## 2 Mathematical Modeling and Equations  
In a Cartesian coordinate system \(O\)-\(xyz\), a horizontal cable subjected to thermal conditions is supported, as illustrated in Figure 1. Thermal stresses arise as a result of the structure's expansion and contraction caused by temperature fields, which in turn affect the static and dynamic behaviors of the suspended cable. Therefore, a distinguishable thermal static profile of the cable can be noted in thermal environments [37]. Here, the suspended cable's three configurations can be distinguished: the static, thermal-stressed, and dynamic configurations. Assuming uniform temperature variation along the cable's length and cross-sectional area, the thermal dynamics of the system exhibit a significantly slower time scale compared to the structural dynamics. The span of the cable between two horizontal supports, denoted by \(O\) and \(B\), is measured as \(L\). The original sag and thermal-stressed sag are represented by \(b\) and \(b_{\Delta T}\), respectively. Under the external excitation, the longitudinal, vertical, and out-of-plane displacements along three directions are expressed by \(\bar{u}(x,t)\), \(\bar{v}(x,t)\), and \(\bar{w}(x,t)\). The partial differential equations of motion in the in-plane and out-of-plane directions are derived using the classical condensed mathematical model [35; 38; 39]. \[ \begin{split}& m\bar{v}+c_{v}\bar{v}-\chi_{\Delta T}^{2}H\bar{v}^{ \prime\prime}-\frac{EA}{L}\left(\frac{y^{\prime\prime}}{\chi_{\Delta T}^{2}}+ \bar{v}^{\prime\prime}\right)\int_{0}^{L}\left[\frac{y^{\prime}\bar{v}^{ \prime}}{\chi_{\Delta T}^{2}}+\frac{1}{2}\Big{(}\bar{v}^{\prime 2}+\bar{w}^{ \prime 2}\Big{)}\right]\mathrm{d}x=F_{v}\cos(\Omega_{v}t+\theta_{v}),\\ & m\bar{w}+c_{w}\bar{w}-\chi_{\Delta T}^{2}H\bar{v}^{\prime \prime}-\frac{EA}{L}\bar{w}^{\prime\prime}\int_{0}^{L}\left[\frac{y^{\prime} \bar{v}^{\prime}}{\chi_{\Delta T}^{2}}+\frac{1}{2}\Big{(}\bar{v}^{\prime 2}+ \bar{w}^{\prime 2}\Big{)}\right]\mathrm{d}x=F_{w}\cos(\Omega_{w}t+\theta_{w}), \end{split}  \tag{1}\]  
where the suspended cable is subject to two external harmonic loads, \(F_{v}\cos(\Omega_{v}t+\theta_{v})\) and \(F_{w}\cos(\Omega_{w}t+\theta_{w})\), with in-plane (out-of-plane) excitation amplitudes, frequencies, and phases represented by \(F_{v}\) (\(F_{w}\)), \(\Omega_{v}\) (\(\Omega_{w}\)), and \(\theta_{v}\) (\(\theta_{w}\)), respectively. These loads apply a distributed force to the cable, with the cable's Young's modulus, cross-sectional area, and mass per length represented by \(E\), \(A\), and \(m\), respectively. A notable deviation between the classical and original differential equations of motion governing the suspended cable and the current model is the introduction of a new tension variation factor \(\chi_{\Delta T}^{2}\)[35]  
\[\chi_{\Delta T}^{2}=\frac{H_{\Delta T}}{H}=\frac{b}{b_{\Delta T}}, \tag{2}\]  
where \(H_{\Delta T}\) (\(H\)) and \(b_{\Delta T}\) (\(b\)) are the cable tension force and sag with (without) thermal effects, respectively. Here, once the temperature changes are neglected (\(\chi_{\Delta T}^{2}=1\)), Equation (1) are reduced to the classical original partial differential equations in [10]. The following dimensionless variables and parameters are also defined  
\[ \begin{split}\bar{v}^{*}&=\frac{\bar{v}}{L},\,\bar {w}^{*}=\frac{\bar{w}}{L},\,x^{*}=\frac{x}{L}\,,\,y^{*}=\frac{y}{L}\,,\,f^{*}= \frac{b}{L}\,,\,t^{*}=\sqrt{\frac{S}{8b}}\,t\,,\\ c_{v,w}^{*}&=\sqrt{\frac{8b}{g}}\frac{c_{v,w}}{m}\,, \Theta=\frac{EA}{H},\,F_{v,w}^{*}=\frac{F_{v,w}L}{H},\,\Omega_{v,w}^{*}= \sqrt{\frac{8b}{g}}\Omega_{v,w}.\end{split}  \tag{3}\]  
By substituting Equation (3) into (1) and removing the asterisk notations for simplicity, we obtain a set of dimensionless equations governing the in-plane and out-of-plane motions  
\[