# Nonlinear Dynamics of an Elastic Stop System and Its Application in a Rotor System
## 2 Model and Solving Method
### Model
Since there is a nonlinear functional relationship between the nonlinear force and displacement in the time domain, there is a nonlinear functional relationship among \(\mathbf{P}_{k}\),\(\mathbf{Q}_{k}\), \(\mathbf{A}_{k}\) and \(\mathbf{B}_{k}\). However, for the generic nonlinear force, this inner relationship is not explicit in the frequency domain; thus, the time-frequency transformation technique is applied, and the principle is shown in Figure 2.  
First, given the coefficients of a specific order harmonic term, define the number of time samples of the IFFT (IDFT) in one cycle as \(N_{ts}\), then the value of the response of the nonlinear system at the \(r^{st}\) discrete-time can be calculated as follows:  
\[\widetilde{\mathbf{x}}_{r}=real\Bigg{\{}\sum_{k=0}^{l}\big{(}\mathbf{A}_{k}+j \mathbf{B}_{k}\big{)}\bigg{(}\cos\frac{2\pi kr}{N_{st}}+j\sin\frac{2\pi kr}{N_ {st}}\bigg{)}\Bigg{\}},\tau=0,1,2,\cdots,N_{st}-1,j=\sqrt{-1} \tag{10}\]  
Then, the nonlinear force is  
\[\widetilde{\mathbf{g}}_{r}=\mathbf{g}\Big{(}\widetilde{\mathbf{x}}_{r}, \dot{\widetilde{\mathbf{x}}}_{r}\Big{)} \tag{11}\]  
Finally, the values of the nonlinear forces at all times are calculated, and the results of the nonlinear forces on the first-order harmonics in the frequency domain are obtained according to the DFT as  
\[\widetilde{\mathbf{g}}_{k}=\frac{\Psi}{N_{st}}\sum_{r=0}^{N_{st}-1}\widetilde {\mathbf{g}}_{r}e^{j(-\frac{\pi\delta kr}{N_{st}})},\Psi= \begin{cases}1,&k=0\\ 2,&k\
eq 0\end{cases}  \tag{12}\]  
Therefore, the implicit functional relationship between the harmonic coefficients in the frequency domain can be determined. Figure 2: Principle of alternating frequency–time technique. For the balancing process, substituting Equations (11) and (12) into Equation (9) and balancing each harmonic coefficient, the nonlinear algebraic equations with \(\mathbf{A}_{k}\) and \(\mathbf{B}_{k}\) is organized as:  
\[\sum_{k=1}^{l}\left\{ \begin{array}{l}\left(-k^{2}\mathbf{MA}_{k}+k\mathbf{DB}_ {k}+\mathbf{P}_{k}\right)\cos k\theta\\ +\left(\left(-k\mathbf{DA}_{k}-k^{2}\mathbf{MB}_{k}+\mathbf{Q}_{k}\right)\sin k \theta\right)\end{array} \right\}+\mathbf{KA}_{0}+\mathbf{P}_{0}=\mathbf{F}_{0} \sin\
u\theta \tag{13}\]  
Balance each term, so that constant terms:  
\[\mathbf{KA}_{0}+\mathbf{P}_{0}=0 \tag{14}\]  
Cosine terms:  
\[-k^{2}\mathbf{MA}_{k}+k\mathbf{DB}_{k}+\mathbf{P}_{k}=0,k=1,2,\cdots,l, \tag{15}\]  
Sinusoidal terms:  
\[-k\mathbf{DA}_{k}-k^{2}\mathbf{MB}_{k}+\mathbf{Q}_{k}=\phi\mathbf{F}_{0},\phi = \begin{cases}1,&k=\
u\\ 0,&k\
eq\
u^{\prime}\end{cases}  \tag{16}\]  
For the sake of easy programming, Equations (14)-(16) are presented in matrix form as  
\[\left[ \begin{array}{ccccc}\mathbf{K}&&&&\\ &\mathbf{\Gamma}_{1}&&&\\ &&\ddots&&\\ &&&\mathbf{\Gamma}_{k}&&\\ &&&\ddots&\\ &&&&\mathbf{\Gamma}_{l}\end{array} \right]\left[ \begin{array}{c}\mathbf{A}_{0 }\\ \mathbf{Z}_{1}\\ \vdots\\ \mathbf{Z}_{k}\\ \vdots\\ \vdots\\ \mathbf{Z}_{l}\end{array} \right]=\left[ \begin{array}{c}-\mathbf{P}_{0}\\ \mathbf{\Theta}_{1}\\ \vdots\\ \mathbf{\Theta}_{k}\\ \vdots\\ \mathbf{\Theta}_{l}\end{array} \right] \tag{17}\]  
in which  
\[\mathbf{\Gamma}_{k}=\left[ \begin{array}{cc}-k^{2}\mathbf{M}&k\mathbf{D}\\ -k\mathbf{D}&-k^{2}\mathbf{M}\end{array} \right],\mathbf{Z}_{k}=\left[ \begin{array}{c}\mathbf{A}_{k}\\ \mathbf{B}_{k}\end{array} \right],\mathbf{\Theta}_{k}=\left[ \begin{array}{c} -\mathbf{P}_{k}\\ -\mathbf{Q}_{k}+\phi\mathbf{F}_{0}\end{array} \right].\]  
As for Equation (17), there are many solving methods, for example, the Newton-type method, which is not introduced in detail here. As for the stability analysis method, the Floquet theory [20] is applied. Solving the above nonlinear algebraic equations is the process of obtaining the fixed point of the function \(\mathbf{Z}_{k}\). In addition, the Newton-Raphson iteration method can be applied.