Computation of periodic orbits for piecewise linear oscillator by harmonic balance methods. (English) Zbl 1491.65161
Summary: In this paper, Harmonic Balance based methods, namely Incremental Harmonic Balance Method and the method of Harmonic Balance with Alternating Frequency and Time traditionally used to compute periodic orbits of smooth nonlinear dynamical systems, are employed to investigate the dynamics of a non-smooth system, specifically a piecewise linear oscillator with a play. The Incremental Harmonic Balance Method was used to compute the period one orbits, including those exhibiting grazing and large impacts. The method of Harmonic Balance with Alternating Frequency and Time was implemented to calculate more complex orbits and multi stability. A good agreement between obtained approximate solutions and numerically calculated responses indicates robustness of the implemented HBMs, which should allow to effectively study the global dynamics of non-smooth systems.
MSC:
| 65P40 | Numerical nonlinear stabilities in dynamical systems |
| 37M05 | Simulation of dynamical systems |
Keywords:
piecewise linear oscillator with play; incremental harmonic balance method; harmonic balance with alternative frequency and time; computation of periodic orbitsReferences:
| [1] | Shaw, S. W.; Holmes, P. J., A periodically forced piecewise linear oscillator, J Sound Vib, 90, 1, 129-155 (1983) · Zbl 0561.70022 |
| [2] | Kleczka, M.; Kreuzer, E.; Wilmers, C., Crisis in mechanical systems, (Proc. int. conf. nonlinear dynamics in engineering (1990), Springer), 141-148 · Zbl 0722.34039 |
| [3] | Luo, A. C.J.; Menon, S., Global chaos in a periodically forced, linear system with a dead-zone restoring force, Chaos Solitons Fractals, 19, 1189-1199 (2004) · Zbl 1069.37027 |
| [4] | Wiercigroch, M., Bifurcation analysis of harmonically excited linear oscillator with clearance, Chaos Solitons Fractals, 4, 2, 297-303 (1994) · Zbl 0794.70011 |
| [5] | Kaharaman, A.; Singh, R., Nonlinear dynamics of a spur gear pair, J Sound Vib, 142, 1, 49-75 (1990) |
| [6] | Theodossiades, S.; Natsiavas, S., Non-linear dynamics of gear-pair systems with periodic stiffness and backlash, J Sound Vib, 229, 2, 287-310 (2000) |
| [7] | de Souza, S. L.T.; Caldas, I. L.; Viana, R. L.; Balthazar, J. M., Sudden changes in chaotic attractors and transient basins in a model for rattling in gear boxes, Chaos Solitons Fractals, 21, 763-772 (2004) · Zbl 1122.70346 |
| [8] | Natsiavas, S., Periodic response and stability of oscillators with simetric trilinear restoring force, J Sound Vib, 134, 2, 315-331 (1989) · Zbl 1235.70094 |
| [9] | Natsiavas, S., Analytical modeling of discrete mechanical systems involving contact, impact, and friction, Appl Mech Rev, 71, 5, Article 050802 pp. (2019) |
| [10] | Nordmark, A. B., Non-periodic motion caused by grazing incidence in an impact oscillator, J Sound Vib, 145, 2, 279-297 (1991) |
| [11] | Dankowicz, H.; Nordmark, A. B., On the origin and bifurcations of stick-slip oscillations, Physica D, 136, 3-4, 280-302 (2000) · Zbl 0963.70016 |
| [12] | Molenaar, J.; De Weger, J. G.; Van De Water, W., Mappings of grazing-impact oscillators, Nonlinearity, 14, 2, 301-321 (2001) · Zbl 0991.34035 |
| [13] | Dankowicz, H.; Zhao, X., Local analysis of co-dimension-one and co-dimension-two grazing bifurcations in impact microactuators, Physica D, 202, 238-257 (2005) · Zbl 1083.34032 |
| [14] | di Bernardo, M.; Feigin, M. I.; Hogan, S. J.; Homer, M. E., Local analysis of c-bifurcations in \(n\)-dimensional piecewise smooth dynamical systems, Chaos Solitons Fractals, 10, 11, 1881-1908 (1999) · Zbl 0967.37030 |
| [15] | Nusse, H.; Ott, E.; Yorke, J., Border collision bifurcations: An explanation for observed bifurcation phenomena, Phys Rev E, 49, 1073-1076 (1994) |
| [16] | Banerjee, S.; Grebogi, C., Border collision bifurcations in two-dimensional piecewise smooth maps, Phys Rev E, 59, 4052-4061 (1999) |
| [17] | di Bernardo, M.; Budd, C. J.; Champneys, A. R., Normal form maps for grazing bifurcations in \(n\)-dimensional piecewise-smooth dynamical systems, Physica D, 160, 3-4, 222-254 (2001) · Zbl 1067.37063 |
| [18] | di Bernardo, M.; Kowalczyk, P.; Nordmark, A. B., Bifurcations of dynamical systems with sliding: Derivation of normal-form mappings, Physica D, 170, 3-4, 175-205 (2002) · Zbl 1008.37029 |
| [19] | Ma, Y.; Agarwal, M.; Banerjee, S., Border collision bifurcations in a soft impact system, Phys Lett A, 354, 281-287 (2006) |
| [20] | Ma, Y.; Ing, J.; Banerjee, S.; Wiercigroch, M.; Pavlovskaia, E., The nature of the normal form map for soft impacting systems, Int J Nonlinear Mech, 43, 504-513 (2008) |
| [21] | Stensson, A.; Nordmark, A. B., Experimental investigation of some consequences of low velocity impacts in the chaotic dynamics of a mechanical system, Philos Trans R Soc Lond A, 347, 439-448 (1994) |
| [22] | Piiroinen, T. P.; Virgin, L. N.; Champneys, A. R., Chaos and period-adding: Experimental and numerical verification of the grazing bifurcation, J Nonlinear Sci, 14, 383-404 (2004) · Zbl 1125.70015 |
| [23] | Ing, J.; Pavlovskaia, E.; Wiercigroch, M., Dynamics of a nearly symmetrical piecewise linear oscillator close to grazing incidence: modelling and experimental verification, Nonlinear Dynam, 46, 225-238 (2006) · Zbl 1170.70302 |
| [24] | Ing, J.; Pavlovskaia, E.; Wiercigroch, M.; Banerjee, S., Experimental study of impact oscillator with one-sided elastic constraint, Phil Trans R Soc A, 366, 1866, 679-704 (2008) · Zbl 1153.74302 |
| [25] | Banerjee, S.; Ing, J.; Pavlovskaia, E.; Wiercigroch, M.; Reddy, R. K., Invisible grazings and dangerous bifurcations in impacting systems: The problem of narrow-band chaos, Phys Rev E, 79, 037201 (2009) |
| [26] | Pavlovskaia, E.; Ing, J.; Wiercigroch, M.; Banerjee, S., Complex dynamics of bilinear oscillator close to grazing, Int J Bifurcation Chaos, 20, 11, 3801-3817 (2010) |
| [27] | Chong, A. S.E.; Yue, Y.; Pavlovskaia, E.; Wiercigroch, M., Global dynamics of a harmonically excited oscillator with a play: Numerical studies, Int J Non-Linear Mech, 94, 98-108 (2017) |
| [28] | Chong, A. S.E., Numerical modelling and stability analysis of non-smooth dynamical systems via abespol (2016), University of Aberdeen, (Ph.D. thesis) |
| [29] | Dankowicz, H.; Schilder, F., Recipes for continuation (2013), SIAM · Zbl 1277.65037 |
| [30] | Tien, M.-H.; D’Souza, K., Analyzing bilinear systems using a new hybrid symbolic-numeric computational method, J Vib Acoust Trans ASME, 141, 3, Article 31008 pp. (2019) |
| [31] | Woiwode, L.; Balaji, N. N.; Kappauf, J.; Tubita, F.; Guillot, L.; Vergez, C.; Cochelin, B.; Grolet, A.; Krack, M., Comparison of two algorithms for Harmonic Balance and path continuation, Mech Syst Signal Process, 136, Article 106503 pp. (2020) |
| [32] | Miguel, L. P.; Teloli, R. D.O.; da Silva, S., Some practical regards on the application of the harmonic balance method for hysteresis models, Mech Syst Signal Process, 143, Article 106842 pp. (2020) |
| [33] | Choi, Y. S.; Noah, S. T., Forced periodic vibration of unsymmetric piecewise-linear systems, J Sound Vib, 121, 3, 117-126 (1988) · Zbl 1235.70086 |
| [34] | Maezawa, S.; Kumano, H.; Minakuchi, Y., Forced vibrations in an unsymmetrical piece-wise linear system excited by general periodic force function, Bull JSME, 23, 68-75 (1980) |
| [35] | Nayfeh, A. H., Introduction to perturbation techniques (1981), John Wiley & Sons: John Wiley & Sons New York · Zbl 0449.34001 |
| [36] | Sanders, J. A.; Verhulst, F., Averaging methods in nonlinear dynamical systems (1985), Springer: Springer New York · Zbl 0586.34040 |
| [37] | Lau, S. L.; Cheung, Y. K., Amplitude incremental variational principle for nonlinear vibration of elastic systems, ASME J Appl Mech, 48, 959-964 (1981) · Zbl 0468.73066 |
| [38] | Lau, S. L.; Yuen, S. W., Solution diagram of non-linear dynamic systems by the IHB method, J Sound Vib, 167, 303-316 (1993) · Zbl 0925.70218 |
| [39] | Raghothama, A.; Narayanan, S., Bifurcation and chaos in escape equation model by incremental harmonic balancing, Chaos Solitons Fractals, 11, 1349-1363 (2000) · Zbl 0962.70023 |
| [40] | Raghothama, A.; Narayanan, S., Bifurcation and chaos of an articulated loading platform with piecewise non-linear stiffness using the incremental harmonic balance method, Ocean Eng, 27, 1087-1107 (2000) |
| [41] | Woo, K.-C.; Rodger, A. A.; Neilson, R. D.; Wiercigroch, M., Application of the harmonic balance method to ground moling machines operating in periodic regimes, Chaos Solitons Fractals, 11, 15, 2515-2525 (2000) · Zbl 0955.70509 |
| [42] | Shen, J. H.; Lin, K. C.; Chen, S. H., Bifurcation and route-to-chaos analyses for Mathieu-Duffing oscillator by the incremental harmonic balance method, Nonliear Dyn, 52, 403-414 (2008) · Zbl 1170.70367 |
| [43] | Shen, Y. J.; Yang, S. P.; Liu, X. D., Nonlinear dynamics of a spur gear pair with time-varying stiffness and backlash based on incremental harmonic balance method, Int J Mech Sci, 48, 1256-1263 (2006) · Zbl 1192.74148 |
| [44] | Hayes, R.; Marques, S. P., Prediction of limit cycle oscillations under uncertainty using a Harmonic Balance method, Comput Struct, 148, 1-13 (2015) |
| [45] | Sun, G.; Fu, X., Discontinuous dynamics of a class of oscillators with strongly nonlinear asymmetric damping under a periodic excitation, Commun Nonlinear Sci Numer Simul, 61, 230-247 (2018) · Zbl 1473.70049 |
| [46] | Charroyer, L.; Chiello, O.; Sinou, J.-J., Self-excited vibrations of a non-smooth contact dynamical system with planar friction based on the shooting method, Int J Mech Sci, 144, 90-101 (2018) |
| [47] | Pei, L.; Wang, S., Dynamics and the periodic solutions of the delayed non-smooth internet TCP-RED congestion control system via HB-AFT, Appl Math Comput, 361, 689-702 (2019) · Zbl 1428.49016 |
| [48] | Dai, W.; Yang, J.; Shi, B., Vibration transmission and power flow in impact oscillators with linear and nonlinear constraints, Int J Mech Sci, 168, Article 105234 pp. (2020) |
| [49] | 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, 958-964 (1985) · Zbl 0584.73109 |
| [50] | 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) |
| [51] | Lau, S. L.; Zhang, W. S., Nonlinear vibrations of piecewise-linear systems by incremental harmonic balance method, J Appl Mech, 59, 153-160 (1992) · Zbl 0850.70247 |
| [52] | Xu, L.; Lu, M. W.; Cao, Q., Nonlinear vibrations of dynamical systems with a general form of piecewise-inear viscous damping by incremental harmonic balance method, Phys Lett A, 301, 65-73 (2002) · Zbl 0997.70022 |
| [53] | Liu, L.; Earl, H. D.; Kenneth, C. H., A novel harmonic balance analysis for the Van Der Pol oscillator, Int J Non-Linear Mech, 42, 1, 2-12 (2007) · Zbl 1200.34041 |
| [54] | Liu, L.; Kalmár-Nagy, T., High-dimensional harmonic balance analysis for second-order delay-differential equations, J Vib Control, 16, 7-8, 1189-1208 (2010) · Zbl 1269.70026 |
| [55] | Ju, R.; Fan, W.; Zhu, W. D., Comparison between the incremental harmonic balance method and alternating frequency/time-domain method, J Vib Acoust, 143, 2, Article 024501 pp. (2021) |
| [56] | Nusse, H. E.; Yorke, J. A., Dynamics: Numerical explorations (1998), Springer-Verlag: Springer-Verlag New York · Zbl 0895.58001 |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
