5. Forced vibration of power conductor

High voltage transmission lines are exposed to loading from the wind. The actual system representation through system simulation strategy considers a case of distributed load through the span of the conductor. In order to simplify simulations, the external force acting on the conductor is represented as a point load. In Eqs. (41)–(43), the equation of motion is solved with an excitation force in order to evaluate the actual response of high voltage transmission lines under aeolian vibration [1]. Hence,

$$EI\frac{\partial^4 y(\mathbf{x},t)}{\partial \mathbf{x}^4} - S\frac{\partial^2 y(\mathbf{x},t)}{\partial \mathbf{x}^2} + \beta I \frac{\partial^5 y(\mathbf{x},t)}{\partial \mathbf{x}^4 \partial t} + C\frac{\partial y(\mathbf{x},t)}{\partial t} + \rho A \frac{\partial^2 y(\mathbf{x},t)}{\partial t^2} = f(\mathbf{x},t) \tag{41}$$

$$\begin{aligned} \sin\left(\frac{n\pi x}{l}\right) \left[ \begin{aligned} \left[ EI \left( \frac{n\pi}{l} \right)^4 T(t) + \mathcal{S} \left( \frac{n\pi}{l} \right)^2 \dot{T}(t) \\ &+ \beta I \left( \frac{n\pi}{l} \right)^4 T(t) + \mathcal{C} \dot{T}(t) + \rho A \ddot{T}(t) \end{aligned} \right] = F \sin w\_{dr} t \end{aligned} \tag{42}$$

$$\ddot{T}(t) + \left[\frac{\beta I}{\rho A} \left(\frac{n\pi}{l}\right)^4 + \frac{\mathcal{C}}{\rho A}\right] \dot{T}(t) + \left[\frac{\mathcal{S}}{\rho A} \left(\frac{n\pi}{l}\right)^2 + \frac{EI}{\rho A} \left(\frac{n\pi}{l}\right)^4\right] T(t) = F\sin\omega\_{dr}t\tag{43}$$

Expressing the model as a multi-degree system yields Eq. (44):

$$T(t) = Ae^{-\check{\zeta}\omega\_n t} \sin\left(\omega\_d t + \phi\right) + X\cos\left(\omega t - \theta\right) \tag{44}$$

The natural frequency of the power conductor under forced vibration is expressed in Eqs. (45) and (46) as:

$$
\omega\_n^2 = \frac{S}{\rho A} \left(\frac{n\pi}{l}\right)^2 + \frac{EI}{\rho A} \left(\frac{n\pi}{l}\right)^4 \tag{45}
$$

$$\left\|2\xi\omega\_n\right\|^2 = \left[\frac{\beta I}{\rho A}\left(\frac{n\pi}{l}\right)^4 + \frac{\mathcal{C}}{\rho A}\right] \tag{46}$$

The solution to the equation of motion under forced vibration is expressed in Eq. (47) as:

$$y(\mathbf{x},t) = \sin\frac{n\pi\mathbf{x}}{l}\left[Ae^{-\zeta\omega t}\sin\left(\sin\omega t + \phi\right) + X\cos\left(\omega t - \theta\right)\right] \tag{47}$$
