A half-century of stochastic equivalent linearization

Structural Control and Health Monitoring

Volumn 13, Issue 1, 2006, Pages 27-40

  Crandall, Stephen H ^a  

^a MASSACHUSETTS INSTITUTE OF TECHNOLOGY   (United States)

Author keywords

Random vibration; Statistical linearization; Stochastic equivalent linearization

Indexed keywords

APPROXIMATION THEORY; EMBEDDED SYSTEMS; NONLINEAR SYSTEMS; PROBABILITY; RESPONSE TIME (COMPUTER SYSTEMS); STOCHASTIC CONTROL SYSTEMS;

RANDOM VIBRATION; STATISTICAL LINEARIZATION; STOCHASTIC EQUIVALENT LINEARIZATION;

LINEARIZATION;

EID: 33144456348     PISSN: 15452255     EISSN: None     Source Type: Journal    
DOI: 10.1002/stc.129     Document Type: Review

References (27)

1. Booton RC. The analysis of non-linear control systems with random inputs. In Proceedings of the Symposium on Nonlinear Circuit Analysis. Fox J (ed.). Sponsored by the Polytechnic Institute of Brooklyn, Edwards Brothers, Inc., Ann Arbor, Mich, 1953; 369-391
2. (More easily obtained is: Booton RC. Nonlinear control systems with random inputs. IEEE Transactions on Circuit Theory 1954; CT-1: 9-18).
3. Kazakov IE. An approximate method for the statistical investigation of nonlinear systems. Trudi Voenna-Vozdushnoi Inzheneroi Akademii imeni Professora N.E. Zhukovskogo 1954; 394:1-52 (in Russian).
4. Caughey TK. Equivalent linearization techniques. Journal of the Acoustical Society of America 1963; 35:1706-1711 (Reference is made to presentations of the procedure in lectures delivered in 1953 at the California Institute of Technology).
5. Elishakoff I, Colajanni P. Booton's problem re-examined. Journal of Sound and Vibration 1998; 210:683-691.
6. Proppe C, Pradlwarter HJ, Schuëller GI. Equivalent linearization and Monte Carlo simulation in stochastic dynamics. Probabilistic Engineering Mechanics 2003; 18:1-15.
7. Kazakov IE. Approximate probabilistic analysis of the accuracy of operation of essentially nonlinear systems. Avtomatika I Telemekhanika 1956; 17:385-409 (in Russian)
8. (Available in English translation: Automation and Remote Control 1956; 17:423-450).
9. Zhang XT, Elishakoff I, Zhang RC. A new stochastic linearization technique based on minimum mean square deviation of potential energies. In Stochastic Structural Dynamics - New Theoretical Developments, Lin YK, Elishakoff I (eds). Springer: Berlin, 1991; 327-338.
10. Kozin F. The method of statistical linearizations for non-linear stochastic vibrations. In Nonlinear Stochastic Dynamic Engineering Systems. Ziegler F, Schuëller GI (eds). IUTAM Symposium: Innsbruck/Igls, Austria, 1988; 45-56.
11. Crandall SH. On using non-Gaussian distributions to perform statistical linearization. In Advances in Stochastic Structural Dynamics, Proceedings of the 5th International Conference on Stochastic Structural Dynamics, Zhu WQ, Cai GQ, Zhang RC (eds). CRC Press: Boca Raton, 2003; 47-62
12. (Reprinted in International Journal of Non-Linear Mechanics 2004; 39:1395-1406).
13. Pradlwarter HJ, Schuëller GI, Chen XW. Consideration of non-Gaussian response properties by use of stochastic equivalent linearization. In Proceedings of the 3rd International Conference on Recent Advances in Structural Dynamics, vol. II, Petyt M, Wolfe HF, Mei C (eds). University of Southampton: Southampton, 1988; 737-752.
14. Crandall SH. Is stochastic equivalent linearization a subtly flawed process? Probabilistic Engineering Mechanics 2001; 16:169-176.
15. Crandall SH. On statistical linearization for nonlinear oscillators. In Problems of the Asymptotic Theory of Nonlinear Oscillations. Koroliuk VS (ed.). Akademiya Nauk Ukraine, Naukova Dumka: Kiev, 1977; 115-122
16. (Reprinted In Nonlinear System Analysis and Synthesis, Ramnath RV, Hedrick JK, Paynter HM (eds). American Society of Mechanical Engineers: New York, 1980; 115-122).
17. Atalik TS, Utku S. Stochastic linearization of multi-degree-of-freedom non-linear systems. Earthquake Engineering and Structural Dynamics 1976; 4:411-420.
18. Kramers HA. Brownian motion in a field of force and the diffusion model of chemical reactions. Physica 1940; 7(4):284-304.
19. Maxwell JC. Illustrations of the dynamical theory of gases. In The Scientific Papers of James Clerk Maxwell, vol. I, Niven WD (ed.). Dover Publications: New York, 1965; 377-409.
20. Wang MC, Uhlenbeck GE. Theory of Brownian motion. Reviews of Modern Physics 1945; 17:323-342.
21. Crandall SH. On the statistical linearization methods of I.E. Kazakov. Problemy Machinostroeniya i Avtomatizatsii 2004; (4):41-47.
22. Lin YK. Probabilistic Theory of Structural Dynamics. McGraw-Hill: New York, 1967.
23. Socha L, Pawleta M. Corrected equivalent linearization of stochastic dynamic systems. Machine Dynamics Problems 1994; 7:149-161.
24. Socha L, Pawleta M. Some remarks on equivalent linearization of stochastic dynamic systems. In Stochastic Structural Dynamics, Spencer BF, Johnson EA (eds). Balkema: Rotterdam, 1999; 106-112.
25. Elishakoff IE, Colajanni P. Stochastic linearization critically re-examined. Chaos, Solitons and Fractals 1997; 8:1957-1972.
26. Colajanni P, Elishakoff IE. A subtle error in the stochastic linearization technique. Chaos, Solitons and Fractals 1998; 9:479-491.
27. Colajanni P, Elishakoff IE. A new look at the stochastic linearization technique for hyperbolic tangent oscillator. Chaos, Solitons and Fractals 1998; 9:1611-1623.