**Author details**

From the geometry of the deformation, and using Hooke's law *σ<sup>x</sup>* ¼ *Eεx*, one can

*w x*ð Þ , *t <sup>∂</sup>x*<sup>2</sup> <sup>1</sup> � <sup>3</sup>

> *∂*2 *w x*ð Þ , *t ∂x*<sup>2</sup>

*<sup>∂</sup>w x*ð Þ , *<sup>t</sup> ∂x* � �<sup>2</sup>

> *<sup>∂</sup>w x*ð Þ , *<sup>t</sup> <sup>∂</sup><sup>t</sup>* � <sup>3</sup> 2 *EI <sup>∂</sup>*<sup>2</sup> *∂x*<sup>2</sup>

d*x* þ *QF*ð Þ¼ *x*, *t Fad*ð Þþ *x*, *t f x*ð Þ , *t*

*εiδ x* � *xi*ð Þ *t* � *ti* ½ �

� �*w x*ð Þ , *<sup>t</sup> <sup>δ</sup> <sup>x</sup>* � *jL*

*<sup>k</sup> <sup>j</sup>* <sup>þ</sup> *cjD<sup>α</sup> <sup>j</sup> t*

*<sup>∂</sup>w x*ð Þ , *<sup>t</sup>*

2

� �<sup>2</sup> " #

*<sup>∂</sup>w x*ð Þ , *<sup>t</sup> ∂x* � �<sup>2</sup> " #

> *<sup>∂</sup>w x*ð Þ , *<sup>t</sup> ∂x*

<sup>þ</sup> *<sup>O</sup> <sup>∂</sup>w x*ð Þ , *<sup>t</sup> ∂x* � �2 !

*:*

(45)

*R* � � up to

*<sup>∂</sup>w x*ð Þ , *<sup>t</sup> ∂x*

(47)

,

� �<sup>2</sup> " #

<sup>þ</sup> *<sup>O</sup> <sup>∂</sup><sup>w</sup> ∂x* � �2 !

d*x* (46)

*<sup>∂</sup><sup>t</sup>* ), putting Eq. (44), Eq. (45)

*∂*2 *w x*ð Þ , *t ∂x*<sup>2</sup>

*NP* þ 1 � �

show that (see reference [19]):

*∂*2 *w x*ð Þ , *t ∂x*<sup>2</sup>

<sup>1</sup> <sup>þ</sup> *<sup>∂</sup>w x*ð Þ , *<sup>t</sup> ∂x* � �<sup>2</sup> � �<sup>3</sup>

details of their derivation in Ref.[19]).

<sup>≈</sup> � *EI <sup>∂</sup>*<sup>4</sup>*w x*ð Þ , *<sup>t</sup>*

Finally taking into account the dissipation (*μ*

*<sup>∂</sup>x*<sup>2</sup>*∂t*<sup>2</sup> <sup>þ</sup> *EI <sup>∂</sup>*<sup>4</sup>*w x*ð Þ , *<sup>t</sup>*

*<sup>∂</sup>*<sup>4</sup>*w x*ð Þ , *<sup>t</sup>*

*<sup>∂</sup>w x*ð Þ , *<sup>t</sup> ∂x* � �<sup>2</sup>

*f x*ð Þ¼ , *t P*

*QF*ð Þ¼ *<sup>x</sup>*, *<sup>t</sup>* <sup>X</sup>

X *N*�1

*i*¼0

*NP*

*j*¼1

ð *L*

0

2

*∂x*<sup>4</sup> þ

<sup>≈</sup> � *EI <sup>∂</sup>*<sup>2</sup>

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

3 2 *EI <sup>∂</sup>*<sup>2</sup> *∂x*<sup>2</sup>

*<sup>T</sup>* <sup>¼</sup> *ES* 2*L* ð *L*

where the Taylor expansion of the inverse of the radius of curvature <sup>1</sup>

0

and Eq. (46) together gives the new desired result (Eq. (1) of the manuscript)

*<sup>∂</sup>x*<sup>4</sup> <sup>þ</sup> *<sup>μ</sup>*

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

*<sup>M</sup>* <sup>¼</sup> *EI*

*ρS ∂*2 *w x*ð Þ , *t <sup>∂</sup>t*<sup>2</sup> � *<sup>ρ</sup><sup>I</sup>*

� *ES* 2*L ∂*2 *w x*ð Þ , *t ∂x*<sup>2</sup>

**28**

where:

*<sup>R</sup>* ¼ �*EI*

Lionel Merveil Anague Tabejieu<sup>1</sup> \*, Blaise Roméo Nana Nbendjo<sup>2</sup> and Giovanni Filatrella<sup>3</sup>

1 Department of Mechanical Engineering, National Higher Polytechnic School of Douala, University of Douala, Douala, Cameroon

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

© 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, provided the original work is properly cited.
