Abstract
The reduction in vibration in gear transmission systems is an engineering task. Particle damping technology attenuates vibration by means of friction and inelastic collisions between damping particles. This study proposes a dynamic model for a spur gear transmission system that contains damping particles inside the holes on gear bodies, using two-way coupling with multi-body dynamics and discrete element method. The equations of motion for the multi-body system are derived using Euler–Lagrange formalism. The discrete element method with a soft contact approach is used to model the dynamic behavior of damping particles. Hertzian contact theory and Coulomb friction theory are applied to modeling contacts. The effects of particle radius, coefficient of friction and restitution coefficient on the dynamic characteristics are explored. Numerical results show that vibration in the transmission is appreciably attenuated by the particle damping mechanism and that the contact friction, and not contact damping, dominates the energy dissipation of the multi-body system in such a centrifugal scenario.
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Acknowledgements
The authors are very grateful to the ministry of science and technology (MOST) of Taiwan for financial support under Project Numbers: MOST 107-2221-E-008-052-MY2 and MOST 105-2221-E-008-048-MY2. The authors also greatly appreciate the valuable discussion with Professor W. Q. Xiao at Xiamen University in China and the technical support of Professor C. K. Lin at National Central University in Taiwan.
Funding
The study was funded by the ministry of science and technology (MOST) of Taiwan (Grant Numbers: 105-2221-E-008-048-MY2 and 107-2221-E-008-052-MY2). The authors also greatly appreciate the valuable discussion with Professor W. Q. Xiao at Xiamen University in China and the technical support of Professor C. K. Lin at National Central University in Taiwan.
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Appendix
Appendix
Two benchmark tests were performed to verify the validity of Eq. (23). The first benchmark test is an elastic normal impact between a steel bead with a radius of 1.5 mm and a rigid flat surface. The input parameters for the steel bead are the same as those in the MBD–DEM modeling, and they are also listed in Table 2. The restitution coefficient is set to unity for elastic impact (a damping ratio of 0), and the incoming velocity is set to \(0.2\hbox { m/s}\). For an elastic sphere impacting with an incoming velocity \(V_\mathrm{in} \), the force–displacement relation during the collision can be described using Hertz contact theory. The complete solution for the elastic normal impact can be found in Timoshenko and Goodier [46]. The maximum normal contact displacement and force are expressed in Eq. (A1), and the contact force–displacement relation is expressed in Eq. (A2).
where E is the sphere’s Young’s modulus, \(\upsilon \) is the sphere’s Poisson’s ratio, \(\rho \) is the sphere’s density and r is the sphere’s radius. Figure 21 shows, respectively, the force–displacement and force–time curves for this elastic normal impact between a steel bead and a rigid flat surface. The simplified Hertz–Mindlin contact force model involving Eq. (23) is used in the DEM simulations. The DEM results and the analytic solutions are plotted in this figure. The DEM result in Fig. 21a shows no energy dissipation from the loading and unloading paths during the collision on account of a damping ratio of 0. Figure 21b shows that the variation of the normal contact force against time is symmetric due to an elastic normal contact. The DEM results are exactly in agreement with the analytical solutions.
The second benchmark test, as shown in Fig. 22, considers a steel bead obliquely impacting a rigid flat surface with a constant resultant velocity but at different incident angles. This test is to verify the calculations of the normal and tangential contact forces. The steel bead is used as in the MBD–DEM modeling, and the corresponding input parameters are also listed in Table 2. The bead-wall restitution coefficient is set to 0.8, and the bead-wall coefficient of friction is set to 0.2. The constant resultant velocity (\(V^{*})\) is set to 2.0 m/s, and the incident angle is varied between \(0^{\circ }\) and \(85^{\circ }\). The simplified Hertz–Mindlin contact force model with Eq. (23) is used to model the oblique bead-wall collision. The DEM results are compared with the corresponding analytical solutions derived by Chung and Ooi [38]. The post-collision angular velocity (\(\omega ^{\prime }\)) of the steel bead is expressed as Eq. (A3).
where \(\mu _{\mathrm{w}}\) is the bead-wall coefficient of friction, \(e_{\mathrm{w}}\) is the bead-wall restitution coefficient, \(\theta ^{*}\) is the incident angle and r is the bead radius. The positive value for \(\omega ^{\prime }\) implies that the steel bead rotates clockwise after impact. The recoil angle on the contact path \((\varphi )\) is a function of incident angle and is expressed as Eq. (A4).
The post-collision angular velocity \(({\omega }')\) is plotted against the incident angle \((\theta ^{*})\) in Fig. 23a. The DEM result matches with the analytical solution in both sliding and sticking regimes, as expressed by Eq. (A3). The recoil angle \((\varphi )\) is also plotted against the incident angles in Fig. 23b. Again, the DEM result matches with the theoretical solution of Eq. (A4). It can be expected that the impact for the incident angle greater than a critical value occurs in a sliding condition. Figure 23 shows that the DEM results predict a critical value of approximately \(45^{\circ }\). Accordingly, Eq. (23) in the simplified Hertz–Mindlin contact force model, verified by the above two benchmark tests, is hence valid.
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Chung, YC., Wu, YR. Dynamic modeling of a gear transmission system containing damping particles using coupled multi-body dynamics and discrete element method. Nonlinear Dyn 98, 129–149 (2019). https://doi.org/10.1007/s11071-019-05177-1
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DOI: https://doi.org/10.1007/s11071-019-05177-1