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

comparison between the results from the mathematical analysis (curve with dotted line) and the results obtained from numerical simulation of Eq. (9) using Eq. (39). The match between the results shows a good level of precision of the approximation made in obtaining Eq. (24). This figure also reveals that the vibration amplitude of the beam decreases and the resonance frequency of the system increases as the number of bearings increases. **Figure 2(b)**, shows the effect of loads number on the beam response. It is observed that as the value of *Nv* increases, the amplitude of vibration at the resonant state merely increases.

Looking at the effects of the order of the fractional derivative *α* on the amplitude of the beam, we obtain the graph of **Figure 3**. Small value of *α* leads to large value of the maximum vibration amplitude. It is also clearly shown that the system is more stable for the highest order of the derivative. The multivalued solution appears for small order and disappears progressively as the order increases. The resonance (a peak of the amplitude) appears as the parameter *k*<sup>0</sup> <sup>0</sup> increases, see **Figure 3(a)-(d)**. The good match between the analytical and the numerical results gives a validation of the approximations made.

Also, the stochastic analysis has allowed to estimate the probabilistic distribution as a consequence of the wind random effects. The beam response, and more specifically the stationary probability density function *Ps*ð Þ *a* of its amplitude *a*, can also be retrieved (**Figures 4** and **5**). This type of analysis indicates that as the additive wind turbulence parameter increases, the peak value of the probability density function

**Figure 3.**

*Effect of the fractional order on the amplitude of the beam A when the dimensionless stiffness k*<sup>0</sup> *<sup>o</sup> varying: Analytical curves (-), numerical curve (o). All The parameters are given in [21].*

where *N*<sup>0</sup> is a normalization constant and

*η* 2 X *Np*

*j*¼1

X *Np*

*j*¼1

*Dα <sup>τ</sup> χ τ<sup>n</sup> <sup>f</sup>* h i � �

*Cα*

where *h* is the integration step and the coefficients *C<sup>α</sup>*

<sup>0</sup> <sup>¼</sup> 1, *<sup>C</sup><sup>α</sup>*

*cjω<sup>α</sup> <sup>j</sup>*

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

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

<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>*

All the parameters concerning the chosen models of beam, of foundation and of the aerodynamic force are presented in Ref. [21]. Theses parameters clearly help to calculated the dimensionless coefficients defined in Eq. (7). It is well know that the validation of the results obtained through analytical investigation is guarantee by the perfect match with the results obtained through numerical simulations. thus, The numerical scheme used in this chapter is based on the Grunwald-Letnikov definition of the fractional order derivative Eq. (39). [37–40] and the Newton-

≈*h*�*<sup>α</sup>*X*<sup>n</sup> <sup>f</sup>*

*l*¼0 *Cα <sup>l</sup> χ τ<sup>n</sup> <sup>f</sup>* �*<sup>l</sup>* � �

*<sup>l</sup>* <sup>¼</sup> <sup>1</sup> � <sup>1</sup> <sup>þ</sup> *<sup>α</sup>*

Now we display in some figures the effects of the main parameters of the proposed model. For example, **Figure 2(a)** shows the effect of the number of the bearings on the amplitude of vibration of the beam. This graph also shows a

*Effects of the number of the bearings Np (a) and number of moving loads Nv (b) on the amplitude response of the beam when driving frequency Ω: Analytical curves (-), numerical curves (o). All The parameters are given*

*l* � �

*Cα*

*Np* þ 1 � �

2

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

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

*Np* þ 1

, *<sup>d</sup>*<sup>0</sup> ¼ � *<sup>Γ</sup>*0*Γ*<sup>3</sup> *Γ*0 2 ,

*:*

, (39)

*<sup>l</sup>* satisfy the following

*<sup>l</sup>*�<sup>1</sup>*:* (40)

(38)

ð Þ� 2*λ* � *ϑ*<sup>1</sup>

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

**4. Numerical analysis of the model**

Leipnik algorithm [37, 38]

recursive relations:

**Figure 2.**

*in [21].*

**22**

*Γ*0 <sup>2</sup> ¼ � <sup>1</sup> 2

*<sup>Γ</sup>*<sup>3</sup> <sup>¼</sup> <sup>1</sup> 2*Ω*

#### **Figure 4.**

*Stationary probability distribution function as function of the amplitude for different values of the additive wind turbulence parameter. Analytical curves (-), numerical curve (o). All The parameters are given in [21].*

decreases and progressively shifts toward larger amplitude values, while the average center position stays in the same position. Thus, the additive (*θ*<sup>0</sup> ) and parametric (*θ*<sup>1</sup> ) wind turbulence decreases the chance for the beam to quickly reach the amplitude resonance. It is also demonstrated that the PDF has only one maximum situated in the vicinity of *am* ¼ 0*:*2.

We have plotted curves **Figure 6(a)** and **(b)** that presents the stationary probability distribution function *Ps*ð Þ *a*, *ϕ* versus the amplitude *a* and the phase *ϕ*. This graph just confirm the results obtained in **Figures 4** and **5** and the highest value of the PDF is more visible.

**Figure 7** presents the times histories of the maximum vibration of the beam. The case where the beam is subjected to moving loads (*a*), to wind actions (*b*) and to the both wind and loads (*c*).
