## 4 Dynamic Characteristics of the Robot-Cable System
### Analysis of the Natural Frequency of the Cable  
The string vibration equation is used to describe cable vibration (Figure 10). Owing to the robot climbing on a long cable, which has certain influences on cable vibration, the robot is regarded as a point mass to establish a string vibration equation with this lumped mass:  
\[\left\{ \begin{array}{ll}u_{tt}-au_{xx}=0&x\
eq x_{0}\\ u_{tt}-a_{x0}u_{xx}=0&x=x_{0}\end{array} \right. \tag{18}\] \begin{table}
\begin{tabular}{c c} \hline \hline
**Item** & **Value** \\ \hline Length & 370–400 mm \\ Width & 236 mm \\ Height & 410–440 mm \\ Material & Aluminum Alloy \\ Gravity & 11.7 kg \\ \hline \hline \end{tabular}
\end{table} 
Table 1: The parameters of the robot. Figure 9: Finite element analysis of the robot. (**a**) Finite element model of the robot; (**b**) Finite element mesh. where \(x_{0}\), \(l\) and \(u\) represent the position of the robot, the length of the cable and the normal displacement of the cable from the initial position, respectively. In addition, \(a\) denotes the ratio of tension \(T\) of the cable to the linear density \(\rho\) of the cable, namely \(\frac{T}{\rho}\), while \(a_{x0}\) represents the ratio of \(T\) to the linear density \(\rho_{x0}\) of the cable at the position of the robot, namely, \(\frac{T}{\rho_{x0}}\). It is difficult to solve string vibration problems when they include a lumped mass. This section adopts an approximate method, namely, the use of the Rayleigh quotient, which can directly estimate the natural frequency of the system without solving its vibration equation. The Rayleigh quotient is expressed in the following form:  
\[R=\frac{V_{\max}}{T_{ref}}=\omega_{n}^{2} \tag{19}\]  
\(V_{\max}\) and \(T_{ref}\) represent the maximum potential energy and kinetic energy for reference of the cable respectively, which indicate quantities relating to potential energy and kinetic energy. \(\omega_{n}\) denotes the \(n\)th-order natural frequency. \(V_{\max}\) and \(T_{ref}\) can be written as:  
\[\left\{ \begin{array}{l}V_{\max}=\frac{1}{2}\int_{0}^{l}T(W_{n}^{\prime}(x)) ^{2}dx\\ T_{ref}=\frac{1}{2}\int_{0}^{l}\rho W_{n}^{2}(x)dx\end{array} \right. \tag{20}\]  
\(W_{n}(x)\) denotes the nth-order vibration mode function relating to \(x\). Free vibration of a tensile string is a linear combination of infinite multi-order vibrations. In general, the higher the order is, the smaller the influence on the spectrum, so it is important to study the first-order vibration. Experience shows that the first-order vibration mode of the cable is similar to static deformation under inertial loads. The first-order natural frequency can represent the frequency of the cable, so it is selected as the static deformation of the cable under the self-weight of the robot. Figure 10b shows that the following equation can be established through the force balance and geometric relationships:  
\[\cos(\arctan\frac{x_{0}}{\varepsilon})+\cos(\arctan\frac{l-x_{0}}{\varepsilon })=\frac{F}{T} \tag{21}\]  
where \(F\) and \(\varepsilon\) represent the component of the self-weight of the robot in the vertical direction of the cable and the maximum static deformation, respectively. Because the distance from the position of maximum deformation to the end on the cable changes linearly, the vibration mode function can be obtained as long as \(\varepsilon\) is known from the following equation:  
\[W(x)=\left\{ \begin{array}{ll}\varepsilon\frac{x}{x_{0}}&,0<x<x_{0}\\ \varepsilon\frac{l-x_{0}}{\varepsilon\frac{l-x_{0}}{\varepsilon}}&,x_{0}<x<l \end{array} \right. \tag{22}\]  
Figure 10: Robot-cable dynamic analysis. (**a**) Schematic diagram of cable vibration; (**b**) Force applied on the robot. Therefore, the relationship between \(x_{0}\) and \(\varepsilon\) is established according to Equation (21).