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Dynamic modeling of a gear transmission system containing damping particles using coupled multi-body dynamics and discrete element method

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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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References

  1. Velex, P.: On the modelling of spur and helical gear dynamic behaviour. Mech. Eng. 75–106. ISBN: 978-953-51-0505-3, (2012)

  2. Fernandez del Rincon, A., Viadero, F., Iglesias, M., Garcia, P., de-Juan, A., Sancibrian, R.: A model for the study of meshing stiffness in spur gear transmissions. Mech. Mach. Theory 61, 30–58 (2013)

    Article  Google Scholar 

  3. Ma, H., Li, Z.W., Feng, M.J., Feng, R.J., Wen, B.C.: Time-varying mesh stiffness calculation of spur gears with spalling defect. Eng. Fail. Anal. 66, 166–176 (2016)

    Article  Google Scholar 

  4. Sánchez, M.B., Pleguezuelos, M., Pedrero, J.I.: Approximate equations for the meshing stiffness and the load sharing ratio of spur gears including Hertzian effects. Mech. Mach. Theory 109, 231–249 (2017)

    Article  Google Scholar 

  5. Xu, Z., Wang, M.Y., Chen, T.: An experimental study of particle damping for beams and plates. J. Vib. Acoust 1 26(1), 141–148 (2004)

    Article  Google Scholar 

  6. Lu, Z., Lu, X.L., Masri, S.F.: Studies of the performance of particle dampers under dynamic loads. J. Sound Vib. 329(26), 5415–5433 (2010)

    Article  Google Scholar 

  7. Lu, Z., Lu, X., Lu, W., Masri, S.F.: Experimental studies of the effects of buffered particle dampers attached to a multi-degree-of-freedom system under dynamic loads. J. Sound Vib. 331, 2007–2022 (2012)

    Article  Google Scholar 

  8. Moore, J.J., Palazzolo, A.B., Gadangi, R., Nale, T.A., Klusman, S.A., Brown, G.V., Kascak, A.F.: A forced response analysis and application of impact dampers to rotor dynamic vibration suppression in a cryogenic environment. J. Vib. Acoust. 117, 300 (1995)

    Article  Google Scholar 

  9. Wong, C.X., Daniel, M.C., Rongong, J.A.: Energy dissipation prediction of particle dampers. J Sound Vib. 319(1–2), 91–118 (2009)

    Article  Google Scholar 

  10. Yao, B., Chen, Q.: Investigation on zero-gravity behavior of particle dampers. J. Vib. Control 21, 124–133 (2013)

    Article  Google Scholar 

  11. Ahmad, N., Ranganath, R., Ghosal, A.: Modeling and experimental study of a honeycomb beam filled with damping particles. J. Sound Vib. 391, 20–34 (2017)

    Article  Google Scholar 

  12. Xiao, W.Q., Huang, Y.X., Jiang, H., Lin, H., Li, J.N.: Energy dissipation mechanism and experiment of particle dampers for gear transmission under centrifugal loads. Particuology 27, 40–50 (2016)

    Article  Google Scholar 

  13. Xiao, W.Q., Li, J.N.: Investigation into the influence of particles’ friction coefficient on vibration suppression in gear transmission. Mech. Mach. Theory 108, 217–230 (2017)

    Article  Google Scholar 

  14. MSC.Software: Using ADAMS/Solver. Mechanical Dynamics, Inc., Ann Arbor, Michigan (1997)

  15. McConville, J.B., McGrath, J.F.: Introduction to ADAMS Theory. Mechanical Dynamics Inc., Michigan (1998)

    Google Scholar 

  16. Shabana, A.A.: Dynamics of Multibody Systems, 3rd edn. Cambridge University Press, Cambridge (2005)

    Book  Google Scholar 

  17. Schiehlen, W.: Computational dynamics: theory and applications of multibody systems. Eur. J. Mech. A/Solids 25(4), 566–594 (2006)

    Article  MathSciNet  Google Scholar 

  18. Flores, P., Ambrósio, J., Claro, J.C.P., Lankarani, H.M.: Kinematics and Dynamics of Multibody Systems with Imperfect Joints. Springer, Berlin (2008)

    MATH  Google Scholar 

  19. Xu, L.X.: An approach for calculating the dynamic load of deep groove ball bearing joints in planar multibody systems. Nonlinear Dyn. 70, 2145–2161 (2012)

    Article  MathSciNet  Google Scholar 

  20. Langerholc, M., Cesnik, M., Slavic, J., Boltear, M.: Experimental validation of a complex, large-scale, rigid-body mechanism. Eng. Struct. 36, 220–227 (2012)

    Article  Google Scholar 

  21. Cundall, P.A., Strack, O.D.L.: Discrete numerical-model for granular assemblies. Geotechnique 29, 47–65 (1979)

    Article  Google Scholar 

  22. Coetzee, C., Els, D., Dymond, G.: Discrete element parameter calibration and the modelling of dragline bucket filling. J. Terramechanics 47, 33–44 (2010)

    Article  Google Scholar 

  23. Barrios, G.K., Tavares, L.M.: A preliminary model of high pressure roll grinding using the discrete element method and multibody dynamics coupling. Int. J. Miner. Process. 156, 32–42 (2016)

    Article  Google Scholar 

  24. Lommen, S., Lodewijks, G., Schott, D.L.: Co-simulation framework of discrete element method and multibody dynamics models. Eng. Comput. 35(3), 1481–1499 (2018)

    Article  Google Scholar 

  25. Johnson, K.L.: Contact Mechanics. Cambridge University Press, Cambridge (1985)

    Book  Google Scholar 

  26. Coulomb, P.C.A.: Theorie des Machines Simples. Bachelier, Paris (1821)

    Google Scholar 

  27. Lankarani, H.M., Nikravesh, P.E.: A contact force model with hysteresis damping for impact analysis of multibody systems. J. Mech. Des.-T. ASME 112(3), 369–376 (1990)

    Article  Google Scholar 

  28. McDevitt, T.: Treatment of frictional contact problems in MSC.ADAMS. In: Proceedings of 2002 North American MSC.ADAMS Users Conference. Scottsdale, AZ (2002)

  29. Radzevich, S.P.: Dudley’s Handbook of Practical Gear Design and Manufacture, 3rd edn. CRC Press, New York (2016)

    Book  Google Scholar 

  30. International Standard BS ISO 6336-1: Calculation of Load Capacity of Spur and Helical Gears-Part I: Basic Principles, Introduction and Influence Factors. 70 (2006)

  31. Lin, H.H., Liou, C.H.: A parametric study of spur gear dynamics. NASA CR-1998-206598 (1998)

  32. EL-Sayed, H.R.: Stiffness of deep-groove ball bearings. Wear 63, 89–94 (1980)

    Article  Google Scholar 

  33. Friswell, M.I., Penny, J.E.T., Garvey, S.D., Lees, A.W.: Dynamics of Rotating Machines. Cambridge University Press, Cambridge (2010)

    Book  Google Scholar 

  34. Pennestrì, E., Rossi, V., Salvini, P., Valentini, P.P.: Review and comparison of dry friction force models. Nonlinear Dyn. 83(4), 1785–1801 (2016)

    Article  Google Scholar 

  35. Geonea, I., Dumitru, N., Dumitru, I.: Experimental and theoretical study of friction torque from radial ball bearings. IOP Conf. Ser.: Mater. Sci. Eng. 252, 012048 (2017)

    Article  Google Scholar 

  36. Tsuji, Y., Tanaka, T., Ishida, T.: Lagrangian numerical simulation of plug flow of cohesionless particles in a horizontal pipe. Powder Technol. 71, 239–250 (1992)

    Article  Google Scholar 

  37. Chung, Y.C., Wu, C.W., Kuo, C.Y., Hsiau, S.S.: A rapid granular chute avalanche impinging on a small fixed obstacle: DEM modeling, experimental validation and exploration of granular stress. Appl. Math. Model. 74, 540–568 (2019)

    Article  MathSciNet  Google Scholar 

  38. Chung, Y.C., Ooi, J.Y.: Benchmark tests for verifying discrete element modelling codes at particle impact level. Granul. Matter 13, 643–656 (2011)

    Article  Google Scholar 

  39. Baumgarte, J.: Stabilization of constraints and integral of motion in dynamical systems. Comput. Methods Appl. Mech. Eng. 1(1), 1–16 (1972)

    Article  MathSciNet  Google Scholar 

  40. Lin, S.T., Huang, J.N.: Stabilization of Baumgarte’s method using the Runge–Kutta approach. J. Appl. Mech.-T. ASME 124, 633–641 (2002)

    Article  Google Scholar 

  41. Thornton, C., Randall, C.W.: Applications of theoretical contact mechanics to solid particle system simulation. In: Satake, M., Jenkins, J.T. (eds.) Micromechanics of Granular Materials, pp. 133–142. Elsevier, Amsterdam (1988)

    Google Scholar 

  42. Chou, H.T., Lee, C.F., Chung, Y.C., Hsiau, S.S.: Discrete element modelling and experimental validation for the falling process of dry granular steps. Powder Technol. 231, 122–134 (2012)

    Google Scholar 

  43. Chung, Y.C., Liao, H.H., Hsiau, S.S.: Convection behavior of non-spherical particles in a vibrating bed: discrete element modeling and experimental validation. Powder Technol. 237, 53–66 (2013)

    Article  Google Scholar 

  44. Chung, Y.C., Lin, C.K., Chou, P.H., Hsiau, S.S.: Mechanical behavior of a granular solid and its contacting deformable structure under uni-axial compression-Part I: joint DEM-FEM modelling and experimental validation. Chem. Eng. Sci. 144, 404–420 (2016)

    Article  Google Scholar 

  45. SKF: General Catalogue. SKF Group, 287–404 (2003)

  46. Timoshenko, S.P., Goodier, J.N.: Theory of Elasticity, 3rd edn. McGraw-Hill, New York (1970)

    MATH  Google Scholar 

Download references

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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Correspondence to Yu-Ren Wu.

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Appendix

Appendix

Fig. 21
figure 21

Elastic normal impact between a steel bead and a rigid flat surface: a force–displacement curve; b force–time cure

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).

$$\begin{aligned} \alpha _{\max }= & {} \left[ {5\sqrt{2}\frac{\pi \rho \left( {1-\upsilon ^{2}} \right) }{E}} \right] ^{\frac{2}{5}}rV_\mathrm{in}^{\frac{4}{5}} , \nonumber \\ P_{\max }= & {} \left[ {\frac{2}{9}\frac{rE^{2}}{\left( {1-\upsilon ^{2}} \right) ^{2}}} \right] ^{\frac{1}{2}}\alpha _{\max }^{\frac{3}{2}} \end{aligned}$$
(A1)
$$\begin{aligned} P= & {} \left[ {\frac{2}{9}\frac{rE^{2}}{\left( {1-\upsilon ^{2}} \right) ^{2}}} \right] ^{\frac{1}{2}}\alpha ^{\frac{3}{2}} \end{aligned}$$
(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.

Fig. 22
figure 22

A steel bead impacting a rigid flat surface at incident angle \(\theta ^{*}\) [38]

Fig. 23
figure 23

Oblique impact of a steel bead with a rigid flat surface: a post-collision angular velocity \({\omega }'\) at different incident angles \(\theta ^{*}\); b rebound angle \(\varphi \) at different incident angles \(\theta ^{*}\)

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).

$$\begin{aligned} {\omega }'\le \frac{5}{2}\frac{\mu _\mathrm{w} (1+e_\mathrm{w} )V^{*}\cos \theta ^{*}}{r} \end{aligned}$$
(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).

$$\begin{aligned} \tan ^{-1}\left[ {\frac{\tan \theta ^{*}}{e_\mathrm{w} }-\frac{7}{2}\mu _\mathrm{w} \left( 1+\frac{1}{e_\mathrm{w} }\right) } \right] \le \varphi \end{aligned}$$
(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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