# Research on Nonlinear Vibration Characteristics of Internal Beveloid Gear Transmission System
## 4 Harmonic Balance Analysis
### Solution of Harmonic Balance Method
\begin{array}{l}-\omega^{2}\mathbf{M}\mathbf{U}^{\mathrm{s}}\sin(\omega t)- \omega^{2}\mathbf{M}\mathbf{U}^{\mathrm{c}}\cos(\omega t)+\omega\mathbf{C} \mathbf{U}^{\mathrm{s}}\cos(\omega t)-\omega\mathbf{C}\mathbf{U}^{\mathrm{c}} \sin(\omega t)+\mathbf{K}^{\prime\prime}\mathbf{U}\\ +\mathbf{K}\mathbf{U}^{\mathrm{s}}\sin(\omega t)+\mathbf{K}\mathbf{U}^{\mathrm{ c}}\cos(\omega t)=\omega\mathbf{C}^{\prime}\mathbf{E}^{\mathrm{s}}\cos(\omega t)+ \mathbf{K}^{\prime}\mathbf{E}^{\mathrm{m}}+\mathbf{K}^{\prime}\mathbf{E}^{ \mathrm{s}}\sin(\omega t)\end{array}  \tag{37}\]  
According to the principle of the harmonic balance approach, the coefficients of the same harmonic term of the matrix equation are set to zero, resulting in a nonlinear differential equation with six equations matching the sine and cosine terms. The matrix is represented as follows:  
\[\left\{ \begin{array}{l}-\omega^{2}\mathbf{M}\mathbf{U}^{\mathrm{s}}-\omega \mathbf{C}\mathbf{U}^{\mathrm{c}}+\mathbf{K}^{\prime\prime}\mathbf{U}^{ \mathrm{s}}+\mathbf{K}f(\mathbf{U}^{\mathrm{s}},b)=\mathbf{K}^{\prime}\mathbf{ E}^{\mathrm{s}}\\ -\omega^{2}\mathbf{M}\mathbf{U}^{\mathrm{c}}+\omega\mathbf{C}\mathbf{U}^{ \mathrm{s}}+\mathbf{K}^{\prime\prime}\mathbf{U}^{\mathrm{c}}+\mathbf{K}f( \mathbf{U}^{\mathrm{c}},b)=\omega\mathbf{C}^{\prime}\mathbf{E}^{\mathrm{s}} \end{array} \right. \tag{38}\]  
The backlash function is identical to the algebras \(a\), \(b\) for combining the \(\mathbf{K}^{\prime\prime}\) term with the original \(\mathbf{K}\) term to generate a new \(\mathbf{K}\) term in order to derive the sine and cosine components of the solution, as shown below:  
\[\left\{\begin{array}{l}\mathbf{U}^{\mathrm{s}}=\left[\mathbf{K}-\omega^{2} \mathbf{M}+\omega^{2}\mathbf{C}\big{(}\mathbf{K}-\omega^{2}\mathbf{M}\big{)} ^{-1}\mathbf{C}\right]^{-1}\!\!\left[\mathbf{  
Considering that the influence of bearing backlash is ignored at this time, and the clearance functions of both are represented by first\(-\)order fitting, there is a certain gap compared with the actual situation. Considering that the analytical method generally fits the backlash function with a polynomial function, the higher the fitting order, the greater the precision. When the harmonic balance method is used to solve the problem, it is found that when the gap function is considered in the numerical solution, the scanning curves of the two have a certain difference. When the backlash is small, the vibration and amplitude of the two are basically the same, and the error is small; when there is a backlash gap, the numerical solution is quite different from the equivalent analytical solution of first\(-\)order processing. Figure 12: Time domain diagram of gear speed and acceleration based on harmonic balance (**a**,**b**) the radial and axial speed of internal gear; (**c**,**d**) radial and axial acceleration of internal gear. Figure 13: Velocity comparison of numerical solution and analytical solution time domain diagram (**a**,**b**) radial and axial vibration velocity comparison. For the general harmonic method, if you want to obtain a more accurate solution, you need to increase the equivalent order of the approximate solution and fit the gap function to a higher\(-\)order series, which leads to a great increase in the difficulty of solving the harmonic method. Therefore, it is difficult to solve the harmonic method when it is necessary to analyze the characteristic influence caused by the gear backlash with high precision.