Sage Journals: Discover world-class research

Abstract

There is a nonlinear disturbance problem in the operation of large inertia load space-driving mechanism, which seriously affects the normal operation of the system. A 14 degree-of-freedom nonlinear time-varying dynamic model was established for a two-stage spur gear system. The dynamic equations were solved numerically based on the Runge–Kutta method. The correctness of the dynamic model was verified through experiments. In the author’s previous research, the transmission error and dynamic response of gear system was analyzed. After the establishment of the dynamic model, a comparative analysis of the load response under different inertia was performed to illustrate the importance of studying large inertia loads. A large inertia load transmission error experimental device was set up to collect and process the transmission error data under different load inertia and different speeds. Comparing experimental results with numerical results, the correctness of the numerical model was verified, and the reasons for the differences between the two were explained. The analysis of the experimental results shows that for the transmission error of large inertia load gear transmission system, the influence of stiffness excitation on the transmission error amplitude is dominant. For the high-speed gear system, the pitch error plays a dominant role.

Keywords

Transmission error large inertia load gear system dynamic model stiffness excitation

Introduction

The gearbox transmission system is widely used in the fields of machine tools, aerospace, ships and vehicles, which plays an important part of mechanical transmission. It is of great significance to conduct fault diagnosis on gears^1,2 and bearings.^3–6 With the gradual development of fault diagnosis technology, many signal processing methods^7–9 and dynamic models^10,11 for gears and bearings have been proposed and established by scholars. With the development of gear transmission reduction mechanism to high precision and large load, the vibration phenomena generated by the gear transmission system under various excitations cannot be ignored. The vibration of most mechanical systems comes from the gear system. The vibration and gear system wear and tear cause the actual trajectory of the gear to deviate from the theoretically calculated value, resulting in transmission errors (TEs), lowering the transmission accuracy, and even causing serious damage to the gear system and other mechanical structures.

There is less research on large inertia load. Currently, it mainly focuses on simulation analysis. There is relatively little information on theoretical research. Z Xu¹² proposed an equivalent simulation method for oversized inertial loads, simulated the huge moment of inertia required by the space station manipulator, and verified the effectiveness of the method through numerical simulation under different working conditions. T Yang et al.¹³ considered the influence of large reduction ratios and studied the dynamic characteristics of space manipulators, but did not analyze the characteristics of large inertia loads. S Chen et al.¹⁴ established the nonlinear dynamic model of bending and torsion coupling for low-speed heavy-load planetary gears with friction, and studied the influence of friction on the nonlinear dynamic characteristics of low-speed heavy-duty gear transmissions. Large inertia load system will produce vibration during acceleration and deceleration. During the speed change process, the system’s steady state will be broken, which will lead to the reduction of transmission accuracy, noise, and the reduction of the service life of some components. The space-driven mechanism studied in this article needs to be started and stopped frequently during the work process, and there is a clear disturbance phenomenon when starting and stopping. Therefore, the influence of large inertia load on the system is analyzed. A Hammami¹⁵ developed a planetary gear-twisted lumped parameter model with power recirculation to study nonlinear behavior under variable load conditions.

Studying the TE of the gear system, analyzing the impact of error excitation on the gear transmission system, and understanding the relationship between tooth profile deviation and pitch deviation and load transmission error (LTE) provide theoretical guidance for gear design, machining, and installation. Fernández et al.^16,17 extended the tooth profile deviation and tooth pitch deviation in the dynamic model, and carried out profile modification. MA Hotait and A Kahraman¹⁸ described a set of gear system dynamics testing device, proposed unmodified and modified spur gear dynamic factor, and dynamic transmission error (DTE) measurement method, and proved their relationship through experiments. C Xun et al.¹⁹ established a nonlinear stochastic model to describe the dynamic behavior of the planetary gear train (PGT), taking into account time-varying meshing stiffness²⁰ and random tooth profile error. Bozca²¹ proposed the optimization of the geometric design parameters of the gearbox based on the TE model to reduce the gear noise in the automotive transmission. T Lin and Z He²² calculated the DTE of the gear system through the finite element method, and considered the assembly error and machining error in the model. W Guangjian et al.²³ considered the error excitation in the model and analyzed the influence of the backlash on the TE.

The experimental research on the gear transmission system can be repeated several times under the same experimental conditions to help researchers discover some new phenomena. The experimental results can be used to verify the correctness of the gear dynamic model and to optimize the insufficiency of the dynamics model. NK Raghuwanshi and P Anand²⁴ used the laser displacement sensor to measure the spur gear tooth deflection along the line of action. Then the gear mesh stiffness is calculated. The experiment is also performed on cracked tooth pair to measure the mesh stiffness. An encoder-based method is applied to measure vibration acceleration of the gear pair, and accelerometer-based measurement systems are employed to obtain the dynamic responses of the housing by L Chao et al.²⁵ L Yu et al.²⁶ presented the dual-eccentricity model for calculating the TE caused by eccentricity errors of gears. Then the experiments are carried out at different input speed and initial starts for acquiring TE of the designed backlash compensation device. X Cao et al.²⁷ verified the real contact patterns of gears using the gear-rolling test. The root mean square values of the first three orders of frequency of the dynamic transfer error under different working conditions were analyzed, and the jump of the gear, periodic motion, complex long-period motion, chaotic motion, and other complex nonlinear vibration forms transmission system were discovered. Subsequently, Kahraman and colleagues^28–30 used the same device to study the influence of the degree of overlap and profile modification on the dynamic characteristics of spur gears and helical gears. Pramono³¹ analyzed the influence of four different tooth shapes on the dynamic characteristics of the gear in the nonresonant region. Velex and Ajmi³² used a test method for helical gears under quasi-static conditions to analyze the approximate relationship between static transfer errors and dynamic factors. Recently, Hotait and Kahraman¹⁸ used experimental methods to analyze the relationship between the DTE and the dynamic factor of the modified gear and the unmodified gear.

In the previous research of the author, the nonlinear dynamics of 14 degrees of freedom were established by considering the factors such as time-varying stiffness, backlash, and pitch error for the nonlinear disturbance problem of the large inertia load space-driven mechanism. Based on the model, the TE and load response were analyzed, and an optimized design was proposed based on the analysis results. This article aims to verify the correctness of the dynamic model and TE simulation analysis. An experimental platform for experimental analysis was proposed. At the same time, the experimental analysis has drawn some conclusions, which enriched the research on large inertia loads, and provided an important reference for the actual design of the gear system.

Dynamic modeling and simulation analysis

Model establishment and solution

Regarding the dynamic modeling of the two-stage spur gear deceleration system of the inertial load space drive mechanism, it is assumed that the mass, the moment of inertia, the radius, and the average meshing stiffness of each gear are evenly distributed along the center gear, and the system damping is the elastic damping. The two-stage spur gear transmission system is simplified and modeled by dynamics, as shown in Figure 1.

Figure 1.

Dynamic model of a two-stage gear transmission system.

Considering the influence of the time-varying meshing stiffness of the two-stage gear system, the gear radial displacement, tooth profile and pitch error on the system, the dynamic equations of the two-stage spur gear transmission system is built based on Newton’s second law and related dynamic knowledge. The dynamic equations are as follows

$m_{1} {\overset{\cdot\cdot}{x}}_{1} = - k_{x 1} x_{1} - c_{x 1} {\overset{\cdot}{x}}_{1}$ (1)

$m_{2} {\overset{\cdot\cdot}{x}}_{2} = - k_{x 2} x_{2} - c_{x 2} {\overset{\cdot}{x}}_{2}$ (2)

$m_{3} {\overset{\cdot\cdot}{x}}_{3} = - k_{x 3} x_{3} - c_{x 3} {\overset{\cdot}{x}}_{3}$ (3)

$m_{4} {\overset{\cdot\cdot}{x}}_{4} = - k_{x 4} x_{4} - c_{x 4} {\overset{\cdot}{x}}_{4}$ (4)

$\begin{matrix} m_{1} {\overset{\cdot\cdot}{y}}_{1} = - k_{y 1} y_{1} - c_{y 1} {\overset{\cdot}{y}}_{1} + k_{t 1} (r_{b 1} θ_{1} - r_{b 2} θ_{2} - y_{1} + y_{2} + e_{t 1}) \\ + c_{t 1} (r_{b 1} {\overset{\cdot}{θ}}_{1} - r_{b 2} {\overset{\cdot}{θ}}_{2} - {\overset{\cdot}{y}}_{1} + {\overset{\cdot}{y}}_{2} + {\overset{\cdot}{e}}_{t 1}) \end{matrix}$ (5)

$\begin{matrix} m_{2} {\overset{\cdot\cdot}{y}}_{2} = - k_{y 2} y_{2} - c_{y 2} {\overset{\cdot}{y}}_{2} - k_{t 1} (r_{b 1} θ_{1} - r_{b 2} θ_{2} - y_{1} + y_{2} + e_{t 1}) \\ - c_{t 1} (r_{b 1} {\overset{\cdot}{θ}}_{1} - r_{b 2} {\overset{\cdot}{θ}}_{2} - {\overset{\cdot}{y}}_{1} + {\overset{\cdot}{y}}_{2} + {\overset{\cdot}{e}}_{t 1}) \end{matrix}$ (6)

$\begin{matrix} m_{3} {\overset{\cdot\cdot}{y}}_{3} = - k_{y 3} y_{3} - c_{y 3} {\overset{\cdot}{y}}_{3} + k_{t 2} (r_{b 3} θ_{3} - r_{b 4} θ_{4} - y_{3} + y_{4} + e_{t 2}) \\ + c_{t 2} (r_{b 3} {\overset{\cdot}{θ}}_{3} - r_{b 4} {\overset{\cdot}{θ}}_{4} - {\overset{\cdot}{y}}_{3} + {\overset{\cdot}{y}}_{4} + {\overset{\cdot}{e}}_{t 2}) \end{matrix}$ (7)

$\begin{matrix} m_{4} {\overset{\cdot\cdot}{y}}_{4} = - k_{y 4} y_{4} - c_{y 4} {\overset{\cdot}{y}}_{4} - k_{t 2} (r_{b 3} θ_{3} - r_{b 4} θ_{4} - y_{3} + y_{4} + e_{t 2}) \\ - c_{t 2} (r_{b 3} {\overset{\cdot}{θ}}_{3} - r_{b 4} {\overset{\cdot}{θ}}_{4} - {\overset{\cdot}{y}}_{3} + {\overset{\cdot}{y}}_{4} + {\overset{\cdot}{e}}_{t 2}) \end{matrix}$ (8)

$J_{0} {\overset{\cdot\cdot}{θ}}_{0} = M_{0} - k_{1} (θ_{0} - θ_{1}) - c_{1} ({\overset{\cdot}{θ}}_{0} - {\overset{\cdot}{θ}}_{1})$ (9)

$\begin{matrix} J_{1} {\overset{\cdot\cdot}{θ}}_{1} = k_{1} (θ_{0} - θ_{1}) + c_{1} ({\overset{\cdot}{θ}}_{0} - {\overset{\cdot}{θ}}_{1}) - r_{b 1} \\ [\begin{matrix} k_{t 1} (r_{b 1} θ_{1} - r_{b 2} θ_{2} - y_{1} + y_{2} + e_{t 1}) \\ + c_{t 1} (r_{b 1} {\overset{\cdot}{θ}}_{1} - r_{b 2} {\overset{\cdot}{θ}}_{2} - {\overset{\cdot}{y}}_{1} + {\overset{\cdot}{y}}_{2} + {\overset{\cdot}{e}}_{t 1}) \end{matrix}] \end{matrix}$ (10)

$\begin{matrix} J_{2} {\overset{\cdot\cdot}{θ}}_{2} = - k_{2} (θ_{2} - θ_{3}) - c_{2} ({\overset{\cdot}{θ}}_{2} - {\overset{\cdot}{θ}}_{3}) + r_{b 2} \\ [\begin{matrix} k_{t 1} (r_{b 1} θ_{1} - r_{b 2} θ_{2} - y_{1} + y_{2} + e_{t 1}) \\ + c_{t 1} (r_{b 1} {\overset{\cdot}{θ}}_{1} - r_{b 2} {\overset{\cdot}{θ}}_{2} - {\overset{\cdot}{y}}_{1} + {\overset{\cdot}{y}}_{2} + {\overset{\cdot}{e}}_{t 1}) \end{matrix}] \end{matrix}$ (11)

$\begin{matrix} J_{3} {\overset{\cdot\cdot}{θ}}_{3} = k_{2} (θ_{2} - θ_{3}) + c_{2} ({\overset{\cdot}{θ}}_{2} - {\overset{\cdot}{θ}}_{3}) - r_{b 3} \\ [\begin{matrix} k_{t 2} (r_{b 3} θ_{3} - r_{b 4} θ_{4} - y_{3} + y_{4} + e_{t 2}) \\ + c_{t 2} (r_{b 3} {\overset{\cdot}{θ}}_{3} - r_{b 4} {\overset{\cdot}{θ}}_{4} - {\overset{\cdot}{y}}_{3} + {\overset{\cdot}{y}}_{4} + {\overset{\cdot}{e}}_{t 2}) \end{matrix}] \end{matrix}$ (12)

$\begin{matrix} J_{4} {\overset{\cdot\cdot}{θ}}_{4} = - k_{3} (θ_{4} - θ_{5}) - c_{3} ({\overset{\cdot}{θ}}_{4} - {\overset{\cdot}{θ}}_{5}) + r_{b 4} \\ [\begin{matrix} k_{t 2} (r_{b 3} θ_{3} - r_{b 4} θ_{4} - y_{3} + y_{4} + e_{t 2}) \\ + c_{t 2} (r_{b 3} {\overset{\cdot}{θ}}_{3} - r_{b 4} {\overset{\cdot}{θ}}_{4} - {\overset{\cdot}{y}}_{3} + {\overset{\cdot}{y}}_{4} + {\overset{\cdot}{e}}_{t 2}) \end{matrix}] \end{matrix}$ (13)

$J_{5} {\overset{\cdot\cdot}{θ}}_{5} = k_{3} (θ_{4} - θ_{5}) + c_{3} ({\overset{\cdot}{θ}}_{4} - {\overset{\cdot}{θ}}_{5}) - M_{5}$ (14)

Detailed description of the dynamic equations can be found in the literature.⁴ There are two main methods for solving the dynamic model. One is the analytical method. This method does not pursue the refinement of the model. The main purpose is to explain the various complex nonlinear dynamic phenomena of gear transmission. Another method is the numerical method, which is used to study systems with more degrees of freedom. Pursuing a refined mathematical model, the analysis results are closer to reality.

The meshing stiffness of the gear system is approximately parabolic by changing of meshing position. For determined by the parameter gear pair, the theoretical position of meshing in, meshing out, and the joint are fixed. Thus, the time-varying stiffness curve of gear pair 1 and gear pair 2 were obtained by the undetermined coefficient method, as illustrated in Figure 2. Then the time-varying stiffness data can be substituted into the dynamics model.

Figure 2.

Time-varying stiffness curve of gear pair 1 and gear pair 2.

Due to the existence of backlash, the two gears engaged with each other cannot always be engaged. Specifically, during the operation of starting procedure, the drive wheel must be rotated by the angle corresponding to the idle path, before engaging with the driven wheel and then the meshing stiffness appears. Therefore, it is necessary to modify the time-varying meshing stiffness in the dynamic equation. The modified time-varying meshing stiffness were

$k'_{t 1} = {\begin{matrix} 0, θ_{1} \leq j_{φ 1} \\ k_{t 1}, θ_{1} > j_{φ 1} \end{matrix}$ (15)

$k'_{t 2} = {\begin{matrix} 0, θ_{3} \leq j_{φ 2} \\ k_{t 2}, θ_{3} > j_{φ 2} \end{matrix}$ (16)

In the formula, the following symbols represent:

$k'_{t 1}, k'_{t 2}$ : Actual meshing stiffness of gear pair 1 and gear pair 2,

$j_{φ 1}, j_{φ 2}$ : The backlash of gear pair 1 and gear pair 2.

Errors can inevitably occur in the production process and installation process of the gear system. Error excitation is a significant factor in the generation of vibration and noise. In this gear system, the tooth profile error and pitch error were considered in the form of a simple harmonic function, as follows

$e_{pf} = a_{1} \sin (α) + b_{1} \sin (2 α) + c_{1} \sin (3 α)$ (17)

$e_{pt} = a_{2} \sin (θ) + b_{2} \sin (2 θ) + c_{2} \sin (3 θ)$ (18)

In the formula, the symbols represent

$e_{pf}, e_{pt}$ : Profile error and pitch error,

$a_{1}, b_{1}, c_{1}$ : Amplitude coefficient of each order harmonic function of profile error,

$a_{2}, b_{2}, c_{2}$ : Amplitude coefficient of each order harmonic function of pitch error,

$α$ : Meshing angular displacement,

$θ$ : Gear rotation angular displacement.

In summary, the gear TE can be expressed as

$e_{t} = e_{pf} + e_{pt}$ (19)

Here, the numerical solution is used to solve the numerical solution of the differential equation of the two-stage gearing dynamic model above. The solution method is mainly to use the ordinary differential equation (ODE) solver in Matlab software. The solver is based on the Runge–Kutta method. In order to solve the computational convenience, when solving the dynamic response of a two-stage gear transmission system, it is assumed that the radial stiffness and damping of the four bearings at the support bearings at both ends of the gear are equal. The angular velocity response curve of each gear was solved, as shown in Figure 3.

Figure 3.

Angular speed curve of each gear.

Given the output angular velocity of the motor $6 ° / s$ , the time interval of the first-stage gear meshing is $T_{1} = 20 ° / 6 ° / s \approx 3.33 s$ . Similarly, the time interval of the second-stage gear tooth change can be obtained, $T_{2} = 16.67 s$ . This means that the gear system will produce an engagement shock each time it passes through $3.33 s$ and $16.67 s$ . From Figure 3, it can be clearly seen that the stiffness impact caused by the change of the meshing gear teeth during gear meshing. The previously analyzed meshing impact interval $T_{1}, T_{2}$ is consistent with the meshing impact period in Figure 3, validating the correctness of the dynamic model.

Simulation analysis

The large inertia load is taken into account in the dynamic model to simulate and control the motor of the driven mechanism. The gear system quickly reach the required angular velocity in a short time, and maintain a constant speed for a certain period of time. When the brake is applied at 35s, the motor speed is quickly reduced to 0, and the load angular velocity response curve is shown in the red curve of Figure 3. To study the influence of large inertia loads on the dynamic response of the drive mechanism, the load inertia was reduced to one-tenth of the original and the load angular velocity response curve was obtained, as shown in the blue curve of Figure 4.

Figure 4.

Comparison of dynamic response under different load inertia.

From Figure 4, it can be seen that when the motor start and brake, the load angular velocity response of the drive mechanism under large inertia loads has a significant inertial hysteresis relative to small inertial loads, which is caused by the time-lag effect of large inertia loads. Large inertia loads cause the system to respond slowly, and the dynamic characteristics of each axis are inconsistent, which inevitably causes the TE of the system. When the motor start and brake, the load angular velocity response of the drive mechanism under large inertia loads fluctuates obviously relative to a small inertia load. At this time, the drive mechanism generates large vibration, which results in poor system stability and seriously affects the service life of the drive mechanism. In normal operation, compared with small inertia loads, the angular velocity response of the large inertia load drive mechanism is less frequent and the operation is more stable. This is because the large inertia load helps to absorb disturbances during operation and has strong antidisturbance capability. By contrast, it is found that there is a big difference between the dynamic response of the gear system under large inertia load and small inertia load. Therefore, when studying the two-stage straight gear system of space drive mechanism, large inertia load should be taken into account. It is of great significance to study large inertia loads. In the subsequent experimental studies, the TEs under different load inertias were analyzed.

Dynamic model experimental verification

Experimental equipment and system

To verify the correctness of the dynamic model of the space-driven gear system, experiments were carried out on an inertial load simulator. The three-dimensional model and the factual picture of the experimental device are shown in Figure 5.

Figure 5.

3D model and physical map of the experiment device.

The experimental device and data acquisition system are shown in Figure 6. The input shaft of the space drive gear system is connected to the stepper motor, and the output shaft is connected to the inertia load simulator. Two eddy current sensors are installed at the radial position of the output shaft to measure the eccentricity of the output shaft. The two eddy current displacement sensor are installed in each of the horizontal and vertical directions of the output shaft section, and the displacement sensor is composed of a probe, a preamplifier, and a signal line. In the static condition, the distance between the front-end surface of each displacement sensor and the test surface is about 1 mm. Solving two derivatives with displacement sensor signal, the output shaft vibration acceleration signal can be obtained. The parameters of the two-stage gear drive system are presented in Table 1.

Figure 6.

Experiment device and data acquisition system.

Table 1.

Two-stage gear drive system parameters.

Gear pair 1		Gear pair 2
Number of teeth of gear 1	18	Number of teeth of gear 3	18
Number of teeth of gear 2	90	Number of teeth of gear 4	360
Modulus of gear 1	0.25 mm	Modulus of gear 3	0.5 mm
Modulus of gear 2	0.25 mm	Modulus of gear 4	0.5 mm
Material of gear 1	Stainless steel	Material of gear 3	Stainless steel
Material of gear 2	Titanium alloy	Material of gear 4	Titanium alloy
Accuracy level	6 h	Accuracy level	6 h
Tolerance of cycle accumulated deviationsof gear 1	0.016 mm	Tolerance of cycle accumulated deviations of gear 3	0.016 mm
Tolerance of cycle accumulated deviationsof gear 2	0.02 mm	Tolerance of cycle accumulated deviations of gear 4	0.032 mm
Tolerance of PE of gear 1	0.008 mm	Tolerance of PE of gear 3	0.008 mm
Tolerance of PE of gear 2	0.008 mm	Tolerance of PE of gear 4	0.006 mm

The hardware of the data acquisition system mainly includes computers, NI acquisition cards, and eddy current displacement sensors. The technical specifications and detailed parameters of each part are shown in Table 2.

Table 2.

The table of data acquisition system equipment.

No.	Device name	Model number	Technical indicators
1	Computer	Lenovo QitianM7150	Core Duo E8400 CPU/intel G41/4G
2	Data acquisition card	USA NI PCI-6221	16 analog inputs, 16-bit resolution, 250 KS/s sampling rate
3	Eddy current displacement sensor	YXS-DWA	Range 0–2 mm, frequency response 0–10kHz,Full output 5000mvDC

The eccentricity test principle of output shaft is shown in Figure 7. Point C is the center point when the output shaft is stationary, and point E is the center point when the output shaft is offset. The two eddy current sensors are arranged vertically. Both sensors pass through the axis C in the initial state. $x_{m}$ and $y_{m}$ are the displacements of the output shaft in both directions measured by the test, $x_{e}$ and $y_{e}$ are the actual offsets of the shaft center in two directions, and $R$ is the output shaft radius.

Figure 7.

Principle of output shaft eccentricity test.

From Figure 7 we can see $AB = CD + AC - BD$ , that is

$x_{m} = x_{e} + R - \sqrt{R^{2} - {y_{e}}^{2}}$ (20)

Similarly it can be obtained that

$y_{m} = y_{e} + R - \sqrt{R^{2} - {x_{e}}^{2}}$ (21)

Simultaneous formulae (20) and (21), the output shaft eccentricity can be obtained

$e = \sqrt{{x_{e}}^{2} + {y_{e}}^{2}}$

During the experiment, the stepping motor drives the gear system and the input rotation speed is $6 ° / s$ . Apply load inertia of 25.8 Kg m², 19.35 Kg m², 12.9 Kg m², 6.45 Kg m², and 2.58 Kg m² to the output shaft of the gear system via an inertial load simulator. The data acquisition system collects the displacement and vibration data of the output shaft during gear operation.

Experimental results analysis and verification

When the load inertia is the default value, 25.8 Kg m², according to the data measured by the eddy current displacement sensor, the time domain curve of the eccentricity of the output shaft is calculated, as shown in Figure 8. The time domain curve of the data measured by the vibration acceleration sensor is shown in Figure 9. The data are Fourier-transformed to obtain the frequency domain curve, and compared with the frequency domain and the curve obtained by solving with Matlab, as shown in Figures 8 and 9.

Figure 8.

Time domain and frequency domain diagram of output shaft eccentricity.

Figure 9.

Time domain and frequency domain diagram of output shaft vibration acceleration.

From the frequency domain curves in Figures 8 and 9, it can be seen that the experimental results are in good agreement with the results of the numerical solution. The frequency components are mainly concentrated on the frequency of the input shaft, intermediate shaft, and the meshing frequency of the two-stage gear pair and its frequency multiplier. There are still some frequency components that cannot be analyzed in the experimental results.

According to the experimental results, the peak-to-peak value of the output shaft eccentricity and vibration acceleration are calculated separately. A comparison between the experimental and the Matlab solution results was made, as shown in Tables 3 and 4. According to the data in the table, the comparison result between the experimental result and the Matlab solution result is shown in Figure 10. From Figure 10, we can see that with the increase of load inertia, the peak-to-peak value of the output shaft eccentricity gradually increases, and the peak-to-peak value of the vibration acceleration gradually decreases. The experimental value and the Matlab simulation value trend is basically consistent, and there is a certain relative error between the two. The correctness of the dynamic model was verified through experiments, which verified the correctness of the dynamic response analysis and TE simulation analysis of previous studies.

Table 3.

Peak-to-peak comparison of output shaft eccentricity.

Load inertia,Kg m²	Experimentalresults, mm	Simulationresults, mm	Error %
2.58	0.022	0.016	−27.27
6.45	0.054	0.033	−38.89
12.9	0.070	0.053	−24.29
19.35	0.091	0.066	−27.47
25.8	0.103	0.074	−28.16

Table 4.

Comparison of output shaft vibration acceleration.

Load inertia, Kg m²	Experimentalresults, mm	Simulation results, mm	Error %
2.58	0.165	0.150	−9.09
6.45	0.138	0.127	−7.97
12.9	0.115	0.111	−3.48
19.35	0.092	0.088	−4.35
25.8	0.056	0.044	−21.43

Figure 10.

Comparison of experimental results and Matlab results. (a) Output shaft eccentricity, (b) vibration acceleration.

The main reason for differences between simulation results and experimental results are as follows. Due to the actual processing and installation errors of the gears, the friction and wear of the tooth surfaces, the lubrication of the oil film between the teeth, and so on, the parameter excitations related to the meshing frequency, such as the meshing stiffness and the tooth shape error, are continuously changed. In the model, only two-stage spur gear transmission systems are modeled dynamically, and the influence of components such as bearings and gearboxes is neglected. Vibration of the above-mentioned components in the experimental process will inevitably affect the vibration of the gear transmission system. Numerical simulation of the input torque and speed are taken as a fixed value. The output torque and speed of stepper motor cannot always be constant, while the output torque and speed go through couplings and bearing housings and other components, which have a certain influence on the actual input of the two-stage spur gear transmission system. There is a certain deviation between the actual installation position and the theoretical installation position of the sensor. There is white noise in the actual experimental system.

TE analysis

TE experimental

The TE experimental device is installed on the inertial load simulator. The schematic diagram is shown in Figure 11. The input and output rotation angles of the gear system are measured in real time by two encoders, and the overall TE of the gear system of the space drive mechanism is obtained through calculation. The input part of the gear system is a closed-loop stepper motor with its own high-precision encoder. The input angle of the gear system can be measured by reading the pulse accumulated value of the encoder. To measure the output rotation angle, the high-precision extension shaft is extended outside the housing, and the extension shaft is coupled with the encoder through the flexible coupling.

Figure 11.

TE experimental device.

The experimental system includes two high-precision encoders, one NI data acquisition card, and one computer. The encoder pulse signal is directly input into the data acquisition card. The experimental device is the same as the data acquisition card and computer used in the experimental device in part 3.

TE under different load inertia

The input speed of gear system is $6 ° / s$ driven by a stepper motor. Through the inertial load simulator, the load inertia of the gear system output shaft is respectively applied to 25.8 Kg m², 19.35 Kg m², 12.9 Kg m², 6.45 Kg m², and 2.58 Kg m². Through the data acquisition system, the angular displacements of the input shaft and the output shaft during gear operation are collected, and then the TE of the gear system is calculated. The time domain and frequency domain diagrams of the TE under different load inertias are shown in Figure 12. The red curve in the figure is the simulation result obtained through numerical solution. The blue curve is the experimental result.

Figure 12.

Time domain and frequency domain plots of TE under different load inertias.

In the gear transmission processing, as the load inertia decreases, the gear meshing frequency and its frequency multiplication amplitude gradually decrease, and the input shaft, while intermediate shaft and output shaft rotational frequency and its frequency multiplication amplitude gradually increase. The variation period of the time-varying stiffness and the tooth profile deviation are the same with the meshing period of the gears. The rotation frequency of the shaft is consistent with the period of change of pitch deviation. Therefore, with the reduction of the load inertia, the TE caused by the stiffness excitation and the tooth profile deviation is reduced, and the TE caused by the pitch deviation is aggravated. It can be seen that for the large inertial load space-driven deceleration mechanism, to reduce the TE of the system and improve the transmission accuracy, the tooth profile can be modified to effectively reduce the TE caused by stiffness excitation and profile deviation.

There are some differences between the simulation results and the experimental results. To perform a clearer analysis, the peak-to-peak value of TE of the simulation results and the experimental results are extracted and compared, as shown in Figure 13. It can be seen from Figure 13 that as the load inertia increases, the peak-to-peak value of TE first becomes smaller and then becomes larger. The reason may be that within a certain range, the TE due to deformation increases with the increase of load inertia, which partially compensated for TEs caused by manufacturing and other factors. When the load inertia continues to increase, TEs caused by deformation plays a leading role.

Figure 13.

Comparison of experimental and numerical results.

It can be seen from Figure 13 that within a certain range of load inertia, the numerical solution results are consistent with the experimental results, which shows the correctness of the numerical model, and also prove that the author’s previous simulation analysis and other work has a certain practical value. When the load inertia is small, the peak-to-peak value of the TE obtained by the numerical solution have little difference with the experimental result. But when the load inertia is large, the peak-to-peak value of the TE obtained by the numerical solution differ slightly bigger with the experimental result. The theoretically calculated data do not fully reflect the basic conditions of the actual TE, but it has a certain reference value. The reason may be that under large inertia, the coupling effect between the first-stage transmission and the second-stage transmission has a great influence. Therefore, the dynamic model needs to be optimized afterward.

TE under different input speeds

The inertia load simulator applies a load inertia of 25.8 Kg m² to the output shaft of the gear system, via a stepper motor driving the gear system, the input speed are set to $6 ° / s$ , $15 ° / s$ , $30 ° / s$ , $45 ° / s$ , and $60 ° / s$ . Through the data acquisition system, the angular displacements of the input shaft and the output shaft during gear operation are collected, and then the TE of the gear system is calculated. The time domain and frequency domain diagrams of the TE at different speeds are shown in Figure 14. The red curve in the figure is the simulation result obtained by numerical solution. The blue curve is the experimental result.

Figure 14.

Time domain and frequency domain plots of TE at different speeds.

It can be seen from Figure 14 that with the increase of the rotational speed, the TE caused by the pitch deviation is intensified, and the stiffness excitation has less effect on TE. It can be seen that for high-speed operation of the gear system, the main influence on transmission accuracy is pitch deviation. The TE caused by the pitch deviation can be reduced by increasing the machining accuracy and gear pairing. In addition, although the pulsation impact is generated at the node due to the abrupt change of the direction of friction when the gear meshes, the effect of this abrupt change is not seen in the actual results. This shows that the impact of the frictional force on dynamic responds to the gear system under normal conditions is relatively small.

The peak-to-peak values of TE at different input speeds are extracted. The comparison between the experimental results and the simulation results is shown in Figure 15. With the increase of rotating speed, the LTE increases linearly. The experimental results are consistent with the simulation results and the increasing trend is consistent, which proves the correctness of the numerical model. There is a certain difference between the experimental results and the simulation results in Figure 15, which have already been introduced in the foregoing and will not be repeated.

Figure 15.

Comparison of experimental and numerical results.

By analyzing the time-domain diagram of the TE of the motion process of the gear system, the factors that may cause the difference between the calculated value and the experimental value of the TE are as follows: In this theoretically calculated model, the time-varying stiffness is assumed to be a parabola, but the actual gear pair meshing stiffness is complex and varied. In the simulation, the coupling between the spur gear pair was not considered. Other factors such as motor input torque and load inertia fluctuations also have a great influence on the experimental value.

Conclusion

This article takes the large inertial load space-driven mechanism as the research object, establishes the nonlinear time-varying dynamics model, solves the dynamic equations through the Runge–Kutta method in Matlab, and verifies the correctness of the dynamic model through experiments. The TE under different loads and speeds are collected and processed respectively. The results are compared with the Matlab results and the correctness of the numerical model is verified.

In a certain range of load inertia, the Matlab solution results are consistent with the experimental results. When the load inertia is large, the TE peak-to-peak value obtained by the Matlab solution is quite different from the experimental result. The theoretically calculated data cannot fully reflect the basic status of the actual TE, but it has a certain reference value, and the dynamic model needs to be performed subsequently.

For gear systems with large inertia loads, the effect of stiffness excitation on the amplitude of TEs dominates, and the controlling of stiffness (tooth profile modification) is the key to improve the accuracy and dynamic stability of gear transmission. For a high-speed gear system, pitch deviation dominates, and the TE can be reduced by increasing gear machining accuracy.

Footnotes

Handling Editor: James Baldwin

Data availability

The data used to support the findings of this study are available from the corresponding author upon request.

Declaration of conflicting interests

The author(s) declared no potential conflicts of interest with respect to the research,authorship,and/or publication of this article.

Funding

The author(s) disclosed receipt of the following financial support for the research,authorship,and/or publication of this article: This work was supported by the National Natural Science Foundation of China (51575007) and the Municipal Education Commission project of Beijing (JC001011201601).

ORCID iDs

Chao Li

Dongxing Cao

References

Cui

Yao

Zhang

et al . Application of pattern recognition in gear faults based on the matching pursuit of a characteristic waveform. Measurement 2017; 104: 212–222.

Cui

. Transmission error analysis and disturbance optimization of two-stage spur gear space driven mechanism with large inertia load. Shock Vib 2018; 2018: 6863176.

Cui

Zhang

et al . Vibration response mechanism of faulty outer race rolling element bearings for quantitative analysis. J Sound Vib 2016; 364: 67–76.

Hao

Song

Wang

et al . Underdetermined source separation of bearing faults based on optimized intrinsic characteristic-scale decomposition and local non-negative matrix factorization. IEEE Access 2019; 7: 11427–11435.

Cui

et al . Quantitative trend fault diagnosis of a rolling bearing based on Sparsogram and Lempel-Ziv. Measurement 2018; 128: 410–418.

Wang

Ren

Song

. A novel weighted sparse representation classification strategy based on dictionary learning for rotating machinery. IEEE T Instrum Meas. Epub ahead of print 15 April 2019. DOI: 10.1109/TIM.2019.2906334

Cui

Jin

Huang

et al . Fault severity classification and size estimation for ball bearings based on vibration mechanism. IEEE Access. Epub ahead of print 15 April 2019. DOI: 10.1109/ACCESS.2019.2911323

Zhao

Jia

. Fault diagnosis of rolling bearing based on feature reduction with global-local margin Fisher analysis. Neurocomputing 2018; 315: 447–464.

Cui

Wang

et al . Improved fault size estimation method for rolling element bearings based on concatenation dictionary. IEEE Access 2019; 7: 22710–22718.

10.

Luo

et al . Nonlinear dynamic response analysis of two-stage spur gear space driving mechanism with large inertia load. J Vibroeng 2018; 20: 86–102.

11.

Cui

Huang

Zhai

et al . Research on the meshing stiffness and vibration response of fault gears under an angle-changing crack based on the universal equation of gear profile. Mech Mach Theory 2016; 105: 554–567.

12.

. Load equivalent simulation method of ultra-large moment of inertia of space station redocking manipulator. J Inform Comput Sci 2015; 12: 431–441.

13.

Yang

Yan

Han

. Nonlinear model of space manipulator joint considering time-variant stiffness and backlash. J Sound Vib 2015; 341: 246–259.

14.

Chen

Zhao

et al . Nonlinear dynamic model and governing equations of low speed and high load planetary gear train with respect to friction. In: Proceedings of the international conference on consumer electronics, communications and networks, Xianning, China, 16–18 April 2011. New York: IEEE.

15.

Hammami

Rincon

AFD

Chaari

et al . Effects of variable loading conditions on the dynamic behaviour of planetary gear with power recirculation. Measurement 2016; 94: 306–315.

16.

Fernández

Iglesias

De-Juan

et al . Gear transmission dynamics: effects of index and run out errors. Appl Acoust 2016; 108: 63–83.

17.

Fernández

Iglesias

De-Juan

et al . Gear transmission dynamic: effects of tooth profile deviations and support flexibility. Appl Acoust 2014; 77: 138–149.

18.

Hotait

Kahraman

. Experiments on the relationship between the dynamic transmission error and the dynamic stress factor of spur gear pairs. Mech Mach Theory 2013; 70: 116–128.

19.

Xun

Long

Hua

. Effects of random tooth profile errors on the dynamic behaviors of planetary gears. J Sound Vib 2018; 415: 91–110.

20.

Cui

Zhai

Zhang

. Research on the meshing stiffness and vibration response of cracked gears based on the universal equation of gear profile. Mech Mach Theory 2015; 94: 80–95.

21.

Bozca

. Transmission error model-based optimisation of the geometric design parameters of an automotive transmission gearbox to reduce gear-rattle noise. Appl Acoust 2018; 130: 247–259.

22.

Lin

. Analytical method for coupled transmission error of helical gear system with machining errors, assembly errors and tooth modifications. Mech Syst Signal Pr 2017; 91: 167–182.

23.

Guangjian

Lin

et al . Research on the dynamic transmission error of a spur gear pair with eccentricities by finite element method. Mech Mach Theory 2017; 109: 1–13.

24.

Raghuwanshi

Anand

. Experimental measurement of mesh stiffness by laser displacement sensor technique. Measurement 2018; 128: 63–70.

25.

Chao

Zongde

Feng

. An improved model for dynamic analysis of a double-helical gear reduction unit by hybrid user-defined elements: experimental and numerical validation. Mech Mach Theory 2018; 127: 96–111.

26.

Wang

Zou

. The experimental research on gear eccentricity error of backlash-compensation gear device based on transmission error. Int J Precis Eng Manuf 2018; 19: 5–12.

27.

Cao

Deng

Wei

. A novel method for gear tooth contact analysis and experimental validation. Mech Mach Theory 2018; 126: 1–13.

28.

Kahraman

Blankenship

Kahraman

et al . Effect of involute contact ratio on spur gear dynamics. J Mech Design 1999; 121: 112–118.

29.

Kahraman

Blankenship

Kahraman

et al . Effect of involute tip relief on dynamic response of spur gear pairs. J Mech Design 1999; 121: 313–315.

30.

Kubur

Kahraman

. Dynamic analysis of a multi-shaft helical gear transmission by finite elements: model and experiment. J Vib Acoust 2004; 126: 398–406.

31.

Pramono

. Experimental study of the influence of tooth profile modifications on spur gear dynamic strain and vibration. Key Eng Mater 2006; 306–308: 79–84.

32.

Velex

Ajmi

. Dynamic tooth loads and quasi—static transmission errors in helical gears—approximate dynamic factor formulae. Mech Mach Theory 2007; 42: 1512–1526.

Experimental study on transmission error of space-driven gear system under large inertia load

Abstract

Keywords

Introduction

Dynamic modeling and simulation analysis

Model establishment and solution

Simulation analysis

Dynamic model experimental verification

Experimental equipment and system

Experimental results analysis and verification

TE analysis

TE experimental

TE under different load inertia

TE under different input speeds

Conclusion

Footnotes

Data availability

Declaration of conflicting interests

Funding

ORCID iDs

References