The influence of random vibration on the design of mechanical components has been considered within the framework of the linear theory of small oscillations. However, in some important cases this theory is inadequate and fails to predict some complex response characteristics that have been observed experimentally and which can only be predicted by nonlinear analyses. This paper describes some recent developments in the theory of nonlinear random vibration based on Markov methods and related problems in the design of dynamical systems. Research efforts have been focused on stability/bifurcation conditions, response statistics and reliability problems. Significant progress has been made in developing new analytical methods and conducting experimental testing. These developments have helped to resolve some controversies, and to enhance our understanding of difficult issues. Experimental and numerical simulations have revealed new phenomena that were not predicted analytically. These include on-off intermittency, snap-through phenomena, and the dependence of the response bandwidth on the excitation level. The main results of studying the responses of nonlinear single-and two-degree-of-freedom systems to random excitations obtained by the author and others are discussed in this paper.
Issue Section:
Research Papers
1.
Ariaratnam, S. T., 1981, “Bifurcation in Nonlinear Stochastic Systems,” New Approaches in Nonlinear Problems in Dynamics, P. Holmes, ed., SIAM Philadelphia, pp. 470–474.
2.
Arnold, L., and Wihstutz, V., eds., 1986, Liapunov Exponents, Lecture Notes in Mathematics, Vol. 1186, Springer-Verlag, New York.
3.
Bogdanoff, J. L., and Kozin, F., 1985, Probabilistic Models of Cumulative Damage, Wiley-Interscience, New York.
4.
Bolotin, V. V., 1989, Prediction of Service Life for Machines and Structures, ASME Press, New York.
5.
Brissaud
A.
Frisch
1974
, “Solving Linear Stochastic Differential Equations
,” J. Math. Phys.
, Vol. 15
, No. 5
, pp. 524
–534
.6.
Cai
G. Q.
Lin
Y. K.
1988
, “On Exact Stationary Solutions of Equivalent Nonlinear Stochastic Systems
,” Int. J. Nonlinear Mech.
, Vol. 23
, No. 4
, pp. 315
–325
.7.
Caughey
T. K.
1960
, “Random Excitation of a Loaded Nonlinear String
,” ASME Journal of Applied Mechanics
, Vol. 27
, pp. 575
–578
.8.
Caughey
T. K.
1963
, “Equivalent Linearization Technique
,” J. Acoust. Soc. Amer.
, Vol. 35
, pp. 1706
–1711
.9.
Caughey
T. K.
1971
, “Nonlinear Theory of Random Vibrations
,” Advances in Applied Mechanics
, Vol. 11
, pp. 209
–253
.10.
Caughey
T. K.
1986
, “On the Response of Nonlinear Oscillators to Stochastic Excitation
,” Probabilistic Engineering Mechanics
, Vol. 1
, No. 1
, pp. 2
–4
.11.
Caughey
T. K.
Ma
F.
1982
, “The Steady State Response of a Class of Dynamical System to Stochastic Excitation
,” ASME Journal of Applied Mechanics
, Vol. 49
, pp. 622
–632
.12.
Caughey
T. K.
Ma
F.
1983
, “The Exact Steady State Solution of a Class of Nonlinear Stochastic Systems
,” Int. J. Nonlinear Mechanics
, Vol. 17
, No. 3
, pp. 137
–142
.13.
Chang, W. K., Ibrahim, R. A., and Afaneh, A., 1994, “Planar and Non-Planar Nonlinear Dynamics of Suspended Cables under Random In-Plane Loading,” ASME WAM Symposium on Nonlinear Dynamics and Stochastic Mechanics.
14.
Clarkson, B. L., 1994, “Review of Sonic Fatigue Technology,” NASA CR 4587, Langley Research Center, Hampton, Virginia.
15.
Crandall
S. H.
1963
, “Perturbation Technique for Random Vibration of Nonlinear Systems
,” J. Acoust. Soc. Amer.
, Vol. 35
, No. 11
, pp. 1700
–1705
.16.
Davies
H. G.
Liu
Q.
1990
, “The Response Envelop Probability Density Function of a Duffing Oscillator with Random Narrow Band Excitation
,” J. Sound Vib.
, Vol. 139
, pp. 1
–8
.17.
Davies
H. G.
Liu
Q.
1992
, “On the Narrow Band Random Response Distribution Function of a Non-Linear Oscillator
,” Int. J. Non-Linear Mech.
, Vol. 27
, No. 5
, pp. 805
–816
.18.
Davies
H. G.
Nandlall
D.
1986
, “Phase Plane for Narrow Band Random Excitation of a Duffing Oscillator
,” J. Sound Vib.
, Vol. 104
, No. 2
, pp. 277
–283
.19.
Davies
H. G.
Rajan
S.
1988
, “Random Superharmonic and Subharmonic Response: Multiple Time Scaling of a Duffing Oscillator
,” J. Sound Vib.
, Vol. 126
, No. 2
, pp. 195
–208
.20.
Dimentberg
M. F.
1971
, “Oscillations of a System with Nonlinear Cubic Characteristics under Narrow Band Random Excitation
,” Mechanics of Solids
, Vol. 6
, pp. 142
–146
.21.
Di Paola, M., 1993, “Stochastic Differential Calculus,” Chapter 2 in Dynamic Motion: Chaotic and Stochastic Behavior, F. Casciati, ed., CISM Courses and Lectures, No. 340, pp. 29–92, Springer-Verlag, New York.
22.
Di Paola
M.
Falsone
G.
1993
, “Stochastic Dynamics of Nonlinear Systems Driven by Non-Normal Delta-Correlated Processes
,” ASME Journal of Applied Mechanics
, Vol. 60
, pp. 141
–148
.23.
Fang
T.
Dowell
E. H.
1987
, “Numerical Simulations of Jump Phenomena in Stable Duffing Systems
,” Int. J. Non-Linear Mechanics
, Vol. 22
, No. 3
, pp. 267
–274
.24.
Gray
A. H.
Caughey
T. K.
1965
, “A Controversy in Problems Involving Random Parametric Excitation
,” J. Math. Phys.
, Vol. 44
, pp. 288
–295
.25.
Grebogi
C.
Ott
E.
Yorke
J. A.
1982
, “Chaotic Attractors in Crisis
,” Physical Review Letters
, Vol. 48
, No. 22
, pp. 1507
–1510
.26.
Grebogi
C.
Ott
E.
Romeiras
F.
Yorke
J. A.
1987
, “Critical Exponents for Crisis-Induced Intermittency
,” Physical Review A
, Vol. 36
, pp. 5365
–5380
.27.
Hammel, S. M., and Platt, N., 1994, “From Intermittency to Stability Extended Turbulence,” ONR Workshop on Reliability and Nonlinear Vibration of Ocean Systems, Arlington, July 18–19.
28.
Heagy
J. F.
Platt
N.
Hammel
S. M.
1994
, “Characterization of On-Off Intermittency
,” Physical Review E
, Vol. 49
, No. 2
, pp. 1140
–1150
.29.
Ibrahim, R. A., 1985, Parametric Random Vibration, John Wiley, New York.
30.
Ibrahim
R. A.
Heinrich
1988
, “Experimental Investigation of Liquid Sloshing under Parametric Random Excitation
,” ASME J. Appl. Mech
, Vol. 55
, pp. 467
–473
.31.
Ibrahim
R. A.
Heo
H.
1986
, “Autoparametric Vibration of Coupled Beams under Random Support Motion
,” ASME Journal of Vibration, Acoustics, Stress, and Reliability in Design
, Vol. 108
, No. 4
, pp. 421
–426
.32.
Ibrahim
R. A.
Lee
B. H.
Afaneh
A. H.
1993
, “Structural Modal Multifurcation with Internal Resonance, Part II: Stochastic Approach
,” ASME Journal of Vibration and Acoustics
, Vol. 115
, No. 2
, pp. 193
–201
.33.
Ibrahim
R. A.
Li
W.
1988
, “Structural Modal Interaction with Combination Internal Resonance under Wide Band Random Excitation
,” J. Sound Vib.
, Vol. 123
, No. 2
, pp. 473
–495
.34.
Ibrahim
R. A.
Roberts
J. W.
1976
, “Broad Band Random Excitation of a Two Degree-of-Freedom System with Autoparametric Coupling
,” J. Sound Vib.
, Vol. 44
, No. 3
, pp. 335
–348
.35.
Ibrahim
R. A.
Roberts
J. W.
1977
, “Stochastic Stability of the Stationary Response of a System with Autoparametric Coupling
,” Zeitschrift fur Ang. Math. und Mech.
, Vol. 57
, No. 9
, pp. 643
–649
.36.
Ibrahim
R. A.
Soundararajan
A.
1983
, “Nonlinear Parametric Liquid Sloshing under Wide Band Random Excitation
,” J. Sound Vib.
, Vol. 91
, pp. 119
–134
.37.
Ibrahim
R. A.
Soundararajan
A.
1985
, “An Improved Approach for Random Parametric Response of Dynamic Systems with Nonlinear Inertia
,” Int. J. Nonlin. Mech.
, Vol. 20
, pp. 309
–344
.38.
Ibrahim
R. A.
Soundararajan
A.
Heo
H.
1985
, “Stochastic Response of Nonlinear Dynamic Systems Based on a Non-Gaussian Closure
,” ASME Journal of Applied Mechanics
, Vol. 52
, No. 4
, pp. 965
–970
.39.
Ibrahim
R. A.
Sullivan
D. G.
1990
, “Experimental Investigation of Structural Autoparametric Interaction under Random Excitation
,” AIAA Journal
, Vol. 28
, No. 2
, pp. 338
–344
.40.
Ibrahim
R. A.
Yoon
Y. J.
Evans
M. G.
1990
, “Random Excitation of Nonlinear Coupled Oscillators
,” Nonlinear Dynamics
, Vol. 1
, No. 1
, pp. 91
–116
.41.
Ibrahim, R. A., and Yoon, Y. J., 1993, “Response Statistics of Nonlinear Hamiltonian Systems to Parametric Filtered White Noise,” Int. Conference on Nonlinear Dynamics and Stochastic Mechanics, University of Waterloo, August 29-September 1.
42.
Ilin
A. M.
Khas’miniskii
R. Z.
1964
, “On Equations of Brownian Motion
,” Theory Probab. Appl.
, Vol. 9
, pp. 421
–444
.43.
Iyengar
R. N.
1986
, “Stochastic Response and Stability of the Duffing Oscillator under Narrow Band Excitation
,” J. Sound Vib.
, Vol. 126
, pp. 255
–263
.44.
Iyengar
R. N.
1988
, “Higher Order Linearization in Non-Linear Random Vibration
,” Int. J. Non-Linear Mech.
, Vol. 23
, pp. 385
–391
.45.
Iyengar
R. N.
1989
, “Response of Nonlinear Systems to Narrow Band Excitation
,” Structural Safety
, Vol. 6
, pp. 177
–185
.46.
Iyengar, R. N., 1992,. “Approximate Analysis of Nonlinear Systems under Narrow Band Random Inputs,” Nonlinear Stochastic Mechanics, IUTAM Symposium Turin, N. Bellomo and F. Casciati, eds., Springer-Verlag.
47.
Kantorovich, L. V., and Krylov, V. I., 1958, Approximate Methods of Higher Analysis, Nordhoff, Netherlands.
48.
Kapitaniak
T.
1985
, “Stochastic Response with Bifurcation to Nonlinear Duffing’s Oscillators
,” J. Sound Vib.
, Vol. 102
, No. 3
, pp. 440
–441
.49.
Khasminskii, R. Z., 1980, Stochastic Stability of Differential Equations, Sijthoff Noordhoff, The Netherlands.
50.
Koliopulos
P. K.
Bishop
S. R.
1993
, “Quasi-Harmonic Analysis of the Behavior of Hardening Duffing Oscillator Subjected to Filtered White Noise
,” Nonlinear Dynamics
, Vol. 4
, pp. 279
–288
.51.
Koliopulos
P. K.
Langley
R. S.
1993
, “Improved Stability Analysis of the Response of a Duffing Oscillator under Filtered White Noise
,” Int. J. Non-Linear Mech.
, Vol. 28
, No. 2
, pp. 145
–155
.52.
Kozin, F., and Sugimoto, S., 1977, “Decision Criteria for Stability of Stochastic Systems from Observed Data,” Stochastic Problems in Dynamics, B. L. Clarkson, ed., Pitman, London, pp. 8–35.
53.
Kushner, H., 1967, Stochastic Stability and Control, Academic Press, New York.
54.
Kushner
H. J.
1969
, “The Cauchy Problem for a Class of Degenerate Parabolic Equations and Asymptotic Properties of the Related Diffusion Processes
,” J. Diff. Equations
, Vol. 6
, No. 2
, pp. 209
–231
.55.
Lee
B. H.
Ibrahim
R. A.
1994
, “Stochastic Bifurcation in Nonlinear Structural Systems Near 1:1 Internal Resonance
,” Probabilistic Engineering Mechanics
, Vol. 9
, pp. 23
–32
.56.
Lennox
W. C.
Kuak
Y. C.
1976
, “Narrow Band Excitation of a Nonlinear Oscillator
,” ASME Journal of Applied Mechanics
, Vol. 43
, pp. 340
–344
.57.
Li
W.
Ibrahim
R. A.
1989
, “Principal Internal Resonances in 3-DOF Systems Subjected to Wide-Band Random Excitation
,” J. Sound Vib.
, Vol. 130
, No. 2
, pp. 305
–321
.58.
Li
W.
Ibrahim
R. A.
1990
, “Monte Carlo Simulation of Coupled Nonlinear Oscillators under Random Excitations
,” ASME Journal of Applied Mechanics
, Vol. 57
, pp. 1097
–1099
.59.
Lin
Y. K.
Cai
G. Q.
1988
, “Exact Stationary Response Solution for Second Order Nonlinear Systems under Parametric and External White-Noise Excitations: Part II
,” ASME Journal of Applied Mechanics
, Vol. 55
, pp. 702
–705
.60.
Lyon
R. H.
1960
, “Response of a Nonlinear String to Random Excitation
,” J. Acoust. Soc. Amer.
, Vol. 28
, pp. 953
–960
.61.
Lyon
R. H.
1965
, “Response of Strings to Random Noise Fields
,” J. Acoust. Soc. Amer.
, Vol. 32
, pp. 391
–398
.62.
Lyon
R. H.
Heckl
M.
Hazelgrove
C. B.
1961
, “Response of Hard-Spring Oscillator to Narrow Band Excitation
,” The J. Acoustical Soc. Amer.
, Vol. 33
, pp. 1404
–1411
.63.
Mei, C., and Wentz, K. R., 1982, “Analytical and Experimental Nonlinear Response of Rectangular Panels to Acoustic Excitation,” AIAA 23rd Structures, Structural Dynamics and Materials Conference, New Orleans, LA, pp. 514–520.
64.
Minorsky, N., 1962, Nonlinear Oscillation, Van Nostrand, New York.
65.
Moshchuk
N. K.
Sinitsyn
I. N.
1991
, “On Stationary Distributions in Nonlinear Stochastic Differential System
,” Quarterly J. Inter. Appl. Math.
, Vol. 44
, No. 4
, pp. 571
–579
.66.
Moyer, E. T., Jr., 1988, “Time Domain Simulation of the Response of Geometrically Nonlinear Panels Subjected to Random Loading,” Proc. 29th AIAA Structures, Structural Dynamics and Materials Conference, Paper No. 88-2236, pp. 210–218.
67.
Muscolino, G., 1993, “Response of Linear and Nonlinear Structural Systems under Gaussian and Non-Gaussian Filtered Input,” Chapter 6 in Dynamic Motion: Chaotic and Stochastic Behavior, F. Casciati, ed., CISM Courses and Lectures, Springer-Verlag, New York, No. 340, pp. 203–299.
68.
Nayfeh
A. H.
Serhan
S. J.
1991
, “Response Moments of Dynamic Systems with Modal Interactions
,” J. Sound Vib.
, Vol. 151
, No. 2
, pp. 291
–310
.69.
Newland
D. E.
1965
, “Energy Sharing in Random Vibration of Nonlinearly Coupled Modes
,” J. Inst. Math. Appl
, Vol. 1
, No. 3
, pp. 199
–207
.70.
Pinsky
M.
1974
, “Stochastic Stability and the Dirichlet Problem
,” Comm. Pure Appl. Math.
, Vol. 27
, pp. 311
–350
.71.
Platt
N.
Spiegel
E. A.
Tresser
C.
1993
, “On-Off Intermittency: A Mechanism for Bursting
,” Physical Review Letters
, Vol. 70
, No. 3
, pp. 279
–282
.72.
Platt
N.
Hammel
S. M.
Heagy
J. F.
1994
, “Effects of Additive Noise on On-Off Intermittency
,” Physical Review Letters
, Vol. 72
, No. 22
, pp. 3498
–3501
.73.
Pomeau
Y.
Manneville
P.
1980
, “Intermittent Transition to Turbulence in Dissipative Dynamical Systems
,” Commun. Math. Phys.
, Vol. 74
, pp. 189
–197
.74.
Red-Horse
J. R.
Spanos
P. D.
1992
, “A Generalization to Stochastic Averaging in Random Vibration
,” Int. J. Nonlinear Mech.
, Vol. 27
, No. 1
, pp. 85
–101
.75.
Reinhall
P. G.
Miles
R. N.
1989
, “Effect of Damping and Stiffness on the Random Vibration of Nonlinear Periodic Plates
,” J. Sound and Vibration
, Vol. 132
, No. 1
, pp. 33
–42
.76.
Roberts
J. B.
1991
, “Multiple Solutions Generated by Statistical Linearization and their Physical Significance
,” Int. J. Non-Linear Mech.
, Vol. 26
, pp. 945
–959
.77.
Roberts
J. B.
Spanos
P-T. D.
1986
, “Stochastic Averaging: An Approximate Method of Solving Random Vibration Problems
,” Intl. J. Non-Linear Mech.
, Vol. 21
, No. 2
, pp. 111
–134
.78.
Roberts, J. B., and Spanos, P-T. D., 1990, Random Vibrations and Statistical Linearization, John Wiley & Sons, New York.
79.
Roberts
J. W.
1980
, “Random Excitation of a Vibratory System with Autoparametric Interaction
,” J. Sound Vib.
, Vol. 69
, No. 1
, pp. 101
–116
.80.
Roy
R. V.
1994
, “Stochastic Averaging of Oscillators Excited by Colored Gaussian Processes
,” Int. J. Nonlinear Mech.
, Vol. 29
, No. 4
, pp. 463
–475
.81.
Schetzen, M., 1980, The Volterra and Wiener Theories of Nonlinear Systems, John Wiley, New York.
82.
Schmidt, G., 1977a, “Probability Densities of Parametrically Excited Random Vibration,” Stochastic Problems in Dynamics, B. L. Clarkson, ed., Pitman, London, pp. 197–213.
83.
Schmidt, G., 1977b, “Vibration Mechanical Systems with Random Parametric Excitation, in Theoretical and Applied Mechanics, W. T. Koiter, ed., North-Halland, Amsterdam, pp. 439–450.
84.
Sobczyk, K., and Spencer, B., 1992, Random Fatigue, Academic Press, New York.
85.
Soize
C.
1991
, “Exact Stationary Response of Multi-Dimensional Nonlinear Hamiltonian Dynamical Systems Under Parametric and External Stochastic Excitations
,” J. Sound Vib.
, Vol. 149
, No. 1
, pp. 1
–24
.86.
Soundararajan
A.
Ibrahim
R. A.
1988
, “Parametric and Autoparametric Vibrations of an Elevated Water Tower, Part III: Random Response
,” J. Sound Vib.
, Vol. 121
, No. 3
, pp. 445
–462
.87.
Spanos
P. D.
1981
, “Stochastic Linearization in Structural Dynamics
,” Appl. Mech. Rev.
, Vol. 34
, pp. 1
–8
.88.
Sun
J. Q.
Hsu
C. S.
1987
, “Cumulant-Neglect Closure Method for Nonlinear Systems under Parametric Excitation
,” ASME Journal of Applied Mechanics
, Vol. 54
, pp. 649
–655
.89.
Tagata
G.
1978
, “Analysis of a Randomly Excited Nonlinear Stretching String
,” J. Sound Vib.
, Vol. 58
, No. 1
, pp. 95
–107
.90.
Tagata
G.
1989
, “Nonlinear String Random Vibration
,” J. Sound Vib.
, Vol. 129
, No. 3
, pp. 361
–384
.91.
Townsond, A. A., 1976, The Structure of Turbulent Shear Flow, Cambridge University Press, Cambridge.
92.
Yong
Y.
Lin
Y. K.
1987
, “Exact Stationary Response Solution for Second Order Nonlinear Systems Under Parametric and External White-Noise Excitations
,” ASME Journal of Applied Mechanics
, Vol. 54
, pp. 414
–418
.
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