**3.4 Approximate solution of the beam responses subjected to the both moving loads**

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

$$\begin{split} a &= \left[ - (2\lambda - \theta\_1) \frac{a}{2} - \frac{1}{2} \eta a \sum\_{j=1}^{N\_p} c\_j a\_0^{a\_j} \Omega^{a\_j - 1} \sin^2 \left( \frac{j \pi}{N\_p + 1} \right) \sin \left( \frac{a\_j \pi}{2} \right) + \frac{\Gamma\_0}{a} \right] d\tau \\ &+ \frac{1}{2\Omega} [G\_{0N} \sin \varphi - F\_{0N} \cos \varphi] d\tau + \sqrt{\Gamma\_0} dW\_1(\tau), \end{split} \tag{33}$$

and

*Θ*<sup>2</sup> *α <sup>j</sup>* � � 2Ω � �<sup>2</sup>

*Θ*<sup>2</sup> *α <sup>j</sup>* � � 2Ω � �<sup>2</sup>

*da* ¼ �ð Þ 2*λ* � *ϑ*<sup>1</sup>

and

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

*<sup>∂</sup>P a*ð Þ , *<sup>τ</sup> <sup>∂</sup><sup>τ</sup>* ¼ � *<sup>∂</sup>*

> � *∂ ∂a πθ*<sup>2</sup> 1*a*

where

*<sup>Γ</sup>*<sup>0</sup> <sup>¼</sup> *πθ*<sup>2</sup> 0

*<sup>Γ</sup>*<sup>2</sup> ¼ � <sup>1</sup> 2

**20**

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

þ *πθ*<sup>2</sup> 0 2Ω<sup>2</sup> *a*

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

� *πθ*<sup>2</sup> 1 4

þ 1 4Ω

þ 1 4Ω

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

*Sξ*ð Þ Ω *dτ* þ

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

Ψ*ξ*ð Þ 2Ω *dτ* þ

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

In the stationary case, *<sup>∂</sup>P a*ð Þ , *<sup>τ</sup>*

*<sup>Ω</sup>*<sup>2</sup> *<sup>S</sup>ξ*ð Þ *<sup>Ω</sup>* , *<sup>Γ</sup>*<sup>1</sup> <sup>¼</sup> *πθ*<sup>2</sup>

ð Þ� 2*λ* � *ϑ*<sup>1</sup>

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

<sup>8</sup> ½ � <sup>3</sup>*Sξ*ð Þþ <sup>2</sup>*<sup>Ω</sup>* <sup>2</sup>*Sξ*ð Þ <sup>0</sup> *P a*ð Þ , *<sup>τ</sup>* � �

1

*j*¼1

*cjωα <sup>j</sup>*

1 2 *η* X *Np*

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

s

3*β* 4

> 3*β* 4

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

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

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

� 9*β* 4

� 9*β* 4

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:

*Np* þ 1 � �

1*a*2

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

1*a*2 4

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

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:

<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>∂</sup><sup>τ</sup>* ¼ 0, the solution of Eq. (30) is:

*Γ*1

*Np* þ 1 � �

2*Γ*<sup>1</sup>

,

sin *πα <sup>j</sup>* 2 � �

!

" #

( ) � �

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

( )

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

<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>

� � � �

*Sξ*ð Þ 2Ω

*Np* þ 1 � �

1*a*2

� � *∂*<sup>2</sup>

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

2

*dW*2ð Þ*τ :*

sin *<sup>α</sup> <sup>j</sup><sup>π</sup>* 2 � � þ *πθ*<sup>2</sup> 0 2*Ω*<sup>2</sup> *a Sξ*ð Þ *Ω*

� ��ð Þ *<sup>Q</sup>*þ<sup>1</sup> , (31)

þ *πθ*<sup>2</sup> 1

<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>

<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>*

X *Np*

*j*¼1

þ 1 2 *πθ*<sup>2</sup> 0 *<sup>Ω</sup>*<sup>2</sup> *<sup>S</sup>ξ*ð Þþ *<sup>Ω</sup> πθ*<sup>2</sup>

*Ps*ð Þ¼ *<sup>a</sup> Na <sup>Γ</sup>*<sup>0</sup> <sup>þ</sup> *<sup>a</sup>*<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> , *<sup>Q</sup>* <sup>¼</sup> *<sup>Γ</sup>*<sup>1</sup> � <sup>2</sup>*Γ*<sup>2</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>a</sup>*<sup>2</sup> *<sup>S</sup>ξ*ð Þþ <sup>Ω</sup> *πθ*<sup>2</sup>

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

*πθ*<sup>2</sup> 0 Ω2

s

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

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

>0, (26)

<0*:* (27)

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

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

*Np* þ 1

*dW*1ð Þ*τ* ,

*dτ*

*dτ*

(28)

(29)

*P a*ð Þ , *τ*

*P a*ð Þ , *τ <sup>∂</sup>a*<sup>2</sup> *:* (30)

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

(32)

$$\begin{split} d\boldsymbol{\rho} &= \frac{1}{2\Omega} \Bigg[ \mathbf{1} - \Omega^2 + \eta \sum\_{j=1}^{N\_p} \left( k\_j + c\_j \boldsymbol{\omega}\_0^{\boldsymbol{\alpha}\_j} \Omega^{\boldsymbol{\alpha}\_j} \cos\left( \frac{\boldsymbol{\alpha}\_j \pi}{2} \right) \right) \sin^2\left( \frac{j\pi}{N\_p + 1} \right) \Bigg] d\boldsymbol{\tau} \\ &+ \frac{1}{2\Omega a} [\boldsymbol{G}\_{0N} \cos \boldsymbol{\rho} + \boldsymbol{F}\_{0N} \sin \boldsymbol{\rho}] d\boldsymbol{\tau} + \frac{1}{a} \sqrt{\boldsymbol{\Gamma}\_0} \, d\boldsymbol{W}\_2(\boldsymbol{\tau}). \end{split} \tag{34}$$

The averaged Fokker-Planck-Kolmogorov equation associated with the previous Itô Eqs. (33) and (34) is

$$\begin{split} \frac{\partial P(a,\phi,\tau)}{\partial \tau} = -\frac{\partial}{\partial a} (\overline{a}\_1 P(a,\phi,\tau)) - \frac{\partial}{\partial \phi} (\overline{a}\_2 P(a,\phi,\tau)) + \frac{1}{2} \frac{\partial^2}{\partial a^2} (\overline{b}\_{11} P(a,\phi,\tau)) \\ + \frac{1}{2} \frac{\partial^2}{\partial \phi^2} (\overline{b}\_{22} P(a,\phi,\tau)), \end{split} \tag{35}$$

where

$$\begin{split} \overline{a}\_{1} &= -\left(2\boldsymbol{\ell} - \theta\_{1}\right)\frac{\boldsymbol{a}}{2} - \frac{1}{2}\eta\mu\sum\_{j=1}^{N\_{p}}c\_{j}\boldsymbol{\alpha}\_{0}^{a\_{j}}\boldsymbol{\Omega}^{a\_{j}-1}\sin^{2}\left(\frac{j\pi}{N\_{p}+1}\right)\sin\left(\frac{\alpha\_{j}\pi}{2}\right) + \frac{1}{2\boldsymbol{\Omega}}[G\_{0N}\sin\phi - F\_{0N}\cos\phi] + \frac{\Gamma\_{0}}{a} \\ \overline{a}\_{2} &= \frac{1}{2\boldsymbol{\Omega}}\left[1 - \boldsymbol{\Omega}^{2} + \eta\sum\_{j=1}^{N\_{p}}\left(b\_{j} + c\_{j}\boldsymbol{\alpha}\_{0}^{a\_{j}}\boldsymbol{\Omega}^{a\_{j}}\cos\left(\frac{\alpha\_{j}\pi}{2}\right)\right)\sin^{2}\left(\frac{j\pi}{N\_{p}+1}\right) + \frac{1}{a}[G\_{0N}\cos\phi + F\_{0N}\sin\phi]\right] \\ \overline{b}\_{11} &= F\_{0} \\ \overline{b}\_{22} &= \frac{F\_{0}}{a^{2}}. \end{split} \tag{36}$$

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

$$P\_s(a,\phi) = N^l a \exp\left\{ \frac{\Gamma\_2'}{\Gamma\_0} a^2 - \frac{a}{\Omega(\Gamma\_0^2 + d\_0^2)} [(d\_0 G\_{0N} + F\_{0N} \Gamma\_0) \cos \phi + (d\_0 F\_{0N} - G\_{0N} \Gamma\_0) \sin \phi] \right\} \tag{37}$$

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

$$\begin{aligned} I\_2' &= -\frac{1}{2} (2\lambda - \theta\_1) - \frac{\eta}{2} \sum\_{j=1}^{N\_p} c\_j a\_0^{a\_j} \mathcal{Q}^{a\_j - 1} \sin^2 \left( \frac{j\pi}{N\_p + 1} \right) \sin \left( \frac{a\_j \pi}{2} \right), \\\ I\_3 &= \frac{1}{2\Omega} \left[ 1 - \mathcal{Q}^2 + \eta \sum\_{j=1}^{N\_p} \left( k\_j + c\_j a\_0^{a\_j} \mathcal{Q}^{a\_j} \cos \left( \frac{a\_j \pi}{2} \right) \right) \sin^2 \left( \frac{j\pi}{N\_p + 1} \right) \right]. \end{aligned} \tag{38}$$
