Effect of dynamic backlash and rotational speed is investigated on the six-degree-of-freedom model of the gear-bearing system with the time-varying meshing stiffness. The relationship between dynamic backlash and center distance can be defined clearly. The nonlinear differential equations of the model are solved by the Newmark-
Gear transmission system is called as one of the most important mechanisms used to transmit power and motion in the modern machinery industry, whose mechanism and complex nonlinear phenomenon have been the focus. The lifetime of gear transmission system is reduced by a series of nonlinear factors and nonlinear excitation, including dynamic backlash, time-varying meshing stiffness, dynamic transmission error, and gear eccentricity. Therefore, it is necessarily analyzed for the dynamic behaviors of the gear transmission system and effect of the parameters on the system. In order to greatly enhance performance, reliability, and dynamic stability of the gear system, the effects of dynamic parameters and excitation have been the focus on research.
Plentiful research achievements have been accomplished by many outstanding scientists. S Theodossiades and S Natsiavas [
There scholars have made contributions to research of gear transmission system with nonlinear factors. However, in order to be more tally with the actual situation, more and more scholars developed dynamic factors and the time-varying parameters on gear-bearing system or gear-rotor-bearing system. Cao Z et al. [
However, some facts are ignored. Some scientists studied considering more factors. Chen Q [
Although some dynamic factors of the gear system are discussed, few researches have investigated the effect of the rotational speed on a dynamic model of the gear-bearing transmission system with dynamic center distance and backlash. In this paper, a 6-DOF dynamic model of the gear-bearing transmission system is established. The coupled relationship between the dynamic backlash and the vibration responses of gear system is researched by analyzing effect of the rotational speed and dynamic backlash variation. The differential equations of motion are solved using the Newmark-
Figure
6-DOF model of the gear-bearing system.
The nonlinear model of the gear system with gear inertia
The meshing stiffness.
Figure
Simplified illustration of dynamic center distance considering gear eccentricity.
Sketch of backlash.
The equations of motion of the gear-bearing system can be expressed as
The parameters of gear transmission are listed in Table
Parameters of the gear-bearing system.
Parameter | Symbol | Numerical value |
---|---|---|
Pressure angle | | 20° |
Moment of inertia | | 0.041/0.079 kg·m2 |
Mass | | 1.53/3.01 kg |
Stiffness of bearings | | 2/2×108 N·m−1 |
Meshing stiffness | | 7.74×108 N·m−1 |
Teeth | | 55/75 |
Modulus | | 2 |
Phase angle of meshing stiffness coefficient | | 0 |
Phase angle of the transmission error | | 0 |
Transmission error | | 1×10−5 m |
Half-backlash value | | 2×10−5 m |
Driving torque | | 22 N·m |
Loading torque | | 30 N·m |
Meshing damping ratio | | 0.07 |
Contact damping ratio | | 0.07 |
For the gear system with dynamic backlash, it is first for dynamic backlash and dynamic meshing angle to make comparison with different
The dynamic backlash variation with different frequency: (a)
The dynamic meshing angle variation with different frequency: (a)
In Figures
The dynamic meshing angle of the gear system is shown in Figures
Amplitude variation with respect to frequency: (a) dynamic backlash, (b) dynamic meshing angle, forward sweep frequency (denoted by blue point), and reverse sweep frequency (denoted by red point).
Firstly, for the nonlinear system with constant backlash, meshing frequency
The nonlinear characteristics of gear-bearing system with constant backlash. Phase map: (a)
Firstly, for the sake of developing the nonlinear response of the multi-degree-of-freedom model, comparing constant backlash model with dynamic backlash model, the different characteristics are described by bifurcation map and 3D frequency spectrum as shown in Figures
Amplitude vibration of the six-degree-of-freedom system: (a) constant backlash; (b) dynamic backlash.
3D frequency spectrum using
Bifurcation map of vibration displacement U of the six-degree-of-freedom system. (a) Constant backlash and (b) dynamic backlash.
Here, the amplitude maps of vibration are showed in Figure
Afterword, the tendency causes explosive increase in the range of
In the dynamic model, 3D frequency spectrum diagrams are depicted by Figure
The bifurcation diagram of the system of constant backlash is shown in Figure
The bifurcation diagram of the system of dynamic backlash is described in Figure
In this paper, the nonlinear dynamic model of gear-bearing transmission system is established as the key to explore the nonlinear responses. In detail, the effects of the constant and dynamic backlash on gear-bearing transmission system have been developed. The relationship between gear center distance and the dynamic backlash is defined considering the gear and bearing factors. To adopt by bifurcation diagram, 3D frequency spectrum, and phase diagram, the influence of the frequency
All the data is in the manuscript. If the researchers are interested in obtaining the numerical solution files, please contact email address: 1274186512@qq.com.
The authors declared no potential conflicts of interest with respect to the research, authorship, and publication of this article.
The authors gratefully acknowledge the financial support provided by Natural Science Foundation of China (No. 51675350).