# Research on Nonlinear Vibration Characteristics of Internal Beveloid Gear Transmission System
## 4 Harmonic Balance Analysis
### Solution of Harmonic Balance Method  
The harmonic balance method assumes that the variables in the gear transmission system can be approximated as a superposition of multiple harmonics. The approximate harmonic superposition is then substituted into the original parameter equations. By integrating the values corresponding to each harmonic term, a system of equations is obtained to solve for the harmonic coefficients based on the assumed solution. Finally, by substituting the obtained harmonic coefficients into the original variables, an approximate solution for the variables in terms of harmonic superposition can be obtained. In the majority of instances, the primary determinants of a gear transmission system are its mean value and its first\(-\)order harmonic component. Consequently, the solution of the original equation can be represented by the mean value and the first\(-\)order harmonic component corresponding to it. Radial, axial, and torsional vibrations of the gear are expressed approximately by their respective first\(-\)order harmonic components:  
\[y_{i}=y_{i}^{\mathrm{g}}\sin(\omega t)+y_{i}^{\mathrm{c}}\cos(\omega t) \tag{27}\]  
\[z_{i}=z_{i}^{\mathrm{g}}\sin(\omega t)+z_{i}^{\mathrm{c}}\cos(\omega t) \tag{28}\]  
\[\theta_{i}=\theta_{i}^{\mathrm{g}}\sin(\omega t)+\theta_{i}^{\mathrm{c}}\cos( \omega t) \tag{29}\]  
where \(y_{i}\), \(z_{i}\), \(\theta_{i}\) are radial, axial, and torsional displacements of gear vibration, respectively; \(y_{i}^{\mathrm{g}}\), \(z_{i}^{\mathrm{s}}\), \(\theta_{i}^{\mathrm{s}}\) are the sinusoidal components of the radial, axial, and torsional displacement of the gear vibration, respectively; \(y_{i}^{\mathrm{c}}\), \(z_{i}^{\mathrm{c}}\), \(\theta_{i}^{\mathrm{c}}\) are the cosine components of the radial, axial, and torsional displacement of the gear vibration, respectively. To aid calculation and analysis, the radial, axial, and torsion of the transmission system are written as matrices, and their first\(-\) and second\(-\)order derivatives are solved:  
\[\mathbf{U}=\mathbf{U}^{\mathrm{s}}\sin(\omega t)+\mathbf{U}^{\mathrm{c}}\cos( \omega t) \tag{30}\]  
\[\dot{\mathbf{U}}=\omega\mathbf{U}^{\mathrm{s}}\cos(\omega t)-\omega\mathbf{U} ^{\mathrm{c}}\sin(\omega t) \tag{31}\]  
\[\ddot{\mathbf{U}}=-\omega^{2}\mathbf{U}^{\mathrm{s}}\sin(\omega t)-\omega^{2} \mathbf{U}^{\mathrm{c}}\cos(\omega t) \tag{32}\]  
In the original analysis, the mean value of the gear transmission error is zero, which is described by the harmonic function as:  
\[e=e^{\mathrm{s}}\sin(\omega t) \tag{33}\]  
where \(e\), \(e^{\mathrm{s}}\) are the gear transmission error and the sinusoidal component of the error, respectively. Simultaneously, the gear transmission error is expressed in matrix form, and its first\(-\)order derivative is calculated:  
\[\mathbf{E}=\mathbf{E}^{\mathrm{s}}\sin(\omega t) \tag{34}\]  
\[\dot{\mathbf{E}}=\omega\mathbf{E}^{\mathrm{s}}\cos(\omega t) \tag{35}\]  
Substitute Equations (27)-(35) into differential Equations (23)-(26) of single\(-\)stage internal gear vibration, obtain 6 equations, and organize them into matrix form:  
\[ \begin{array}{l}-\omega^{2}\mathbf{M}\mathbf{U}^{\mathrm{s}}\sin(\omega t)- \omega^{2}\mathbf{M}\mathbf{U}^{\mathrm{c}}\cos(\omega t)+\omega\mathbf{C} \mathbf{U}^{\mathrm{s}}\cos(\omega t)\\ -\omega\mathbf{C}\mathbf{U}^{\mathrm{c}}\sin(\omega t)+\mathbf{K}f(\mathbf{U},b)+\mathbf{K}^{\prime\prime}f_{b}(\mathbf{U}^{\prime\prime},bb)=0\end{array}  \tag{36}\]  
where \(f(\mathbf{U},\ b),f_{b}(\mathbf{U}^{\prime\prime},\ bb)\) represent bearing backlash and tooth backlash functions, respectively. In order to assist calculation, the tooth backlash function is stated as a first\(-\)order polynomial, the radial and axial backlash coefficients are \(a\) and \(b\), respectively, and the meshing stiffness is chosen as the average stiffness. The matrix form is represented as follows:  
\[