0
REVIEW ARTICLES

Linearization in Analysis of Nonlinear Stochastic Systems: Recent Results—Part I: Theory

[+] Author and Article Information
L. Socha

Faculty of Mathematics and Natural Sciences, Collage of Sciences, Cardinal Stefan Wyszynski University, ul.Dewajtis 5, 01-815 Warsaw, Polande-mail: leslawsocha@poczta.onet.pl

Appl. Mech. Rev 58(3), 178-205 (May 27, 2005) (28 pages) doi:10.1115/1.1896368 History: Online May 27, 2005
Copyright © 2005 by ASME
Your Session has timed out. Please sign back in to continue.

References

Booton, R. C., 1953, “The Analysis of Nonlinear Central Systems With Random Inputs,” Proc. Symp. on Nonlinear Circuit Analysis, Polytechnic Institute of Brooklyn, and also (1954), IRE Transactions on Circuit Theory1 , pp. 32–34.
Kazakov,  I. E., 1954, “An Approximate Method for the Statistical Investigation of Nonlinear Systems” (in Russian), Trudi Voenna-Vozdushnoi Inzhenernoi Akademii imeni Professora N. E. Zhukowskogo, 394, pp. 1–52.
Kazakov,  I. E., 1956, “Approximate Probabilistic Analysis of the Accuracy of the Operation of Essentially Nonlinear Systems,” Avtom. Telemekh., 17, pp. 423–450.
Caughey,  T. K., 1959, “Response of Nonlinear String to Random Loading,” ASME J. Appl. Mech., 26, pp. 341–344.
Caughey,  T. K., 1963, “Equivalent Linearization Techniques,” J. Acoust. Soc. Am., 35, pp. 1706–1711. Reference is made to presentations in Cal Tech Lectures in 1953.
Spano,  P. T., 1981, “Stochastic Linearization in Structural Dynamics,” Appl. Mech. Rev., 34, pp. 1–8.
Roberts,  J. B., 1981, “Response of Nonlinear Mechanical Systems to Random Excitation, Part 2: Equivalent Linearization and Other Methods,” Shock Vib. Dig., 13, pp. 15–29.
Crandall,  S. H., and Zhu,  W. Q., 1983, “Random Vibration: A Survey of Recent Developments,” ASME J. Appl. Mech., 50, pp. 953–962.
Roberts, J. B., and Spanos, P. D. T., 1990, Random Vibration and Statistical Linearization, Wiley, Chichester.
Socha,  L., and Soon,  T. T., 1991, “Linearization in Analysis of Nonlinear Stochastic Systems,” Appl. Mech. Rev., 44, pp. 399–422.
Newland, D. E., 1993, “An Introduction to Random Vibrations, Spectral and Wavelet Analysis,” 3rd Ed., Longman Scientific and Technical, Burnt Mill, Harlow, Essex.
Soong, T. T., and Grigoriu, M., 1993, Random Vibration of Mechanical and Structural Systems, PTR, Prentice-Hall, Englewood Cliffs, NJ.
Nigam, N. C., and Narayanan, S., 1994, Applications of Random Vibrations, Narosa Publishing House, New Dehli.
Lin, Y. K., and Cai, G. Q., 1995, Probabilistic Structural Dynamics Advances Theory and Applications, McGraw-Hill, New York.
Lutes, L. D., and Sarkani, S., 1997, Stochastic Analysis of Structural and Mechanical Vibrations, Prentice-Hall, Upper Saddle River, NJ.
Solnes, J., 1997, Stochastic Processes and Random Vibrations Theory and Practice, Wiley, Chichester.
Elishakoff,  I., 1995, “Random Vibration of Structures: A Personal Perspective,” Appl. Mech. Rev., 48, pp. 809–825.
Elishakoff,  I., 2000, “Stochastic Linearization Technique: A New Interpretation and a Selective Review,” Shock Vib. Dig., 32, pp. 179–188.
Proppe,  C., Pradlwarter,  H. J., and Schueller,  G. I., 2003, “Equivalent Linearization and Monte Carlo Simulation in Stochastic Dynamics,” Probab. Eng. Mech., 18, pp. 1–15.
Socha,  L., 2004, “Linearization in Analysis of Nonlinear Stochastic Systems: Recent Results—Part II: Applications,” Appl. Mech. Rev., (in press).
Socha,  L., and Pawleta,  M., 1994, “Corrected Equivalent Linearization,” Machine Dynamics Problems, 7, pp. 149–161.
Elishakoff,  I., and Colojani,  P., 1997, “Stochastic Linearization Critically Re-examined,” Chaos, Solitons Fractals, 8, pp. 1957–1972.
Colojani,  P., and Elishakoff,  I., 1998, “A Subtle Error in Conventional Stochastic Linearization Techniques,” Chaos, Solitons Fractals, 9, pp. 479–491.
Colojani,  P., and Elishakoff,  I., 1998, “A New Look at the Stochastic Linearization Technique for Hyperbolic Tangent Oscillator,” Chaos, Solitons Fractals, 9, pp. 1611–1623.
Elishakoff,  I., and Colojani,  P., 1998, “Booton’s Problem Re-examined,” J. Sound Vib., 210, pp. 683–691.
Bernard,  P., and Wu,  L., 1998, “Stochastic Linearization: The Theory,” J. Appl. Probab., 35, pp. 718–730.
Bernard,  P., 1998, “Stochastic Linearization: What is Available What is not,” Computers and Structures, 67, pp. 9–18.
Socha, L., and Pawleta, M., 1999, “Some Remarks on Equivalent Linearization of Stochastic Systems,” Stochastic Structural Dynamics, B. F. Spencer and E. A. Johnson, eds., Balkema, Rotterdam, pp. 105–112.
Crandall,  S. H., 2001, “Is Stochastic Equivalent Linearization a Subtly Flawed Procedure,” Probab. Eng. Mech, 16, pp. 169–176.
Crandall, S. H., 1980, “On Statistical Linearization For Nonlinear Oscillators,” Nonlinear System Analysis and Synthesis, R. V. Rammath, J. K. Hedrick and H. M. Paynter, eds., Am. Soc. Mech. Engrg., New York, pp. 199–209.
Crandall, S. H., 2003, “On Using Non-Gaussian Distributions to Perform Statistical Linearization,” Advances in Stochastic Structural Dynamics, W. Q. Zhu, G. Q. Cai and R. C. Zhang, eds., CRC Press, Boca Raton, pp. 49–62.
Bernard, P., 2003, “Stochastic Linearization: True Standard or Gaussian,” Computational Stochastic Mechanics, P. D. Spanos and G. Deodatis, eds., Millpress, Rotterdam, pp. 59–66.
Socha,  L., and Pawleta,  M., 2001, “Are Statistical Linearization and Standard Equivalent Linearization the Same Methods in the Analysis of Stochastic Dynamic Systems,” J. Sound Vib., 248, pp. 387–394.
Anh,  N. D., and Hung,  L. X., 2003, “An Improved Criterion of Gaussian Equivalent Linearization for Analysis of Nonlinear Stochastic Systems,” J. Sound Vib., 268, pp. 177–200.
Pawleta,  M., 1991, “Stochastic Linearization of Composite Dynamic Systems,” (in Polish) J. Theor. Appl. Mech., 29, pp. 355–375.
Anh, N. D., and DiPaola, M., 1995, “Some Extensions of Gaussian Equivalent Linearization,” Proc. of Int. Conference on Nonlinear Stochastic Dynamics, Hanoi, Vietnam, pp. 5–15.
Pradlwarter,  H. J., 2001, “Nonlinear Stochastic Response Distributions by Local Statistical Linearization,” Int. J. Non-Linear Mech., 36, pp. 1135–1151.
Kazakov,  I. E., 1998, “An Extension of the Method of Statistical Linearization,” Avtomatika i Telemekhanika, 59, pp. 220–224.
Lee,  J., 1995, “Improving the Equivalent Linearization Technique for Stochastic Duffing Oscillators,” J. Sound Vib., 186, pp. 846–855.
Grundmann,  H., Hartmann,  C., and Waubke,  H., 1998, “Structures Subjected to Stationary Stochastic Loadings. Preliminary Assessment by Statistical Linearization Combined With an Evolutionary Algorithm,” Comput. Struct., 67, pp. 53–64.
Suzuki, Y., and Minai, R., 1987, Application of Stochastic Differential Equations to Seismic Reliability Analysis of Hysteretic Structures, (Lecture Notes in Eng., Vol. 32 ), Springer, New York.
Davis, G. L., and Spanos, P. D., 1999, “Developing Force-Limited Random Vibration Test Specifications for Nonlinear Systems Using the Frequency-Shift Method,” Proc. of 40th AIAA/SDM Conference, St. Louis, 4/12-15/99, pp. 1688–1698.
Isidori, A., 1989, Nonlinear Control Systems: An Introduction, 2nd Ed., Springer, Berlin.
Socha,  L., 1994, “Some Remarks on Exact Linearization of a Class of Stochastic Dynamical Systems,” IEEE Trans. Autom. Control, 39, pp. 1980–1984.
Naess,  A., Galeazzi,  F., and Dogliani,  M., 1992, “Extreme Response Predictions of Nonlinear Compliant Offshore Structures by Stochastic Linearization,” Appl. Aciean. Res., 14, pp. 71–81.
Anh,  N. D., and Schiehlen,  W., 1999, “A Technique for Obtaining Approximate Solutions in Gaussian Equivalent Linearization,” Comput. Methods Appl. Mech. Eng., 168, pp. 113–119.
Anh,  N. D., and Schiehlen,  W., 1997, “New Criterion for Gaussian Equivalent Linearization,” Eur. J. Mech. A/Solids, 16, pp. 1025–1039.
Beaman,  J. J., and Hedrick,  J. K., 1981, “Improved Statistical Linearization for Analysis of Control of Nonlinear Stochastic Systems: Part I: An Extended Statistical Linearization Technique,” ASME J. Dyn. Syst., Meas., Control, 101, pp. 14–21.
Chen, G., 1999, “Cascade Linearization of Nonlinear System Subjected to Gaussian Excitations,” Stochastic Structural Dynamics, B. F. Spencer and E. A. Johnson, eds., Balkema, Rotterdam, pp. 69–76.
Duval, L., et al. 1999, “Zero and Non-Zero Mean Analysis of MDOF Hysteretic Systems Via Direct Linearization,” Stochastic Structural Dynamics, B. F. Spencer and E. A. Johnson, Balkema, Rotterdam, pp. 77–84.
Molnar,  A. J., Vaschi,  K. M., and Gay,  C. W., 1976, “Application of Normal Mode Theory of Pseudo Force Methods to Solve Problems With Nonlinearities,” ASME J. Pressure Vessel Technol., 98, pp. 151–156.
Benfratello,  S., 1996, “Pseudo-Force Method for a Stochastic Analysis of Nonlinear Systems,” Probab. Eng. Mech., 11, pp. 113–123.
Uchino,  E., Ohta,  M., and Takata,  H., 1993, “A New State Estimation Method for a Quantized Stochastic Sound System Based on a Generalized Statistical Linearization,” J. Sound Vib., 160, pp. 193–203.
Iyengar,  R. N., and Roy,  D., 1996, “Conditional Linearization in Nonlinear Random Vibration,” J. Eng. Mech., 119, pp. 197–200.
Spanos,  P. T. D., and Iwan,  W. D., 1979, “Harmonic Analysis of Dynamic Systems With Non-Symmetric Nonlinearities,” ASME J. Dyn. Syst., Meas., Control, 101, pp. 31–36.
Iyengar,  R. N, and Roy,  D., 1998, “New Approaches for the Study of Nonlinear Oscillators,” J. Sound Vib., 211, pp. 843–875.
Iyengar,  R. N., and Roy,  D., 1998, “Extensions of the Phase Space Linearization Method,” J. Sound Vib., 211, pp. 877–906.
Roy,  D., 2000, “Exploration of the Phase-Space Linearization Method for Deterministic and Stochastic Nonlinear Dynamical Systems,” Nonlinear Dyn., 23, pp. 225–258.
Casciati,  F., Faravelli,  L., and Hasofer,  A. M., 1993, “A New Philosophy for Stochastic Equivalent Linearization,” Probab. Eng. Mech., 8, pp. 179–185.
Bouc,  R., 1994, “The Power Spectral Density of Response for a Strongly Nonlinear Random Oscillator,” J. Sound Vib., 175, pp. 317–331.
Ismaili,  M. A., and Bernard,  P., 1997, “Asymptotic Analysis and Linearization of the Randomly Perturbed Two-Wells Duffing Oscillator,” Probab. Eng. Mech., 12, pp. 171–178.
Ricciardi,  G., and Elishakoff,  I., 2002, “A Novel Local Stochastic Linearization Method Via Two Extremum Entropy Principles,” Int. J. Non-Linear Mech., 37, pp. 785–800.
Socha,  L., 1995, “Moment Equivalent Linearization Technique,” Z. Angew. Math. Mech., 75, SII, pp. 577–578.
Cheded,  L., 2003, “Invariance Property: Higher-Order Extension and Application to Gaussian Random Variables,” Signal Process., 83, pp. 1545–1551.
Zhang, X. T., Elishakoff, I., and Zhang, R. Ch., 1991, “A New Stochastic Linearization Technique Based on Minimum Mean-Square Deviation of Potential Energies,” Stochastic Structural Dynamics—New Theoretical Developments, Y. K. Lin and I. Elishakoff, eds., Springer-Verlag, Berlin, pp. 327–338.
Elishakoff, I., and Falsone, G., 1993, “Some Recent Developments in Stochastic Linearization Technique,” Computational Stochastic Mechanics, A. H. Cheng and C. Y. Yang, eds., Computational Mechanics Publication, Southampton, N. Y., pp. 175–194.
Falsone,  G., and Elishakoff,  I., 1994, “Modified Stochastic Linearization Technique for Coloured Noise Excitation of Duffing Oscillator,” Int. J. Non-Linear Mech., 29, pp. 65–69.
Elishakoff, I., and Zhang, R. C., 1991, Comparison of the New Energy-Based Version of the Stochastic Linearization Technique, Nonlinear Stochastic Mechanics, N. Bellomo and F. Casciati, eds., Springer, Berlin, pp. 201–212.
Elishakoff,  I., and Zhang,  X. T., 1992, “An Appraisal of Different Stochastic Linearization Criteria,” J. Sound Vib., 153, pp. 370–375.
Elishakoff,  I., and Colombi,  P., 1993, “Successful Combination of the Stochastic Linearization and Monte Carlo Methods,” J. Sound Vib., 160, pp. 554–558.
Elishakoff, I., 1991, “Method of Stochastic Linearization: Revisited and Improved,” Computational Stochastic Mechanics, P. D. Spanos and C. A. Brebbia, eds., Computational Mechanics Publication and Elsevier Applied Science, London, pp. 101–111.
Elishakoff, I., 1995, “Some Results in Stochastic Linearization of Nonlinear Systems,” Nonlinear Dynamics and Stochastic Mechanics, W. H. Klieman and N. Namachchivaya, CRC Press, Boca Raton, pp. 259–281.
Zhang,  X. T., Zhang,  R. C., and Xu,  Y. L., 1993, “Analysis on Control of Flow-Induced Vibration by Tuned Liquid Damper With Crossed TubeLike Containers,” J. Wind. Eng. Ind. Aerodyn., 50, pp. 351–360.
Zhang,  R., Elishakoff,  I., and Shinozuka,  M., 1994, “Analysis of Nonlinear Sliding Structures by Modified Stochastic Linearization Methods,” Nonlinear Dyn., 5, pp. 299–312.
Fang,  J., Elishakoff,  I., and Caimi,  R., 1995, “Nonlinear Response of a Beam Under Stationary Random Excitation by Improved Stochastic Linearization Method,” Appl. Math. Model., 19, pp. 106–111.
Elishakoff,  I., Fang,  J., and Caimi,  R., 1995, “Random Vibration of a Nonlinearly Deformed Beam by a New Stochastic Linearization Technique,” Int. J. Solids Struct., 32, pp. 1571–1584.
Elishakoff, I., and Bert, C., 1999, “Complementary Energy Criterion in Nonlinear Stochastic Dynamics,” Application of Stochastic and Probability, R. L. Melchers and M. G. Steward, eds., A. A. Balkema, Rotterdam, pp. 821–825.
Elishakoff,  I., 2000, “Multiple Combinations of the Stochastic Linearization Criteria by the Moment Approach,” J. Sound Vib., 237, pp. 550–559.
Zhang, X. T., and Zhang, R. C., 1999, “Energy-Based Stochastic Equivalent Linearization With Optimized Power,” Stochastic Structural Dynamics, B. F. Spencer and E. A. Johnson, eds., Balkema, Rotterdam, pp. 113–117.
Zhang,  R. C., 2000, “Work/Energy-Based Stochastic Equivalent Linearization With Optimized Power,” J. Sound Vib., 230, pp. 468–475.
Wang,  G. Y., and Dai,  M., 2001, “Equivalent Linearization Method Based on Energy-to cth-Power Difference Criterion in Nonlinear Stochastic Vibration Analysis of Multi-Degree-of-Freedom Systems,” Appl. Math. Mech., 22, pp. 947–955.
Crandall,  S., 1973, “Correlations and Spectra of Nonlinear System Response,” Nonlinear Vibration Problems, 14, pp. 39–53.
Apetaur,  M., and Opicka,  F., 1983, “Linearization of Nonlinear Stochastically Excited Dynamic Systems,” J. Sound Vib., 86, pp. 563–585.
Bellizzi, S., and Bouc, R., 1995, “Spectral Response of Asymetrical Random Oscillators,” Computational Stochastic Mechanics, P. D. Spanos, eds., A. A. Balkema, Rotterdam, pp. 77–86.
Bellizzi,  S., and Bouc,  R., 1996, “Spectral Response of Asymetrical Random Oscillators,” Probab. Eng. Mech., 11, pp. 51–59.
Bellizzi,  S., and Bouc,  R., 1999, “Analysis of Multi-Degree of Freedom Strongly Nonlinear Mechanical Systems With Random Input: Part II: Equivalent Linear System With Random Matrices and Spectral Density Matrix,” Probab. Eng. Mech., 14, pp. 245–256.
Bernard, P., 1991, “About Stochastic Linearization,” Nonlinear Stochastic Mechanics, N. Bellomo and F. Casciati, eds., Springer, Berlin, pp. 61–70.
Bernard,  P., and Taazount,  M., 1994, “Random Dynamics of Structures With Gaps: Simulation and Spectral Linearization,” Nonlinear Dyn., 5, pp. 313–335.
Ismaili,  M. A., 1996, “Design of a System of Linear Oscillators Spectrally Equivalent to a Nonlinear One,” Int. J. Non-Linear Mech., 31, pp. 573–580.
Socha, L., 1995, “Application of Probability Metrics to the Linearization and Sensitivity Analysis of Stochastic Dynamic Systems,” Proc. of Int. Conference on Nonlinear Stochastic Dynamics, Hanoi, Vietnam, pp. 193–202.
Socha,  L., 1998, “Probability Density Equivalent Linearization Technique for Nonlinear Oscillator With Stochastic Excitations,” Z. Angew. Math. Mech., 78(S3), pp. 1087–1088.
Socha,  L., 1999, “Probability Density Equivalent Linearization and Nonlinearization Techniques,” Arch. Mech., 51, pp. 487–507.
Socha,  L., 1999, “Statistical and Equivalent Linearization Techniques with Probability Density Criteria,” J. Theor. Appl. Mech., 37, pp. 369–382.
Socha,  L., 2002, “Probability Density Statistical and Equivalent Linearization Techniques,” Int. J. Syst. Sci., 33, pp. 107–127.
Socha, L., 1999, “Application of Probability Metrics to the Nonlinearization Analysis,” Stochastic Structural Dynamics, B. F. Spencer and E. A. Johnson, eds., Balkema, Rotterdam, pp. 99–104.
Socha, L., and Błachuta, M., 2000, “Application of Linearization Methods With Probability Density Criteria in Control Problems,” Proc. of American Control Conference, Chicago, CD-ROM 4, Danvers, MA, pp. 2775–2779.
Socha,  L., 2001, “Statistical Linearization of the Duffing Oscillator Under Non-Gaussian Excitations in Probability Density Functions Space,” Z. Angew. Math. Mech., 81(S3), pp. 647–648.
Lutes,  L. D., 1970, “Approximate Technique for Treating Random Vibration of Hysteretic Systems,” J. Acoust. Soc. Am., 48, pp. 299–306.
To,  C. W. S., and Li,  D. M., 1991, “Equivalent Nonlinearization of Nonlinear Systems to Random Excitations,” Probab. Eng. Mech., 6, pp. 184–192.
Cai,  C. Q., and Lin,  Y. K., 1988, “A New Approximate Solution Technique for Randomly Excited Nonlinear Oscillators,” Int. J. Non-Linear Mech., 23, pp. 409–420.
Elishakoff,  I., and Cai,  G. Q., 1993, “Approximate Solution for Nonlinear Random Vibration Problems by Partial Stochastic Linearization,” Probab. Eng. Mech., 8, pp. 233–237.
To,  C. W. S., 1993, “A Statistical Nonlinearization Technique in Structural Dynamics,” J. Sound Vib., 161, pp. 543–548.
Polidori,  D. C., and Beck,  J. L., 1996, “Approximate Solutions for Nonlinear Random Vibration Problems,” Probab. Eng. Mech., 11, pp. 179–185.
Polidori,  D. C., Beck,  J. L., and Papadimitriou  C., 2000, “A New Stationary PDF Approximation for Nonlinear Oscillators,” Int. J. Non-Linear Mech., 35, pp. 657–673.
Lei,  Z., and Qiu,  C., 1996, “A New Equivalent Nonlinearization Method for Random Vibrations of Nonlinear Systems,” Mech. Res. Commun., 23, pp. 131–136.
Lei,  Z., and Qiu,  C., 1997, “An Equivalent Nonlinearization Method for Analyzing Response of Nonlinear Systems to Random Excitations,” Appl. Math. Mech., 18, pp. 551–561.
Cavaleri,  L., and Di-Paola,  M., 2000, “Statistic Moments of the Total Energy of Potential Systems and Application to Equivalent Nonlinearization,” Int. J. Non-Linear Mech., 35, pp. 573–587.
Wang,  R., Kusumoto,  S., and Zhang,  Z., 1996, “A New Equivalent Nonlinearization Technique,” Probab. Eng. Mech., 11, pp. 129–137.
Zhu,  W. Q., Huang,  Z. L., and Suzuki,  Y., 2001, “Equivalent Nonlinear System Method for Stochastically Excited and Dissipated Partially Integrable Hamiltonian Systems,” Int. J. Non-Linear Mech., 36, pp. 773–786.
Zhao,  L., and Chen,  Q., 1997, “An Equivalent Nonlinearization Method for Analyzing Response of Nonlinear Systems to Random Excitations,” Appl. Math. Mech., 18, pp. 551–561.
Zhao,  L., and Chen,  Q., 1996, “A New Equivalent Nonlinearization Method for Random Vibrations of Nonlinear Systems,” Mech. Res. Commun., 23, pp. 131–136.
Donley, M. G., and Spanos, P. D., 1990, Dynamic Analysis of Nonlinear Structures by the Method of Statistical Quadratization (Lecture Notes in Engineering, Vol. 37 ), Springer, New York, pp. 1–172.
Spanos,  P. D., and Donley,  M. G., 1991, “Equivalent Statistical Quadratization for Nonlinear System,” J. Eng. Mech., 117, pp. 1289–1310.
Spanos,  P. D., and Donley,  M. G., 1992, “Nonlinear Multi-Degree-of-Freedom System Random Vibration by Equivalent Statistical Quadratization,” Int. J. Non-Linear Mech., 27, pp. 735–748.
Tognarelli,  M. A., Zhao,  J., Rao,  K. B., and Kareem,  A., 1997, “Equivalent Statistical Quadratization and Qubicization for Nonlinear System,” J. Eng. Mech., 123, pp. 512–523.
Quek,  S. T., Li,  X. M., and Koh,  C. G., 1994, “Stochastic Response of Jack-Up Platform by the Method for Statistical Quadratization,” Appl. Ocean. Res., 16, pp. 113–122.
Li,  X. M., Quek,  S. T., and Koh,  C. G., 1995, “Stochastic Response of Offshore Platform by Statistical Cubization,” J. Eng. Mech., 121, pp. 1056–1068.
Fatica,  G., and Floris,  C., 2003, “Moment Equation Analysis of Base-isolated Buldings Subjected to Support Motion,” J. Eng. Mech., 129, pp. 94–106.
Kareem,  A., and Zhao,  J., 1994, “Analysis of Non-Gaussian Surge Response of Tension Leg Platforms Under Wind Loads,” J. Offshore Mech. Arctic Eng., 116, pp. 137–144.
Kareem,  A., Zhao,  J., and Tognarelli,  M. A., 1995, “Surge Response Statistics of Tension Leg Platforms Under Wind and Wave Loads: Statistical Quadratization Approach,” Probab. Eng. Mech., 10, pp. 225–240.
Bedrosian,  E., and Rice,  S., 1971, “The Outpost Properties of Volterra Systems (Nonlinear Systems With Memory) Driven by Harmonic and Gaussian Inputs,” Proc. IEEE, 59, pp. 1688–1707.
Tognarelli,  M. A., Zhao,  J., and Kareem,  A., 1997, “Equivalent Statistical Qubicization for System and Forcing Nonlinearization,” J. Eng. Mech., 123, pp. 890–893.
Spanos,  P. D., Di Paola,  M., and Failla,  G., 2002, “A Galerkin Approach for Power Spectrum Determination for Nonlinear Oscillators,” Meccanica, 37, pp. 51–65.
Krenk,  S., and Roberts,  J. B., 1999, “Local Similarity in Nonlinear Random Vibration,” ASME J. Appl. Mech., 66, pp. 225–235.
Rudinger,  F., and Krenk,  S., 2003, “Spectral Density of an Oscillator With Power Law Damping Excited by White Noise,” J. Sound Vib., 261, pp. 365–371.
Caughey,  T. K., 1986, “On the Response of Nonlinear Oscillators to Stochastic Excitation,” Probab. Eng. Mech., 1, pp. 2–4.
Chang,  R. J., 1991, “A Practical Technique for Spectral Analysis of Nonlinear Systems Under Stochastic Parametric and External Excitations,” ASME J. Vibr. Acoust., 113, pp. 516–522.
Young,  G. E., and Chang,  R. J., 1987, “Prediction of the Response of Nonlinear Oscillators Under Stochastic Parametric and External Excitations,” Int. J. Non-Linear Mech., 22, pp. 151–160.
Chang,  R. J., 1992, “Non-Gaussian Linearization Method for Stochastic Parametrically and Externally Excited Nonlinear Systems,” ASME J. Dyn. Syst., Meas., Control, 114, pp. 20–26.
Sobiechowski, C., and Sperling, L., 1996, “An Iterative Statistical Linearization Technique for MDOF Systems,” EUROMECH-2nd European Nonlinear Oscillation Conference, L. Pust and F. Peterka, eds., Prague, September 9–13, pp. 419–422.
Sperling,  L., 1986, “Approximate Analysis of Nonlinear Stochastic Differential Equations Using Certain Generalized Quasi-Moment Functions,” Acta Mech., 59, pp. 183–200.
Falsone, G., 1991, “Stochastic Linearization for the Response of MDOF Systems Subjected to External and Parametric Gaussian Excitations,” Computational Stochastic Mechanics, P. D. Spanos and C. A. Brebbia, eds., Computational Mechanics Publication and Elsevier Applied Science, London, pp. 303–314.
Falsone,  G., 1992, “Stochastic Linearization for the Response of MDOF Systems Under Parametric Gaussian Excitations,” Int. J. Non-Linear Mech., 27, pp. 1025–1037.
Soize,  C., 1995, “Stochastic Linearization Method With Random Parameters for SDOF Nonlinear Dynamical Systems: Prediction and Identification Procedures,” Probab. Eng. Mech., 10, pp. 143–152.
Soize,  C., and Le Fur,  O., 1997, “Modal Identification of Weakly Nonlinear Multidimensional Dynamical Systems Using Stochastic Linearization Method With Random Coefficients,” Mech. Syst. Signal Process., 11, pp. 37–49.
Naprstek,  J., and Fischer,  O., 1997, “Spectral Properties of a Nonlinear Self-Excited System With Random Perturbations of the Parameters,” Z. Angew. Math. Mech., 77, S1, pp. 241–242.
Tylikowski,  A., and Marowski,  W., 1986, “Vibration of a Nonlinear Single Degree of Freedom System Due to Poissonian Impulse Excitation,” Int. J. Non-Linear Mech., 21, pp. 229–238.
Grigoriu, M., 1995, Applied Non-Gaussian Processes: Examples, Theory, Simulation, Linear Random Vibration, and MATLAB Solutions, Prentice-Hall, Englewood Cliffs, NJ.
Grigoriu,  M., 1995, “Linear and Nonlinear Systems With Non-Gaussian White Noise Input,” Probab. Eng. Mech., 10, pp. 171–180.
Grigoriu,  M., 1995, “Equivalent Linearization for Poisson White Noise Input,” Probab. Eng. Mech., 10, pp. 45–51.
Grigoriu,  M., 2000, “Equivalent Linearization for Systems Driven by Levy White Noise,” Probab. Eng. Mech., 15, pp. 185–190.
Sobiechowski, C., and Socha, L., 1998, “Statistical Linearization of the Duffing Oscillator Under Non-Gaussian External Excitation,” Proc. of Int. Conference on Computational Stochastic Mechanics, P. D. Spanos, ed., Balkema, Rotterdam, pp. 125–133.
Sobiechowski,  C., and Socha,  L., 2000, “Statistical Linearization of the Duffing Oscillator Under Non-Gaussian External Excitation,” J. Sound Vib., 231, pp. 19–35.
Atalik,  T. S., and Utku,  S., 1976, “Stochastic Linearization of Multi-Degree-of-Freedom Nonlinear Systems,” Earthquake Eng. Struct. Dyn., 4, pp. 411–420.
Grigoriu,  M., and Ariaratnam,  S., 1988, “Response of Linear Systems to Polynomials of Gaussian Process,” ASME J. Appl. Mech., 55, pp. 905–910.
Krenk, S., and Gluver, H., 1988, “An Algorithm for Moments of Response From Non-Normal Excitation of Linear Systems,” Stochastic Structural Dynamics, S. T. Ariaratnam, G. I. Schueller and I. Elishakoff, eds., Elsevier, London, pp. 181–195.
Muscolino,  G., 1995, “Linear Systems Excited by Polynomial Forms of Non-Gaussian Filtered Process,” Probab. Eng. Mech., 10, pp. 35–44.
Iyengar,  R. N., and Jaiswal,  O. R., 1993, “A New Model for non-Gaussian Random Excitations,” Probab. Eng. Mech., 8, pp. 281–287.
Proppe,  C., 2002, “Equivalent Linearization of MDOF Systems Under External Poisson White Noise Excitation,” Probab. Eng. Mech., 17, pp. 393–399.
Proppe,  C., 2003, “Stochastic Linearization of Dynamical Systems Under Parametric Poisson White Noise Excitation,” Int. J. Non-Linear Mech., 38, pp. 543–555.
Hou, Z., Noori, M. N., Wang, Y., and Duval, L., 1999, “Dynamic Behavior of Duffing Oscillators Under a Disordered Periodic Excitation,” Stochastic Structural Dynamics, B. F. Spencer and E. A. Johnson, eds., Balkema, Rotterdam, pp. 93–98.
Sobiechowski,  C., 1999, “Statistical Linearization of Dynamical Systems Under Parametric Delta-Correlated Excitation,” Z. Angew. Math. Mech., 79, S2, pp. 315–316.
Falsone, G., and Pirrotta, A., 1995, “A New Stochastic Linearization Approach,” Computational Stochastic Mechanics, P. D. Spanos, ed., A. A. Balkema, Rotterdam, pp. 105–112.
Di Paola,  M., and Falsone,  G., 1993, “Stochastic Dynamics of Nonlinear Systems Driven by Non-Normal Delta-Correlated Processes,” ASME J. Appl. Mech., 60, pp. 141–148.
Iwan,  W. D., and Krousgrill,  R. G., 1983, “Equivalent Linearization for Continuous Dynamical Systems,” ASME J. Appl. Mech., 50, pp. 415–420.
Iwan,  W. D., and Whirley,  R. G., 1993, “Nonstationary Equivalent Linearization of Nonlinear Continuous Systems,” Probab. Eng. Mech., 8, pp. 273–280.
Sireteanu, T., 1996, “Effect of System Nonlinearities Obtained by Statistical Linearization Methods,” EUROMECH-2nd European Nonlinear Oscillation Conference, L. Pust and F. Peterda, eds., Prague, pp. 415–417.
Lin,  V. H. L., and Cheng,  V. H. L., 1991, “Statistical Linearization for Multi-Input/Multi-Output Nonlinearities,” J. Guid. Control, 14, pp. 1315–1318.
Vasta,  M., and Schueller,  G. I., 2000, “Phase Space Reduction in Stochastic Dynamics,” J. Eng. Mech., 126, pp. 626–632.
Smyth,  A. W., and Masri,  S. F., 2002, “Nonstationary Response of Nonlinear Systems Using Equivalent Linearization With a Compact Analytical Form of the Excitation Process,” Probab. Eng. Mech., 17, pp. 97–108.
Ohtori,  T., and Spencer,  B. F., 2002, “Semi-Implicit Integration Algorithm for Stochastic Analysis of Multi-Degree-of-Freedom Structures,” J. Eng. Mech., 128, pp. 635–643.
Chernyshev,  K. R., 2002, “Using Informational Measures of Dependence in Statistical Linearization,” Autom. Remote Control (Engl. Transl.), 63, pp. 1439–1447.
Grigoriu,  M., 1991, “Statistically Equivalent Solutions of Stochastic Mechanics Problems,” J. Eng. Mech., 117, pp. 1906–1918.
Socha,  L., 1999, “Poli-Criterial Equivalent Linearization Technique,” Z. Angew. Math. Mech., 78(SII), pp. 317–318.
Silva,  F. L., and Ruiz,  S. E., 2000, “Calibration of the Equivalent Linearization Gaussian Approach Applied to Simple for Hysteretic Systems Subjected to Narrow Band Seismic Motions,” Struct. Safety, 22, pp. 211–231.
Foliente,  G. C., Singh,  M. P., and Noori,  M. N., 1996, “Equivalent Linearization of Generally Pinching Hysteretic, Degrading Systems,” Earthquake Eng. Struct. Dyn., 25, pp. 611–629.
Dobson,  S., Noori,  M., Hou,  Z., and Dimentberg,  M., 1998, “Direct Implementation of Stochastic Linearization for SDOF Systems with General Hysteresis,” Struct. Engrg. Mech., 6, pp. 473–484.
Yan,  X., and Nie,  J., 2000, “Response of SMA Superelastic Systems Under Random Excitation,” J. Sound Vib., 238, pp. 893–901.
Kimura,  K., Yasumuro,  H., and Sakata,  M., 1994, “Non-Gaussian Equivalent Linearization for Non-Stationary Random Vibration of Hysteretic Systems,” Probab. Eng. Mech., 9, pp. 15–22.
Takewaki,  I., 2002, “Critical Excitation for Elastic-Plastic Structures Via Statistical Equivalent Linearization,” Probab. Eng. Mech., 17, pp. 73–84.
Hurtado,  J. E., and Barbat,  A. H., 1996, “Improved Stochastic Linearization Method Using Mixed Distributions,” Struct. Safety, 18, pp. 49–62.
Hurtado,  J. E., and Barbat,  A. H., 2000, “Equivalent Linearization of the Bouc-Wen Hysteretic Model,” Eng. Struct., 22, pp. 1121–1132.
Spanos,  P. D., and Tsavachidis,  S., 2001, “Deterministic and Stochastic Analyses of Nonlinear System With a Biot Visco-Elastic Elements,” Earthquake Eng. Struct. Dyn., 30, pp. 595–612.
Ni,  Y. Q., Ying,  Z. G., and Zhu  W. Q., 2002, “Random Response of Integrable Duhem Hysteretic Systems Under Non-White Excitation,” Int. J. Non-Linear Mech., 37, pp. 1407–1419.
Bouc,  R., and Boussaa,  D., 2002, “Drifting Response of Hysteretic Oscillators to Stochastic Excitation,” Int. J. Non-Linear Mech., 37, pp. 1397–1406.
Ying,  Z. G., 2003, “Response Analysis of Randomly Excited Nonlinear Systems With Symmetric Weighting Preisach Hysteresis,” Acta Mech. Sin., 19, pp. 365–370.
Pradlwarter,  H. J., and Li,  W., 1991, “On the Computations of the Stochastic Response of Highly Nonlinear Large MDOF-Systems Modeled by Finite Elements,” Probab. Eng. Mech., 6, pp. 109–116.
Locke,  J. E., 1994, “Finite-Element Nonlinear Random Response of Beams,” J. Sound Vib., 178, pp. 201–210.
Muravyov,  A. A., and Rizzi,  S. A., 2003, “Determination of Nonlinear Stiffness With Application to Random Vibration of Geometrically Nonlinear Structures,” Comput. Struct., 15, pp. 1513–1523.
Chen,  R. P., Mei,  C., and Wolfe,  H. F., 1996, “Comparison of Finite Element Nonlinear Beam Random Response With Experimental Results,” J. Sound Vib., 195, pp. 719–737.
Cheng,  G. F., Lee,  Y. Y., and Mei,  C., 2003, “Nonlinear Random Response of Internally Hinged Beams,” Finite Elem. Anal. Design, 39, pp. 487–504.
Schueller,  G. I., and Pradlwarter,  H. J., 1999, “On the Stochastic Response of Nonlinear FE Models,” Arch. Appl. Mech., 69, pp. 765–784.
Emam, H. H., Pradlwarter, H. J., and Schueller, G. I., 1999, “On the Computational Implementation of EQL in FE-Analysis,” Stochastic Structural Dynamics, B. F. Spencer and E. A. Johnson, eds., Balkema, Rotterdam, pp. 85–91.
Emam,  H. H., Pradlwarter,  H. J., and Schueller,  G. I., 2000, “A Computational Procedure for Implementation of Equivalent Linearization in Finite Element Analysis,” Earthquake Eng. Struct. Dyn., 29, pp. 1–17.
Pradlwarter,  H. J., Schueller,  G. I., and Schenk,  C. A., 2003, “A Computational Procedure to Estimate the Stochastic Dynamic Response of Large Nonlinear FE-Models,” Comput. Methods Appl. Mech. Eng., 192, pp. 777–801.
Pradlwarter,  H. J., 2002, “Deterministic Integration Algorithms for Stochastic Response Computations of FE Systems,” Comput. Struct., 80, pp. 1489–1502.
To,  C. W. S., 1986, “The Stochastic Central Difference Method in Structural Dynamics,” Comput. Struct., 23, pp. 813–818.
To,  C. W. S., 1988, “Recursive Expression for Random Response of Nonlinear Systems,” Comput. Struct., 29, pp. 451–457.
To,  C. W. S., and Liu,  M. L., 1993, “Recursive Expression for Time Dependent Means and Mean Square Responses of a Multi-Degree-of-Freedom Nonlinear Systems,” Comput. Struct., 48, pp. 993–1000.
Wonham,  W. M., and Cashman,  W. F., 1969, “A Computational Approach to Optimal Control of Stochastic Saturating Systems,” Int. J. Control, 10, pp. 77–98.
Kwakernaak, H., and Sivan, R., 1972, Linear Optimal Control Systems, Wiley, New York.
Beaman,  J. J., 1984, “Nonlinear Quadratic Gaussian Control,” Int. J. Control, 39, pp. 343–361.
Kazakov,  I. E., 1984, “Analytical Synthesis of a Quasi-Optimal Additive Control in a Nonlinear Stochastic System,” Avtom. Telemekh., 45, pp. 34–46.
Yoshida,  K., 1984, “A Method of Optimal Control of Nonlinear Stochastic Systems With Non-Quadratic Criteria,” Int. J. Control, 39, pp. 279–291.
Gokcek,  C., Kabamba,  P. T., and Meerkov,  S. M., 2000, “Disturbance Rejection in Control Systems With Saturating Actuators,” Nonlinear Analysis, 40, pp. 213–226.
Gokcek,  C., Kabamba,  P. T., and Meerkov,  S. M., 2000, “Optimization of Disturbance Rejection in Systems With Saturating Actuators,” J. Math. Anal. Appl., 249, pp. 135–159.
Gokcek,  C., Kabamba,  P. T., and Meerkov,  S. M., 2001, “An LQR/LQG Theory for Systems With Saturating Actuators,” IEEE Trans. Autom. Control, 46, pp. 1529–1542.
Han,  S. I., and Kim,  J. S., 2003, “Nonlinear Quadratic Gaussian Control With Loop Transfer Recovery,” Mechatronics, 13, pp. 273–293.
Kazakov,  I. E., 1994, “Gaussian Approximation When Optimizing a Control in a Stochastic Nonlinear System,” J. Comp. Systems Sci. Int., 32, pp. 118–122.
Kazakov,  I. E., 1995, “Optimization of Control in a Nonlinear Stochastic System by a Local Criterion,” J. Comp. Systems Sci. Int., 35, pp. 940–947.
Kazakov,  I. E., and Proppe,  C., “Analytical Construction of a Conditionally Optimal Control in a Nonlinear Stochastic System by a Complex Local Functional,” Avtom. Telemekh., 56, pp. 34–46.
Yang,  J. N., Li,  Z., and Vongchavalitkul,  S., 1994, “Stochastic Hybrid Control of Hysteretic Structures,” Probab. Eng. Mech., 9, pp. 125–133.
Suzuki,  Y., 1995, “Stochastic Control of Hysteretic Structural Systems,” Sadhana: Proc., Indian Acad. Sci., 20, pp. 475–488.
Raju,  G. V., and Narayanan,  S., 1995, “Active Control of Nonstationary Response of a 2-Degree of Freedom Vehicle Model With Nonlinear Suspension,” Sadhana: Proc., Indian Acad. Sci., 20, pp. 489–499.
Narayanan,  S., and Senthil,  S., 1998, “Stochastic Optimal Active Control of a 2-DOF Quarter Car Model With Nonlinear Passive Suspension Elements,” J. Sound Vib., 211, pp. 495–506.
Socha,  L., 2000, “Application of Statistical Linearization Techniques to Design of Quasi-Optimal Active Control of Nonlinear Systems,” J. Theor. Appl. Mech., 38, pp. 591–605.
Socha,  L., and Proppe,  C., 2002, “Control of the Duffing Oscillator Under Non-Gaussian External Excitation,” Eur. J. Mech. A/Solids, 6, pp. 1069–1082.
Koliopulos,  P. K., and Langley,  R. S., 1993, “Improved Stability Analysis of the Response of a Duffing Oscillator Under Filtered White Noise,” Int. J. Non-Linear Mech., 28, pp. 145–155.
Socha,  L., and Soong,  T. T., 1992, “Sensitivity and Linearization Techniques in Analysis of Nonlinear Stochastic Systems,” J. Sound Vib., 156, pp. 79–97.
Proppe, C., and Schueller, G. I., 2000, “Equivalent Linearization Revisited,” Proc. of Int. Conference on Advances in Structural Dynamics, J. M. Ko and Y. L. Xu, eds., Hong Kong, Dec. 13–15, Elsevier, New York, pp. 1207–1214.
Socha, L., and Zasucha, G., 1991, “The Sensitivity Analysis of Stochastic Hysteretic Dynamic Systems,” Computational Stochastic Mechanics, P. D. Spanos and C. A. Brebbia, eds., Computational Mechanics Publication and Elsevier Applied Science, London, pp. 71–79.
Huang,  C. D., and Soong,  T. T., 1994, “Stochastic Sensitivity Analysis of Nonlinear Primary-Secondary Structural Systems,” Eng. Struct., 16, pp. 91–96.
Park,  Y. J., 1994, “Equivalent Linearization for Seismic Responses. I: Formulation and Error Analysis,” J. Eng. Mech., 118, pp. 2207–2226.
Micaletti,  R. C., Cakmak,  A. S., Nielsen,  S. R. K., and Koyluoglu,  H. U., 1998, “Error Analysis of Statistical Linearization With Gaussian Closure for Large-Degree-of-Freedom Systems,” Probab. Eng. Mech., 13, pp. 77–84.
Skrzypczyk,  J., 1994, “Accuracy Analysis of Statistical Linearization Methods Applied to Nonlinear Continuous Systems Described by Random Integral Equations,” J. Theor. Appl. Mech., 32, pp. 841–865.
Skrzypczyk,  J., 1995, “Accuracy Analysis of Statistical Linearization Methods Applied to Nonlinear Dynamical Systems,” Rep. Math. Phys., 36, pp. 1–20.
Casciati,  F., and Venini  P., 1994, “Equivalent Linearization for Seismic Responses, I: Formulation and Error Analysis-Discussion,” J. Eng. Mech., 120, pp. 676–678.

Figures

Tables

Errata

Discussions

Some tools below are only available to our subscribers or users with an online account.

Related Content

Customize your page view by dragging and repositioning the boxes below.

Related Journal Articles
Related eBook Content
Topic Collections

Sorry! You do not have access to this content. For assistance or to subscribe, please contact us:

  • TELEPHONE: 1-800-843-2763 (Toll-free in the USA)
  • EMAIL: asmedigitalcollection@asme.org
Sign In