## 3 Two-Layer Singular Value Decomposition
### Number Characteristic of Singular Values for Multi-Sinusoid FM Rubbing Fault Response  
When \(f_{d_{q}}/f_{m_{q}}<<1\), the rubbing fault response can be approximately formulated as Equation (9) by neglecting the minor frequency components with low amplitudes [4]. The two instantaneous frequency components with the largest amplitude and the two instantaneous amplitude components with the largest amplitude are considered. \[ \begin{split} s(t)&=A(t)\cos[2\pi f_{c}t+\sum\limits_{q= 1}^{Q}\frac{f_{d_{q}}}{f_{m_{q}}}\sin(2\pi f_{m_{q}}t+\theta_{q})+\gamma]\\ &\approx[b+a\cos(2\pi f_{G}t+\psi_{A})+e\cos(2\pi f_{E}t+\psi_{E})] \times\\ &\sum\limits_{|n_{1}|+|n_{2}|+|n_{3}|<3}[\prod\limits_{q=1}^{3}f_{n_{q}}( \frac{f_{d_{q}}}{f_{m_{q}}})]\cos[\sum\limits_{q=1}^{3}n_{q}(2\pi f_{m_{q}}t+ \theta_{q})+2\pi f_{C}t+\gamma]\end{split}  \tag{11}\]in which initial phases for signal \(s(t)\) is \(\gamma\); \(A(t)\) is instantaneous amplitude (IA)  
\[A(t)=[b+a\cos(2\pi f_{A}t+\psi_{A})+e\cos(2\pi f_{E}t+\psi_{E})] \tag{12}\]  
in which \(f_{A}\) and \(f_{E}\) are modulation frequencies of IA; \(\psi_{A}\) and \(\psi_{E}\) are initial phases; \(a\) and \(e\) are the oscillating amplitudes of IA; \(b\) is a constant. \(F(t)\) is instantaneous frequency (IF)  
\[f_{c}+\sum_{q=1}^{Q}f_{d_{q}}\cos(2\pi f_{m_{q}}t+\theta_{q}) \tag{13}\]  
in which fundamental frequency is \(f_{c}\); \(f_{d}\) and \(f_{m}\) are the oscillating amplitude and modulation frequency of IF; initial phases for IF is \(\theta\); \(J_{n}(. )\) is the \(n\)th-order Bessel function of the first kind. \[J_{n}(\rho)=\frac{j^{-n}}{2\pi}\int\limits_{0}^{2\pi}\exp[j(\rho\cos\theta+n \theta)]d\theta \tag{14}\]  
Define the equation \(g_{n_{1},n_{2},n_{3}}(t)\) as  
\[ \begin{array}{ll}g_{n_{1},n_{2},n_{3}}(t)&=[b+a\cos(2\pi f_{A}t+\psi_{A})+e \cos(2\pi f_{E}t+\psi_{E})]\\ &\times[\prod\limits_{q=1}^{3}I_{n_{q}}(\frac{f_{d_{q}}}{f_{m_{q}}})]\cos[ \sum\limits_{q=1}^{3}n_{q}(2\pi f_{m_{q}}t+\theta_{q})+2\pi f_{C}t+\gamma]\\ &\\ s(t)\approx\sum\limits_{|n_{1}|+|n_{2}|+|n_{3}|<3}g_{n_{1},n_{2},n_{3}}(t)\\ \end{array}  \tag{15}\]  
When \(n_{1}=1,n_{2}=1,n_{3}=0\)  
\[ \begin{array}{ll}g_{1,1,0}(t)=b[\prod\limits_{q=1}^{3}J_{n_{q}}(\frac{f_{d_ {q}}}{f_{m_{q}}})]\cos(2\pi f_{m_{1}}t+\theta_{1}+2\pi f_{m2}t+\theta_{2}+2\pi f _{C}t+\gamma)\\ +0.5a[\prod\limits_{q=1}^{3}J_{n_{q}}(\frac{f_{d_{q}}}{f_{m_{q}}})][\cos(2\pi f _{m_{1}}t+\theta_{1}+2\pi f_{m2}t+\theta_{2}+2\pi f_{C}t+\gamma-2\pi f_{A}t- \psi_{A})+\\ \cos(2\pi f_{m_{1}}t+\theta_{1}+2\pi f_{m2}t+\theta_{2}+2\pi f_{C}t+\gamma+2 \pi f_{A}t+\psi_{A})]\\ +0.5e[\prod\limits_{q=1}^{3}J_{n_{q}}(\frac{f_{d_{q}}}{f_{m_{q}}})][\cos(2 \pi f_{m_{1}}t+\theta_{1}+2\pi f_{m2}t+\theta_{2}+2\pi f_{C}t+\gamma-2\pi f_{E }t-\psi_{E})+\\ \cos(2\pi f_{m_{1}}t+\theta_{1}+2\pi f_{m2}t+\theta_{2}+2\pi f_{C}t+\gamma+2 \pi f_{E}t+\psi_{E})]\\ \end{array}  \tag{17}\]  
Signal \(g_{1,1,0}(t)\) are discretized by sampling interval \(T_{s}\). Hankel matrices can be constructed. \[\mathbf{H}_{110}=\mathbf{H}_{aa}+\mathbf{H}_{ab}+\mathbf{H}_{ac} \tag{18}\]  
in which \(\mathbf{H}_{110}\) is the Hankel matrix of \(g_{1,1,0}(t)\), \(\mathbf{H}_{aa}\) is the Hankel matrix of the first monomial of \(g_{1,1,0}(t)\), \(\mathbf{H}_{ab}\) is the second monomial of \(g_{1,1,0}(t)\), \(\mathbf{H}_{ac}\) is the third monomial of \(g_{1,1,0}(t)\).