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.
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Nayfeh, A.H., Balachandran, B.: Applied Nonlinear Dynamics: Analytical, Computational, and Experimental Methods. Wiley Series in Nonlinear Science. Wiley, New York (1995)
Lau, S.L., Cheung, Y.K.: Amplitude incremental variational principle for nonlinear vibration of elastic systems. J. Appl. Mech. 48(4), 959–964 (1981)
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)
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)
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)
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)
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)
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)
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)
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)
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)
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
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)
Raghothama, A., Narayanan, S.: Periodic response and chaos in nonlinear systems with parametric excitation and time delay. Nonlinear Dyn. 27(4), 341–365 (2002)
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)
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)
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
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)
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)
Pun, D., Liu, Y.B.: On the design of the piecewise linear vibration absorber. Nonlinear Dyn. 22(4), 393–413 (2000)
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)
Raghothama, A., Narayanan, S.: Bifurcation and chaos in escape equation model by incremental harmonic balancing. Chaos Solitons Fractals 11(9), 1349–1363 (2000)
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
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)
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)
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)
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)
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)
Chung, J., Ro, D.S.: Dynamic analysis of an automatic dynamic balancer for rotating mechanisms. J. Sound Vib. 228(5), 1035–1056 (1999)
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)
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)
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)
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
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)
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)
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)
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)
Lu, C.J.: Stability analysis of a single-ball automatic balancer. J. Vib. Acoust. 128(1), 122–125 (2006)
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Lu, CJ., Lin, YM. A modified incremental harmonic balance method for rotary periodic motions. Nonlinear Dyn 66, 781–788 (2011). https://doi.org/10.1007/s11071-011-9950-4
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DOI: https://doi.org/10.1007/s11071-011-9950-4