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Nonlinear Dynamics

, Volume 66, Issue 4, pp 781–788 | Cite as

A modified incremental harmonic balance method for rotary periodic motions

  • Chung-Jen Lu
  • Yu-Min Lin
Original Paper

Abstract

The determination of periodic solutions is an essential step in the study of dynamic systems. If some of the generalized coordinates describing the configuration of a system are angular positions relative to certain reference axes, the associated periodic motions divide into two types: oscillatory and rotary periodic motions. For an oscillatory periodic motion, all the generalized coordinates are periodic in time. On the other hand, for a rotary periodic motion, some angular coordinates may have unbounded magnitude due to the persistent circulation about their pivots. In this case, although the behaviour of the system is periodic physically, those angular coordinates are not periodic in time. Although various effective methods have been developed for the determination of oscillatory periodic motion, the rotary periodic motion can only be determined by brute force integration. In this paper, the incremental harmonic balance (IHB) method is modified so that rotary periodic motions can be determined as well as oscillatory periodic motions in a unified formulation. This modified IHB method is applied to a practical device, a rotating disk equipped with a ball-type balancer, to show its effectiveness.

Keywords

Harmonic balance Periodic motion Unbalance vibration 

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References

  1. 1.
    Nayfeh, A.H., Balachandran, B.: Applied Nonlinear Dynamics: Analytical, Computational, and Experimental Methods. Wiley Series in Nonlinear Science. Wiley, New York (1995) MATHCrossRefGoogle Scholar
  2. 2.
    Lau, S.L., Cheung, Y.K.: Amplitude incremental variational principle for nonlinear vibration of elastic systems. J. Appl. Mech. 48(4), 959–964 (1981) MATHCrossRefGoogle Scholar
  3. 3.
    Lau, S.L., Cheung, Y.K., Wu, S.Y.: A variable parameter incrementation method for dynamic instability of linear and nonlinear elastic systems. J. Appl. Mech. 49(4), 849–853 (1982) MATHCrossRefGoogle Scholar
  4. 4.
    Cheung, Y.K., Lau, S.L.: Incremental time-space finite strip method for non-linear structural vibrations. Earthquake Eng. Struct. Dyn. 10(2), 239–253 (1982) CrossRefGoogle Scholar
  5. 5.
    Lau, S.L., Cheung, Y.K., Wu, S.Y.: Nonlinear vibration of thin elastic plates. Part 2: Internal resonance by amplitude-incremental finite element. J. Appl. Mech. 51(4), 845–851 (1984) CrossRefGoogle Scholar
  6. 6.
    Pierre, C., Dowell, E.H.: A study of dynamic instability of plates by an extended incremental harmonic-balance method. J. Appl. Mech. 52(3), 693–698 (1985) MATHCrossRefGoogle Scholar
  7. 7.
    Cheung, Y.K., Chen, S.H., Lau, S.L.: Application of the incremental harmonic-balance method to cubic nonlinearity systems. J. Sound Vib. 140(2), 273–286 (1990) MathSciNetCrossRefGoogle Scholar
  8. 8.
    Yuen, S.W., Lau, S.L.: Effects of inplane load on nonlinear panel flutter by incremental harmonic-balance method. AIAA J. 29(9), 1472–1479 (1991) CrossRefGoogle Scholar
  9. 9.
    Leung, A.Y.T., Chui, S.K.: Nonlinear vibration of coupled duffing oscillators by an improved incremental harmonic-balance method. J. Sound Vib. 181(4), 619–633 (1995) MathSciNetCrossRefGoogle Scholar
  10. 10.
    Raghothama, A., Narayanan, S.: Non-linear dynamics of a two-dimensional airfoil by incremental harmonic balance method. J. Sound Vib. 226(3), 493–517 (1999) CrossRefGoogle Scholar
  11. 11.
    Chen, S.H., Cheung, Y.K., Xing, H.X.: Nonlinear vibration of plane structures by finite element and incremental harmonic balance method. Nonlinear Dyn. 26(1), 87–104 (2001) MATHCrossRefGoogle Scholar
  12. 12.
    Sze, K.Y., Chen, S.H., Huang, J.L.: The incremental harmonic balance method for nonlinear vibration of axially moving beams. J. Sound Vib. 281(3–5), 611–626 (2005). doi: 10.1016/j.jsv.2004.01.012. ISSN 0022-460X CrossRefGoogle Scholar
  13. 13.
    Zheng, G., Ko, J.M., Ni, Y.Q.: Super-harmonic and internal resonances of a suspended cable with nearly commensurable natural frequencies. Nonlinear Dyn. 30(1), 55–70 (2002) MATHCrossRefGoogle Scholar
  14. 14.
    Raghothama, A., Narayanan, S.: Periodic response and chaos in nonlinear systems with parametric excitation and time delay. Nonlinear Dyn. 27(4), 341–365 (2002) MathSciNetMATHCrossRefGoogle Scholar
  15. 15.
    Lau, S.L., Cheung, Y.K., Wu, S.Y.: Incremental harmonic-balance method with multiple time scales for aperiodic vibration of non-linear systems. J. Appl. Mech. 50(4A), 871–876 (1983) MathSciNetMATHCrossRefGoogle Scholar
  16. 16.
    Pierre, C., Ferri, A.A., Dowell, E.H.: Multi-harmonic analysis of dry friction damped systems using an incremental harmonic-balance method. J. Appl. Mech. 52(4), 958–964 (1985) MATHCrossRefGoogle Scholar
  17. 17.
    Zhou, J.X., Zhang, L.: Incremental harmonic balance method for predicting amplitudes of a multi-d.O.F. Non-linear wheel shimmy system with combined coulomb and quadratic damping. J. Sound Vib. 279(1–2), 403–416 (2005). doi: 10.1016/j.jsv.2003.11.005. ISSN 0022-460X CrossRefGoogle Scholar
  18. 18.
    Wong, C.W., Zhang, W.S., Lau, S.L.: Periodic forced vibration of unsymmetrical piecewise-linear systems by incremental harmonic-balance method. J. Sound Vib. 149(1), 91–105 (1991) CrossRefGoogle Scholar
  19. 19.
    Lau, S.L., Zhang, W.S.: Nonlinear vibrations of piecewise-linear systems by incremental harmonic-balance method. J. Appl. Mech. 59(1), 153–160 (1992) MathSciNetMATHCrossRefGoogle Scholar
  20. 20.
    Pun, D., Liu, Y.B.: On the design of the piecewise linear vibration absorber. Nonlinear Dyn. 22(4), 393–413 (2000) MATHCrossRefGoogle Scholar
  21. 21.
    Lau, S.L., Yuen, S.W.: The Hopf-bifurcation and limit-cycle by the incremental harmonic-balance method. Comput. Methods Appl. Mech. Eng. 91(1–3), 1109–1121 (1991) MathSciNetCrossRefGoogle Scholar
  22. 22.
    Raghothama, A., Narayanan, S.: Bifurcation and chaos in escape equation model by incremental harmonic balancing. Chaos Solitons Fractals 11(9), 1349–1363 (2000) MathSciNetMATHCrossRefGoogle Scholar
  23. 23.
    Shen, J.H., Lin, K.C., Chen, S.H., Sze, K.Y.: Bifurcation and route-to-chaos analyses for Mathieu–Duffing oscillator by the incremental harmonic balance method. Nonlinear Dyn. 52(4), 403–414 (2008). doi: 10.1007/s11071-007-9289-z. ISSN 0924-090X MATHCrossRefGoogle Scholar
  24. 24.
    Lee, J., Moorham, W.K.V.: Analytical and experimental analysis of a self-compensating dynamic balancer in a rotating mechanism. J. Dyn. Syst. Meas. Control 118, 468–475 (1996) MATHCrossRefGoogle Scholar
  25. 25.
    Kim, W., Chung, J.: Performance of automatic ball balancers on optical disc drives. Proc. Inst. Mech. Eng., C J. Mech. Eng. Sci. 216(11), 1071–1080 (2002) CrossRefGoogle Scholar
  26. 26.
    Chao, P.C.P., Sung, C.-K., Wang, C.-C.: Dynamic analysis of the optical disk drives equipped with an automatic ball balancer with consideration of torsional motions. J. Appl. Mech. 72(6), 826–842 (2005) MATHCrossRefGoogle Scholar
  27. 27.
    Kim, W., Lee, D.-J., Chung, J.: Three-dimensional modelling and dynamic analysis of an automatic ball balancer in an optical disk drive. J. Sound Vib. 285(3), 547–569 (2005) CrossRefGoogle Scholar
  28. 28.
    Chao, P.C.P., Sung, C.K., Wu, S.T., Huang, J.S.: Nonplanar modeling and experimental validation of a spindle-disk system equipped with an automatic balancer system in optical disk drives. Microsyst. Technol. 13(8–10), 1227–1239 (2007) CrossRefGoogle Scholar
  29. 29.
    Chung, J., Ro, D.S.: Dynamic analysis of an automatic dynamic balancer for rotating mechanisms. J. Sound Vib. 228(5), 1035–1056 (1999) CrossRefGoogle Scholar
  30. 30.
    Kang, J.R., Chao, C.P., Huang, C.L., Sung, C.K.: The dynamics of a ball-type balancer system equipped with a pair of free-moving balancing masses. J. Vib. Acoust. 123(4), 456–465 (2001) CrossRefGoogle Scholar
  31. 31.
    Chung, J., Jang, I.: Dynamic response and stability analysis of an automatic ball balancer for a flexible rotor. J. Sound Vib. 259(1), 31–43 (2003) MathSciNetCrossRefGoogle Scholar
  32. 32.
    Chao, P.C.P., Sung, C.-K., Leu, H.-C.: Effects of rolling friction of the balancing balls on the automatic ball balancer for optical disk drives. J. Tribol. 127(4), 845–856 (2005) CrossRefGoogle Scholar
  33. 33.
    Lu, C.J., Wang, M.C., Huang, S.H.: Analytical study of the stability of a two-ball automatic balancer. Mech. Syst. Signal Process. 23(3), 884–896 (2009). doi: 10.1016/j.ymssp.2008.06.008 CrossRefGoogle Scholar
  34. 34.
    Green, K., Champneys, A.R., Friswell, M.I.: Analysis of the transient response of an automatic dynamic balancer for eccentric rotors. Int. J. Mech. Sci. 48(3), 274–293 (2006) MATHCrossRefGoogle Scholar
  35. 35.
    Green, K., Champneys, A.R., Lieven, N.J.: Bifurcation analysis of an automatic dynamic balancing mechanism for eccentric rotors. J. Sound Vib. 291(3–5), 861–881 (2006) CrossRefGoogle Scholar
  36. 36.
    Rajalingham, C., Bhat, R.B., Rakheja, S.: Automatic balancing of flexible vertical rotors using a guided ball. Int. J. Mech. Sci. 40(9), 825–834 (1998) MATHCrossRefGoogle Scholar
  37. 37.
    Huang, W.Y., Chao, C.P., Kang, J.R., Sung, C.K.: The application of ball-type balancers for radial vibration reduction of high-speed optic disk drives. J. Sound Vib. 250(3), 415–430 (2002) CrossRefGoogle Scholar
  38. 38.
    Lu, C.J.: Stability analysis of a single-ball automatic balancer. J. Vib. Acoust. 128(1), 122–125 (2006) CrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media B.V. 2011

Authors and Affiliations

  1. 1.Department of Mechanical EngineeringNational Taiwan UniversityTaipeiTaiwan

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