## 2 Derivation of Theoretical Formulae for Flexible Hanger  
To illustrate the universality of the formula, take any of the inclined flexible hanger cables shown in the following Figure 1: \(L\) represents the length of the hanger, \(T\) denotes the tension applied to the hanger in the \(x\)-axis direction, \(\theta\) indicates the angle of inclination of the hanger, \(\theta\) is 0 when the hanger is vertical, \(s\) signifies the droop, and \(L_{0}\) is defined as the vertical span of tie. According to [PERSON]'s [21] method based on the law of conservation of energy, the following can be shown:  
\[E_{0}+V_{0}=C=E_{\max}=V_{\max}, \tag{1}\]  
where \(E_{0}\) and \(V_{0}\) are the kinetic and potential energies at a certain time, \(C\) is a constant, and \(E_{max}\) and \(V_{max}\) are the maximum values of kinetic and potential energy. Assuming the first-order vibration function of the structure:  
\[y(x,t)=\varphi(x)\cos(\omega t+\theta), \tag{2}\]  
where \(\varphi(x)\) is the mode shape function, and \(\omega\) is the natural frequency. Equation (2) takes the first-order derivative with respect to \(t\), which yields the following:  
\[y^{\prime}(x,t)=-\omega\varphi(x)\cos(\omega t+\theta) \tag{3}\]  
The total kinetic energy of the structure is the following:  
\[E_{0}=\frac{1}{2}{\int_{0}^{1}{m(y^{\prime})}^{2}}\,dx, \tag{4}\]  
where \(m\) is the linear density of the hanger. The total potential energy of the structure is the following:  
\[V_{0}=\frac{1}{2}{\int_{0}^{1}{EI[\varphi^{\prime\prime}(x)]}^{2}}\,dx+\frac{1 }{2}{\int_{0}^{1}{(T+mgx\sin\theta[\varphi^{\prime}(x)]}^{2}}\,dx \tag{5}\]  
Figure 1: Basic characteristic parameters of hanger. Then, the maximum value of kinetic energy is the following:  
\[E_{\max}=\frac{1}{2}\omega^{2}\int_{0}^{l}m\left[\rho(x)^{2}\right]dx \tag{6}\]  
The maximum value of the potential energy of the structure is the following:  
\[V_{\max}=\frac{1}{2}\int_{0}^{l}EI[\varphi^{\prime\prime}(x)]^{2}dx+\frac{1}{2} \int_{0}^{l}T\left[\varphi^{\prime}(x)\right]^{2}dx \tag{7}\]  
The joint Equations (1), (4) and (5) are obtained as follows:  
\[\omega^{2}=\frac{\int_{0}^{l}EI[\varphi(x)]^{2}dx+\int_{0}^{l}\left(T+mgx\sin \theta\right)[\varphi^{\prime}(x)]^{2}dx}{\int_{0}^{l}m\left[\varphi(x)\right] ^{2}dx} \tag{8}\]  
When the boundary conditions at both ends are different, the vibration mode function of the structure is also different, and the formula for calculating the cable force under different boundary conditions is analyzed as follows:  
(1) Articulated at both ends  
The vibration function is similar to that of a simply supported structure and can be expressed as follows:  
\[E_{0}+V_{0}=C=E_{\max}=V_{\max}, \tag{1}\]0  
By substituting \(\omega_{n}=2\pi f_{n}\) and the first and second derivatives of Equation (9) with respect to \(x\) into Equation (8) yields the following:  
\[E_{0}+V_{0}=C=E_{\max}=V_{\max}, \tag{1}\]1  
where \(f_{n}\) is the \(n\)th-order fundamental frequency. (2) Articulated at one end and fixed at the other  
Since it is difficult to find the higher-order vibration pattern of the structure, the first-order vibration pattern is set as [22]:  
\[\varphi_{1}(x)=\left\{ \begin{array}{l}1-\cos\frac{\pi x}{l};x\in\left[0, \frac{l}{2}\right]\\ \sin\frac{\pi x}{l};x\in\left[\frac{l}{2},l\right]\end{array} \right. \tag{11}\]  
Substitute Equation (11) into Equation (8) to obtain the following:  
\[T=\frac{4 mf_{1}^{2}l^{2}}{1.376}-EI\frac{\pi^{2}}{l^{2}}-1.203g\sin\theta\frac {l}{2} \tag{12}\]  
(3) Fixed at both extremities  
The first-order mode can be assumed as follows:  
\[\varphi_{1}(x)=\left\{ \begin{array}{l}1-\cos\frac{\pi x}{l};x\in\left[0, \frac{l}{2}\right]\\ 1+\cos\frac{\pi x}{l};x\in\left[\frac{l}{2},l\right]\end{array} \right. \tag{13}\]  
Substituting Equation (13) into Equation (8) yields the following:  
\[T=\frac{4 mf_{1}^{2}l^{2}}{2.205}-EI\frac{\pi^{2}}{l^{2}}-g\sin\theta\frac{l} {2} \tag{14}\]Equations (10), (12) and (14) are suitable for most engineering requirements.