Investigation of vibration characteristics of gear-bearing transmission system considering the effect of profile shifted coefficient

Abstract

In this paper, a dynamics model of the gear-bearing transmission system with bending and torsion coupling is established, and the influence of gear shifted on the vibration characteristics of the gear system is considered. First, the time-varying mesh stiffness of the profile shifted gear is derived by using the potential energy method. Then, based on the Lagrange equation numerical method, the vibration differential equation of the model is obtained, and the differential equation is solved by the fourth-order Runge-Kutta method. Finally, the influence of the profile shifted coefficient and input torque on the vibration characteristics of the gear transmission system is analyzed by using the time domain graph and the frequency domain graph. The conclusions drawn from this study can help engineers better understand the running performance and vibration characteristics of profile shifted gear.

Introduction

Gear transmission is widely utilized in aerospace, wind power generation, and other industries as one of the most frequently used mechanical transmission technologies. External excitations such as rotational speed and torque, together with internal excitations such as time-varying meshing stiffness, tooth side clearance, and transmission faults, cause the gear system’s dynamic behavior to become more complex. As a result, gear system dynamics and the factors that influence them have become a major research topic.

Time-varying meshing stiffness is one of the most important internal excitations that cause vibration and noise in gear transmission systems. At present, many scholars have used finite element method, analytical method, and loaded tooth contact method ¹ to calculate the time-varying mesh stiffness of gears. Baishun Zhao et al. ² established a dynamic model considering the influence of geometric eccentricity by using the load-tooth contact method. Qibin Wang ³ and Kangkang Chen ⁴ have established a finite element model of time-varying mesh stiffness of cylindrical spur gears considering the effect of complex matrix and cracks to investigate the effect of the web width and bore diameter on the time-varying mesh stiffness of gears. Shuo Feng et al. ⁵ combined finite element method and analytical method to solve time-varying meshing stiffness of inner gear pair and investigated its influence on ring thickness. Through finite element modeling, Siar Deniz Yavuz et al. ⁶ studied the complex dynamic behavior of the transmission system including parallel shaft gear and cross shaft gear. Jiaxing Zhang et al. ⁷ developed a finite element method for determining time-varying meshing stiffness of gears based on some commercial software, and analyzed the influence of gear teeth misalignment on stiffness. Considering the influence of gear cracks and drive shaft cracks on the gear transmission system, Yong Shen et al. ⁸ calculated the time-varying meshing stiffness of gear by finite element method and analyzed the dynamic response of different faults. Although the finite element method is more accurate in solving the time-varying gear mesh stiffness, however, it takes too long to calculate the solution and causes unnecessary waste of time, so most scholars prefer to use the analytical method. Del Rincon A F et al. ⁹ presented a method to calculate the meshing stiffness and loaded transmission error. At the same time, the friction and the possibility of contact were considered. Hui Ma et al. ¹⁰ proposed an analytical model of time-varying meshing stiffness of cylindrical spur gears considering extended meshing and modified fillet-base stiffness and analyzed the effects of different modification parameters on the stiffness of the gear system. Chongyang Xie et al. ¹¹ established a time-varying meshing stiffness model with tooth profile error by considering the influence of coupling flexibility between adjacent gear teeth. Zaigang Chen et al. ¹² established an improved gear analysis model by considering the effects of tooth deformation and tooth profile deviation and revealed the mechanism of the effect of tooth body structure coupling on gear tooth meshing stiffness. Jinhai Wang et al. ¹³ proposed a time-varying meshing stiffness calculation model for displacement gears and investigated the effect of displacement coefficient on gear tooth stiffness. Zaigang Chen et al. ¹⁴ established a time-varying meshing stiffness model for internal profile shifted gear pairs based on the potential energy principle. Zhifang Zhao et al. ¹⁵ established a fractal contact model considering the influence of tooth surface roughness on gear pair and studied the influence of roughness and tooth surface sliding friction on time-varying meshing stiffness. Diez-Ibarbia A et al. ¹⁶ proposed a model to assess the energy efficiency of the spur gear and investigated the effect of the profile shifted coefficient on efficiency.

The dynamic characteristics of the gear system are very complicated, so it is significant to study the mechanism of its influence on the healthy operation of the system. Zhao Weiqiang et al. ¹⁷ investigated the vibration characteristics of a pitting fault gear transmission system considering the effects of eccentricity and friction. Zheng Cao et al. ¹⁸ proposed a cylindrical spur gear dynamics model considering the effects of time-varying meshing stiffness, tooth clearance and tooth misalignment. Weiwei Liu et al. ¹⁹ established the high-speed gear transmission system of wind turbine considering the influence of electromagnetic torque disturbance and eccentricity, and analyzed the influence of electromagnetic torque disturbance on the dynamic response of the system. Sha Wei et al. ²⁰ established an uncertain single-stage gear dynamic model to investigate the effect of uncertain parameters to the system, and the results were verified by experiment. Yong Yi et al. ²¹ established a nonlinear model of gear transmission system considering time-varying pressure angle and tooth gap, and the dynamic characteristics of the system were investigated with tooth gap as the control variable. Wenchao Mo et al. ²² established a dynamic model of a helical gear transmission system considering the influence of sliding friction and gear error and studied the influence of gear error on the vibration characteristics of the system. Considered the gear eccentricity and nonlinear bearing force, Xiang et al. ²³ investigated the effect of gear eccentricity and rotational speed on the gear system.

In summary, the majority of recent research has focused on the normal cylindrical spur gear system, whereas with no consideration of the effect of gear profile shifted on the gear transmission system. In this paper, a nonlinear dynamics model of a 16-degree-of-freedom gear-rotor-bearing transmission system is established, the time-varying mesh stiffness of the profile shifted gear is computed by using the potential energy principle, the nonlinear differential equations are obtained by using Lagrange’s equation, and the differential equations are solved using the fourth-order Langer-Kutta method in this paper. The effects of the profile shifted coefficient and input torque on the dynamic characteristics of the gear system are investigated.

Time-varying meshing stiffness of profile shifted gears

The most dominant source of internal excitation for gear vibration noise is time-varying meshing stiffness, which has a straightforward impact on gear operational life. As a result, accurate estimation of time-varying meshing stiffness is critical for researching gear dynamic characteristics. In this paper, based on the potential energy principle, the gear tooth is simplified to a variable-section cantilever beam. As illustrated in Figure 1, its bending energy, shear energy, and axial compression energy are computed as follows:

$\{\begin{array}{l}{U}_{\mathrm{b}}=\frac{F^2}{2k_{\mathrm{b}}}={\int }_0^{\mathrm{d}}\frac{{\left[F_v\left(d-x\right)-F_{p}h\right]}^2}{2E{I}_x}\mathrm{d}x\\ U_{\mathrm{s}}=\frac{F^2}{2k_{\mathrm{s}}}={\int }_0^{\mathrm{d}}\frac{1.2F_v^2}{2G{A}_x}\mathrm{d}x\\ U_{\mathrm{a}}=\frac{F^2}{2k_{\mathrm{a}}}={\int }_0^{\mathrm{d}}\frac{F_p^2}{2E{A}_x}\mathrm{d}x\end{array}$

(1)

Where, $k_{\mathrm{b}}$ , $k_{\mathrm{s}}$ , and $k_{\mathrm{a}}$ are, respectively, represent the bending stiffness, the shear stiffness and the axial compression stiffness, $F_p$ and $F_v$ are the tangential and radial components of the tooth surface contact force F, E, and G are the modulus of elasticity and shear modulus, respectively, $I_x$ and $A_x$ are the moment of inertia and cross-sectional area at any point of the involute, respectively, d is the distance between the root of the tooth and the meshing point, h is the distance from the centerline of the gear to the meshing point, and the relevant quantities are calculated as follows:

$\{\begin{array}{l}h=R_b\left[\left({\alpha }_1+{\alpha }_2\right)\cos {\alpha }_1-\sin {\alpha }_1\right]\\ h_x=R_b\left[\left(\alpha +{\alpha }_2\right)\cos \alpha -\sin \alpha \right]\\ d=R_b\left[\left({\alpha }_1+{\alpha }_2\right)\sin {\alpha }_1+\cos {\alpha }_1-\cos {\alpha }_2\right]-R_r\cos {\alpha }_3\\ x=R_b\left[\cos \alpha -\left({\alpha }_2-\alpha \right)\sin \alpha -\cos {\alpha }_2\right]-R_r\cos {\alpha }_3\\ I_x=\frac{2}{3}{h}_x^{3}L\\ A_x=2h_{x}L\end{array}$

(2)

where α is the rotation angle of gear, α₂ represents the half tooth angle of the base circle, ${\alpha }_3$ is the half tooth angle of the root circle, $R_{\mathrm{b}}$ is the base circle radius, $R_{\mathrm{r}}$ is the root circle radius, L is the tooth width.

$\{\begin{array}{l}{\alpha }_2=\frac{\pi }{2z_i}+\tan {\alpha }_0-{\alpha }_0\left(i=\mathrm{1,2}\right)\\ {\alpha }_2^{\prime }=\frac{\pi }{2z_i}+\tan {\alpha }_0-{\alpha }_0+\frac{2X_i\tan \hspace{0.17em}{\alpha }_0}{z_i}\left(i=\mathrm{1,2}\right)\end{array}$

(3)

Figure 1.

Spur gear cantilever beam model.

When the gear is shifted, ${\alpha }_2^{\prime }$ will replace α₂, at the same time, the meshing angle of the gear will change after the shift and it is no longer equal to the pressure angle of gear reference circle, which is calculated as follows:

$inv{\alpha }_0^{\prime }=2\hspace{0.17em}\tan \hspace{0.17em}{\alpha }_0\left(\frac{X_1+X_2}{z_1+z_2}\right)+inv{\alpha }_0$

(4)

where $X_i\left(i=\mathrm{1,2}\right)$ represents the profile shifted coefficient of the gear and the pinion, respectively. “inv” is the involute function which defined as $inv\beta =\tan \beta -\beta$ . According to the formula for calculating the minimum profile shifted coefficient of the gear without root cut, the minimum profile shifted coefficient of the pinion can be determined as −0.17. Figure 2 show the variation relationship between gear profile shifted coefficient and meshing angle, and the range of displacement coefficient is selected from −0.15 to 0.15 in this study.

Figure 2.

Engagement angle of the profile shifted gear.

The center distance of the gear will also change accordingly with the change of the profile shifted coefficient, and the center distance of the gear after the shifting is as follows:

$a^{\prime }=m\left(\frac{z_1+z_2}{2}+Y\right)$

(5)

where, Y is the center distance variation coefficient, which is calculated as follows:

$Y=\frac{z_1+z_2}{2}\left(\frac{\cos {\alpha }_0}{\cos {\alpha }_0^{\prime }}-1\right)$

(6)

The time-varying mesh stiffness of gears is solved based on the energy method, bending stiffness k_b, shear stiffness $k_{\mathrm{s}}$ and axial compression stiffness k_a are calculated as follows: ¹⁷

$\begin{array}{l}\frac{1}{k_{\mathrm{b}}}=\frac{{\left[1-\frac{\left(Z-2.5\right)\cos {\alpha }_1\cos {\alpha }_3}{Z\cos {\alpha }_0^{\prime }}\right]}^3-{\left(1-\cos {\alpha }_1\cos {\alpha }_2^{\prime }\right)}^3}{2EL\cos {\alpha }_1{\sin }^3{\alpha }_2^{\prime }}\\ +{\int }_{-{\alpha }_1}^{{\alpha }_2^{\prime }}\frac{3{\left\{1+\cos {\alpha }_1\left[\left({\alpha }_2^{\prime }-\alpha \right)\sin \alpha -\cos \alpha \right]\right\}}^2\left({\alpha }_2^{\prime }-\alpha \right)\cos \alpha }{2EL\left({\left[\sin \alpha +\left({\alpha }_2^{\prime }-\alpha \right)\cos \alpha \right]}^3\right)}\mathrm{d}\alpha \end{array}$

(7)

$\begin{array}{l}\frac{1}{k_{\mathrm{s}}}=\frac{1.2\left(1+\nu \right){\cos }^2{\alpha }_1\left(\cos {\alpha }_2^{\prime }-\frac{Z-2.5}{Z\cos {\alpha }_0^{\prime }}\cos {\alpha }_3\right)}{EL\sin {\alpha }_2^{\prime }}\\ +{\int }_{-{\alpha }_1}^{{\alpha }_2^{\prime }}\frac{1.2\left(1+\nu \right)\left({\alpha }_2^{\prime }-\alpha \right)\cos \alpha \hspace{0.17em}{\cos }^2{\alpha }_1}{EL\left(\left[\sin \alpha +\left({\alpha }_2^{\prime }-\alpha \right)\cos \alpha \right]\right)}\mathrm{d}\alpha \end{array}$

(8)

$\begin{array}{l}\frac{1}{k_{\mathrm{a}}}=\frac{{\sin }^2{\alpha }_1\left(\cos {\alpha }_2^{\prime }-\frac{Z-2.5}{Z\cos {\alpha }_0^{\prime }}\cos {\alpha }_3\right)}{2EL\sin {\alpha }_2^{\prime }}\\ +{\int }_{-{\alpha }_1}^{{\alpha }_2^{\prime }}\frac{\left({\alpha }_2^{\prime }-\alpha \right)\cos \alpha {\sin }^2{\alpha }_1}{2EL\left(\left[\sin \alpha +\left({\alpha }_2^{\prime }-\alpha \right)\cos \alpha \right]\right)}\mathrm{d}\alpha \end{array}$

(9)

Due to the phenomenon of alternating single and double tooth meshing in the process of gear meshing, therefore, ${\alpha }_1={\alpha }_{i,j}\left(i,j=\mathrm{1,2}\right)$ , the calculation is as follows:

${\alpha }_{\mathrm{1,1}}={\theta }_1-\frac{\pi }{2z_1}-inv{\alpha }_0^{\prime }+\psi$

(10)

${\alpha }_{\mathrm{1,2}}={\alpha }_{\mathrm{1,1}}+\frac{2\pi }{z_1}$

(11)

${\alpha }_{\mathrm{2,1}}=\tan \left(\arccos \frac{z_2\cos {\alpha }_0^{\prime }}{z_2+2+2\times x_2}\right)-\frac{\pi }{2z_2}-inv{\alpha }_0^{\prime }-\frac{z_1}{z_2}{\theta }_1$

(12)

${\alpha }_{\mathrm{2,2}}={\alpha }_{\mathrm{2,1}}-\frac{2\pi }{z_2}$

(13)

$\psi =\tan \left[\arccos \frac{z_1\cos {\alpha }_0^{\prime }}{\sqrt{{(z_2+2+2\times X_2)}^2+(z_1+z_2+2\times Y{)}^2-2(z_2+2+2\times X_2)(z_1+z_2+2\times Y\times \cos (\arccos \frac{z_2\cos {\alpha }_0^{\prime }}{z_2+2+2\times X_2}-{\alpha }_0^{\prime })}}\right]$

(14)

The Hertz contact stiffness of the gear pair is calculated as follow: ²⁴

$k_h=\frac{\pi EL}{4\left(1-{\nu }^2\right)}$

(15)

where $\nu$ is the Poisson ratio.

Combining equations (7)–(15), the total time-varying meshing stiffness of the gear pair is calculated as follows:

$\{\begin{array}{cc}{k}_t=\frac{1}{\frac{1}{k_{\mathrm{h}}}+\frac{1}{k_{\mathrm{b}1}}+\frac{1}{k_{\mathrm{s}1}}+\frac{1}{k_{\mathrm{a}1}}+\frac{1}{k_{\mathrm{b}2}}+\frac{1}{k_{\mathrm{s}2}}+\frac{1}{k_{\mathrm{a}2}}}& \mathrm{S}\mathrm{i}\mathrm{n}\mathrm{g}\mathrm{l}\mathrm{e}‐\mathrm{t}\mathrm{o}\mathrm{o}\mathrm{t}\mathrm{h}‐\mathrm{p}\mathrm{a}\mathrm{i}\mathrm{r}\\ k_t={\displaystyle \sum }_{i=1}^2\frac{1}{\frac{1}{k_{\mathrm{h},i}}+\frac{1}{k_{\mathrm{b}1,i}}+\frac{1}{k_{\mathrm{s}1,i}}+\frac{1}{k_{\mathrm{a}1,i}}+\frac{1}{k_{\mathrm{b}2.i}}+\frac{1}{k_{\mathrm{s}2,i}}+\frac{1}{k_{\mathrm{a}2,i}}}& \mathrm{D}\mathrm{o}\mathrm{u}\mathrm{b}\mathrm{l}\mathrm{e}‐\mathrm{t}\mathrm{o}\mathrm{o}\mathrm{t}\mathrm{h}‐\mathrm{p}\mathrm{a}\mathrm{i}\mathrm{r}\end{array}$

(16)

The time-varying meshing stiffness of the profile shifted gear is shown in Figure 3, from which it can be seen that the time-varying meshing stiffness increases with the increase of the shifting coefficient when the gears take positive shifting, and the changes is nonlinear, while the double-tooth meshing duration decreases and the single-tooth meshing duration increases. This is because the gear overlap ratio decreases when the gear is in positive transmission, and the tooth thickness of the positive transmission gear increases compared to the standard gear. The opposite result is obtained when the gears are negatively shifted, and the magnitude of the time-varying meshing stiffness decreases as the absolute value of the shifting coefficient increases, while the double meshing duration increases and the single meshing duration decreases.

Figure 3.

Time-varying meshing stiffness of profile shifted gears. (a) Positive shifted transmission. (b) Negative shifted transmission.

Dynamic model

The dynamic model of the gear system

For the convenience to illustrate, a diagram of the structure of the gear-bearing transmission system is shown in Figure 4. The system is symmetrically arranged with the gear shaft, utilizing deep groove ball bearings as the support of the gear shaft, and ignoring the effects of installation errors and thermal deformation of each part. For this structural diagram, a multi-degree-freedom bending-torsion coupled gear-rotor-bearing transmission system centralized mass dynamics model considering profile shifted and friction is established in this paper, as shown in Figure 4.

Figure 4.

The structural diagram of the gear-rotor-bearing transmission system.

In Figure 4, O₁, O₂ are, respectively, represent the theoretical rotational center of the driving gear and driven gear, $B_i\left(i=\mathrm{1,2,3,4}\right)$ are the rotational center of the bearings. $J_d$ and $J_g$ are the rotational inertia of the input and the output.

In Figure 5, m₁ and m₂ are the mass of the driving and driven gear, G₁ (x_g1, y_g1) and G₂ (x_g2, y_g2) are the center-of-mass coordinate of the gear. Eccentric distances of gear pair are ρ₁ and ρ₂, in addition, ρ₂ = 2ρ₁. r_b1 and r_b2 are the radii of the base circles. The moment of inertia with reference to the center of mass is J₁ and J₂. $m_{bi}\left(i=\mathrm{1,2,3,4}\right)$ represent the mass of bearings.

Figure 5.

Dynamics model of bent-torsion coupled gear-rotor-bearing transmission system.

As shown in Figure 5, The angular displacement generated by the flexible vibration of the gear, including the input/output, driving/driven gear, are assumed to be ${\theta }_d$ , ${\theta }_g$ , ${\theta }_1$ and ${\theta }_2$ , the rotational angle are given by

$\begin{array}{l}{\phi }_{\mathrm{d}}={\omega }_{1}t+{\theta }_{\mathrm{d}},{\phi }_{\mathrm{g}}={\omega }_{2}t+{\theta }_{\mathrm{g}}\\ {\phi }_1={\omega }_{1}t+{\theta }_1,{\phi }_2={\omega }_{2}t+{\theta }_2\end{array}$

(17)

where ${\omega }_i\left(i=\mathrm{1,2}\right)$ is the angular velocity of the driving and driven gear. t is the meshing time. The relationship between the actual center of the gear and the center of mass can be written as

$\{\begin{array}{l}{x}_{\mathrm{g}1}=x_1+{\rho }_1\cos {\phi }_1,y_{\mathrm{g}1}=y_1+{\rho }_1\sin {\phi }_1\\ x_{\mathrm{g}2}=x_2+{\rho }_2\cos {\phi }_2,y_{\mathrm{g}2}=y_2+{\rho }_2\sin {\phi }_2\end{array}$

(18)

The elastic deformation of the gear drive shafts in the direction of rotation as follow:

$\{\begin{array}{l}{\delta }_{x1}=x_1-{\xi }_2{x}_{\mathrm{b}1}-{\xi }_1{x}_{\mathrm{b}2},{\delta }_{y1}=y_1-{\xi }_2{y}_{\mathrm{b}1}-{\xi }_1{y}_{\mathrm{b}2}\\ {\delta }_{x2}=x_2-{\xi }_3{x}_{\mathrm{b}4}-{\xi }_4{x}_{\mathrm{b}3},{\delta }_{y2}=y_2-{\xi }_3{y}_{\mathrm{b}4}-{\xi }_4{y}_{\mathrm{b}3}\\ {\xi }_1=l_{11}/l_1,{\xi }_2=l_{12}/l_1,{\xi }_3=l_{23}/l_2,{\xi }_4=l_{24}/l_2\end{array}$

(19)

Dynamic transmission error between two meshing gears is defined as

$\{\begin{array}{l}\delta \left(t\right)=\left(r_{\mathrm{b}1}{\phi }_1-r_{\mathrm{b}2}{\phi }_2\right)+\left(y_{\mathrm{g}1}-y_{\mathrm{g}2}\right)-e\left(t\right)\\ =\left(r_{\mathrm{b}1}{\phi }_1-r_{\mathrm{b}2}{\phi }_2\right)+\left(y_1-y_2+{\rho }_1\sin {\phi }_1-{\rho }_2\sin {\phi }_2\right)-e\left(t\right)\\ e\left(t\right)=e_0+e_r\sin \left({\omega }_{m}t+{\phi }_m\right)\end{array}$

(20)

where $e\left(t\right)$ was static transmission error which generated from manufacturing and assembling. e₀ and e_r are the mean as well as the amplitude of the meshing errors, respectively.

During the gear meshing operation, the dynamic meshing force F_m was specified as:

$F_m={\displaystyle \sum }_{i=1}^j{F}_{mi}={\displaystyle \sum }_{i=1}^j{k}_{mi}f\left(\delta \hspace{0.17em}\left(t\right)\right)+c_{mi}{\delta }^{\prime }\left(t\right)$

(21)

where j = 1,2, only one pair of teeth is involved in the meshing process of the gear at j = 1, otherwise, when there are two pairs of gear teeth involved in the meshing, j = 2. $f\left(\delta \left(t\right)\right)$ is the nonlinear function which can be written as follows:

$f\left(\delta \left(t\right)\right)=\{\begin{array}{l}\delta \left(t\right)-\left(1-\tau \right)b \delta \left(t\right)>b\\ \tau b‐b<\delta \left(t\right)<b,0\le \tau \le 1\\ \delta \left(t\right)+\left(1-\tau \right)b \delta \left(t\right)<-b\end{array}$

(22)

where, $\tau$ is the coefficient of the nonlinear function that can be defined as 0 in this paper. ¹⁷ b is the gear backlash.

Figure 6 is a diagram of the gear pair meshing. In Figure 6, the meshing point is M, $\overline{N_{1}M}$ and $\overline{N_{2}M}$ are the friction force arms. The direction of frictional force changed at the pitch point during the meshing process. F_f1 and F_f2 represented the frictional forces of the gears and have the same magnitude and opposite direction. In this paper, the direction of the friction force was defined in Figure 5. It is assumed that the friction coefficient does not change with tooth meshing, and the friction force and friction moment of the gear can be calculated as follows:

$\{\begin{array}{l}\overline{N_{1}M}=\left(r_{b1}+r_{b2}\right)\tan \hspace{0.17em}\alpha -\sqrt{r_{a2}^2-r_{b2}^2+r_{b1}{\omega }_{1}t}\\ \overline{N_{2}M}=\left(r_{b1}+r_{b2}\right)\tan \hspace{0.17em}\alpha -\overline{N_{1}M}\end{array}$

(23)

$\{\begin{array}{l}{F}_f=\mu \lambda \left(v_s\right)F_m\\ T_{f1}=F_{f1}\overline{N_{1}M},T_{f2}=F_{f2}\overline{N_{2}M}\end{array}$

(24)

where, $\mu$ is the friction coefficient, v_s is the slide velocity on the meshing point, $r_{bi}\left(i=\mathrm{1,2}\right)$ represent base circle radius of the driving gear and driven gear, r_a2 is the addendum circle radius of the driven gear, $\lambda \left(v_s\right)$ is the direction coefficient of slide velocity and can be seen as follow:

$\lambda \left(v_s\right)=\{\begin{array}{l}1 v_s>0\\ 0 v_s=0\\ -1 v_s<0\end{array}$

(25)

Figure 6.

Diagram of the gear pair meshing.

The dynamic of the ball bearing

Figure 7 is the diagram of the ball bearing. The outer ring of the bearing has interference fit on the bearing seat, and the inner ring has also interference fit with the shaft, and the balls are evenly arranged between the inner and outer rings. It can be ignored the centrifugal force of the balls. The deformation of the ball can be express as

${\eta }_j=x_{bi}\cos {\varphi }_j+y_{bi}\sin {\varphi }_j-\gamma ,\left(i=\mathrm{1,2,3,4}.j=\mathrm{1,2,3},...,N_h\right)$

(26)

where $x_{bi}$ and $y_{bi}$ are the vibration displacements along the x and y directions generated by the bearing during the running process. $\gamma$ is the bearing backlash. The force between the bearing ball and ball races can be solved according to the Hertzian contact principle that can be seen as follow:

$f_j=K_b{\left({\eta }_j\right)}^{3/2}\cdot H\left({\eta }_j\right)$

(27)

where $K_b$ is the bearing contact stiffness, and $H\left({\eta }_j\right)$ represent the Heaviside function.

Figure 7.

The diagram of the ball bearing.

The dynamic bearing force is decomposed into component forces along the x, y directions. The two component forces are written as

$\{\begin{array}{l}{F}_{xi}={\displaystyle \sum }_{j=1}^{N_b}{f}_j\cos \hspace{0.17em}{\varphi }_j\\ F_{yi}={\displaystyle \sum }_{j=1}^{N_b}{f}_j\sin \hspace{0.17em}{\varphi }_j\end{array}\left(i=\mathrm{1,2,3,4}\right)$

(28)

Mathematical model of gear system

The vibration differential equation for the 16 degrees of freedom of the gear-bearing transmission system can be derived from the Lagrangian equation. The differential equation of the system is obtained by carrying equations (30)–(34) into the Lagrangian equation (29). The Lagrangian equation is as follows:

$\frac{\partial }{\partial t}\left(\frac{\partial T}{\partial X^{\prime }}\right)-\frac{\partial T}{\partial X}+\frac{\partial U}{\partial X}+\frac{\partial R}{\partial X^{\prime }}=F$

(29)

The displacement vector of the gear system is shown below:

$\mathbf{X}=\left[{\theta }_{\mathrm{d}} {\theta }_{\mathrm{g}} x_1 y_1 {\theta }_1 x_2 y_2 {\theta }_2 x_{\mathrm{b}1} y_{\mathrm{b}1} x_{\mathrm{b}2} y_{\mathrm{b}2} x_{\mathrm{b}3} y_{\mathrm{b}3} x_{\mathrm{b}4} y_{\mathrm{b}4}\right]$

(30)

The system takes into account the effects of friction, load, time-varying meshing stiffness, etc. The kinetic energy of the system T, the potential energy U due to the deformation of gears, drive shafts and bearings, and the dissipation energy R taking into account the damping effect of the components are all calculated. It can be described as

$T=\frac{1}{2}\left[m_i\left({x^{\prime }}_{g1}^2+{y^{\prime }}_{g1}^2\right)+m_{bi}\left({x^{\prime }}_{b1}^2+{y^{\prime }}_{b1}^2\right)+J_d{{\phi }^{\prime }}_d^2+J_1{{\phi }^{\prime }}_1^2+J_2{{\phi }^{\prime }}_2^2+J_g{{\phi }^{\prime }}_g^2\right]$

(31)

$U=\frac{1}{2}[k_{\mathrm{s}i}\left({\delta }_{xi}^2+{\delta }_{yi}^2\right)+k_{\mathrm{t}1}{\left({\phi }_1-{\phi }_{\mathrm{d}}\right)}^2+k_{\mathrm{t}2}{\left({\phi }_{\mathrm{g}}-{\phi }_2\right)}^2]$

(32)

$R=\frac{1}{2}\left[c_{si}\left({{\delta }^{\prime }}_{xi}^2+{{\delta }^{\prime }}_{yi}^2\right)+c_{t1}{\left({\phi }_1^{\prime }-{\phi }_d^{\prime }\right)}^2+c_{t2}{\left({\phi }_g^{\prime }-{\phi }_2^{\prime }\right)}^2+c_{bxi}{x^{\prime }}_{bi}^2+c_{byi}{y^{\prime }}_{bi}^2\right]$

(33)

The generalized force vector is given below:

$\begin{array}{l}\mathbf{F}=[\begin{array}{cccccccc}{T}_{\mathrm{d}}& -T_{\mathrm{g}}& -F_{\mathrm{f}}& -F_{\mathrm{m}}-m_{1}g& -F_{\mathrm{m}}{r}_{\mathrm{b}1}+T_{\mathrm{f}1}& F_{\mathrm{f}}& -F_{\mathrm{m}}-m_{2}g& -F_{\mathrm{m}}{r}_{\mathrm{b}2}-T_{\mathrm{f}2}\end{array}\\ \begin{array}{cccccccc}{F}_{x1}& F_{y1}-m_{\mathrm{b}1}g& F_{x2}& F_{y2}-m_{\mathrm{b}2}g& F_{x3}& F_{y3}-m_{\mathrm{b}3}g& F_{x4}& F_{y4}-m_{\mathrm{b}4}g\end{array}]\end{array}$

(34)

where $F_x$ and $F_y$ are the bearing forces along the x and y directions during the operation of the bearing, g is the acceleration of gravity, and it equals to 9.8 m/s². Combining the above equations, the vibration differential equation can be obtained.

Differential equations of vibration at the input and output of the system are seen in below:

$\{\begin{array}{l}{J}_d:J_d{\theta }_d^{″}+c_{t1}\left({\theta }_d^{\prime }-{\theta }_1^{\prime }\right)+k_{t1}\left({\theta }_d-{\theta }_1\right)=T_d\\ J_g:J_g{\theta }_g^{″}+c_{t2}\left({\theta }_g^{\prime }-{\theta }_2^{\prime }\right)+k_{t2}\left({\theta }_g-{\theta }_2\right)=-T_g\end{array}$

(35)

Differential equations for the vibration of the driving and driven gears are seen in below:

$\begin{array}{l}\{\begin{array}{cc}{x}_1:& \begin{array}{l}{m}_1{x}_1^{″}+c_{\mathrm{s}1}\left(x_1^{\prime }-{\xi }_2{x}_{\mathrm{b}1}^{\prime }-{\xi }_1{x}_{\mathrm{b}2}^{\prime }\right)+k_{\mathrm{s}1}\left(x_1-{\xi }_2{x}_{\mathrm{b}1}-{\xi }_1{x}_{\mathrm{b}2}\right)\\ =m_1{\rho }_1{\theta }_1^{″}\sin \left({\omega }_{1}t+{\theta }_1\right)+m_1{\rho }_1{\left({\omega }_1+{\theta }_1^{\prime }\right)}^2\cos \left({\omega }_{1}t+{\theta }_1\right)-F_{\mathrm{f}}\end{array}\\ y_1:& \begin{array}{l}{m}_2{x}_2^{″}+c_{\mathrm{s}2}\left(x_2^{\prime }-{\xi }_4{x}_{\mathrm{b}3}^{\prime }-{\xi }_3{x}_{\mathrm{b}4}^{\prime }\right)+k_{s2}\left(x_2-{\xi }_4{x}_{\mathrm{b}3}-{\xi }_3{x}_{\mathrm{b}4}\right)\\ =m_2{\rho }_2{\theta }_2^{″}\sin \left({\omega }_{2}t+{\theta }_2\right)+m_2{\rho }_2{\left({\omega }_2+{\theta }_2^{\prime }\right)}^2\cos \left({\omega }_{2}t+{\theta }_2\right)-F_{\mathrm{m}}-m_{1}g\end{array}\\ {\theta }_1:& \begin{array}{l}\left(J_1+m_1{\rho }_1^2\right){\theta }_1^{″}+c_{\mathrm{t}1}\left({\theta }_1^{\prime }-{\theta }_{\mathrm{d}}^{\prime }\right)+k_{\mathrm{t}1}\left({\theta }_1-{\theta }_{\mathrm{d}}\right)\\ =m_1{\rho }_1\sin \left({\omega }_{1}t+{\theta }_1\right)x_1^{″}-m_1{\rho }_1\cos \left({\omega }_{1}t+{\theta }_1\right)y_1^{″}-F_m{r}_{\mathrm{b}1}+T_{\mathrm{f}1}\end{array}\end{array}\\ \{\begin{array}{cc}{x}_2:& \begin{array}{l}{m}_2{x}_2^{″}+c_{\mathrm{s}2}\left(x_2^{\prime }-{\xi }_4{x}_{\mathrm{b}3}^{\prime }-{\xi }_3{x}_{\mathrm{b}4}^{\prime }\right)+k_{\mathrm{s}2}\left(x_2-{\xi }_4{x}_{\mathrm{b}3}-{\xi }_3{x}_{\mathrm{b}4}\right)\\ =m_2{\rho }_2{\theta }_2^{″}\sin \left({\omega }_{2}t+{\theta }_2\right)+m_2{\rho }_2{\left({\omega }_2+{\theta }_2^{\prime }\right)}^2\cos \left({\omega }_{2}t+{\theta }_2\right)+F_{\mathrm{f}}\end{array}\\ y_2:& \begin{array}{l}{m}_2{y}_2^{″}+c_{\mathrm{S}2}\left(y_2^{\prime }-{\xi }_4{y}_{\mathrm{b}3}^{\prime }-{\xi }_3{y}_{\mathrm{b}4}^{\prime }\right)+k_{s1}\left(y_2-{\xi }_4{y}_{\mathrm{b}3}-{\xi }_3{y}_{\mathrm{b}4}\right)\\ =m_2{\rho }_2{\left({\omega }_2+{\theta }_2^{\prime }\right)}^2\sin \left({\omega }_{2}t+{\theta }_2\right)-m_2{\rho }_2{\theta }_2^{″}\cos \left({\omega }_{2}t+{\theta }_2\right)-F_m-m_{2}g\end{array}\\ {\theta }_2:& \begin{array}{l}\left(J_2+m_2{\rho }_2^2\right){\theta }_2^{″}+c_{\mathrm{t}2}\left({\theta }_2^{\prime }-{\theta }_{\mathrm{g}}^{\prime }\right)+k_{t2}\left({\theta }_2-{\theta }_{\mathrm{g}}\right)\\ =m_2{\rho }_2\sin \left({\omega }_{2}t+{\theta }_2\right)x_2^{″}-m_2{\rho }_2\cos \left({\omega }_{2}t+{\theta }_2\right)y_2^{″}-F_{\mathrm{m}}{r}_{\mathrm{b}2}-T_{\mathrm{f}2}\end{array}\end{array}\end{array}$

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The differential equation of vibration of the bearing is as follows:

$\{\begin{array}{c}{x}_{\mathrm{b}1}:\\ y_{\mathrm{b}1}:\\ x_{\mathrm{b}2}:\\ y_{\mathrm{b}2}:\\ x_{\mathrm{b}3}:\\ y_{\mathrm{b}3}:\\ x_{\mathrm{b}4}:\\ y_{\mathrm{b}4}:\end{array}\begin{array}{c}{m}_{\mathrm{b}1}{x}_{\mathrm{b}1}^{″}+c_{\mathrm{s}1}{\xi }_2\left(-x_1^{\prime }+{\xi }_2{x}_{\mathrm{b}1}^{\prime }+{\xi }_1{x}_{\mathrm{b}2}^{\prime }\right)+c_{x1}{x}_{\mathrm{b}1}^{\prime }+k_{\mathrm{s}1}{\xi }_2\left(-x_1+{\xi }_2{x}_{\mathrm{b}1}+{\xi }_1{x}_{\mathrm{b}2}\right)=F_{x1}\\ m_{\mathrm{b}1}{y}_{\mathrm{b}1}^{″}+c_{\mathrm{s}1}{\xi }_2\left(-y_1^{\prime }+{\xi }_2{y}_{\mathrm{b}1}^{\prime }+{\xi }_1{y}_{\mathrm{b}2}^{\prime }\right)+c_{y1}{y}_{\mathrm{b}1}^{\prime }+k_{\mathrm{s}1}{\xi }_2\left(-y_1+{\xi }_2{y}_{\mathrm{b}1}+{\xi }_1{y}_{\mathrm{b}2}\right)=F_{y1}-m_{\mathrm{b}1}g\\ m_{\mathrm{b}2}{x}_{\mathrm{b}2}^{″}+c_{\mathrm{s}1}{\xi }_1\left(-x_1^{\prime }+{\xi }_2{x}_{\mathrm{b}1}^{\prime }+{\xi }_1{x}_{\mathrm{b}2}^{\prime }\right)+c_{x2}{x}_{\mathrm{b}2}^{\prime }+k_{\mathrm{s}1}{\xi }_1\left(-x_1+{\xi }_2{x}_{\mathrm{b}1}+{\xi }_1{x}_{\mathrm{b}2}\right)=F_{x2}\\ m_{\mathrm{b}2}{y}_{\mathrm{b}2}^{″}+c_{\mathrm{s}1}{\xi }_1\left(-y_1^{\prime }+{\xi }_2{y}_{\mathrm{b}1}^{\prime }+{\xi }_1{y}_{\mathrm{b}2}^{\prime }\right)+c_{y2}{y}_{\mathrm{b}1}^{\prime }+k_{\mathrm{s}1}{\xi }_1\left(-y_1+{\xi }_2{y}_{\mathrm{b}1}+{\xi }_1{y}_{\mathrm{b}2}\right)=F_{y2}-m_{\mathrm{b}2}g\\ m_{\mathrm{b}3}{x}_{\mathrm{b}3}^{″}+c_{\mathrm{s}2}{\xi }_4\left(-x_2^{\prime }+{\xi }_4{x}_{\mathrm{b}3}^{\prime }+{\xi }_3{x}_{\mathrm{b}4}^{\prime }\right)+c_{x3}{x}_{\mathrm{b}3}^{\prime }+k_{\mathrm{s}2}{\xi }_4\left(-x_2+{\xi }_4{x}_{\mathrm{b}3}+{\xi }_3{x}_{\mathrm{b}4}\right)=F_{x3}\\ \begin{array}{c}{m}_{\mathrm{b}3}{y}_{\mathrm{b}3}^{″}+c_{\mathrm{s}2}{\xi }_4\left(-y_2^{\prime }+{\xi }_4{y}_{\mathrm{b}3}^{\prime }+{\xi }_3{y}_{\mathrm{b}4}^{\prime }\right)+c_{y3}{y}_{\mathrm{b}3}^{\prime }+k_{\mathrm{s}2}{\xi }_4\left(-y_2+{\xi }_4{y}_{\mathrm{b}3}+{\xi }_3{y}_{\mathrm{b}4}\right)=F_{y3}-m_{\mathrm{b}3}g\\ \begin{array}{c}{m}_{\mathrm{b}4}{x}_{\mathrm{b}4}^{″}+c_{\mathrm{s}2}{\xi }_3\left(-x_2^{\prime }+{\xi }_4{x}_{\mathrm{b}3}^{\prime }+{\xi }_3{x}_{\mathrm{b}4}^{\prime }\right)+c_{x4}{x}_{\mathrm{b}4}^{\prime }+k_{\mathrm{s}2}{\xi }_3\left(-x_2+{\xi }_4{x}_{\mathrm{b}3}+{\xi }_3{x}_{\mathrm{b}4}\right)=F_{x4}\\ m_{\mathrm{b}4}{y}_{\mathrm{b}4}^{″}+c_{\mathrm{s}2}{\xi }_3\left(-y_2^{\prime }+{\xi }_4{y}_{\mathrm{b}3}^{\prime }+{\xi }_3{y}_{\mathrm{b}4}^{\prime }\right)+c_{y4}{y}_{\mathrm{b}4}^{\prime }+k_{\mathrm{s}2}{\xi }_3\left(-y_2+{\xi }_4{y}_{\mathrm{b}3}+{\xi }_3{y}_{\mathrm{b}4}\right)=F_{y4}-m_{\mathrm{b}4}g\end{array}\end{array}\end{array}$

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Result

The gear-rotor-bearing transmission system has a number of nonlinear factors as well as a complicated dynamic behavior. The vibration differential equations are solved using the fourth-order Lundgren-Kutta method in this paper, and the effects of the profile shifted coefficient and input torque on the system’s vibration characteristics are investigated in the sections that follow. Tables 1–3 indicate the gear, driving shaft, and bearing parameters, respectively.

Table 1.

Parameters of the gears.

Parameter	Symbol	Numerical value
Number of teeth	$z_1,z_2$	30, 20
Modulus	$M$	4 mm
Tooth width	$L$	38 mm
Moment of inertial	$J_1/J_2$	3.15 × 10−3/6.22 × 10−4 kg·m2
Backlash	b	2 × 10−5 m

Table 2.

Parameters of the shafts.

Parameter	Symbol	Numerical value
Moment of inertia of input/output	$J_d/J_g$	8.25 × 10−4/1.75 × 10−3 kg·m2
Input/output torque	$T_d/T_g$	240/160 N·m
Torsional stiffness of the shaft	$k_{s1}/k_{s2}$	5 × 107/4 × 107 N/m
Bending stiffness of the shaft	$k_{t1}/k_{t2}$	3 × 106/1 × 106 N/m

Table 3.

Parameters of the bearings.

Parameter	Symbol	Numerical value
Mass of the bearings	$m_{bi}/m_{bj}\left(i=\mathrm{1,2}.j=\mathrm{3,4}\right)$	0.94/0.54 kg
Outer ring radius of bearing	$R_i/R_j$	0.031/0.021 m
Inner ring radius of bearing	$r_i/r_j$	0.02/0.015 m
Contact stiffness of the bearings	$P_{bi}/P_{bj}$	13.34 × 108/10.56 × 108 N/m
Ball number	$N_i/N_j$	14/18

Effect of profile shifted on vibration characteristics

In this section, on the basis of the time-varying meshing stiffness calculation of the above-mentioned profile shifted gears, the effect of different profile shifted coefficients on the vibration characteristics of the gear transmission system was investigated. The speed of the driving gear was set to 3000 r/min and the tooth surface friction coefficient is 0.1, the pinion shifting coefficient was selected as 0, 0.1, 0.15, −0.1, −0.15, and the other parameters were taken as the same values.

Figure 8 shows the time domain waveforms of the pinion with various profile shifted coefficients in the x₂, y₂, and θ₂ directions. The profile shifted coefficient has varying degrees of influence on the gear’s vibration in different directions. The vibration displacement in the θ₂ direction is significantly large. The vibration displacement in the x₂ direction is most clearly affected by the profile shifted coefficient variation in Figure 8(a), as well as the vibration displacement decreasing when a negative profile shifted coefficient is employed and increasing when a positive profile shifted coefficient is used. The amplitude of the vibration displacement in the y₂, θ₂ directions does not change considerably in Figure 8(b) and (c), but the change in the θ₂ direction is more obvious. Because the profile shifted coefficient has no effect on the gear system’s vibration waveform, the gear shifted has no effect on the gear system’s initial motion form.

Figure 8.

The time domain waveform at different profile shifted: (a) The displacement of x₂ direction; (b) The displacement of y₂ direction; (c) The displacement of θ₂ direction.

The peak-peak value (P-P) of the time domain waveform in the different directions is shown in Figure 9. It indicates the degree of changes in the vibration of the gear system. P-P has a tendency to decrease and then increase in the x₂ and y₂ direction. In the θ₂ direction, it changes in alternating forms of increase and decrease. According to the changing trend of the P-P, the minimum value appears in the profile shifted coefficient −0.06,818. The vibration displacement of the gear system is minimal in this case.

Figure 9.

The peak-peak value of the time domain waveform: (a) In x₂ direction; (b) In y₂ direction; (c) In θ₂ direction.

The spectrum plots of standard gearing in the x₂, y₂, and θ₂ directions are shown in Figure 10(a)–(c). As shown in the figure the gears have approximately the same frequency components in all three directions, and the meshing frequency $f_{\mathrm{m}}$ ( $\hspace{0.17em}{f}_{\mathrm{m}}=\left(n_1\times z_1/60=1500\mathrm{H}\mathrm{z}\right)$ ) is the dominant frequency, much larger than the other frequency components. In addition, multiples of the meshing frequency can be observed that is $1/16f_{\mathrm{m}},1/8f_{\mathrm{m}},1/4f_{\mathrm{m}},1/2f_{\mathrm{m}},2f_{\mathrm{m}},3f_{\mathrm{m}}$ , etc. It can also be observed that the bearing variable stiffness frequency $f_{\mathrm{b}2}$ ( $N_2{r}_2{n}_2/\left[60\times \left(R_2+r_2\right)\right]=563.4\mathrm{H}\mathrm{z}$ ) and its harmonics $0.5f_{\mathrm{b}2}$ , $1.5f_{\mathrm{b}2}$ , $2f_{\mathrm{b}2}$ and $2.5f_{\mathrm{b}2}$ , as well as the system rotation frequency $f_{\mathrm{r}2}$ ( $n_2/60=75\mathrm{H}\mathrm{z}$ ). What is different is that in Figure 10(c), the bearing variable stiffness frequency $f_{\mathrm{b}1}$ ( $N_1{r}_1{n}_1/\left[60\times \left(R_1+r_1\right)\right]=274.3\mathrm{H}\mathrm{z}$ ) as well as the rotation frequency $f_{\mathrm{r}1}$ ( $n_1/60=50\mathrm{H}\mathrm{z}$ ) appear in the θ₂ direction. The combined frequencies $0.5\left(\hspace{0.17em}{f}_{\mathrm{m}}-f_{\mathrm{b}2}\right)$ , $f_{\mathrm{m}}-f_{\mathrm{b}2}$ , can also be observed in the spectrum plots.

Figure 10.

The frequency spectrum of the system at no profile shifted: (a) The spectrum of x₂ direction; (b) The spectrum of y₂ direction; (c) The spectrum of θ₂ direction.

The effect of gear profile shifted on gear vibration in the y, θ direction is slight in the frequency spectrum, so this work only makes analysis for the variation in the x direction. Figure 11 shows the spectrum of the shifted gear in the x₂ direction with different shifting coefficients, where (a) and (b) are negative transmission and (c) and (d) are positive transmission. Compared with Figure 10(a), the magnitudes at $f_{\mathrm{m}}$ and $1/2f_{\mathrm{m}}$ in Figure 11(a) and (b) both decrease relatively, but the decrease of $1/2f_{\mathrm{m}}$ is larger when the profile shifted coefficient is −0.15, and the magnitudes of the higher harmonics of the meshing frequency and the rest of the frequencies do not change obviously. In Figure 11(c) and (d), the amplitude of $f_{\mathrm{m}}$ increases with the increase of the profile shifted coefficient, but the amplitude of the $1/2f_{\mathrm{m}}$ still decreases, and similarly, the rest of the frequency components change slightly. As shown in Table 4, compared with the standard gear, the amplitude at the meshing frequency decreases by 16.8% and 26.4% when the profile shifted coefficient is −0.1, −0.15, and increases by 12.9% and 17.8% when the profile shifted coefficient is 0.1 and 0.15, respectively.

Figure 11.

The frequency spectrum of the system at different profile shifted: (a) At profile shifted in −0.1; (b) At profile shifted in −0.15; (c) At profile shifted in 0.1; (d) At profile shifted in 0.15.

Table 4.

The amplitude of the frequency.

X2	fm(m)	1/2fm(m)
0.15	1.597 × 10−6	3.17 × 10−7
0.1	1.533 × 10−6	3.315 × 10−7
0	1.362 × 10−6	1.018 × 10−6
−0.1	1.14 × 10−6	6.506 × 10−7
−0.15	1.014 × 10−6	2.083 × 10−7

Figure 12 shows the dynamic meshing force of the gears with different profile shifted coefficients. As shown in the figure, the dynamic meshing force of the gearing system is maximum when the gear shifting coefficient $X=0$ , and decreases when the gear profile shifted coefficient $X\ne 0$ . However, the curve is shifted to the left when the gear profile shifted coefficient $X>0$ , on the contrary, the curve is shifted to the right when the gear profile shifted coefficient $X<0$ . This indicates that the time of vibration shock generated by alternating single and double tooth meshing of positive transmission gear train is advanced with the increase of the profile shifted coefficient, and the time of vibration shock generated by alternating single and double tooth meshing of the negative transmission gear train is delayed with the increase of the absolute value of the profile shifted coefficient.

Figure 12.

Gear dynamic meshing force at different profile shifted: (a) Dynamic meshing force; (b) Regional enlarged drawing.

Effect of input torque on vibration characteristics

The effect of the input torque on the dynamic characteristics of the gear system is investigated in this work. Figure 13 shows the time domain waveform with input torque as the control parameter at profile shifted coefficient −0.06,818. It can be seen in Figure 13 that the gear system’s vibration will be deflected as the input torque change. Meanwhile, the vibration displacement of the gear system increases as the input torque increases.

Figure 13.

The time domain waveform at different torque: (a) The displacement of x₂ direction; (b) The displacement of y₂ direction; (c) The displacement of θ₂ direction.

The frequency spectra of the gear system under different input torques in x₂, y₂, and θ₂ directions are shown in Figure 14. It is obviously that the meshing frequency f_m is the dominant frequency and its amplitude much larger than other frequency components. The multiples of the meshing frequency 2f_m can be seen in the figure. In the x₂ and y₂ directions, the amplitude of meshing frequency tends to increase with a change in the input torque. It is worth noting that there is a significant increase in 0.5f_m in Figure 14(b). There is an apparent change of frequency amplitude 0.5 (f_m − f_b2) in the θ₂ direction.

Figure 14.

The frequency spectrum of the system under different input torques: (a) T_d = 90 N·m; (b) T_d = 180 N·m; (c) T_d = 360 N·m.

Conclusions

In this paper, a dynamic model of a 16-degree-of-freedom bending-torsional coupling profile shifted gear transmission system is established to study the influence of the profile shifted coefficient on the system. The potential energy method is used to derive and solve the time-varying mesh stiffness of the profile shifted gear. The vibration characteristics of the profile shifted gear transmission system are studied through the time domain and frequency domain diagrams. The main conclusions drawn from this study are as follows:

1. Considering the Hertzian contact stiffness, bending stiffness, shear stiffness and axial compression stiffness, the pinion is shifted, and the time-varying meshing stiffness of the profile shifted gear is solved by numerical simulation. For positive transmission, the time-varying meshing stiffness of the gears increases and the double-tooth meshing interval decreases. On the contrary, for negative transmission, the time-varying mesh stiffness of the gears decreases, but the double-tooth meshing interval increases.

2. Through the analysis of the time domain diagram, the vibration displacement undergoes a complex variation process as the profile shifted coefficient changes, moreover, the vibration displacement can be effectively reduced when a suitable coefficient is used. Meanwhile, when the positive transmission is adopted, the meshing frequency component increases and the vibration shock advances; when the negative transmission is adopted, the meshing frequency component decreases and the vibration shock lags.

3. With the increase of the input torque, the vibration displacement will be intensified and deflected, and the amplitude of frequency has an obvious change.

In all, the model established in this paper provides assistance in the investigation of the vibration of the profile shifted gear system. Conclusions can be used for applications on gear vibration and noise reduction.