**2.2 Reduced model equation**

According to the Galerkin's method [27, 28] and by taking into account the boundary conditions of the beam, the solution of the partial differential Eq. (1) is given by:

$$w(\mathbf{x},t) = \sum\_{n=1}^{\infty} q\_n(t) \sin\left(\frac{n\pi\mathbf{x}}{L}\right),\tag{4}$$

Where

introduced:

where

� 1 þ *η*

X *Np*

*j*¼1

*ϕ* could be solved as:

*βa*<sup>3</sup>

time interval [0, T].

**17**

<sup>0</sup> þ 1 þ

3 2

*M*2ð Þ¼� *a*, *ψ η*

*F*0*<sup>N</sup>* ¼ *P*<sup>0</sup> 1 þ

*DOI: http://dx.doi.org/10.5772/intechopen.96878*

2 sin~*τ*<sup>0</sup> sin ð Þ ð Þ *N* � 1 ~*τ*<sup>0</sup> 1 � cos 2ð Þ ~*τ*<sup>0</sup>

*<sup>G</sup>*0*<sup>N</sup>* <sup>¼</sup> <sup>2</sup>*P*<sup>0</sup> sin~*τ*<sup>0</sup> sin ð Þ ð Þ *<sup>N</sup>* � <sup>1</sup> <sup>~</sup>*τ*<sup>0</sup> 1 � cos 2ð Þ ~*τ*<sup>0</sup>

**3. Analytical explanation of the model**

**3.1 Effective analytical solution of the problem**

Substituting Eq. (11) into Eq. (9) we obtain:

8 < :

*<sup>k</sup> <sup>j</sup>* sin <sup>2</sup> *<sup>j</sup><sup>π</sup>*

X *Np*

*j*¼1

*Np* þ 1 � �

*cjω<sup>α</sup> <sup>j</sup>*

8 >>><

>>>:

" #

<sup>0</sup> sin <sup>2</sup> *<sup>j</sup><sup>π</sup>*

*<sup>a</sup>*\_ ¼ � <sup>1</sup>

*<sup>a</sup>ϕ*\_ ¼ � <sup>1</sup>

X *Np*

*<sup>k</sup> <sup>j</sup>* sin <sup>2</sup> *<sup>j</sup><sup>π</sup>*

" # � � *<sup>a</sup>*<sup>0</sup> <sup>¼</sup> *<sup>ϑ</sup>*<sup>0</sup> � <sup>1</sup>

*Np* þ 1

Then, one could apply the stochastic averaging method [13–15] to Eq. (15) in

*j*¼1

*a*<sup>0</sup> satisfies the following non-linear equation:

*<sup>β</sup>a*<sup>2</sup> <sup>þ</sup> *<sup>η</sup>*

*a*\_ cos *ψ* � *aφ*\_ sin *ψ* ¼ 0 *<sup>a</sup>*\_ sin *<sup>ψ</sup>* � *<sup>a</sup>φ*\_ cos *<sup>ψ</sup>* ¼ � <sup>1</sup>

*<sup>M</sup>*1ð Þ¼ *<sup>a</sup>*, *<sup>ψ</sup> <sup>F</sup>*0*<sup>N</sup>* sin ð Þ� *<sup>ψ</sup>* � *<sup>φ</sup> <sup>G</sup>*0*<sup>N</sup>* cosð Þþ *<sup>ψ</sup>* � *<sup>φ</sup>* ð Þ <sup>2</sup>*<sup>λ</sup>* <sup>þ</sup> *<sup>ϑ</sup>*<sup>1</sup> *<sup>a</sup>*<sup>Ω</sup> sin *<sup>ψ</sup><sup>a</sup>* � <sup>1</sup>

<sup>þ</sup> <sup>3</sup>*βa*<sup>2</sup> <sup>0</sup> þ 3

*Np* þ 1 � �*D<sup>α</sup> <sup>j</sup>*

� �, <sup>~</sup>*τ*<sup>0</sup> <sup>¼</sup> *<sup>d</sup><sup>π</sup>*

*Vibrations of an Elastic Beam Subjected by Two Kinds of Moving Loads and Positioned on…*

In order to directly evaluate the response of the beam, the stochastic averaging method [13–15] is first applied to Eq. (6), then the following change in variables is

*χ τ*ð Þ¼ *a*<sup>0</sup> þ *a*ð Þ*τ* cos *ψ*, *χ τ* \_ð Þ¼�Ω*a*ð Þ*τ* sin *ψ*, *ψ* ¼ Ω*τ* þ *ϕ τ*ð Þ, (11)

<sup>4</sup> *<sup>β</sup>a*<sup>2</sup> � <sup>Ω</sup><sup>2</sup>

According to Eq. (13) The derivatives of the generalized amplitude *a* and phase

<sup>Ω</sup> ½ � *<sup>M</sup>*1ð Þþ *<sup>a</sup>*, *<sup>ψ</sup> <sup>M</sup>*2ð Þ *<sup>a</sup>*, *<sup>ψ</sup>* sin *<sup>ψ</sup>*

<sup>Ω</sup> ½ � *<sup>M</sup>*1ð Þþ *<sup>a</sup>*, *<sup>ψ</sup> <sup>M</sup>*2ð Þ *<sup>a</sup>*, *<sup>ψ</sup>* cos *<sup>ψ</sup>*

<sup>Ω</sup> ½ � *<sup>M</sup>*1ð Þþ *<sup>a</sup>*, *<sup>ψ</sup> <sup>M</sup>*2ð Þ *<sup>a</sup>*, *<sup>ψ</sup> :*

cos *ψ* þ

*<sup>τ</sup>* ð Þþ *a* cos *ψ* ð Þ *θ*<sup>0</sup> � *θ*1Ω*a* sin *ψ ξ τ*ð Þ*:*

*a*2

2 *ϑ*2Ω<sup>2</sup> *a*2

<sup>2</sup> *<sup>ϑ</sup>*2Ω<sup>2</sup> � <sup>3</sup>*βa*<sup>0</sup>

cosð Þ *N*~*τ*<sup>0</sup>

sin ð Þ *N*~*τ*<sup>0</sup> *:*

2*L* ,

(10)

(12)

(13)

(14)

*:* (15)

<sup>4</sup> *<sup>β</sup>a*<sup>3</sup> cos 3*<sup>ψ</sup>*

� � cos 2*ψ*,

where *qn*ð Þ*t* are the amplitudes of vibration and sin ð Þ *nπx=L* are modal functions solutions of the beam linear natural equation with the associated boundary conditions. It is convenient to adopt the following dimensionless variables:

$$\chi\_n = \frac{q\_n}{l\_r}, \ \tau = \alpha\_0 t, \ \xi = \frac{u}{u\_\varepsilon},\tag{5}$$

the single one-dimensional modal equation with *χ<sup>n</sup>* ¼ *χ τ*ð Þ is given as:

$$\ddot{\boldsymbol{\chi}}(\tau) + (2\boldsymbol{\lambda} + \boldsymbol{\theta}\_1)\dot{\boldsymbol{\chi}}(\tau) + \boldsymbol{\chi}(\tau) + \boldsymbol{\theta}\boldsymbol{\chi}^3(\tau) + \eta \sum\_{j=1}^{N\_p} (\mathbf{k}\_j + c\_j a\_0^{a\_j} D\_\tau^{a\_j}) \boldsymbol{\chi}(\tau) \sin^2 \left(\frac{j\pi}{N\_p + 1}\right)$$

$$= \boldsymbol{\theta}\_2 \dot{\boldsymbol{\chi}}^2(\tau) + \boldsymbol{\theta}\_0 + (\boldsymbol{\theta}\_0 + \boldsymbol{\theta}\_1 \dot{\boldsymbol{\chi}}(\tau)) \boldsymbol{\xi}(\tau) + f\_0 \sum\_{i=0}^{N-1} c\_i \sin \boldsymbol{\Omega} \left[\tau - i \frac{d a\_0}{v}\right]. \tag{6}$$

With

$$\begin{aligned} \label{eq:1} \mathfrak{Q} = \frac{\pi \upsilon}{L \rho\_0}, \ f\_0 = \frac{2PL^3}{l\_r EI \pi^4}, \ \eta = \frac{2L^3}{EI\pi^4}, \ \beta = \frac{l\_r^2}{4} \left[\frac{\mathcal{S}}{I} - \frac{3}{2} \left(\frac{\pi}{L}\right)^2\right], \\\\ \label{eq:1} \theta\_1 = \frac{\rho\_a bL^3 A\_1 \overline{U}}{2\pi^2 \sqrt{EI\rho \left[L^2 S + I\pi^2\right]}}, \ \;\theta\_0 = \frac{2\rho\_a bA\_0 \overline{U}^2 L^4}{EIl\_r \pi^5}, \ \;\theta\_2 = \frac{4\rho\_a bL^2 A\_2 l\_r}{3\rho \left[L^2 S + I\pi^2\right]}, \\\\ \label{eq:1} \theta\_0 = \frac{2\overline{U}\rho\_a bL^4 U\_c A\_0}{EIl\_r \pi^6}, \ \;\theta\_1 = \frac{\rho\_a bL^3 A\_1 U\_c}{2\pi^2 \sqrt{EI\rho \left[L^2 S + I\pi^2\right]}}, \ \;\lambda = \frac{\mu L^3}{2\pi^2 \sqrt{EI\rho \left[L^2 S + I\pi^2\right]}}. \end{aligned}$$

And

$$
\rho a\_0 = \frac{\pi^2}{L} \sqrt{\frac{EI}{\rho \left(L^2 \mathbb{S} + I\pi^2\right)}}, \quad l\_r = \frac{L}{2}. \tag{8}
$$

According to Refs.[29, 30], Eq. (6) becomes:

$$\ddot{\boldsymbol{\chi}}(\tau) + (2\boldsymbol{\lambda} + \boldsymbol{\theta}\_1)\dot{\boldsymbol{\chi}}(\tau) + \boldsymbol{\chi}(\tau) + \boldsymbol{\theta}\boldsymbol{\chi}^3(\tau) + \eta \sum\_{j=1}^{N\_p} (\boldsymbol{k}\_j + c\_j a\_0^{a\_j} D\_\tau^{a\_j}) \boldsymbol{\chi}(\tau) \sin^2 \left(\frac{j\pi}{N\_p + 1}\right), \tag{9}$$

$$\ddot{\boldsymbol{\chi}} = \boldsymbol{\theta}\_2 \dot{\boldsymbol{\chi}}^2(\tau) + \boldsymbol{\theta}\_0 + (\boldsymbol{\theta}\_0 + \boldsymbol{\theta}\_1 \dot{\boldsymbol{\chi}}(\tau)) \boldsymbol{\xi}(\tau) + F\_{0N} \sin\left(\boldsymbol{\Omega}\tau\right) - G\_{0N} \cos\left(\boldsymbol{\Omega}\tau\right). \tag{9}$$

*Vibrations of an Elastic Beam Subjected by Two Kinds of Moving Loads and Positioned on… DOI: http://dx.doi.org/10.5772/intechopen.96878*

Where

**2.2 Reduced model equation**

€*χ τ*ð Þþ ð Þ <sup>2</sup>*<sup>λ</sup>* <sup>þ</sup> *<sup>ϑ</sup>*<sup>1</sup> *χ τ* \_ð Þþ *χ τ*ð Þþ *βχ*<sup>3</sup>

, *<sup>f</sup>* <sup>0</sup> <sup>¼</sup> <sup>2</sup>*PL*<sup>3</sup>

*A*1*U*

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi *EIρ L*<sup>2</sup>

*UcA*<sup>0</sup>

*S* þ *Iπ*<sup>2</sup>

¼ *ϑ*2*χ*\_ 2

With

And

¼ *ϑ*2*χ*\_ 2

**16**

<sup>Ω</sup> <sup>¼</sup> *<sup>π</sup><sup>v</sup> Lω*<sup>0</sup>

*<sup>ϑ</sup>*<sup>1</sup> <sup>¼</sup> *<sup>ρ</sup>abL*<sup>3</sup>

2*π*<sup>2</sup>

*<sup>θ</sup>*<sup>0</sup> <sup>¼</sup> <sup>2</sup>*UρabL*<sup>4</sup>

given by:

According to the Galerkin's method [27, 28] and by taking into account the boundary conditions of the beam, the solution of the partial differential Eq. (1) is

*qn*ð Þ*<sup>t</sup>* sin *<sup>n</sup>π<sup>x</sup>*

where *qn*ð Þ*t* are the amplitudes of vibration and sin ð Þ *nπx=L* are modal functions solutions of the beam linear natural equation with the associated boundary condi-

, *<sup>τ</sup>* <sup>¼</sup> *<sup>ω</sup>*0*t*, *<sup>ξ</sup>* <sup>¼</sup> *<sup>u</sup>*

X *Np*

*j*¼1

X *N*�1

*i*¼0

2 *r* 4 *S I* � 3 2 *π L* � �<sup>2</sup> � �

*A*1*Uc*

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi *EIρ L*<sup>2</sup>

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi *EI*

*<sup>S</sup>* <sup>þ</sup> *<sup>I</sup>π*<sup>2</sup> � �

X *Np*

*j*¼1

*ρ L*<sup>2</sup>

ð Þþ *τ η*

ð Þþ *τ ϑ*<sup>0</sup> þ ð Þ *θ*<sup>0</sup> þ *θ*1*χ τ* \_ð Þ *ξ τ*ð Þþ *F*0*<sup>N</sup>* sin ð Þ� Ω*τ G*0*<sup>N</sup>* cosð Þ Ω*τ :*

s

*S* þ *Iπ*<sup>2</sup>

*L*4

*EIlrπ*<sup>5</sup> , *<sup>ϑ</sup>*<sup>2</sup> <sup>¼</sup> <sup>4</sup>*ρabL*<sup>2</sup>

<sup>q</sup> � � , *<sup>λ</sup>* <sup>¼</sup> *<sup>μ</sup>L*<sup>3</sup>

, *lr* <sup>¼</sup> *<sup>L</sup>*

*<sup>k</sup> <sup>j</sup>* <sup>þ</sup> *cjω<sup>α</sup> <sup>j</sup>*

<sup>0</sup> *<sup>D</sup><sup>α</sup> <sup>j</sup> τ* � �*χ τ*ð Þsin <sup>2</sup> *<sup>j</sup><sup>π</sup>*

*L* � �

*uc*

*<sup>k</sup> <sup>j</sup>* <sup>þ</sup> *cjω<sup>α</sup> <sup>j</sup>*

<sup>0</sup> *<sup>D</sup><sup>α</sup> <sup>j</sup> τ* � �*χ τ*ð Þsin <sup>2</sup> *<sup>j</sup><sup>π</sup>*

*ε<sup>i</sup>* sin Ω *τ* � *i*

,

3*ρ L*<sup>2</sup>

2*π*<sup>2</sup>

, (4)

, (5)

*dω*<sup>0</sup> *v* � �

*:*

*A*2*lr*

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi *EIρ L*<sup>2</sup>

<sup>2</sup> *:* (8)

*S* þ *Iπ*<sup>2</sup> <sup>q</sup> � � *:*

> *Np* þ 1 � �

> > (9)

(7)

*<sup>S</sup>* <sup>þ</sup> *<sup>I</sup>π*<sup>2</sup> � � ,

*Np* þ 1 � �

(6)

*w x*ð Þ¼ , *<sup>t</sup>* <sup>X</sup><sup>∞</sup>

*Advances in Dynamical Systems Theory, Models, Algorithms and Applications*

tions. It is convenient to adopt the following dimensionless variables:

*<sup>χ</sup><sup>n</sup>* <sup>¼</sup> *qn lr*

ð Þþ *τ ϑ*<sup>0</sup> þ ð Þ *θ*<sup>0</sup> þ *θ*1*χ τ* \_ð Þ *ξ τ*ð Þþ *f* <sup>0</sup>

*lrEIπ*<sup>4</sup> , *<sup>η</sup>* <sup>¼</sup> <sup>2</sup>*L*<sup>3</sup>

<sup>q</sup> � � , *<sup>ϑ</sup>*<sup>0</sup> <sup>¼</sup> <sup>2</sup>*ρabA*0*U*<sup>2</sup>

*EIlrπ*<sup>6</sup> , *<sup>θ</sup>*<sup>1</sup> <sup>¼</sup> *<sup>ρ</sup>abL*<sup>3</sup>

*<sup>ω</sup>*<sup>0</sup> <sup>¼</sup> *<sup>π</sup>*<sup>2</sup> *L*

According to Refs.[29, 30], Eq. (6) becomes:

€*χ τ*ð Þþ ð Þ <sup>2</sup>*<sup>λ</sup>* <sup>þ</sup> *<sup>ϑ</sup>*<sup>1</sup> *χ τ* \_ð Þþ *χ τ*ð Þþ *βχ*<sup>3</sup>

2*π*<sup>2</sup>

*n*¼1

the single one-dimensional modal equation with *χ<sup>n</sup>* ¼ *χ τ*ð Þ is given as:

ð Þþ *τ η*

*EIπ*<sup>4</sup> , *<sup>β</sup>* <sup>¼</sup> *<sup>l</sup>*

$$\begin{aligned} F\_{0N} &= P\_0 \left[ 1 + \frac{2 \sin \tilde{\tau}\_0 \sin \left( (N - 1)\tilde{\tau}\_0 \right)}{1 - \cos \left( 2\tilde{\tau}\_0 \right)} \cos \left( N\tilde{\tau}\_0 \right) \right], \ \tilde{\tau}\_0 = \frac{d\pi}{2L}, \\\ G\_{0N} &= \frac{2P\_0 \sin \tilde{\tau}\_0 \sin \left( (N - 1)\tilde{\tau}\_0 \right)}{1 - \cos \left( 2\tilde{\tau}\_0 \right)} \sin \left( N\tilde{\tau}\_0 \right). \end{aligned} \tag{10}$$
