Abstract
Gear vibration and noise are important parameters that affect the transmission performance. The vibration and noise control of the gear transmission system has become a very important research field. Presently, the main methods of vibration control of the gear transmission system are increasing coincidence, shape modification, displacement, damping, etc. In this study, a novel method of staggered tooth phase of gears is proposed, the meshing principle of staggered phase tuning gears is studied, the calculation formula of time-varying meshing stiffness of staggered phase tuning gears is derived, and the mapping relationship between staggered phase and vibration response is studied. The experimental scheme of the parallel shaft spur gear pair with different staggering phases is designed, the experimental study of vibration reduction effect of gear pair with different staggering phases is carried out, and the experimental results are compared and analyzed with the simulation results.
Keywords: Time-varying meshing stiffness, Vibration response, Vibration control, Method of gear staggering phase tuning
1. Introduction
Gear vibration and noise significantly impact the performance of transmission systems, making their control a critical research area. Various approaches have been developed to mitigate vibrations, including increasing coincidence, gear tooth profile modification, gear displacement, and damping. However, these methods may have limitations or introduce additional complexity. In this study, a novel approach called “gear staggering phase tuning” is proposed to reduce vibrations in spur gear transmissions. This method involves adjusting the meshing phase of gears by slightly rotating them relative to each other. The principle, theoretical derivation, and experimental validation of this method are presented.
Previous research on gear vibration control has focused on different aspects. Neriya et al. [1] studied the vibration characteristics of helical gear systems considering the multi-degree-of-freedom coupling of gear shafts, nonlinear backlash, and time-varying meshing stiffness. Kahraman [2] established a linear dynamic model for helical gear pairs that accounts for shaft flexibility, bearing float, and dynamic coupling due to gear meshing. Wang et al. [3] investigated the dynamic behavior of helical gear systems using two coupling models: transverse-torsional and transverse-torsional-axial swing, considering the effect of time-varying meshing stiffness (TVMS). Amabili et al. [4] studied the steady-state response and stability of spur gear with low coincidence using a single-degree-of-freedom model that considers the time-varying stiffness and viscous damping proportional to the meshing stiffness. Neubauer et al. [5] derived the non-uniform geometric equations for non-equidistant gears. Eritenel et al. [6] studied the three-dimensional nonlinear vibrations of gear pairs. Wei et al. [7] used numerical simulation to study the effects of involute coincidence, bearing support stiffness, meshing damping, and backlash on the dynamic transmission error and vibration stability of helical gear systems. Liu et al. [8] found that phase differences between multiple planetary gears in an annular gear ring can eliminate large amplitude responses under certain conditions. Liu et al. [9] proposed a dynamics model for herringbone planetary gears applicable during variable speed processes and provided suggestions for avoiding tooth separation and tooth back contact in planetary gear set design to suppress vibrations.
Recently, there has been significant research on meshing phases as an important parameter for studying gear transmission vibration performance. Parker et al. [10-11] conducted rigorous mathematical derivations on meshing forces and analyzed the design principles for suppressing certain harmonics of planetary modal responses through meshing phase tuning in planetary gear transmissions. Zhang et al. [12] studied the parameter design and dynamic response of high-order tuning gears, derived the optimal transmission parameters for tuning gears, and verified the influence of tuning gear phase angle and tuning order on dynamic response.
In this study, the meshing principle of staggered phase tuning gears is explored, the calculation formula for the time-varying meshing stiffness of staggered phase tuning gears is derived, and the mapping relationship between staggered phase and vibration response is studied. An experimental scheme for parallel-shaft spur gear pairs with different staggered phases is designed, and the vibration reduction effect of gear pairs with different staggered phases is experimentally studied. The experimental results are compared and analyzed with simulation results, providing a new approach for vibration control in gear transmission systems.
2. Principle of Gear Staggering Phase Tuning
2.1 Staggered Phase Tuning Gears
For a spur gear with a tooth width of b, it can be divided into two gears with a tooth width of b/n. These two gears, with tooth widths of b/n, are labeled as the first, second, third, …, nth layers, and a certain angle is staggered between each layer. This results in a meshing phase difference between the gears without affecting the gear ratio, satisfying the meshing principle of each layer. By adjusting the meshing phase difference, the gear transmission performance can be improved. This method is referred to as the gear staggering phase tuning method, with the corresponding gears referred to as staggered phase tuning gears, and the corresponding angle referred to as the staggered phase angle.
For a parallel-axis single-stage transmission system with two meshing pairs, if one side of the gears remains unchanged and the other side is rotated a certain angle circumferentially, when rotated by one standard tooth space width πmt/2, the corresponding tooth space angle is π/z. At this point, the tooth space of one gear corresponds to the tooth of the other gear, and the meshing stiffness function curve differs from the standard meshing stiffness function curve by half a meshing period Tm/2. When rotated by one standard tooth pitch πmt, the corresponding tooth space angle is 2π/z, and the meshing stiffness function curve differs from the standard meshing stiffness function curve by one meshing period Tm. Due to the periodicity of the meshing stiffness function, the two function curves completely overlap.
For a parallel-axis single-stage transmission system with two meshing pairs, the relationship between the staggered phase p and the staggered phase angle φ is:
p=φ2π/z(1)
where z is the number of teeth on the driving gear. Due to the periodicity of gear meshing, the staggered phase angle cannot exceed one standard tooth space angle 2π/z, and the relationship between the staggered phase angle φ and the rotation angle φ0 of one side of the gear is:
φ=mod(φ0,2πz)(2)
where mod() is the modulo function.
2.2 Transmission Principle of Staggered Phase Tuning Gears
Since the two adjacent gears are staggered by a certain angle circumferentially, the corresponding teeth on the two meshing gear pairs do not enter the actual meshing region simultaneously. This results in the two meshing gear pairs being in different positions in the multi-tooth and single-tooth meshing regions at the same time.
For standard spur gear, the coincidence is 1<ε<2, and the meshing process alternately occurs between single-tooth and double-tooth meshing in a periodic manner. For tuned spur gear, the two meshing gear pairs are staggered by a certain angle circumferentially, so the teeth of the two meshing gear pairs do not enter the actual meshing region at the same time. Therefore, when one meshing gear pair is in the single-tooth meshing region, by adjusting the staggered phase angle, the other meshing gear pair may be in the double-tooth meshing region, and vice versa. Thus, the two meshing pairs with staggered phases can be considered as two meshing pairs with different initial meshing phases.
For standard spur gear, the two meshing pairs always enter the single-tooth or double-tooth meshing region simultaneously. Assuming the width of one side of the gear is b, the contact line length of the standard spur gear meshing pair fluctuates between 2b and 4b. For staggered phase tuning spur gear pairs, when meshing pair A1 is in the single-tooth meshing region, by adjusting the staggered phase angle, meshing pair A2 may be in the single-tooth meshing region or the double-tooth meshing region. If meshing pair A2 is in the single-tooth meshing region, the combined contact line length for the entire gear pair is 2b. If meshing pair A2 is in the double-tooth meshing region, the combined contact line length for the entire gear pair is 3b. Similarly, if meshing pair A1 is in the double-tooth meshing region, meshing pair A2 may be in the single-tooth meshing region (combined contact line length of 3b) or the double-tooth meshing region (combined contact line length of 4b). For the entire staggered phase tuning gear meshing pair, at least three pairs of teeth participate in meshing. the change in contact line length for two tuned spur gear pairs with different coincidences.
3. Meshing Stiffness of Staggered Phase Tuning Gears
To simplify the formula derivation, the meshing stiffness function is defined as the sum of the average value and the variation:
k(t)=ˉk+Δk(t)(3)
where ˉk is the average value of the meshing stiffness over one period, and Δk(t) is the time-varying part of the meshing stiffness.
For a staggered phase tuning gear pair, assuming the meshing stiffnesses of two adjacent gear pairs are k(1)(t) and k(2)(t), with superscripts (1) and (2) representing the two gear pairs, and defining the initial meshing phases of the two gear pairs as p(1) and p(2), respectively, the meshing stiffnesses of the two gear pairs considering the staggered phase are:
k(1)(t)=ˉk+Δkp(1)(t)k(2)(t)=ˉk+Δkp(2)(t)(4)
where ˉk is the average value of the meshing stiffness of a single gear pair over one meshing period, and Δkp(1) and Δkp(2) are the variation parts of the meshing stiffness of the two gear pairs. Since p(1) ≠ p(2), Δkp(1) ≠ Δkp(2).
A staggered phase tuning gear pair can be considered as two gear pairs meshing simultaneously. Therefore, the meshing stiffness of a single gear pair in a staggered phase tuning gear pair can be obtained by averaging the meshing stiffnesses of the two gear pairs:
kd(t)=12k(1)(t)+k(2)(t)
Further整理yields:
kd(t)=ˉk+12Δkp(1)(t)+Δkp(2)(t)
where ˉk is the average part of the meshing stiffness of the staggered phase tuning gear pair, and Δk is the variation part of the meshing stiffness of the staggered phase tuning gear pair. Through a Fourier series expansion, we have:
Δk=∑n=1∞(ap(1)ncos2nπtT+bp(1)nsin2nπtT)+∑n=1∞(ap(2)ncos2nπtT+bp(2)nsin2nπtT)=∑n=1∞[(ap(1)n+ap(2)n)cos2nπtT+(bp(1)n+bp(2)n)sin2nπtT]=∑n=1∞Jn||sin(2nπtT+φ)(7)
where φ=arctan(ap(1)n/bp(2)n) is the initial meshing phase angle of the staggered phase tuning gear, T is the meshing period, t is the meshing time, and p(1) and p(2) are the initial meshing phases of the two meshing gear pairs. For calculation convenience, Jn represents the amplitude of the nth order:
Jn=(ap(1)n)2+(bp(2)n)2(8)
Substituting into Equation (7), we get:
Δk=∑n=1∞Jn||sin(2nπtT+φ)(9)
Since the initial meshing phases p(1) and p(2) of the staggered phase tuning gears must be different, it is assumed that p(1)=0 and p(2)=p. For a spur gear pair, we have:
ap(1)n=ap(2)n=2(kmin−kmax)nπ{cos(nπα1)+cos[nπ(α1−2p)]}sin(nπα1)bp(1)n=bp(2)n=2(kmin−kmax)nπ{sin(nπα1)+sin[nπ(α1−2p)]}sin(nπα1)(10)
Let Qz=2(kmin−kmax)nπsin(nπα1)(11)
then:
ap(1)n=Qz{cos(nπα1)+cos[nπ(α1−2p)]}bp(2)n=Qz{sin(nπα1)+sin[nπ(α1−2p)]}(12)
The nth order amplitude of the staggered spur gear is:
Jn=(ap(1)n)2+(bp(2)n)2=||2Qzcos(nπp)||(13)
It can be seen that the amplitude Jn of the variation part of the meshing stiffness of the staggered phase tuning gear is related to α1, α2, α3, and p, i.e., the amplitude of the meshing stiffness of the staggered phase tuning gear is not only related to the amount of staggered phase but also to the original gear coincidence. For example, if the coincidence of a spur gear pair is ε0=1.6924, then ε0−1=0.6924≈2/3. When 3rd-order tuning is used, the meshing stiffness fluctuation is minimized. According to Equation (13), by summing the first 18 orders of meshing stiffness amplitude components for tuning gears of different orders, the stiffness amplitude for different tuning orders can be obtained.
It can be seen that compared to traditional spur gear, the meshing stiffness amplitude of tuned gears is reduced, with the 3rd-order tuning having the smallest meshing stiffness amplitude. This phenomenon occurs because one complete meshing cycle of a spur gear consists of a double-tooth meshing region and a single-tooth meshing region. The lengths of the single-tooth and double-tooth meshing regions are directly affected by the coincidence. When the coincidence ratio coefficient is c/3 (c is an integer), the single-tooth and double-tooth meshing regions are in a multiple relationship. The base coincidence ratio coefficient is approximately 2/3, meaning the double-tooth meshing region is nearly twice the length of the single-tooth meshing region. Therefore, one meshing cycle can be divided into three equal time zones.
Three sets of spur gear pair parameters are selected to achieve base coincidences of 1.506, 1.66, and 1.75.
The time-varying meshing stiffness of tuned gears of different orders is solved for the three sets of parameters. When the base coincidence ε0≈1.506, the coincidence ratio coefficient λ=1/2, and 2nd-order tuning results in nearly zero meshing stiffness fluctuation. When ε0≈1.66, λ≈2/3, and 3rd-order tuning results in nearly zero meshing stiffness fluctuation. When ε0≈1.75, λ=3/4, and 4th-order tuning results in nearly zero meshing stiffness fluctuation.
4. Experimental Design for Vibration Reduction Using Staggered Phase Tuning
To validate the vibration reduction effect of staggered phase tuning gear transmission, an experimental scheme for parallel-shaft spur gear pairs with variable staggered phases is designed, and vibration response experiments are conducted.
4.1 Design of Staggered Phase Tuning Scheme
To validate the vibration reduction effect of staggered phases and minimize errors introduced by experimental part processing and assembly, a spline connection is used in this experiment. Three staggered phase angles are selected for experimental study: staggered phase p=0, 0.2, and 0.4. To obtain three sets of tuned gear pairs, since the number of teeth for a pair of gears in the design is 36, the tooth space angle is 360°/36=10°. To match the staggered phase angles, the number of spline teeth is designed to be 30, and different staggered phase tuning gears are obtained by adjusting the spline tooth positions.
When the spline teeth are misaligned by 0 teeth, a 0-staggered phase tuning gear is obtained. When the spline teeth are misaligned by 1 tooth, it is equivalent to rotating the second gear by 360°/30×1=12° on the same shaft while the first gear remains stationary, resulting in a tooth misalignment angle of 2° and a staggered phase of 2°/10°=0.2. When the spline teeth are misaligned by 2 teeth, it is equivalent to rotating the second gear by 360°/30×2=24° on the same shaft, resulting in a tooth misalignment angle of 4° and a staggered phase of 4°/10°=0.4.
When the spline teeth are misaligned by 3 teeth, the tooth misalignment angle is 6°, and the staggered phase is 6°/10°=0.6, which is structurally identical to the case with staggered phase p=0.4. When misaligned by 4 teeth, the angle is 8°, the staggered phase is 0.8, and it is structurally identical to p=0.2. When misaligned by 5 teeth, the angle is 10°, which is one tooth space angle, and it is structurally identical to p=0. When misaligned by 6 teeth, the angle is 12°, and it is structurally identical to p=0.2. Based on the meshing principle of staggered phase tuning, spur gear pairs with different staggered phases are installed.
4.2 Test Rig Principle and Structure
The working principle of the test rig, and the overall structure of the test rig. The test rig mainly includes: 1. Servo drive motor; 2, 4, 7, 9, 12, 14. Couplings; 3, 13. Torque speed sensors; 5, 11. Bearing seats; 6, 10. Grating sensors; 8. Test gearbox; 15. Load motor; 16, 17, 18. Test bench frame.
4.3 Sensor Arrangement
To accurately obtain the vibration parameters of the transmission shaft in the radial direction, it is necessary to fix the vibration displacement sensors on the test rig in two perpendicular directions through a zigzag bracket. The measuring points for vibration displacement are arranged at points (a), (b), (c), and (d) . Vibration acceleration sensors can measure vibration responses in three directions, so only one sensor needs to be placed at the input end and the output end respectively. The measuring points for vibration acceleration are arranged at points (e) and (f) .
5. Experimental Study on Vibration Reduction Using Staggered Phase Tuning Gears
At an input speed of 500 r/min and a load of 20 N·m, the vibration displacements of the input end and output end of the parallel-shaft spur gear pairs with different staggered phases are tested. The vibration displacement results in the x-direction at the input end for staggered phases p=0, p=0.2, and p=0.4. The frequency domain results of the vibration displacements at the input end. The vibration displacement results in the x-direction at the output end for different staggered phases. The frequency domain results of the vibration displacements at the output end. The x-direction is defined as the horizontal vibration displacement, and the y-direction as the vertical vibration displacement. Due to space limitations, only the vibration response results in the x-direction are presented here.
For staggered phases p=0, p=0.2, and p=0.4, the root mean square (RMS) values of the vibration displacements at the input end are 23.0 μm, 11.8 μm, and 10.1 μm, respectively. Compared to the staggered phase p=0 (without tooth staggering), the RMS values of the vibration displacements at p=0.2 and p=0.4 decrease by approximately 48.7% and 56.1%, respectively. At the output end, the RMS values of the vibration displacements are 22.9 μm, 10.3 μm, and 11.9 μm, respectively, showing reductions of about 55.0% and 48.0% compared to p=0 for p=0.2 and p=0.4. the rotation frequency f_z and its harmonics are the main excitation frequencies for vibration displacement. At p=0, the amplitude of the rotation frequency at the input end is 28.9 μm; at p=0.2 and p=0.4, the amplitudes are 15.1 μm and 12.6 μm, respectively, representing decreases of 47.8% and 56.4%. The amplitudes of the harmonics of the rotation frequency are relatively smaller.
At an input speed of 500 r/min and a load of 20 N·m, the vibration accelerations of the input end and output end of the parallel-shaft spur gear pairs with different staggered phases are tested. The vibration acceleration results in the x-direction at the input end for staggered phases p=0, p=0.2, and p=0.4. The frequency domain results of the vibration accelerations at the input end. The vibration acceleration results in the x-direction at the output end for different staggered phases. The frequency domain results of the vibration accelerations at the output end.
For staggered phases p=0, p=0.2, and p=0.4, the RMS values of the vibration accelerations at the input end are 6.89 m/s², 5.17 m/s², and 4.22 m/s², respectively. Compared to the staggered phase p=0, the RMS values of the vibration accelerations at p=0.2 and p=0.4 decrease by approximately 24.9% and 38.8%, respectively. At the output end, the RMS values of the vibration accelerations are 5.85 m/s², 4.98 m/s², and 4.08 m/s², respectively, representing reductions of about 14.9% and 30.3% compared to p=0 for p=0.2 and p=0.4. It can be observed that regardless of the staggered phase value, the vibration acceleration at the input end is slightly greater than that at the output end. This is due to factors such as meshing impacts that inevitably consume energy during the actual operation of the gearbox, resulting in a slightly lower output power and consequently lower vibration acceleration at the output end. the meshing frequency f_m and its harmonics are the main excitation frequencies for vibration acceleration. At p=0.2 and p=0.4, the amplitudes of the meshing frequency and its harmonics are smaller than those at p=0.
Vibration accelerations at the input end and output end of the parallel-shaft spur gear pairs are tested for different load torques at staggered phases p=0, p=0.2, and p=0.4. the experimental results of the RMS values of the vibration accelerations in the x-direction at the input end and output end, respectively, for different load torques. For comparison, the simulation results of the RMS values of the vibration accelerations at the input end and output end for different torques are presented , respectively.
The study finds that regardless of the staggered phase value, the vibration accelerations at both the input end and output end increase with increasing load torque. Both the experimental and simulation results show that the vibration accelerations at staggered phases p=0.2 and p=0.4 are lower than those at p=0. This validates that the theoretical model of vibration reduction using staggered phases is consistent with the experimental results in terms of the vibration reduction trend. For the input end, compared to the experimental results at p=0, the experimental results at p=0.2 and p=0.4 decrease by an average of about 15.9% and 28.6%, respectively, while the simulation results decrease by an average of about 19.4% and 32.3%, respectively. For the output end, the experimental results decrease by an average of 18.2% and 30.4%, and the simulation results decrease by an average of 19.1% and 31.9%. This indicates that the method of staggered phases can reduce the vibration acceleration of gear pairs. The experimental results are slightly larger than the simulation results, mainly due to installation errors, shaft misalignment, bearing clearances, and other factors during the processing and assembly of experimental components.
6. Conclusion
This study investigates the relationship between the meshing stiffness of spur/helical gear pairs with staggered phases and the staggered phase, as well as the vibration reduction effect of the staggered phase method on parallel-shaft spur/helical gear transmission systems. Experimental schemes for parallel-shaft spur gear pairs and helical gear pairs with different staggered phases are designed, and vibration response experiments are conducted under different staggered phases. The experimental results are compared with simulation results. The main conclusions are as follows:
Experimental schemes for parallel-shaft gear pairs with different staggered phases are designed and experiments are conducted. By comparing the simulation model with the experimental results, it is shown that the theoretical model of vibration reduction using staggered phases is consistent with the experimental results in terms of the vibration reduction trend.
A calculation model for the meshing stiffness of gear pairs with staggered phases is established. The results show that the time-varying meshing stiffness of gears with staggered phases is related not only to the contact ratio but also to the staggered phase p.
The vibration responses of parallel-shaft gear pairs under different staggered phases are studied. Based on the vibration responses and time-varying meshing stiffness of gear pairs at different staggered phases, the optimal staggered phase for gear pairs is obtained.