## 4 Research on the Influence of Track Structure Stiffness Change on Track Deformation
### Analysis of Sensitivity Indexes of Vehicle-Track System Dynamics Response  
According to the research results of Sections 3 and 4, parts of the key dynamic performance of heavy-haul train and track structure system changes with track structure stiffness have been clearly analyzed to show the performance of the LVT under the action of a heavy-haul train. This section mainly studies the dynamic response sensitivity indexes of the vehicle-track system under different track structure parameter changes and discusses the degree of influence of different track structure parameters on the dynamic response of the vehicle-track system, which can provide a reference for the optimization design of LVT structure dynamics [27]. In order to accurately and quantitatively describe the changes in the dynamic response of the vehicle-track system, the ratio of the dynamic response change difference of the vehicle-track system to the stiffness change difference of the track structure is introduced as the sensitivity index of the dynamic response parameters, that is, the stiffness \(k\) of the track structure when it changes from \(k_{1}\) to \(k_{2}\), will cause a certain dynamic response parameter \(D\) of the rail system to change from \(D_{1}\) to \(D_{2}\)[30], as shown in Equation (1):  
\[\varepsilon_{D}=\frac{(D_{2}-D_{1})/(D_{2}+D_{1})}{(k_{2}-k_{1})/(k_{2}+k_{1 })} \tag{1}\]  
The world's main heavy-haul train countries and organizations, such as the United Kingdom, the United States, France, Germany, Japan, China and the International Union of Railways (UIC), etc., commonly used vehicle-track system dynamic response evaluation coefficients include wheel-rail lateral force, wheel-rail vertical force, wheel load reduction rate, wheel load reduction rate, wear coefficient, lateral acceleration of vehicle body, and lateral acceleration of vehicle body [26; 31; 32; 33; 34; 35; 36; 37; 38]. In addition, the reference [32] also puts forward the concept of track structure deformation coefficient, including vertical displacement of rail, lateral displacement of rail and gauge expansion. Therefore, this paper analyzes the sensitivity indexes of all the above evaluation coefficients to track structure parameters. The sensitivity indexes of vertical stiffness under rail, lateral stiffness under rail, vertical stiffness under block, and lateral stiffness under block are, respectively, defined as \(\varepsilon_{\text{iD}}\)(i = 1-4). The dynamic response parameters of the rail system include wheel-rail vertical force, wheel-rail lateral force, derailment coefficient, wheel load reduction rate, wear coefficient, rail vertical displacement, rail lateral displacement, gauge expansion, lateral acceleration of vehicle body and vehicle acceleration of vehicle body. Its sensitivity indexes to the track structure parameters can be defined as \(\varepsilon_{\text{iD}}\)(i = 1-4, j = 1-10). In order to measure the degree of track structure parameters to the dynamic response of the vehicle-track system, the maximum response of the dynamic coefficient and the corresponding structural parameter value at this time are calculated. The sensitivity index can be expressed as shown in formula (2):  
\[\varepsilon_{D}=\frac{(D_{\text{max}}-D_{\text{min}})/(D_{\text{max}}+D_{\text {min}})}{(k_{2}-k_{1})/(k_{2}+k_{1})} \tag{2}\]  
The sensitivity index is the ratio of the rate of change of the dynamic response coefficient to the rate of change of the stiffness of the track structure. When k increases so that \(D\) increases, it is a positive value, otherwise it is a negative value. Table 8 summarizes the sensitivity indexes of vertical stiffness under different rails, lateral stiffness under rails, vertical stiffness under blocks, and lateral stiffness under blocks.