1. Introduction
In the field of mechanical transmission, gears play a crucial role. Cylindrical spur gears are widely used in various mechanical systems due to their simple structure and reliable transmission performance. Tooth profile modification is an important means to improve the performance of cylindrical spur gears. It can optimize the meshing process of gears, reduce vibration and noise, and enhance the durability of the gear system. Therefore, studying the meshing dynamic characteristics of cylindrical spur gears with tooth profile modification has important theoretical and practical significance.
1.1 Background and Significance
The development of modern mechanical engineering puts forward higher requirements for the performance of gear transmission systems. The traditional standard gears often cannot meet the needs of high – speed, heavy – load, and precise transmission. Tooth profile modification can effectively improve the meshing quality of gears. For example, in the automotive industry, the application of tooth – profile – modified gears can reduce the noise of the transmission system, improve the comfort of the vehicle, and also enhance the power – transmission efficiency. In industrial machinery, such as large – scale manufacturing equipment and power – generation equipment, modified gears can improve the reliability and service life of the equipment, reducing maintenance costs and production interruptions.
1.2 Research Status
Many scholars have conducted in – depth research on the meshing characteristics of gears. Some researchers focus on the calculation methods of meshing stiffness. For example, Wang et al. proposed an analytical – finite – element method for calculating the mesh stiffness of spur gear pairs with complicated foundation and crack, which combines the advantages of finite – element calculation accuracy and the calculation speed of the gear tooth – surface analysis method. However, most of the existing research mainly focuses on the TVMS algorithm and the dynamic characteristics of general gear transmission, while the research on the geometric and dynamic characteristics of modified spur gears is relatively insufficient. This study aims to fill this gap and provide a more comprehensive understanding of the meshing dynamic characteristics of cylindrical spur gears with tooth profile modification.
2. Analysis Model of Time – Varying Meshing Stiffness (TVMS)
2.1 Stiffness Calculation Based on the Potential Energy Method
The TVMS is a typical periodic excitation source in the gear – transmission system, which directly affects the operating performance and service life of the transmission system. Establishing an effective TVMS model is crucial for accurately predicting the dynamic performance of the spur – gear transmission system.
In this study, based on the principle of the potential – energy method, the gear tooth is simplified as a variable – cross – section cantilever beam. The total potential energy consists of Hertzian contact energy, bending energy, shear energy, axial compression energy, and fillet foundation energy. According to the theory of elasticity, these five types of potential energy can be expressed as shown in Table 1:
| Potential Energy Component | Formula | Explanation |
|---|---|---|
| Hertzian Contact Energy (\(U_{h}\)) | \(U_{h}=\frac{F^{2}}{2k_{h}}\) | F is the meshing force of the gear pair on the contact surface, and \(k_{h}\) is the Hertz contact stiffness |
| Bending Energy (\(U_{b}\)) | \(U_{b}=\frac{F^{2}}{2k_{b}}\) | \(k_{b}\) is the bending stiffness |
| Shear Energy (\(U_{s}\)) | \(U_{s}=\frac{F^{2}}{2k_{s}}\) | \(k_{s}\) is the shear stiffness |
| Axial Compression Energy (\(U_{a}\)) | \(U_{a}=\frac{F^{2}}{2k_{a}}\) | \(k_{a}\) is the axial compression stiffness |
| Fillet Foundation Energy (\(U_{f}\)) | \(U_{f}=\frac{F^{2}}{2k_{f}}\) | \(k_{f}\) is the fillet foundation stiffness |
The total potential energy stored in a single gear can be described by the above – mentioned component stiffnesses. The total TVMS of the meshing gear pair can also be derived accordingly.
It should be noted that when calculating the stiffness components, different geometric parameter expressions are required according to whether the root circle is smaller or larger than the base circle.
2.1.1 Case 1: Root Circle Smaller than the Base Circle
When the root circle is smaller than the base circle, the tooth profile of the gear is an involute curve between the addendum circle and the base circle, and the rest is a transition curve. The geometric parameters used to analyze the stiffness components can be expressed by a series of formulas. For example, the height – related parameter h is calculated by \(h = R_{b}[(\varphi_{1}+\varphi_{2})\cos\varphi_{1}-\sin\varphi_{1}]\), where \(R_{b}\) is the base – circle radius, and \(\varphi_{1}\), \(\varphi_{2}\) are relevant angles. The axial compression, bending, and shear meshing stiffnesses can be calculated by integrating along the tooth profile, as shown in Table 2:
| Stiffness Component | Formula |
|---|---|
| Bending Meshing Stiffness (\(\frac{1}{k_{b}}\)) | \(\frac{1}{k_{b}}=-\int_{\mu_{2}}^{\mu_{1}}\frac{3a_{x}(R_{b}-R_{f}\cos\varphi_{3}\cos\varphi_{1}-a_{x}\varphi\cos\varphi_{1}-b_{x}\cos\varphi_{1})}{2EL[R_{b}\sin\varphi_{2}+r_{f}-\sqrt{r_{f}^{2}-(x – d_{1})^{2}}]^{3}}d\varphi\) |
| Shear Meshing Stiffness (\(\frac{1}{k_{s}}\)) | \(\frac{1}{k_{s}}=-\int_{\varphi_{1}}^{\varphi_{3}}\frac{1.2a_{x}(1 + v)\cos^{2}\varphi_{1}}{EL[R_{b}\sin\varphi_{2}+r_{f}-\sqrt{r_{f}^{2}-(x – d_{1})^{2}}]}d\varphi+\int_{-\varphi_{1}}^{\varphi_{2}}\frac{1.2(1 + v)(\varphi_{2}-\varphi)\cos\varphi\cos^{2}\varphi_{1}}{EL[\sin\varphi_{2}+(\varphi_{2}-\varphi)\cos\varphi]}d\varphi\) |
| Axial Compression Meshing Stiffness (\(\frac{1}{k_{a}}\)) | \(\frac{1}{k_{a}}=-\int_{\varphi_{2}}^{\varphi_{3}}\frac{a_{x}\sin^{2}\varphi_{1}}{2EL[R_{b}\sin\varphi_{2}+r_{f}-\sqrt{r_{f}^{2}-(x – d_{1})^{2}}]}d\varphi+\int_{-\varphi_{1}}^{\varphi_{2}}\frac{(\varphi_{2}-\varphi)\cos\varphi\sin^{2}\varphi_{1}}{EL[\sin\varphi_{2}+(\varphi_{2}-\varphi)\cos\varphi]}d\varphi\) |
where E is the Young’s modulus, L is the tooth width, v is the Poisson’s ratio, and other parameters are geometric parameters related to the gear tooth profile. The Hertz contact stiffness \(k_{h}=\frac{\pi EL}{4(1 – v^{2})}\), and the fillet foundation stiffness \(\frac{1}{k_{f}}=\frac{\cos^{2}\beta}{EL}[L^{*}(\frac{u_{f}}{S_{f}})^{2}+M^{*}(\frac{u_{f}}{S_{f}})+P^{*}(1 + Q^{*}\tan^{2}\varphi_{1})]\), where \(\beta\), \(u_{f}\), \(S_{f}\), \(L^{*}\), \(M^{*}\), \(P^{*}\), \(Q^{*}\) are parameters that can be found in relevant references.
2.1.2 Case 2: Root Circle Larger than the Base Circle
When the root circle is larger than the base circle, the tooth profile is an involute curve from the addendum circle to the root circle. The bending, shear, and axial compression meshing stiffnesses can be calculated by different integral formulas, as shown in Table 3:
| Stiffness Component | Formula |
|---|---|
| Bending Meshing Stiffness (\(\frac{1}{k_{b}}\)) | \(\frac{1}{k_{b}}=\int_{-\infty}^{\infty}\frac{3\{1+\cos\varphi_{1}[(\varphi_{2}-\varphi)\sin\varphi-\cos\varphi]\}^{2}(\varphi_{2}-\varphi)\cos\varphi}{2EL[\sin\varphi_{2}+(\varphi_{2}-\varphi)\cos\varphi]^{3}}d\varphi\) |
| Shear Meshing Stiffness (\(\frac{1}{k_{s}}\)) | \(\frac{1}{k_{s}}=\int_{-\varphi_{1}}^{\varphi_{s}}\frac{1.2(1 + v)(\varphi_{2}-\varphi)\cos\varphi\cos^{2}\varphi_{1}}{EL[\sin\varphi_{2}+(\varphi_{2}-\varphi)\cos\varphi]}d\varphi\) |
| Axial Compression Meshing Stiffness (\(\frac{1}{k_{a}}\)) | \(\frac{1}{k_{a}}=\int_{-\varphi_{1}}^{\varphi_{3}}\frac{(\varphi_{2}-\varphi)\cos\varphi\sin^{2}\varphi_{1}}{EL[\sin\varphi_{2}+(\varphi_{2}-\varphi)\cos\varphi]}d\varphi\) |
where \(\varphi_{4}\) and \(\varphi_{5}\) need to be determined by solving a set of coupled equations \(R_{b}[\cos\varphi_{5}-(\varphi_{2}-\varphi_{5})\sin\varphi_{5}]-R_{f}\cos\varphi_{4}=0\) and \(R_{b}[\sin\varphi_{5}-(\varphi_{2}-\varphi_{5})\cos\varphi_{5}]-R_{f}\sin\varphi_{4}=0\).
2.2 Geometric Relationship Analysis
For a set of modified gears and pinions with equal modification coefficients, the pitch circle and the base circle remain unchanged, and their geometric relationships are similar to those of an equivalent standard gear and pinion set. However, for modified gears with unequal modification coefficients, the geometric relationships need to be re – considered.
In the meshing process of gears, the pitch circle of the gear no longer coincides with the reference circle, and the position of pure rolling changes. Based on the basic theory of modified spur gears, the geometric relationship at the start of gear meshing is shown in Figure 1. When the gear and pinion start to mesh, the contact occurs at point \(A_{p}\) between the line of action and the addendum circle of the gear. As the gear transmission progresses, the gear and pinion finally separate at the intersection of the line of action and the addendum circle of the pinion. The angles \(\varphi_{1}\) of the pinion and the gear can be expressed as \(\varphi_{1,p}=\varphi_{1,p}^{0}+\beta_{p}\) and \(\varphi_{1,g}=\varphi_{1,g}^{0}-i_{g}\beta_{g}\), and the angles \(\varphi_{2}\) can be expressed as \(\varphi_{2,p}=\frac{\pi}{N_{p}}\frac{s_{p}}{s_{p}+e_{p}}+inv\varphi_{0}\) and \(\varphi_{2,g}=\frac{\pi}{N_{p}}\frac{s_{g}}{s_{g}+e_{g}}+inv\varphi_{0}\), where \(i_{s}\) is the transmission ratio, \(N_{p}\) and \(N_{g}\) are the number of teeth of the pinion and the gear respectively, \(s_{p}\), \(s_{g}\) are the tooth thicknesses, \(e_{p}\), \(e_{g}\) are the differences between the pitch and tooth thicknesses, \(\varphi_{0}\) is the reference pressure angle, and inv represents the involute function. The initial values of the angles \(\varphi_{1,p}^{0}\) and \(\varphi_{1,g}^{0}\) also have corresponding mathematical expressions.
3. Research Framework of Modified Spur Gears
3.1 Overall Framework
Based on the established TVMS model, this section examines the modified spur gears. The research framework is shown in Figure 2. First, according to the vibration mode of the modified spur gear, the concentrated – mass model is used to extract its vibration response and statistical characteristics. Then, through the comparison of TVMS, the dynamic characteristics of the gear system under different modification conditions are analyzed. Finally, the dynamic characteristics of the gear system are evaluated by comparing the frequency spectra and statistical indicators.
3.2 Dynamic Model of the Gear System
The dynamic model of the gear and pinion system is a six – degree – of – freedom model. For spur gears, since the meshing stiffness affects the planar vibration of the gear pair rather than the axial vibration, only the motion along the line of action, the motion perpendicular to the line of action, and the rotation are considered. The shaft end is connected to the bearing through a parallel spring – damper device. The control equations of the power system can be written as shown in Table 4:
| Equation | Expression | Explanation |
|---|---|---|
| Equation for Pinion’s Motion along the Line of Action (\(m_{p}\ddot{x}_{p}+c_{b}\dot{x}_{p}+k_{b}x_{p}=-F_{m}\)) | \(m_{p}\ddot{x}_{p}+c_{b}\dot{x}_{p}+k_{b}x_{p}=-F_{m}\) | \(m_{p}\) is the mass of the pinion, \(c_{b}\) is the bearing damping, \(k_{b}\) is the bearing stiffness, \(F_{m}\) is the meshing force, and \(x_{p}\) is the displacement of the pinion along the line of action |
| Equation for Pinion’s Motion Perpendicular to the Line of Action (\(m_{p}\ddot{y}_{p}+c_{b}\dot{y}_{p}+k_{b}y_{p}=-F_{f}\)) | \(m_{p}\ddot{y}_{p}+c_{b}\dot{y}_{p}+k_{b}y_{p}=-F_{f}\) | \(y_{p}\) is the displacement of the pinion perpendicular to the line of action, and \(F_{f}\) is the friction force |
| Equation for Pinion’s Rotation (\(I_{p}\ddot{\beta}_{p}=-F_{m}R_{b,p}-T_{p}\)) | \(I_{p}\ddot{\beta}_{p}=-F_{m}R_{b,p}-T_{p}\) | \(I_{p}\) is the moment of inertia of the pinion, \(\beta_{p}\) is the angular displacement of the pinion, \(R_{b,p}\) is the base – circle radius of the pinion, and \(T_{p}\) is the driving torque |
| Equation for Gear’s Motion along the Line of Action (\(m_{g}\ddot{x}_{g}+c_{b}\dot{x}_{g}+k_{b}x_{g}=F_{m}\)) | \(m_{g}\ddot{x}_{g}+c_{b}\dot{x}_{g}+k_{b}x_{g}=F_{m}\) | \(m_{g}\) is the mass of the gear, \(x_{g}\) is the displacement of the gear along the line of action |
| Equation for Gear’s Motion Perpendicular to the Line of Action (\(m_{g}\ddot{y}_{g}+c_{b}\dot{y}_{g}+k_{b}y_{g}=F_{f}\)) | \(m_{g}\ddot{y}_{g}+c_{b}\dot{y}_{g}+k_{b}y_{g}=F_{f}\) | \(y_{g}\) is the displacement of the gear perpendicular to the line of action |
| Equation for Gear’s Rotation (\(I_{g}\ddot{\beta}_{g}=-F_{m}R_{b,g}-T_{g}\)) | \(I_{g}\ddot{\beta}_{g}=-F_{m}R_{b,g}-T_{g}\) | \(I_{g}\) is the moment of inertia of the gear, \(\beta_{g}\) is the angular displacement of the gear, \(R_{b,g}\) is the base – circle radius of the gear, and \(T_{g}\) is the load torque |
The meshing force \(F_{m}\) and the friction force \(F_{f}\) can be further expressed as \(F_{m}=k(t)[x_{p}-x_{g}+R_{b,g}\beta_{g}-e(t)]+c_{m}[\dot{x}_{p}-\dot{x}_{g}+\dot{R}_{b,g}\dot{\beta}_{g}-\dot{e}(t)]\) and \(F_{f}=-\mu F_{m}\), where \(k(t)\) is the time – varying meshing stiffness, \(c_{m}\) is the meshing damping coefficient, \(e(t)\) is the static transmission error, and \(\mu\) is the friction coefficient. The meshing damping coefficient \(c_{m}=2\zeta\sqrt{\overline{k}_{m}m}\), where \(\zeta\) is the damping ratio, \(\overline{k}_{m}\) is the average meshing stiffness, and m is the equivalent mass (\(m = \frac{m_{p}m_{g}}{m_{p}+m_{g}}\)).
