrations [20].

ð45Þ

ð46Þ

ð47Þ

ð48Þ

ð49Þ

where is an arbitrary constant determined from the initial conditions. Note that relationship

In the case of the one-to-one internal resonance (10b), we seek the solution in the form of Eq. (42) also. Using the procedure for the elimination of secular terms, we obtain the following

where is an unknown function, , is the system's initial energy, , , an overdot denotes differentiation with respect to , and the coefficients and dependent upon the system

(45) is similar to the first integral (26b) for a two-degree-of-freedom system.

set of equations:

82 Nonlinear Systems - Modeling, Estimation, and Stability

parameters [8].

yield

Representing and using Eq. (46) yield

incomplete elliptic integral of the first kind [17].

one of the most beautiful suspension bridges, are available in [19].

4.1. Soliton-like solutions

where is an arbitrary constant determined from the initial conditions.

Eliminating the variable γ in Eq. (48) and in the second equation of (46) and integrating over

where is a value determined by the relative level of the initial amplitudes, and the quantities , , and are the coefficients [11]. The integral in Eq. (49) can be transformed to an

As examples, the nonlinear free vibrations of the Golden Gate Bridge in San Francisco are considered. All geometrical data, as well as natural frequency spectra and mode shapes for this

The solution to Eq. (47) has the form

Under the relationships among the natural frequencies, rad/s and rad/s (a one-to-one internal resonance), the analytical solutions may be found by solving Eq. (49), respectively, as [20]

$$\ln\left|2\xi^{-1}\sqrt{-0.13\xi^2 - 0.195\xi + 0.019} + 0.27\xi^{-1} - 1.4\right|\_{\xi\_0}^{\xi} = 0.431E\_0T\_2,$$

$$\ln\left|2(1-\xi)^{-1}\sqrt{-0.066(1-\xi)^2 + 0.037(1-\xi) + 0.004} + 0.132(1-\xi)^{-1} + 0.536\right|\_{\xi\_0}^{\xi} \qquad(50)$$

$$= -0.036E\_0T\_2,$$

where denotes the evaluation at the upper and lower limits of integration.

In the first case of Eq. (50), the coefficients and in the integral (49) become zero, and the analytical solution corresponding to the separatrix describes a one-sided energy transfer from the vertical vibration to the torsional vibration (a low aperiodic regime), which leads in time to the conversion of the flexural-torsional vibrations to the predominantly torsional vibrations. This regime is the most unfavorable and dangerous for suspension bridges.

In the second case of Eq. (50), the analytical solution corresponding to the separatrix describes a one-sided energy transfer from the torsional vibration to the vertical vibration (an upper aperiodic regime), so that the flexural-torsional vibrations evolve into the predominantly vertical vibrations with time.

The solutions obtained may be interpreted on the phase plane by virtue of streamlines of the phase fluid which is demonstrated in Figures 5 and 7 for solutions (32) and (50), respectively. Digits near curves indicate the magnitudes of the values and corresponding to the streamlines.

The analysis of the phase portraits in terms of the variables ξ and γ for various oscillatory regimes demonstrates that they contain both closed and nonclosed streamlines which are separated by the curves separatrixes. Along the separatrixes, one succeeds in finding analytical solutions that are inherently soliton-like solutions in the theory of vibrations and describe the complete one-sided energy transfer from one subsystem to another.

Note that soliton-like solutions could be found also in an analytical form for the case of free damped vibrations of a suspension bridge, when damping features of the system are described

On the phase plane, these solutions correspond to streamlines which separate closed lines of

Soliton-Like Solutions in the Problems of Vibrations of Nonlinear Mechanical Systems: Survey

http://dx.doi.org/10.5772/intechopen.74434

85

Since soliton-like solution may describe unfavorable vibratory regimes of real mechanical systems, then they should be investigated systematically by virtue of mathematical models of these systems, in order to avoid, wherever possible, such dangerous vibratory regimes when designing and constructing real structures. A thorough analysis of internal resonances in thin

Soliton-like solutions in the cases of combinational internal resonances for systems with an infinite number of degrees-of-freedom, when more than two natural modes of vibration are coupled, could be found in sight as well, and such examples for nonlinear plates and cylindri-

The research described in this publication was made possible in part by the Ministry of

[1] Mandel'shtam L. Lecture Notes on Theory of Vibrations (in Russian). Moscow: Nauka; 1972

[2] Nayfeh AH, Balachandran B. Modal interactions in dynamical and structural systems.

[3] Sado D. Energy transfer in two-degree-of-freedom vibrating systems—A survey.

[4] Vitt AA, Gorelik GA. Vibrations of an elastic pendulum as an example of vibrations of two parametrically coupled linear systems (in Russian). Journal of Technical Physics. 1933;2–3:

[5] Sado D. Analysis of vibration of two-degree of freedom system with inertial coupling.

Education and Science of the Russian Federation under Project # 9.5138.2017/8.9.

phase fluid flow from nonclosed ones. These lines are called separatrixes.

cal shells are presented in [27, 28] respectively.

Yury A. Rossikhin and Marina V. Shitikova\*

\*Address all correspondence to<EMAIL_ADDRESS>
Applied Mechanics Reviews. 1989;42:175-201

Machine Dynamics Problems. 1984;1:67-77

Mechanika Teoretyczna i Stosowana. 1993;31:151-173

Voronezh State Technical University, Voronezh, Russian Federation

Acknowledgements

Author details

References

294-307

plates and cylindrical shells could be found in [23, 24] and [25, 26], respectively.

Figure 7. Phase portraits: (a) , and (b)

by ordinary first-order time derivative [21] or defined by a fractional derivative with a fractional parameter (the order of the fractional derivative) changing from zero to one [22].

#### 5. Conclusions

From the review presented, the following conclusions could be deduced. In all considered vibratory systems—1dof, 2dof, and multi-dof—under certain conditions, there exist solutions that describe irreversible processes of energy transfer from its one type to another. Such solutions are called soliton-like solutions and could be written in an analytical form.

On the phase plane, these solutions correspond to streamlines which separate closed lines of phase fluid flow from nonclosed ones. These lines are called separatrixes.

Since soliton-like solution may describe unfavorable vibratory regimes of real mechanical systems, then they should be investigated systematically by virtue of mathematical models of these systems, in order to avoid, wherever possible, such dangerous vibratory regimes when designing and constructing real structures. A thorough analysis of internal resonances in thin plates and cylindrical shells could be found in [23, 24] and [25, 26], respectively.

Soliton-like solutions in the cases of combinational internal resonances for systems with an infinite number of degrees-of-freedom, when more than two natural modes of vibration are coupled, could be found in sight as well, and such examples for nonlinear plates and cylindrical shells are presented in [27, 28] respectively.

### Acknowledgements

The research described in this publication was made possible in part by the Ministry of Education and Science of the Russian Federation under Project # 9.5138.2017/8.9.

### Author details

Yury A. Rossikhin and Marina V. Shitikova\*

\*Address all correspondence to<EMAIL_ADDRESS>
Voronezh State Technical University, Voronezh, Russian Federation

### References

by ordinary first-order time derivative [21] or defined by a fractional derivative with a fractional parameter (the order of the fractional derivative) changing from zero to one [22].

From the review presented, the following conclusions could be deduced. In all considered vibratory systems—1dof, 2dof, and multi-dof—under certain conditions, there exist solutions that describe irreversible processes of energy transfer from its one type to another. Such

solutions are called soliton-like solutions and could be written in an analytical form.

5. Conclusions

Figure 7. Phase portraits: (a) , and (b)

84 Nonlinear Systems - Modeling, Estimation, and Stability