Accurate solutions and stability criterion for periodic oscillations in hysteretic systems. (English) Zbl 0716.70027
Summary: The study of the periodic response of hysteretic oscillators is reduced to that of nonlinear elastic oscillators by assuming an incremental formulation for the constitutive relationship. The harmonic balance method with many components allows for accurate of computation periodic solutions. The Floquet theory can be used to check stability. Developed equations are applied to the study of frequency response curves of a hysteretic oscillator that, although simple, shows both degrading and nondegrading behaviour. The results reported clearly show the shortcomings of traditional methods; the influence of higher harmonics is far from negligible.
MSC:
| 70J35 | Forced motions in linear vibration theory |
| 74H45 | Vibrations in dynamical problems in solid mechanics |
| 70J25 | Stability for problems in linear vibration theory |
Keywords:
periodic response; hysteretic oscillators; nonlinear elastic oscillators; harmonic balance method; Floquet theory; frequency responseReferences:
| [1] | Caughey T.K.,Sinusoidal excitation of a system with bilinear hysteresis, Journal of AppliedMechanics, Vol. 27, 1960, pp. 640–643. |
| [2] | Jennings P.C.,Response of a general yielding structure, Journal of Engineering Mechanics Division, Vol. 90, No. EM2, 1964, pp. 131–166. |
| [3] | Iwan W.D.,The steady-state response of the double bilinear hysteretic model, Journal of Applied Mechanics, Vol. 32, 1965, pp. 921–925. · Zbl 0131.39106 |
| [4] | Capecchi D., Vestroni F.,Steady-state dynamic analysis of hysteretic systems, Journal of Engineering Mechanics, Vol. 111, 1985, pp. 1515–1531. · doi:10.1061/(ASCE)0733-9399(1985)111:12(1515) |
| [5] | Capecchi D.,Vestroni F.,Periodic response of hysteretic oscillators. To appear in the International Journal of Non-Linear Mechanics. · Zbl 0712.73050 |
| [6] | Nayfeh A.G., Mook D.T.,Nonlinear oscillations, John Wiley and Sons, New York, 1979. · Zbl 0418.70001 |
| [7] | Guckenheimer J., Holmes P.,Nonlinear oscillations, dynamical systems and bifurcations of vector fields, Springer Verlag, New York, 1983. · Zbl 0515.34001 |
| [8] | Hayashi C.,Nonlinear oscillations in physical systems, Mc-Graw-Hill, New York, 1964. · Zbl 0192.50605 |
| [9] | Masri S.F.,Forced vibration of the damped bilinear hysteretic oscillator, Journal of the Acoustical Society of America, Vol. 57, 1975, pp. 105–112. · Zbl 0293.70018 · doi:10.1121/1.380419 |
| [10] | Miller G.R., Butler M.E.,Periodic response of elastic-perfectly plastic SDOF oscillator, Journal of Engineering Mechanics, Vol. 114, 1988, pp. 536–550. · doi:10.1061/(ASCE)0733-9399(1988)114:3(536) |
| [11] | Hayashi C.,The influence of hysteresis on non-linear resonance, Journal of the Franklin Institute, Vol. 281, 1966, pp. 379–386. · doi:10.1016/0016-0032(66)90299-7 |
| [12] | Badrakhan F.,Dynamic analysis of yielding and hysteretic systems by polynomial approximation, Journal of Sound and Vibration, Vol. 125, 1988, pp. 23–42. · Zbl 1235.65016 · doi:10.1016/0022-460X(88)90412-9 |
| [13] | Urabe M., Reiter A.,Numerical computation of nonlinear forced oscillations by Galerkin’s procedure, Journal of Mathematical Analysis and Applications, Vol. 14, 1966, pp. 107–140. · Zbl 0196.49405 · doi:10.1016/0022-247X(66)90066-7 |
| [14] | Ling F.H., Wu X.X.,Fast Galerkin method and its application to determine periodic solutions of nonlinear oscillators, International Journal of Non-Linear Mechanics, Vol. 22, 1987, pp. 89–98. · Zbl 0606.73025 · doi:10.1016/0020-7462(87)90012-6 |
| [15] | Van Dooren R.,On the transition from regular to chaotic behaviour in the Duffing oscillator, Journal of Sound and Vibration, Vol. 123 (2), 1988, pp. 327–339. · Zbl 1235.70136 · doi:10.1016/S0022-460X(88)80115-9 |
| [16] | Iooss G., Joseph D.D.,Elementary stability and bifurcation theory, Springer-Verlag, New-York, 1980. · Zbl 0443.34001 |
| [17] | Kondou T., Tamura M., Sueoka A.,On a method of higher approximation and determination of stability criterion for steady oscillations in nonlinear systems, Bulletin of JSME, Vol. 29, No. 248, 1986, pp. 525–532. |
| [18] | Bazant Z.P., Krizek A.J., Shieh C.L.,Hysteretic endocronic theory for sand, Journal of Engineering Mechanics, Vol. 111, 1983 pp. 1073–1095. |
| [19] | Bouc R.,Modèle mathèmatique d’hystèrèsis, Acustica, Vol. 24, 1971, pp. 16–25. · Zbl 0237.73020 |
| [20] | Swift J.W., Wiensenfeld K.,Suppression of period doubling in symmetric systems, Physical Review Letters, Vol. 52, 1984 pp. 707–708. · doi:10.1103/PhysRevLett.52.705 |
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.
