**3. Analytical explanation of the model**

### **3.1 Effective analytical solution of the problem**

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 introduced:

$$\chi(\tau) = a\_0 + a(\tau)\cos\psi,\ \dot{\chi}(\tau) = -\Omega a(\tau)\sin\psi,\ \text{ $\psi = \Omega\tau + \phi(\tau)$ },\tag{11}$$

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

$$\begin{cases} \dot{a}\cos\psi - a\dot{\rho}\sin\psi = 0\\ \dot{a}\sin\psi - a\dot{\rho}\cos\psi = -\frac{1}{\Omega}[M\_1(a,\,\mu) + M\_2(a,\,\mu)]. \end{cases} \tag{12}$$

where

$$M\_{1}(a,\boldsymbol{\psi}) = F\_{0N}\sin\left(\boldsymbol{\psi} - \boldsymbol{\varrho}\right) - G\_{0N}\cos\left(\boldsymbol{\psi} - \boldsymbol{\varrho}\right) + (2\boldsymbol{\lambda} + \boldsymbol{\theta}\_{1})a\Omega\sin\boldsymbol{\upmu}a - \frac{1}{4}\beta a^{3}\cos 3\boldsymbol{\upmu}\cdot\boldsymbol{\upmu}\,\Delta\boldsymbol{\upmu}\,\Delta\boldsymbol{\upmu}\,\Delta\boldsymbol{\upmu}\,\Delta\boldsymbol{\upmu}\,\Delta\boldsymbol{\upmu}\,\Delta\boldsymbol{\upmu}\,\Delta\boldsymbol{\upmu}\,\Delta\boldsymbol{\upmu}\,\Delta\boldsymbol{\upmu}\,\Delta\boldsymbol{\upmu}\,\Delta\boldsymbol{\upmu}\,\Delta\boldsymbol{\upmu}\,\Delta\boldsymbol{\upmu}\,\Delta\boldsymbol{\upmu}\,\Delta\boldsymbol{\upmu}\,\Delta\boldsymbol{\upmu}\,\Delta\boldsymbol{\upmu}\,\Delta\boldsymbol{\upmu}\,\Delta\boldsymbol{\upmu}\,\Delta\boldsymbol{\upmu}\,\Delta\boldsymbol{\upmu}\,\Delta\boldsymbol{\upmu}\,\Delta\boldsymbol{\upmu}\,\Delta\boldsymbol{\upmu}\,\Delta\boldsymbol{\upmu}\,\Delta\boldsymbol{\upmu}\,\Delta\boldsymbol{\upmu}\,\Delta\boldsymbol{\upmu}\,\Delta\boldsymbol{\upmu}\,\Delta\boldsymbol{\upmu}\,\Delta\boldsymbol{\upmu}\,\Delta\boldsymbol{\upmu}\,\Delta\boldsymbol{\upmu}\,\Delta\boldsymbol{\upmu}\,\Delta\boldsymbol{\upmu}\,\Delta\boldsymbol{\upmu}\,\Delta\boldsymbol{\upmu}\,\Delta\boldsymbol{\upmu}\,\Delta\boldsymbol{\upmu}\,\Delta\boldsymbol{\upmu}\,\Delta\boldsymbol{\upmu}\,\Delta\boldsymbol{\upmu}\,\Delta\boldsymbol{\upmu}\,\Delta\boldsymbol{\upmu}\,\Delta\boldsymbol{\upmu}\,\Delta\boldsymbol{\upmu}\,\Delta\boldsymbol{\upmu}\,\Delta\boldsymbol{\upmu}\,\Delta\boldsymbol{\upmu}\,\Delta\boldsymbol{\upmu}\,\Delta\boldsymbol{\upmu}\,\Delta\boldsymbol{\upmu}\,\Delta\boldsymbol{\upmu}\,\Delta\boldsymbol{\upmu$$

According to Eq. (13) The derivatives of the generalized amplitude *a* and phase *ϕ* could be solved as:

$$\begin{cases} \dot{a} = -\frac{1}{\Omega} [M\_1(a, \psi) + M\_2(a, \psi)] \sin \psi \\\\ a \dot{\phi} = -\frac{1}{\Omega} [M\_1(a, \psi) + M\_2(a, \psi)] \cos \psi \end{cases} \tag{14}$$

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

$$
\rho \theta a\_0^3 + \left[ \mathbf{1} + \frac{\mathbf{3}}{2} \theta a^2 + \eta \sum\_{j=1}^{N\_p} k\_j \sin^2 \left( \frac{j \pi}{N\_p + 1} \right) \right] a\_0 = \theta\_0 - \frac{\mathbf{1}}{2} \theta\_2 \mathbf{1} \mathbf{2}^2 a^2. \tag{15}
$$

Then, one could apply the stochastic averaging method [13–15] to Eq. (15) in time interval [0, T].

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

$$\begin{cases} \dot{a} = -\lim\_{T \to \infty} \frac{1}{T} \prod\_{0}^{T} [M\_1(a, \mu) + M\_2(a, \mu)] \sin \eta dy \\\\ a\dot{\rho} = -\lim\_{T \to \infty} \frac{1}{T} \int\_0^T [M\_1(a, \mu) + M\_2(a, \mu)] \cos \eta dy \end{cases} \tag{16}$$

this model of the turbulent component of the wind *ξ τ*ð Þ amounts to a bounded or

*<sup>i</sup> <sup>ω</sup>*<sup>2</sup> <sup>þ</sup> *<sup>ω</sup>*<sup>2</sup>

The next sections of this chapter will presents the analytical developments that we have made in order to express the beam response as a function of the system parameters. Then, let's start with the case where the beam is subjected to the

*<sup>i</sup>* � *γ*<sup>4</sup> *<sup>i</sup> <sup>=</sup>*<sup>4</sup> � �<sup>2</sup> <sup>þ</sup> *<sup>γ</sup>*<sup>2</sup>

*<sup>i</sup>* <sup>þ</sup> *<sup>γ</sup>*<sup>4</sup> *<sup>i</sup> <sup>=</sup>*<sup>4</sup> � �

*<sup>i</sup> ω*<sup>2</sup> h i *:* (21)

*σ*2 *i γ*2

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

4*π ω*<sup>2</sup> � *ω*<sup>2</sup>

**3.2 Analytical estimate of the beam response under moving loads only**

We first consider system (1) with only deterministic moving loads

*cjω<sup>α</sup> <sup>j</sup>*

X *Np*

*j*¼1

<sup>1</sup> *α <sup>j</sup>* � � <sup>þ</sup> <sup>Θ</sup><sup>2</sup>

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

*πα <sup>j</sup>* 2

By substituting *<sup>a</sup>* <sup>¼</sup> *<sup>A</sup>*, *<sup>ϕ</sup>* <sup>¼</sup> <sup>Φ</sup> and *<sup>a</sup>*\_ <sup>¼</sup> 0, *<sup>ϕ</sup>*\_ <sup>¼</sup> 0 in Eqs. (22) and (23), algebraic manipulations give for the steady-state vibrations of the system response *A* the

(*Fad*ð Þ¼ *x*, *t* 0) neglecting wind effects on the beam. If *ϑ*<sup>1</sup> ¼ *θ*<sup>0</sup> ¼ *θ*<sup>1</sup> ¼ 0, Eqs. (15)

<sup>0</sup> *<sup>Ω</sup><sup>α</sup> <sup>j</sup>*�<sup>1</sup> sin <sup>2</sup> *<sup>j</sup><sup>π</sup>*

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

<sup>2</sup> *α <sup>j</sup>* � � � � *<sup>A</sup>*<sup>2</sup> <sup>¼</sup> *<sup>F</sup>*<sup>2</sup>

<sup>0</sup> Ω*<sup>α</sup> <sup>j</sup>* cos

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

The stability of the steady-state vibration of the system response is investigated by using the method of Andronov and Witt [34] associated to the Routh-Hurwitz criterion [35]. Thus, the steady-state response is asymptotically stable if Eq. (26) is

*πα <sup>j</sup>* 2 � � � � sin <sup>2</sup> *<sup>j</sup><sup>π</sup>*

> *Np* þ 1 � �*:*

" # � �

*Np* þ 1 � � sin

<sup>0</sup> Ω*<sup>α</sup> <sup>j</sup>* cos

<sup>2</sup><sup>Ω</sup> ½ � *<sup>G</sup>*0*<sup>N</sup>* sin *<sup>ϕ</sup>* � *<sup>F</sup>*0*<sup>N</sup>* cos *<sup>ϕ</sup>* , (22)

*πα <sup>j</sup>* 2 � �

*πα <sup>j</sup>* 2 � � � � sin <sup>2</sup> *<sup>j</sup><sup>π</sup>*

<sup>0</sup>*<sup>N</sup>* <sup>þ</sup> *<sup>G</sup>*<sup>2</sup>

*Np* þ 1

<sup>0</sup>*<sup>N</sup>*, (24)

*Np* þ 1 � �, (23)

(25)

cosine-Wiener noise, whose spectral density is given by:

*i*¼1

*Φξ*ð Þ¼ *<sup>ω</sup>* <sup>X</sup>*<sup>m</sup>*

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

moving loads only.

and (16) become:

and

*<sup>a</sup>φ*\_ <sup>¼</sup> *<sup>a</sup>* 2Ω

> þ 1

*<sup>a</sup>*\_ ¼ �*λ<sup>a</sup>* � <sup>1</sup>

þ 1

<sup>1</sup> � <sup>Ω</sup><sup>2</sup> <sup>þ</sup> <sup>3</sup>*βa*<sup>2</sup>

following non-linear equation:

*<sup>A</sup>*<sup>6</sup> � <sup>3</sup> 2 *β*Θ<sup>1</sup> *α <sup>j</sup>*

� � <sup>¼</sup> <sup>Ω</sup><sup>2</sup> � <sup>1</sup> � <sup>3</sup>*βa*<sup>2</sup>

� � <sup>¼</sup> <sup>2</sup>Ω*<sup>λ</sup>* <sup>þ</sup> *<sup>η</sup>*

9 <sup>16</sup> *<sup>β</sup>*<sup>2</sup>

with

Θ<sup>1</sup> *α <sup>j</sup>*

Θ<sup>2</sup> *α <sup>j</sup>*

**19**

<sup>2</sup><sup>Ω</sup> ½ � *<sup>G</sup>*0*<sup>N</sup>* cos *<sup>φ</sup>* <sup>þ</sup> *<sup>F</sup>*0*<sup>N</sup>* sin *<sup>φ</sup> :*

2 *ηa* X *Np*

<sup>0</sup> þ 3 <sup>4</sup> *<sup>β</sup>a*<sup>2</sup> <sup>þ</sup> *<sup>η</sup>*

� �*A*<sup>4</sup> <sup>þ</sup> <sup>Θ</sup><sup>2</sup>

X *Np*

*j*¼1

<sup>0</sup> Ω*<sup>α</sup> <sup>j</sup>* sin

<sup>0</sup> � *η*

*cjω<sup>α</sup> <sup>j</sup>*

X *Np*

*j*¼1

satisfied and unstable if Eq. (27) is satisfied:

*j*¼1

According to this method, one could select the time terminal T as *T* ¼ 2*π=*Ω in the case of periodic function (*M*1ð Þ *a*, *ψ* ), or *T* ¼ ∞ in the case of aperiodic one (*M*2ð Þ *a*, *ψ* ). Accordingly, one could obtain the following pair of first order differential equations for the amplitude *a(τ)* and the phase *ϕ τ*ð Þ:

$$\dot{a} = -(2\lambda - \theta\_1)\frac{a}{2} - \frac{1}{2}\eta a \sum\_{j=1}^{N\_p} c\_j a\_0^{a\_j} \Omega^{a\_j - 1} \sin^2\left(\frac{j\pi}{N\_p + 1}\right) \sin\left(\frac{a\_j \pi}{2}\right) + \frac{1}{2\Omega} [G\_{0N} \sin \varphi - F\_{0N} \cos \varphi],$$

$$+ \frac{\pi \theta\_1^2 a}{8} [3\mathcal{S}\_\xi(2\Omega) + 2\mathcal{S}\_\xi(0)] + \frac{\pi \theta\_0^2}{2\Omega^2 a} \mathcal{S}\_\xi(\Omega) + \sqrt{\frac{\pi \theta\_0^2}{\Omega^2} \mathcal{S}\_\xi(\Omega) + \frac{\pi \theta\_1^2 a^2}{4} [\mathcal{S}\_\xi(2\Omega) + 2\mathcal{S}\_\xi(0)] \xi\_1(\tau)},\tag{17}$$

and

$$a\dot{\rho} = \frac{a}{2\Omega} \left[ 1 + 3\beta a\_0^2 - \Omega^2 + \frac{3}{4}\beta a^2 + \eta \sum\_{j=1}^{N\_p} \left( k\_j + c\_j a b\_0^{a\_j} \Omega^{a\_j} \cos\left(\frac{\alpha\_j \pi}{2}\right) \right) \sin^2\left(\frac{j\pi}{N\_p + 1}\right) \right],\tag{18}$$

$$+ \frac{1}{2\Omega} \left[ G\_{0N} \cos\rho + F\_{0N} \sin\rho \right] - \frac{\pi \theta\_1^2 a}{4} \Psi\_\xi(\mathfrak{Q}\Omega) + \sqrt{\frac{\pi \theta\_0^2}{\Omega^2} S\_\xi(\mathfrak{Q}) + \frac{\pi \theta\_1^2 a^2}{4} S\_\xi(\mathfrak{Q}\Omega)} \,\xi\_2(\tau). \tag{19}$$

Here *Sξ*ð Þ Ω and Ψ*ξ*ð Þ Ω are the cosine and sine power spectral density function, respectively [31]:

$$\begin{aligned} S\_{\xi}(\varOmega) &= \int\_{-\infty}^{+\infty} R(\zeta) \cos \left(\varOmega \tau\right) \mathrm{d}\zeta = 2 \int\_{0}^{+\infty} R(\zeta) \cos \left(\varOmega \tau\right) \mathrm{d}\zeta = 2 \int\_{-\infty}^{0} R(\zeta) \cos \left(\varOmega \tau\right) \mathrm{d}\zeta, \\\Psi\_{\xi}(\varOmega) &= 2 \int\_{0}^{+\infty} R(\zeta) \sin \left(\varOmega \tau\right) \mathrm{d}\zeta = -2 \int\_{-\infty}^{0} R(\zeta) \sin \left(\varOmega \tau\right) \mathrm{d}\zeta, \\\ \int\_{-\infty}^{+\infty} R(\zeta) \sin \left(\varOmega \tau\right) \mathrm{d}\zeta &= \mathbf{0}; \ R(\zeta) = \mathbf{E}[\xi(\tau)\xi(\tau + \zeta)]. \end{aligned} \tag{19}$$

In this work, *ξ τ*ð Þ is assumed to be an harmonic function with constant amplitude *σ<sup>i</sup>* and random phases *γiBi*ð Þþ *τ θi*. So, according to Refs. [31–33] the following model of *ξ τ*ð Þ has been chosen:

$$\xi(\tau) = \sum\_{i=1}^{m} \sigma\_i \cos \left[ a\_i \tau + \gamma\_i B\_i(\tau) + \theta\_i \right], \tag{20}$$

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

this model of the turbulent component of the wind *ξ τ*ð Þ amounts to a bounded or cosine-Wiener noise, whose spectral density is given by:

$$\Phi\_{\xi}(\boldsymbol{\alpha}) = \sum\_{i=1}^{m} \frac{\sigma\_i^2 \chi\_i^2 \left(\boldsymbol{\alpha}^2 + \boldsymbol{\alpha}\_i^2 + \chi\_i^4 / 4\right)}{4\pi \left[\left(\boldsymbol{\alpha}^2 - \boldsymbol{\alpha}\_i^2 - \chi\_i^4 / 4\right)^2 + \chi\_i^2 \boldsymbol{\alpha}^2\right]}.\tag{21}$$

The next sections of this chapter will presents the analytical developments that we have made in order to express the beam response as a function of the system parameters. Then, let's start with the case where the beam is subjected to the moving loads only.

#### **3.2 Analytical estimate of the beam response under moving loads only**

We first consider system (1) with only deterministic moving loads (*Fad*ð Þ¼ *x*, *t* 0) neglecting wind effects on the beam. If *ϑ*<sup>1</sup> ¼ *θ*<sup>0</sup> ¼ *θ*<sup>1</sup> ¼ 0, Eqs. (15) and (16) become:

$$\begin{split} \dot{a} &= -\lambda \mathfrak{a} - \frac{1}{2} \eta \mathfrak{a} \sum\_{j=1}^{N\_p} c\_j a\_0^{a\_j} \mathfrak{Q}^{a\_j - 1} \sin^2 \left( \frac{j \pi}{N\_p + 1} \right) \sin \left( \frac{\pi a\_j}{2} \right) \\ &+ \frac{1}{2 \Omega} [G\_{0N} \sin \phi - F\_{0N} \cos \phi], \end{split} \tag{22}$$

and

*a*\_ ¼ � lim *T*!∞

8

>>>>>>>><

>>>>>>>>:

*a* 2 � 1 2 *ηa* X*Np j*¼1

<sup>8</sup> <sup>½</sup>3*Sξ*ð Þþ <sup>2</sup><sup>Ω</sup> <sup>2</sup>*Sξ*ð Þ <sup>0</sup> � þ *πθ*<sup>2</sup>

<sup>1</sup> <sup>þ</sup> <sup>3</sup>*βa*<sup>2</sup>

<sup>2</sup><sup>Ω</sup> <sup>½</sup>*G*0*<sup>N</sup>* cos *<sup>φ</sup>* <sup>þ</sup> *<sup>F</sup>*0*<sup>N</sup>* sin *<sup>φ</sup>*� � *πθ*<sup>2</sup>

<sup>0</sup> � <sup>Ω</sup><sup>2</sup> <sup>þ</sup>

*a*\_ ¼ �ð Þ 2*λ* � *ϑ*<sup>1</sup>

þ *πθ*<sup>2</sup> 1*a*

and

*<sup>a</sup>φ*\_ <sup>¼</sup> *<sup>a</sup>* 2Ω

respectively [31]:

*Sξ*ð Þ¼ *Ω*

Ψ*ξ*ð Þ¼ *Ω* 2

þ ð∞

�∞

**18**

þ ð∞

�∞

þ ð∞

0

model of *ξ τ*ð Þ has been chosen:

þ 1 *aφ*\_ ¼ � lim *T*!∞

1 *T* ð *T*

0

1 *T* ð *T*

differential equations for the amplitude *a(τ)* and the phase *ϕ τ*ð Þ:

<sup>0</sup> <sup>Ω</sup>*<sup>α</sup> <sup>j</sup>*�<sup>1</sup> sin <sup>2</sup> *<sup>j</sup><sup>π</sup>*

*cjωα <sup>j</sup>*

0 2Ω<sup>2</sup> *a Sξ*ð Þþ Ω

> 3 <sup>4</sup> *<sup>β</sup>a*<sup>2</sup> <sup>þ</sup> *<sup>η</sup>*

*R*ð Þ*ζ* cosð Þ *Ωτ* d*ζ* ¼ 2

*R*ð Þ*ζ* sin ð Þ *Ωτ* d*ζ* ¼ �2

*R*ð Þ*ζ* sin ð Þ *Ωτ* d*ζ* ¼ 0; *R*ð Þ¼ *ζ* E½ � *ξ τ*ð Þ*ξ τ*ð Þ þ *ζ :*

*ξ τ*ð Þ¼ <sup>X</sup>*<sup>m</sup>*

*i*¼1

0

1

According to this method, one could select the time terminal T as *T* ¼ 2*π=*Ω in the case of periodic function (*M*1ð Þ *a*, *ψ* ), or *T* ¼ ∞ in the case of aperiodic one (*M*2ð Þ *a*, *ψ* ). Accordingly, one could obtain the following pair of first order

> *Np* þ 1 � �

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

s

*πθ*<sup>2</sup> 0

*R*ð Þ*ζ* cosð Þ *Ωτ* d*ζ* ¼ 2

*R*ð Þ*ζ* sin ð Þ *Ωτ* d*ζ*,

" # � �

*πθ*<sup>2</sup> 0 <sup>Ω</sup><sup>2</sup> *<sup>S</sup>ξ*ð Þþ <sup>Ω</sup> *πθ*<sup>2</sup>

s

X *Np*

*j*¼1

Ψ*ξ*ð Þþ 2Ω

Here *Sξ*ð Þ Ω and Ψ*ξ*ð Þ Ω are the cosine and sine power spectral density function,

1*a* 4

> þ ð∞

> > 0

ð 0

�∞

In this work, *ξ τ*ð Þ is assumed to be an harmonic function with constant amplitude *σ<sup>i</sup>* and random phases *γiBi*ð Þþ *τ θi*. So, according to Refs. [31–33] the following

1

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

<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>d<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>d<sup>ψ</sup>*

sin *<sup>α</sup> <sup>j</sup><sup>π</sup>* 2 � � þ 1

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

<sup>0</sup> <sup>Ω</sup>*<sup>α</sup> <sup>j</sup>* cos *<sup>α</sup> <sup>j</sup><sup>π</sup>*

<sup>Ω</sup><sup>2</sup> *<sup>S</sup>ξ*ð Þþ <sup>Ω</sup> *πθ*<sup>2</sup>

� � � �

<sup>4</sup> ½ � *<sup>S</sup>ξ*ð Þþ <sup>2</sup><sup>Ω</sup> <sup>2</sup>*Sξ*ð Þ <sup>0</sup>

2

1*a*2 4

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

ð 0

�∞

*σ<sup>i</sup>* cos *ωiτ* þ *γiBi*ð Þþ *τ θ<sup>i</sup>* ½ �, (20)

1*a*2

(16)

<sup>2</sup><sup>Ω</sup> ½ � *<sup>G</sup>*0*<sup>N</sup>* sin *<sup>φ</sup>* � *<sup>F</sup>*0*<sup>N</sup>* cos *<sup>φ</sup>*

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

*Sξ*ð Þ 2Ω

*R*ð Þ*ζ* cosð Þ *Ωτ* d*ζ*,

*Np* þ 1

*ξ*2ð Þ*τ :*

(18)

(19)

*ξ*1ð Þ*τ* ,

(17)

$$\begin{split} a\dot{\rho} &= \frac{a}{2\Omega} \left[ 1 - \Omega^2 + 3\beta a\_0^2 + \frac{3}{4}\beta a^2 + \eta \sum\_{j=1}^{N\_p} \left( k\_j + c\_j a\_0^{a\_j} \Omega^{a\_j} \cos\left(\frac{\pi \alpha\_j}{2}\right) \right) \sin^2\left(\frac{j\pi}{N\_p + 1}\right) \right] \\ &+ \frac{1}{2\Omega} [G\_{0N}\cos\rho + F\_{0N}\sin\rho]. \end{split} \tag{23}$$

By substituting *<sup>a</sup>* <sup>¼</sup> *<sup>A</sup>*, *<sup>ϕ</sup>* <sup>¼</sup> <sup>Φ</sup> and *<sup>a</sup>*\_ <sup>¼</sup> 0, *<sup>ϕ</sup>*\_ <sup>¼</sup> 0 in Eqs. (22) and (23), algebraic manipulations give for the steady-state vibrations of the system response *A* the following non-linear equation:

$$\frac{9}{16}\rho^2 A^6 - \frac{3}{2}\rho\Theta\_1(a\_j)A^4 + \left[\Theta\_1^2(a\_j) + \Theta\_2^2(a\_j)\right]A^2 = F\_{0N}^2 + G\_{0N}^2,\tag{24}$$

with

$$\begin{split} \Theta\_{1}(a\_{j}) &= \Omega^{2} - \mathbf{1} - 3\beta a\_{0}^{2} - \eta \sum\_{j=1}^{N\_{p}} \left( k\_{j} + c\_{j} \boldsymbol{\mu}\_{0}^{a\_{j}} \Omega^{a\_{j}} \cos\left(\frac{\pi a\_{j}}{2}\right) \right) \sin^{2}\left(\frac{j\pi}{N\_{p} + 1}\right), \\ \Theta\_{2}(a\_{j}) &= 2\Omega\lambda + \eta \sum\_{j=1}^{N\_{p}} c\_{j} \boldsymbol{\mu}\_{0}^{a\_{j}} \Omega^{a\_{j}} \sin\left(\frac{\pi a\_{j}}{2}\right) \sin^{2}\left(\frac{j\pi}{N\_{p} + 1}\right). \end{split} \tag{25}$$

The stability of the steady-state vibration of the system response is investigated by using the method of Andronov and Witt [34] associated to the Routh-Hurwitz criterion [35]. Thus, the steady-state response is asymptotically stable if Eq. (26) is satisfied and unstable if Eq. (27) is satisfied:

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

$$
\left(\frac{\Theta\_2(a\_j)}{2\Omega}\right)^2 + \frac{1}{4\Omega} \left[\frac{3\beta}{4}A^2 - \Theta\_1(a\_j)\right] \times \left[\frac{9\beta}{4}A^2 - \Theta\_1(a\_j)\right] > 0,\tag{26}
$$

Above *N* is a normalization constant that guarantees Ð

**3.4 Approximate solution of the beam responses subjected**

*j*¼1

<sup>2</sup><sup>Ω</sup> ½ � *<sup>G</sup>*0*<sup>N</sup>* sin *<sup>φ</sup>* � *<sup>F</sup>*0*<sup>N</sup>* cos *<sup>φ</sup> <sup>d</sup><sup>τ</sup>* <sup>þ</sup> ffiffiffiffiffi

X *Np*

*j*¼1

*<sup>∂</sup><sup>a</sup>* ð Þ� *<sup>a</sup>*1*P a*ð Þ , *<sup>ϕ</sup>*, *<sup>τ</sup> <sup>∂</sup>*

<sup>2</sup>Ω*<sup>a</sup>* ½ � *<sup>G</sup>*0*<sup>N</sup>* cos *<sup>φ</sup>* <sup>þ</sup> *<sup>F</sup>*0*<sup>N</sup>* sin *<sup>φ</sup> <sup>d</sup><sup>τ</sup>* <sup>þ</sup>

*cjω<sup>α</sup> <sup>j</sup>*

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

<sup>0</sup> *Ωα <sup>j</sup>*�<sup>1</sup> sin <sup>2</sup> *<sup>j</sup><sup>π</sup>*

<sup>0</sup> *<sup>Ω</sup><sup>α</sup> <sup>j</sup>* cos *<sup>α</sup> <sup>j</sup><sup>π</sup>*

*Np* þ 1 � � sin *<sup>α</sup> <sup>j</sup><sup>π</sup>*

" #

Applying the solution procedure proposed by Huang *et al.* [36], one obtains the

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

and stochastic wind loads?

**to the both moving loads**

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

Thus, Eqs. (22) and (23) become:

*a* 2 � 1 2 *ηa* X *Np*

<sup>1</sup> � <sup>Ω</sup><sup>2</sup> <sup>þ</sup> *<sup>η</sup>*

*a* ¼ �ð Þ 2*λ* � *ϑ*<sup>1</sup>

þ 1

and

*<sup>d</sup><sup>φ</sup>* <sup>¼</sup> <sup>1</sup> 2Ω

> þ 1

*<sup>∂</sup>P a*ð Þ , *<sup>ϕ</sup>*, *<sup>τ</sup>*

where

*a*<sup>1</sup> ¼ �ð Þ 2*λ* � *ϑ*<sup>1</sup>

*<sup>a</sup>*<sup>2</sup> <sup>¼</sup> <sup>1</sup> 2Ω

*b*<sup>11</sup> ¼ *Γ*<sup>0</sup> *<sup>b</sup>*<sup>22</sup> <sup>¼</sup> *<sup>Γ</sup>*<sup>0</sup> *a*2 *:*

*Ps*ð Þ¼ *a*, *ϕ N*<sup>0</sup>

**21**

Itô Eqs. (33) and (34) is

*<sup>∂</sup><sup>τ</sup>* ¼ � *<sup>∂</sup>*

þ 1 2 *∂*2

*a* 2 � 1 2 *ηa* X*Np j*¼1 *cjωα <sup>j</sup>*

> X*Np j*¼1

following exact stationary solution

*Γ*0 2 *Γ*0

*a* exp

*<sup>k</sup> <sup>j</sup>* <sup>þ</sup> *cjωα <sup>j</sup>*

*<sup>a</sup>*<sup>2</sup> � *<sup>a</sup>* Ω Γ<sup>2</sup> <sup>0</sup> <sup>þ</sup> *<sup>d</sup>*<sup>2</sup> 0

<sup>1</sup> � *<sup>Ω</sup>*<sup>2</sup> <sup>þ</sup> *<sup>η</sup>*

What about the case where the beam is subjected to the both moving vehicles

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

Finally, the case where the beam is subjected to the series of lumped loads and the wind actions is investigated. For the analytical purposes, we assume that the beam is linear and it is submitted to only the additive effects of the wind loads.

<sup>0</sup> <sup>Ω</sup>*<sup>α</sup> <sup>j</sup>*�<sup>1</sup> sin <sup>2</sup> *<sup>j</sup><sup>π</sup>*

<sup>0</sup> <sup>Ω</sup>*<sup>α</sup> <sup>j</sup>* cos *<sup>α</sup> <sup>j</sup><sup>π</sup>*

ffiffiffiffiffiffi Γ0 <sup>p</sup> *dW*2ð Þ*<sup>τ</sup> :*

*<sup>∂</sup><sup>ϕ</sup>* ð Þþ *<sup>a</sup>*2*P a*ð Þ , *<sup>ϕ</sup>*, *<sup>τ</sup>* <sup>1</sup>

*<sup>∂</sup>ϕ*<sup>2</sup> *<sup>b</sup>*22*P a*ð Þ , *<sup>ϕ</sup>*, *<sup>τ</sup>* � �, (35)

2 � � <sup>þ</sup>

*Np* þ 1 � �

2 *∂*2

1

þ 1 *a*

� � ½ � ð Þ *<sup>d</sup>*0*G*0*<sup>N</sup>* <sup>þ</sup> *<sup>F</sup>*0*<sup>N</sup>Γ*<sup>0</sup> cos *<sup>ϕ</sup>* <sup>þ</sup> ð Þ *<sup>d</sup>*0*F*0*<sup>N</sup>* � *<sup>G</sup>*0*<sup>N</sup>Γ*<sup>0</sup> sin *<sup>ϕ</sup>*

( )

" # � � *<sup>d</sup><sup>τ</sup>*

1 *a*

The averaged Fokker-Planck-Kolmogorov equation associated with the previous

Γ0 <sup>p</sup> *dW*1ð Þ*<sup>τ</sup>* ,

" #

*Np* þ 1

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

� � sin *<sup>α</sup> <sup>j</sup><sup>π</sup>*

2 � � <sup>þ</sup>

*Np* þ 1

*<sup>∂</sup>a*<sup>2</sup> *<sup>b</sup>*11*P a*ð Þ , *<sup>ϕ</sup>*, *<sup>τ</sup>* � �

<sup>2</sup>*<sup>Ω</sup>* <sup>½</sup>*G*0*<sup>N</sup>* sin *<sup>ϕ</sup>* � *<sup>F</sup>*0*<sup>N</sup>* cos *<sup>ϕ</sup>*� þ <sup>Γ</sup><sup>0</sup>

½ � *G*0*<sup>N</sup>* cos *ϕ* þ *F*0*<sup>N</sup>* sin *ϕ*

Γ0 *a*

*dτ*

(33)

(34)

*a*

(36)

(37)

∞ 0

*Ps*ð Þ *a* d*a* ¼ 1.

$$
\left(\frac{\Theta\_2(a\_j)}{2\Omega}\right)^2 + \frac{1}{4\Omega} \left[\frac{3\beta}{4}A^2 - \Theta\_1(a\_j)\right] \times \left[\frac{9\beta}{4}A^2 - \Theta\_1(a\_j)\right] < 0. \tag{27}
$$

The trivial solution of Eq. (15) is *a*<sup>0</sup> ¼ 0.

What about the case where the beam is subjected to the stochastic wind loads?

#### **3.3 Approximate solution of the beam response subjected to wind loads only**

In this case (*Fad*ð Þ *x*, *t* 6¼ 0) and *F*0*<sup>N</sup>* ¼ *G*0*<sup>N</sup>* ¼ 0, Eqs. (22) and (23) become:

$$\begin{split} da &= \left\{-\left(2\dot{\iota} - \theta\_{1}\right)\frac{a}{2} - \frac{1}{2}\eta a \sum\_{j=1}^{N\_{p}} c\_{j} \boldsymbol{\mu}\_{0}^{a\_{j}} \boldsymbol{\Omega}^{a\_{j}-1} \sin^{2}\left(\frac{j\pi}{N\_{p}+1}\right) \sin\left(\frac{\alpha\_{1}\pi}{2}\right) + \frac{\pi \theta\_{1}^{2} a}{8} \left[3\boldsymbol{\Sigma}\_{\boldsymbol{\xi}}(2\boldsymbol{\Omega}) + 2\boldsymbol{\Sigma}\_{\boldsymbol{\xi}}(0)\right] \right\} d\tau \\ &+ \frac{\pi \theta\_{0}^{2}}{2\boldsymbol{\Omega}^{2}a} \boldsymbol{\mathcal{S}}\_{\boldsymbol{\xi}}(\boldsymbol{\Omega}) d\tau + \sqrt{\frac{\pi \theta\_{0}^{2}}{\boldsymbol{\Omega}^{2}} \boldsymbol{\mathcal{S}}\_{\boldsymbol{\xi}}(\boldsymbol{\Omega}) + \frac{\pi \theta\_{1}^{2} a^{2}}{4} \left[\boldsymbol{\mathcal{S}}\_{\boldsymbol{\xi}}(2\boldsymbol{\Omega}) + 2\boldsymbol{\mathcal{S}}\_{\boldsymbol{\xi}}(0)\right] d\mathcal{W}\_{1}(\tau)}, \end{split} \tag{28}$$

and

$$d\rho = \frac{1}{2\Omega} \left\{ 1 + 3\theta a\_0^2 - \Omega^2 + \frac{3}{4}\theta a^2 + \eta \sum\_{j=1}^{N\_p} \left( k\_j + c\_j \rho a\_0^{a\_j} \Omega^{a\_j} \cos\left(\frac{\alpha\_j \pi}{2}\right) \right) \sin^2\left(\frac{j\pi}{N\_p + 1}\right) \right\} d\tau. \tag{29}$$

$$ -\frac{\pi \theta\_1^2}{4} \Psi\_\xi(2\Omega) d\tau \right. \tag{20} \\ = \frac{1}{\Omega} \frac{\theta\_0^2}{\omega^2} S\_\xi(\Omega) + \frac{\pi \theta\_1^2 a^2}{4} S\_\xi(2\Omega) \, dW\_2(\tau). \tag{21}$$

Here *W*1ð Þ*τ* and *W*2ð Þ*τ* are independent normalized Weiner processes. In order to evaluate the effects of wind parameters on the system response, we derive an evolution equation for the Probability Density Function (PDF) of the variable amplitude *a*ð Þ*τ* . The Fokker-Planck equation corresponding to the Langevin (Eq. (28)) reads:

$$\begin{split} \frac{\partial P(a,\tau)}{\partial \tau} &= -\frac{\partial}{\partial a} \left[ \left( -(2\lambda - \theta\_{1})\frac{a}{2} - \frac{1}{2}\eta a \sum\_{j=1}^{N\_{\text{f}}} c\_{j} \boldsymbol{\rho}\_{0}^{a} \boldsymbol{\mathscr{Q}}^{a} \boldsymbol{\varepsilon}^{-1} \sin^{2} \left( \frac{j\pi}{N\_{\text{f}} + 1} \right) \sin \left( \frac{a\boldsymbol{\rho}\_{r}\pi}{2} \right) + \frac{\pi \theta\_{0}^{2}}{2\Omega^{2}a} S\_{\text{f}}(\boldsymbol{\Omega}) \right) \boldsymbol{P}(a,\tau) \right] \\ &- \frac{\partial}{\partial a} \left[ \frac{\pi \theta\_{1}^{2} a}{8} [3S\_{\text{f}}(2\Omega) + 2S\_{\text{f}}(\boldsymbol{0})] \boldsymbol{P}(a,\tau) \right] + \frac{1}{2} \left( \frac{\pi \theta\_{0}^{2}}{\Omega^{2}} S\_{\text{f}}(\boldsymbol{\Omega}) + \frac{\pi \theta\_{1}^{2} a^{2}}{4} [S\_{\text{f}}(2\Omega) + 2S\_{\text{f}}(\boldsymbol{0})] \right) \frac{\partial^{2} \boldsymbol{P}(a,\tau)}{\partial a^{2}}. \end{split} \tag{30}$$

In the stationary case, *<sup>∂</sup>P a*ð Þ , *<sup>τ</sup> <sup>∂</sup><sup>τ</sup>* ¼ 0, the solution of Eq. (30) is:

$$P\_s(a) = \text{Na} \left(\Gamma\_0 + a^2 \Gamma\_1\right)^{-(Q+1)},\tag{31}$$

where

$$\begin{aligned} \Gamma\_{0} &= \frac{\pi \theta\_{0}^{2}}{\Omega^{2}} S\_{\xi}(\mathfrak{A}), \Gamma\_{1} = \frac{\pi \theta\_{1}^{2}}{4} [S\_{\xi}(2\mathfrak{A}) + 2S\_{\xi}(\mathfrak{O})], \ Q = \frac{\Gamma\_{1} - 2\Gamma\_{2}}{2\Gamma\_{1}}, \\\Gamma\_{2} &= -\frac{1}{2}(2\lambda - \mathfrak{A}\_{1}) - \frac{1}{2}\eta \sum\_{j=1}^{N\_{p}} c\_{j} \mathfrak{a}\_{0}^{a\_{j}} \mathcal{Q}^{a\_{j} - 1} \sin^{2} \left(\frac{j\pi}{N\_{p} + 1}\right) \sin \left(\frac{\pi a\_{j}}{2}\right) + \frac{\pi \theta\_{1}^{2}}{8} [3\mathcal{S}\_{\xi}(2\mathfrak{A}) + 2\mathcal{S}\_{\xi}(\mathfrak{O})]. \end{aligned} \tag{32}$$

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

Above *N* is a normalization constant that guarantees Ð ∞ 0 *Ps*ð Þ *a* d*a* ¼ 1.

What about the case where the beam is subjected to the both moving vehicles and stochastic wind loads?
