**Appendix A**

the first mode contains the essential information, and using the stochastic averaging method. The analysis has therefore some limitations, namely the limited values of parameters that have been explored, and to have retained the first mode only in the Galerkin's method. Also, the vehicles' train has been (over?) simplified as a simple periodic drive. With this limitations in mind, let us summarize the main findings. To start with, in this framework it is possible to investigate how the main parameters of the moving loads and of the bearings affect the beam response, and especially how the driving frequency, the loads number, the stiffness coefficient, fractional-order of the viscosity term and the number of bearings affect the

*Vibration amplitude of the beam χ(τ) as function of the time τ. All The parameters are given in [21].*

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

dynamic behaviour of the beam. The resonance phenomenon and the stability in the beam system strongly depends on the stiffness and fractional-order of the derivative term of the viscous properties of the bearings. There are a number of quantitative results that are worth mentioning. Firstly, as the number of moving loads increases, the resonant amplitude of the beam increases as well. Secondly, it has been established that as the number of bearings increases, the resonant amplitude decreases and, more importantly, shifts toward larger frequency values. Thirdly, the system response becomes more stable as the order of the derivative increases, for the multivalued solution only appears for the smallest order and quickly disappears as the order increases. All the above results are a consequence of the analysis of the oscillations. However, 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 of its amplitude, can also be retrieved. This type of analysis indicates that as the additive wind turbulence parameter increases, the peak value of the probability density function decreases and progressively shifts toward larger amplitude values, while the average center position stays in the same position. Thus, the additive and parametric wind turbulence decreases the chance for the beam to quickly reach the

Numerical simulations have confirmed these predictions. This behavior is depicted in **Figures 2**–**5**, which have practical implications, on which we would like to comment. To make an example, the beam system frequency *Ω* displays the bridge response as the vehicles speed changes, see Eq. (7). In principle it could be possible to avoid large oscillations by controlling the speed of the freight vehicles,

amplitude resonance.

**26**

**Figure 7.**

To deal with the modelling, let us consider the dynamic equilibrium of a beam element of length *dx*; *w* ¼ *w x*ð Þ , *t* and *θ* ¼ *θ*ð Þ *x*, *t* be the transversal displacement and the angle of rotation of the beam element respectively. We denote the internal bending moment by *M*, the internal shear force by *V*, the inplane tension due to the inplane strain, issue of the assumed negligible longitudinal displacement of the beam by *T*, the foundation-beam interaction force (per unit length of the beam's axis) by *QF*ð Þ *x*, *t* and the external distributed loading by *Fad*ð Þ *x*, *t* and *f x*ð Þ , *t* .

Setting the vertical forces on the element equal to the mass times acceleration gives:

$$\frac{\partial V}{\partial \mathbf{x}} = Q\_F(\mathbf{x}, t) - f(\mathbf{x}, t) - F\_{ad}(\mathbf{x}, t) + \rho \mathbf{S} \frac{\partial^2 w(\mathbf{x}, t)}{\partial t^2} \tag{41}$$

While summing moments produces:

$$\frac{\partial M}{\partial \mathbf{x}} = V - \rho I \frac{\partial^2 \theta(\mathbf{x}, t)}{\partial t^2} - T \frac{\partial w(\mathbf{x}, t)}{\partial \mathbf{x}} \tag{42}$$

For small rotation *<sup>θ</sup>*ð Þ *<sup>x</sup>*, *<sup>t</sup>* <sup>≈</sup>*<sup>∂</sup>w x*ð Þ , *<sup>t</sup> <sup>∂</sup><sup>x</sup>* , Eq. (42) becomes:

$$\frac{\partial M}{\partial \mathbf{x}} = V - \rho I \frac{\partial^3 w(\mathbf{x}, t)}{\partial t^2 \partial \mathbf{x}} - T \frac{\partial w(\mathbf{x}, t)}{\partial \mathbf{x}} \tag{43}$$

Combining Eq. (41) and Eq. (43) then yields:

$$\frac{\partial^2 M}{\partial \mathbf{x}^2} = Q\_F(\mathbf{x}, t) - f(\mathbf{x}, t) - F\_{dd}(\mathbf{x}, t) + \rho \mathbf{S} \frac{\partial^2 w(\mathbf{x}, t)}{\partial t^2} - \rho I \frac{\partial^4 w(\mathbf{x}, t)}{\partial t^2 \partial \mathbf{x}^2} - T \frac{\partial^2 w(\mathbf{x}, t)}{\partial \mathbf{x}^2} \tag{44}$$

From the geometry of the deformation, and using Hooke's law *σ<sup>x</sup>* ¼ *Eεx*, one can show that (see reference [19]):

$$M = \frac{EI}{R} = -EI\frac{\frac{\partial^2 w(\mathbf{x}, t)}{\partial \mathbf{x}^2}}{\left[\mathbf{1} + \left(\frac{\partial w(\mathbf{x}, t)}{\partial \mathbf{x}}\right)^2\right]^{\frac{2}{3}}} \approx -EI\frac{\partial^2 w(\mathbf{x}, t)}{\partial \mathbf{x}^2} \left[\mathbf{1} - \frac{3}{2} \left(\frac{\partial w(\mathbf{x}, t)}{\partial \mathbf{x}}\right)^2\right] + O\left(\left(\frac{\partial w(\mathbf{x}, t)}{\partial \mathbf{x}}\right)^2\right) \tag{4.5}$$

$$\approx -EI\frac{\partial^4 w(\mathbf{x}, t)}{\partial \mathbf{x}^4} + \frac{3}{2}EI\frac{\partial^2}{\partial \mathbf{x}^2} \left[\frac{\partial^2 w(\mathbf{x}, t)}{\partial \mathbf{x}^2} \left(\frac{\partial w(\mathbf{x}, t)}{\partial \mathbf{x}}\right)^2\right] + O\left(\left(\frac{\partial w}{\partial \mathbf{x}}\right)^2\right) \tag{4.6}$$

where the Taylor expansion of the inverse of the radius of curvature <sup>1</sup> *R* � � up to the second order is carried out. According to the assumed negligible longitudinal displacement of the beam, the tension in the beam *T* can be determined as (see the details of their derivation in Ref.[19]).

$$T = \frac{ES}{2L} \int\_0^L \left(\frac{\partial w(\varkappa, t)}{\partial \varkappa}\right)^2 d\varkappa \tag{46}$$

,

**Author details**

**29**

Lionel Merveil Anague Tabejieu<sup>1</sup>

Douala, University of Douala, Douala, Cameroon

provided the original work is properly cited.

and Giovanni Filatrella<sup>3</sup>

\*, Blaise Roméo Nana Nbendjo<sup>2</sup>

1 Department of Mechanical Engineering, National Higher Polytechnic School of

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

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

© 2021 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,

2 Laboratory of Modelling and Simulation in Engineering, Biomimetics and Prototypes, Faculty of Science, University of Yaounde I, Yaounde, Cameroon

3 University of Sannio, Via Port' Arsa 11, Benevento, Sannio, Italy

\*Address all correspondence to: lmanaguetabejieu@gmail.com

Finally taking into account the dissipation (*μ <sup>∂</sup>w x*ð Þ , *<sup>t</sup> <sup>∂</sup><sup>t</sup>* ), putting Eq. (44), Eq. (45) and Eq. (46) together gives the new desired result (Eq. (1) of the manuscript)

$$
\rho S \frac{\partial^2 w(\mathbf{x}, t)}{\partial t^2} - \rho I \frac{\partial^4 w(\mathbf{x}, t)}{\partial \mathbf{x}^2 \partial t^2} + EI \frac{\partial^4 w(\mathbf{x}, t)}{\partial \mathbf{x}^4} + \mu \frac{\partial w(\mathbf{x}, t)}{\partial t} - \frac{3}{2} EI \frac{\partial^2}{\partial \mathbf{x}^2} \left[ \frac{\partial^2 w(\mathbf{x}, t)}{\partial \mathbf{x}^2} \left( \frac{\partial w(\mathbf{x}, t)}{\partial \mathbf{x}} \right)^2 \right]
$$

$$
$$

where:

$$\begin{aligned} f(\mathbf{x}, t) &= P \sum\_{i=0}^{N-1} e\_i \delta[\mathbf{x} - \mathbf{x}\_i(t - t\_i)] \\ Q\_F(\mathbf{x}, t) &= \sum\_{j=1}^{N\_P} (k\_j + c\_j D\_t^{a\_j}) w(\mathbf{x}, t) \delta \left[ \mathbf{x} - \frac{jL}{N\_P + 1} \right] \end{aligned}$$

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