# The Numerical Study for the Effect of Stiffness Matching on Wheel-Rail Curve Squeal Noise
## 2 Methodology
### Finite Element Complex Eigenvalue Analysis of Wheel-Rail System  
Drawing upon the principles of modal coupling theory, it becomes evident that the transverse creep force acting between the wheel tread and the top surface of the rail can be equated to the introduction of a disturbance load to the wheel-rail system. This disturbance load, in turn, engenders an inherent asymmetry in the stiffness matrix of the system. This asymmetry serves as a trigger, exciting unstable modes within the system. To tackle the complex eigenvalue problem arising from the inherent asymmetry of the system's stiffness matrix, one can employ complex eigenvalue analysis. This analytical technique is instrumental in identifying those modes within the system that are predisposed to experiencing friction-induced self-excited vibrations, ultimately leading to the manifestation of squeal noise [30]. Considering the friction between the wheel and the rail when the wheel-rail transverse creep occurs, the dynamics equation of the system is as follows:  
\[[\mathrm{M}]\left\{\ddot{\mathrm{x}}\right\}+[\mathrm{C}]\left\{\dot{\mathrm{ x}}\right\}+[\mathrm{K}]\left\{\mathrm{x}\right\}=\left\{\mathrm{F}_{\mathrm{f}}\right\} \tag{12}\]  
where \([\mathrm{M}],\ [\mathrm{C}],\ [\mathrm{K}]\) are the stiffness matrix, damping matrix, and stiffness matrix of the system, respectively; the displacement vectors of each degree of freedom of the system are the contact friction between the wheels and rails, expressed as the relative displacement of the nodes between the contact surfaces as:  
\[\left\{\mathrm{F}_{\mathrm{f}}\right\}\ =\ [\mathrm{K}_{\mathrm{f}}]\left\{ \mathrm{x}\right\} \tag{13}\]  
where \([\mathrm{K}_{\mathrm{f}}]\) is the friction stiffness matrix, or simply the friction matrix, which relates the friction force to the nodal displacement. Combining Equations (1) and (2), the friction force of the right term of Equation (1) is shifted to the left term to obtain the following second-order sublinear differential equation:  
\[[\mathrm{M}]\left\{\ddot{\mathrm{x}}\right\}+[\mathrm{C}]\left\{\dot{\mathrm{ x}}\right\}+[\mathrm{K}-\mathrm{K}_{\mathrm{f}}]\left\{\mathrm{x}\right\}=\left\{ 0\right\} \tag{14}\]  
Equation (14) is the dynamical equation for a free vibrating system containing a frictional stiffness term, the solution of which takes the following form:  
\[\mathrm{x}=\left\{\Phi\right\}\mathrm{e}^{\lambda\mathrm{t}} \tag{15}\]  
Substituting its differentiation into Equation (14) yields:  
\[([\mathrm{M}]\lambda^{2}+[\mathrm{C}]\lambda+[\mathrm{K}-\mathrm{K}_{\mathrm{ f}}])\left\{\Phi\right\}=\left\{0\right\} \tag{16}\]  
where \(\Phi\) is the eigenvector and \(\lambda\) is the eigenvalue. The friction matrix \([\mathrm{K}_{\mathrm{f}}]\) due to the friction force is asymmetric, which leads to an asymmetric stiffness matrix of the system. Therefore, solving the equation under certain conditions yields conjugate complex eigenvalues including real and imaginary parts:  
\[\lambda_{\mathrm{i}_{1,2}}=\sigma_{\mathrm{i}}\ \pm\ \mathrm{j}\omega_{ \mathrm{i}} \tag{17}\]  
The motion of the system can be described in terms of conjugate complex eigenvalues and eigenvectors as follows:  
\[\left\{\mathrm{x}_{\mathrm{i}}\right\}\ =\ \left\{\Phi_{\mathrm{i}}\right\}\mathrm{e} ^{(\sigma_{\mathrm{i}}+\mathrm{j}\omega_{\mathrm{i}})\mathrm{t}}+\left\{\Phi_{ \mathrm{i}}\right\}\mathrm{e}^{(\sigma_{\mathrm{i}}-\mathrm{j}\omega_{\mathrm{ i}})\mathrm{t}} \tag{18}\]  
The transformation by [PERSON]'s formula yields:  
\[\left\{\mathrm{x}_{\mathrm{i}}\right\}\ =\ \left\{\Phi_{\mathrm{i}}\right\}\mathrm{e} ^{\sigma_{\mathrm{i}}\mathrm{t}}\mathrm{cos}\ \omega_{\mathrm{i}}\mathrm{t} \tag{19}\]For the i-th-order mode, \(\sigma_{\rm i}\) is the real part of its eigenvalue, which represents the damping coefficient of the system, \(\omega_{\rm i}\) is its imaginary part, and c represents the intrinsic frequency of the system. A positive real part causes the vibration to expand and develop into a strong self-excited vibration, at which point the system is unstable.