In the field of mechanical engineering, spur cylindrical gears are widely used in various transmission systems due to their simple structure, high reliability, and efficient power transmission. However, the vibration and noise generated during the operation of gear pairs can affect the performance and service life of the entire system. Therefore, studying the vibration characteristics of spur cylindrical gear pairs is of great significance for improving the reliability and performance of mechanical systems.
Introduction
With the development of gear transmission systems towards high speed and high power density, the problem of vibration and noise reduction has become a primary issue for high-efficiency, low-noise, and high-durability gear transmissions. The random characteristics of the comprehensive transmission error and the micro-topography of the tooth surface of gears, caused by factors such as gear processing, assembly forms, and changes in the operating environment, lead to complex dynamic behaviors in the gear system. In engineering practice, the uncertainty of the transmission error during the processing and assembly of gear transmissions results in a certain randomness in the tooth surface morphology. The tooth surface morphology affects the tooth surface friction force, and the tooth surface friction force increases with the increase of the tooth surface roughness. The gear transmission error and the tooth surface friction parameters interact with each other, and the dynamic characteristics of the gear transmission considering the randomness of the random error and the tooth surface friction parameters are not clear. Therefore, studying the gear dynamics problem considering the random comprehensive transmission error and the tooth surface friction parameters of gears provides important theoretical support for the subsequent reliability analysis and dynamic optimization of gear transmissions.
Many scholars at home and abroad have conducted a large number of theoretical and experimental studies on the vibration characteristics of gear systems under different forms of excitation. Some scholars have considered the influence of random errors on the dynamics of gear systems, but these studies have not taken friction factors into account. Some scholars have conducted research on gear systems in the field of tooth surface friction, but in order to better discuss the relationship between friction and dynamics, scholars have begun to deeply explore the randomness parameters that interfere with the friction characteristics of gears.
In summary, the current research has the following characteristics: firstly, the comprehensive tooth error is synthesized based on the deterministic theoretical method, ignoring the time-varying nature of tooth surface friction; secondly, the existing models do not consider the comprehensive influence of the randomness of errors and tooth surface friction on the dynamic characteristics of gears. Therefore, by combining the probability statistics and the concentrated mass method, the error and the tooth surface friction parameters are mathematically represented in the form of random variables, and a 3-degree-of-freedom bending-torsional coupled vibration model of the spur cylindrical gear pair considering the randomness of the tooth surface friction parameters under random errors is established to analyze the influence of each random parameter on the dynamic response of the system, providing a theoretical basis for the subsequent dynamic optimization of the gear transmission system.
Dynamics Model of Spur Cylindrical Gear Pairs Considering Random Errors and Tooth Surface Friction
A 3-degree-of-freedom bending-torsional coupled dynamics model of the spur cylindrical gear pair is established using the mass concentration method, as shown in Figure 1. The displacement of the gear in the x-direction is xi (i = p, g), the displacement in the y-direction is yi (i = p, g), and the displacement in the direction of rotation around the gear axis is θi (i = p, g). The subscripts p and g represent the driving wheel and the driven wheel, respectively.
The displacement vector of the mass point is:
q = [xp, yp, θp, xg, yg, θg]T
The corresponding bending-torsional vibration equations considering the tooth surface friction, time-varying meshing stiffness, and random errors are:
where kp^x and kg^x are the equivalent support stiffnesses of the driving and driven gears in the x-direction, respectively; kp^y and kg^y are the equivalent support stiffnesses of the driving and driven gears in the y-direction, respectively; km(t) is the time-varying meshing stiffness of the gear; Ti(t) is the torque of the driving and driven gears; Ri(t) is the meshing curvature radius of the driving and driven gears along the contact line; mi is the mass of the driving and driven gears; Ii is the moment of inertia; Ff(t) is the time-varying friction force under the random roughness of the gear teeth; α is the pressure angle of the gear pair; e(t) is the gear error.
Taking a spur cylindrical gear pair in a certain transmission as the research object, its parameters are shown in Table 1.
| Parameter | Driving Wheel | Driven Wheel |
|---|---|---|
| Number of Teeth | 33 | 26 |
| Accuracy Grade | 6GJ | 6GJ |
| Mass (kg) | 10.6 | 7.43 |
| Module (mm) | 7 | 7 |
| Moment of Inertia (kg·mm2) | 147670 | 61426 |
| Pressure Angle (°) | 20 | 20 |
| Tooth Width (mm) | 69 | 69 |
| Input Speed (r/min) | 2000 | – |
| Input Torque (N·m) | 2340.7 | – |
The time-varying meshing stiffness curve of the gear pair is calculated according to the numerical integration formula of Weber’s energy method, as shown in Figure 2.
Randomness Analysis of Errors and Tooth Surface Friction Parameters
Random Errors
The gear error mainly includes the base pitch error and the tooth profile error. The base pitch error and the tooth profile error in each pair of gear pairs are combined into the comprehensive transmission error of the gear. According to the deterministic theory and method in Reference [8], it changes sinusoidally within one meshing cycle.
ei(t) = em + Ei sin(ωt + φi)
where em is the constant value of the gear meshing error; Ei is the amplitude of the meshing error of the gear pair along the meshing line direction; ω is the meshing frequency of the gear pair; φi is the initial phase.
The tooth profile error and the base pitch error are caused by the processing conditions, are independent of each other and change with time. The random error is the superposition of the deterministic harmonic function and the Gaussian white noise, as shown in Equation (5).
e(t) = ei(t) + Δff/cosα + Δfpb/cosα
e(t) = ei(t) + ξ(t)
where Δff is the tooth profile error; Δfpb is the base pitch error; ξ(t) is the Gaussian white noise.
According to the GB/T 10095.1 – 2008 and GB/T 10095.2 – 2008 standards, the relevant parameters of the 6th grade precision gear are selected as the deterministic part of the random error, and the random variable obeys a Gaussian distribution with a mean of 0 and a variance of 0.0005. The Gaussian white noise and the random error of the 6th grade precision gear are shown in Figures 3 and 4.
Tooth Surface Friction Parameters under the Interference of Random Errors
The time-varying friction force model on the gear meshing line is shown in Figure 5, and the corresponding time-varying friction torque Ti(t) is:
i = (p, g)
where μ(t) is the tooth surface friction coefficient.
The random error makes the micro-topography of the tooth surface uncertain, and with the change of the meshing contact state of the gear, the meshing curvature radius, the friction coefficient, and the normal load per unit contact length along the contact line fluctuate randomly and have random characteristics.
Assuming that the gear surface roughness obeys Gaussian random distribution to characterize the randomness of the tooth surface friction coefficient, we have:
μ(t) = μ0 + σμξ(t)
where ξ(t) is a random variable with a mean of 0 and a Gaussian distribution; the critical mean value of the friction coefficient μ0 is 0.109; the standard deviation of the friction coefficient σμ is 0.05.
The friction coefficient curve calculated according to Equation (7) is shown in Figure 6.
The actual meshing state of the gear is shown in Figure 7, and the distance from the node to the instantaneous meshing point is s(t). According to the previous assumption, the surface roughness obeys the Gaussian random distribution, so the random part of s(t) also obeys the Gaussian distribution. We have:
where sμ is the mean value of the distance from the instantaneous meshing point on the tooth profile to the gear pitch circle solved using the geometric relationship, and its calculation model is shown in Figure 8.
By combining Equations (8) and (9), the time-domain response curve of the meshing curvature radius is obtained, as shown in Figure 9.
Dynamic Response Characteristics Analysis
To study the vibration characteristics of the spur cylindrical gear pair considering the randomness of errors and tooth surface friction, the spur cylindrical gear pair shown in Table 1 is taken as the research object, and the fourth-order fixed-step Runge-Kutta method is used to solve Equation (2), with a simulation step size of 0.00015.
Under the interference of random errors, the tooth surface friction coefficient and the meshing radius also have a certain randomness. The solved response characteristics are analyzed, and the vibration responses of the driving and driven gears are shown in Figure 10.
It can be seen from Figure 10 that the bending vibration amplitude of the driven wheel is larger, and the torsional vibration amplitudes of the two gears are similar. For the same gear, the vibration amplitude in the x-direction is smaller than that in the y-direction. Combined with Table 2, it can be seen that under the interference of random errors, the randomness of the response of the driven wheel is stronger. For the same gear, the randomness of the response in the y-direction is stronger than that in the x-direction.
The power spectral density is used to analyze the continuous transient response of random vibration, which reflects the statistical results under the excitation of multiple random variables. The power spectrum of the vibration response is shown in Figure 11.
As shown in Figures 2(b) and 4, the peak frequency of the meshing stiffness corresponds to the meshing frequency of 1100 Hz; the peak frequencies of the random error are 1100 Hz and 2200 Hz, respectively. It can be seen from Figure 11 that under the simulation conditions shown in Table 1, the energy peaks of the torsional vibration signals of the gear transmission are concentrated at 1100 Hz and 2200 Hz, indicating that the gear vibration response signal is the modulation of the meshing frequency and its multiple frequencies by the excitation frequency. Due to the convolution effect between the meshing stiffness, the friction coefficient, and the random error, there are many sidebands near the peak, and the amplitude is relatively large.
Randomness Influence Analysis
Influence of Random Errors on Gear Vibration Characteristics
Since the vibration response trends of the driving wheel and the driven wheel are consistent, the driving wheel is taken as the research object. By comparing and analyzing the time-domain and frequency-domain characteristics of the vibration response, the influence of random errors on the vibration characteristics of the gear transmission is obtained, as shown in Figures 12 and 13.
It can be seen from Figure 12 that the vibration acceleration curve of the gear system without random errors is relatively smooth, while under the interference of random errors, the acceleration of the gear system fluctuates violently. As shown in Table 3, under the interference of random errors, the randomness of the dynamic response of the gear is further enhanced, indicating that the random errors have a strong interference on the dynamic characteristics of the gear.
It can be seen from Figure 13 that the acceleration response curve of the driving gear without random errors is relatively smooth, with peaks at 1100 Hz and 2200 Hz. Due to the interaction between the meshing frequency and the random error frequency, there is a symmetrical frequency divergence at 860 Hz and 1340 Hz on both sides of the meshing frequency. Under the random error, the acceleration response amplitude of the driving gear increases, the randomness is strong, and there is a continuous spectrum with complex fluctuations. The vibration peak is larger than the vibration peak without the action of random errors, and the specific statistical parameters are shown in Table 4.
The phase diagram is the geometric expression of the state trajectory of the dynamic system on the phase plane. The phase diagram of the gear transmission process is shown in Figure 14. The motion form of the gear pair without random error interference is a smooth closed curve, indicating that the gear is performing a simple periodic motion. Due to the intervention of random errors, the phase diagram becomes more chaotic and disorderly, and the phase trajectory line also has obvious randomness.
Influence of the Randomness of Tooth Surface Friction Parameters on Gear Vibration Characteristics
Considering the interference of random errors, the vibration characteristics of the gear transmission are compared and analyzed under two situations: ① without considering the influence of random errors on the tooth surface friction, assuming that the tooth surface friction coefficient, the meshing radius, etc. are constant values; ② considering the interference of random errors, and assuming that the tooth surface friction coefficient, the meshing radius, etc. are random variables disturbed by random errors, as shown in Figure 15.
It can be seen from Figure 15 that the acceleration curve considering the randomness of the gear friction parameters fluctuates more violently than the acceleration curve when the tooth surface friction parameters are set as constant values. Combined with Table 5, when the tooth surface friction parameters are random, the randomness of the gear vibration acceleration increases, indicating that the randomness of the tooth surface friction parameters interferes with the vibration characteristics of the gear transmission to a certain extent.
Through the spectrum analysis of the gear vibration acceleration response considering the randomness of the tooth surface friction parameters, it can be seen from Figure 16 that the gear vibration acceleration has peaks at 1100 Hz and 2200 Hz when the tooth surface friction parameters are random or constant. Due to the superposition of the gear meshing frequency and the friction parameter frequency, symmetric modulation frequencies appear at 860 Hz and 1340 Hz on both sides of the meshing frequency. Combined with Table 6, at the peak, the amplitude of the gear vibration acceleration when the tooth surface friction parameters are random is higher than that when the tooth surface friction parameters are constant.
Conclusion
(1) By using the probability statistics method, the random characteristics of the error and the tooth surface friction parameters are mathematically represented, the existing geometric model is modified, and the mathematical mapping model between the time-varying meshing curvature radius and the tooth surface friction parameters under the interference of random errors is established. The random error and the tooth surface friction parameters are calculated through examples.
(2) By using the concentrated mass method, a 3-degree-of-freedom bending-torsional coupled vibration model of the spur cylindrical gear pair considering the randomness of the error and the tooth surface friction parameters is established, and the dynamic equation is solved numerically. The influence of the randomness of the error and the tooth surface friction on the bending-torsional vibration response of the gear transmission is compared and analyzed.
(3) The simulation analysis of the influence of the randomness of the random error and the tooth surface friction parameters on the vibration characteristics of the gear transmission shows that the random error has an amplification effect on the gear acceleration response, and the randomness of the response increases; the randomness of the tooth surface friction parameters increases the vibration amplitude of the gear acceleration response; these two interference factors make the acceleration response more complex in the frequency domain and the phase diagram. The dynamic instability of the gear system is aggravated. The model established in this paper reflects the actual transmission process of the gear to a certain extent, and the research results provide ideas for the establishment of the gear transmission dynamics model and theoretical support for the subsequent dynamic optimization of the gear transmission.
| Comparison Item | Without Random Error | With Random Error |
|---|---|---|
| Vibration Acceleration Curve | Relatively smooth | Fluctuates violently |
| Randomness of Dynamic Response | Weak | Stronger |
| Influence on Gear Dynamics | Little interference | Strong interference |
| Peak Frequency | 1100 Hz and 2200 Hz | 1100 Hz and 2200 Hz |
| Modulation Frequency | None | Symmetrical at 860 Hz and 1340 Hz |
| Phase Diagram | Smooth closed curve | Chaotic and disorderly |
| Comparison Item | Tooth Surface Friction Parameters as Constant | Tooth Surface Friction Parameters as Random |
|---|---|---|
| Randomness of Gear Vibration Acceleration | Weaker | Stronger |
| Influence on Gear Vibration Characteristics | Less interference | Certain interference |
| Peak Frequency of Vibration Acceleration | 1100 Hz and 2200 Hz | 1100 Hz and 2200 Hz |
| Modulation Frequency | None | Symmetrical at 860 Hz and 1340 Hz |
| Amplitude at Peak Frequency | Lower | Higher |
In future research, we can further explore the influence of other factors on the vibration characteristics of spur cylindrical gear pairs, such as the influence of tooth surface wear, lubrication conditions, and manufacturing errors on the gear dynamics. Additionally, developing more accurate and efficient numerical methods and experimental techniques can provide more reliable results for the analysis and optimization of gear systems. This will contribute to the improvement of the performance, reliability, and durability of gear transmissions in various mechanical applications.