Skip to main content
SpringerLink
Log in

Statistical Linearization of Stochastic Dynamic Systems Under External Excitations

  • Chapter
Book cover Linearization Methods for Stochastic Dynamic Systems

Part of the book series: Lecture Notes in Physics ((LNP,volume 730))

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

Log in via an institution
Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 39.99
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 54.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info
Hardcover Book
USD 54.99
Price excludes VAT (USA)
  • Durable hardcover edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. Ahmadi, G.: Mean square response of a Duffing oscillator to a modulated white noise excitation by the generalized method of equivalent linearization. J. Sound Vib. 71, 9–15, 1980.

    Article  MATH  Google Scholar 

  2. Alexandrov, V., et al.: Study of the accuracy of nonlinear nonstationary systems by the method of statistical linearization. Avtomatika i Telemekhanika 24, 488–495, 1965.

    Google Scholar 

  3. Anh, N. and Schiehlen, W.: A technique for obtaining approximate solutions in Gaussian equivalent linearization. Comput. Methods Appl. Mech. Eng. 168, 113–119, 1999.

    Article  MathSciNet  MATH  Google Scholar 

  4. Beaman, J. and Hedrick, J.: Improved statistical linearization for analysis of control of nonlinear stochastic systems: Part 1: An extended statistical linearization method. Trans. ASME J. Dyn. Syst. Meas. Control 101, 14–21, 1981.

    Article  MathSciNet  Google Scholar 

  5. Booton, R. C.: The analysis of nonlinear central systems with random inputs. IRE Trans. Circuit Theory 1, 32–34, 1954.

    Google Scholar 

  6. Cai, G. and Suzuki, Y.: On statistical quasi-linearization. Int. J. Non-Linear Mech. 40, 1139–1147, 2005.

    Article  MATH  Google Scholar 

  7. Cheded, L.: Invariance property: Higher-order extension and application to Gaussian random variables. Signal Process. 83, 1545–1551, 2003.

    Article  MATH  Google Scholar 

  8. Conte, J. and Peng, B.: An explicit closed form solution for linear systems subjected to nonstationary random excitations. Probab. Eng. Mech. 11, 37–50, 1996.

    Article  Google Scholar 

  9. Crandall, S.: On using non-Gaussian distributions to perform statistical linearization. In: W. Zhu, G. Cai, and R. Zhang (eds.), Advances in Stochastic Structural Dynamics, pp. 49–62. CRC Press, Boca Raton, 2003.

    Google Scholar 

  10. Crandall, S.: On using non-Gaussian distributions to perform statistical linearization. Int. J. Non-Linear Mech. 39, 1395–1406, 2004.

    Article  MATH  Google Scholar 

  11. Csaki, F.: Nonlinear Optimal and Adaptive Systems. Akademiai Kiado, Budapest, 1972 (in Russian).

    Google Scholar 

  12. Davis, G. and Spanos, P.: Developing force-limited random vibration test specifications for nonlinear systems using the frequency-shift method. In: Proc. 40th AIAA/SDM Conference, pp. 1688–1698, St.Louis, MO, 4/12-15/99, 1999.

    Google Scholar 

  13. Dimentberg, M.: Vibration of system with nonlinear cubic characteristic under narrow band random excitation. Izw.AN SSSR, MTT 2, 1971 (in Russian).

    Google Scholar 

  14. Elishakoff, I.: Method of stochastic linearization: Revisited and improved. In: S. P. D. and C. Brebbia (eds.), Computational Stochastic Mechanics, pp. 101–111. Computational Mechanics Publication and Elsevier Applied Science, London, 1991.

    Google Scholar 

  15. Elishakoff, I.: Some results in stochastic linearization of nonlinear systems. In: K. W. H. and N. Namachchivaya (eds.), Nonlinear Dynamics and Stochastic Mechanics, pp. 259–281. CRC Press, Boca Raton, 1995.

    Google Scholar 

  16. Elishakoff, I.: Multiple combinations of the stochastic linearization criteria by the moment approach. J. Sound Vib. 237, 550–559, 2000.

    Article  MathSciNet  Google Scholar 

  17. Elishakoff, I. and Bert, C.: Complementary energy criterion in nonlinear stochastic dynamics. In: R. Melchers and M. Steward (eds.), Application of Stochastic and Probability, pp. 821–825. A. A. Balkema, Rotterdam, 1999.

    Google Scholar 

  18. Elishakoff, I. and Colombi, P.: Successful combination of the stochastic linearization and Monte Carlo methods. J. Sound Vib. 160, 554–558, 1993.

    Article  MATH  Google Scholar 

  19. Elishakoff, I. and Falsone, G.: Some recent developments in stochastic linearization method. In: A.-D. Cheng and C. Yang (eds.), Computational Stochastic Mechanics, pp. 175–194. Computational Mechanics Publication, Southampton, New York, 1993.

    Google Scholar 

  20. Elishakoff, I., Fang, J., and Caimi, R.: Random vibration of a nonlinearly deformed beam by a new stochastic linearization method. Int. J. Solids Struct. 32, 1571–1584, 1995.

    Article  MathSciNet  MATH  Google Scholar 

  21. Elishakoff, I. and Zhang, R.: Comparison of the new energy-based version of the stochastic linearization method. In: N. Bellomo and F. Casciati (eds.), Nonlinear Stochastic Mechanics, pp. 201–212. Springer, Berlin, 1992.

    Google Scholar 

  22. Elishakoff, I. and Zhang, X.: An appraisal of different stochastic linearization criteria. J. Sound Vib. 153, 370–375, 1992.

    Article  MathSciNet  MATH  Google Scholar 

  23. Evlanov, L. and Kazakov, I. E.: Statistical investigation of nonlinear auto-oscillatory systems with non-steady state modes. Avtomatika i Telemekhanika 31, 45–53, 1970.

    Google Scholar 

  24. Evlanov, L. and Kazakov, I. E.: Statistical investigation of nonlinear oscillatory systems in the steady states modes. Avtomatika i Telemekhanika 30, 1902–1910, 1969.

    Google Scholar 

  25. Falsone, G.: An extension of the Kazakov relationship for non-Gaussian random variables and its use in the non-linear stochastic dynamics. Probab. Eng. Mech. 20, 45–56, 2005.

    Article  Google Scholar 

  26. Falsone, G. and Elishakoff, I.: Modified stochastic linearization method for coloured noise excitation of Duffing oscillator. Int. J. Non-Linear Mech. 29, 65–69, 1994.

    Article  MathSciNet  MATH  Google Scholar 

  27. Fang, J., Elishakoff, I., and Caimi, R.: Nonlinear response of a beam under stationary random excitation by improved stochastic linearization method. Appl. Math. Modeling 19, 106–111, 1995.

    Article  MATH  Google Scholar 

  28. Ferrante, F., Arwade, S., and Graham-Brady, L.: A translation model for non-Gaussian random processes. Probab. Eng. Mech. 20, 215–228, 2005.

    Article  Google Scholar 

  29. Friedrich, H.: On a method of equivalent statistical linearization. J. Tech. Phys. 16, 205–212, 1975.

    MATH  Google Scholar 

  30. Grigoriu, M.: Equivalent linearization for Poisson white noise input. Probab. Eng. Mech. 10, 45–51, 1995.

    Article  Google Scholar 

  31. Grigoriu, M.: Linear and nonlinear systems with non-Gaussian white noise input. Probab. Eng. Mech. 10, 171–180, 1995.

    Article  Google Scholar 

  32. Grigoriu, M.: Parametric models of nonstationary Gaussian processes. Probab. Eng. Mech. 10, 95–102, 1995.

    Article  Google Scholar 

  33. Grigoriu, M.: Applied non-Gaussian Processes: Examples, Theory, Simulation, Linear Random Vibration, and MATLAB Solutions. Prentice Hall, Englewood Cliffs, 1995.

    MATH  Google Scholar 

  34. Grigoriu, M.: Equivalent linearization for systems driven by Levy white noise. Probab. Eng. Mech. 15, 185–190, 2000.

    Article  Google Scholar 

  35. Gurley, K., Kareem, A., and Tognarelli, M.: Simulation of a class of non-normal random process. Int. J. Non-Linear Mech. 31, 601–617, 1996.

    Article  MATH  Google Scholar 

  36. Hou, Z., Noori, M., Wang, Y., and Duval, L.: Dynamic behavior of Duffing oscillators under a disordered periodic excitation. In: B. Spencer and E. Johnson (eds.), Stochastic Structural Dynamics, pp. 93–98. Balkema, Rotterdam, 1999.

    Google Scholar 

  37. Indira, R., et al.: Diffusion in a bistable potential: A comparartive study of different methods of solution. J. Stat. Phys. 33, 181–194, 1983.

    Article  Google Scholar 

  38. Isserlis, L.: On a formula for the product-moment coefficient in any number of variables. Biometrica 12, 134–139, 1918.

    Google Scholar 

  39. Iwan, W. and Mason, A.: Equivalent linearization for systems subjected to nonstationary random excitation. Int. J. Non-Linear Mech. 15, 71–82, 1980.

    Article  MathSciNet  MATH  Google Scholar 

  40. Iyengar, R.: Response of nonlinear systems to narrow band excitation. Struct. Safety 6, 177–185, 1989.

    Article  Google Scholar 

  41. Iyengar, R. N. and Jaiswal, O. R.: A new model for non-Gaussian random excitations. Probab. Eng. Mech. 8, 281–287, 1993.

    Article  Google Scholar 

  42. Jahedi, A. and Ahmadi, G.: Application of Wiener-Hermite expansion to nonstationary random vibration of a Duffing oscillator. Trans. ASME J. Appl. Mech. 50, 436–442, 1987.

    MathSciNet  Google Scholar 

  43. Jamshidi, M.: An overview on the solutions of the algebraic matrix Riccati equation and related problems. Large Scale Syst. 1, 167–192, 1980.

    MathSciNet  MATH  Google Scholar 

  44. Kazakov, I. E.: An approximate method for the statistical investigation for nonlinear systems. Trudy VVIA im Prof. N. E. Zhukovskogo 394, 1–52, 1954 (in Russian).

    Google Scholar 

  45. Kazakov, I. E.: Approximate probabilistic analysis of the accuracy of the operation of essentially nonlinear systems. Avtomatika i Telemekhanika 17, 423–450, 1956.

    MathSciNet  Google Scholar 

  46. Kazakov, I. E.: Statistical analysis of systems with multi-dimensional nonlinearities. Avtomatika i Telemekhanika 26, 458–464, 1965.

    MathSciNet  Google Scholar 

  47. Kazakov, I. E.: An extension of the method of statistical linearization. Avtomatika i Telemekhanika 59, 220–224, 1998.

    MathSciNet  Google Scholar 

  48. Kazakov, I.: Statistical Theory of Control Systems in State Space. Nauka, Moskwa, 1975 (in Russian).

    Google Scholar 

  49. Kazakov, I. and Dostupov, B.: Statistical Dynamic of Nonlinear Automatic Systems. Fizmatgiz, Moskwa, 1962 (in Russian).

    Google Scholar 

  50. Kimura, K. and Sakata, M.: Non-stationary responses of a non-symmetric non-linear system subjected to a wide class of random excitation. J. Sound Vib. 76, 261–272, 1981.

    Article  MATH  Google Scholar 

  51. Lee, J.: Improving the equivalent linearization method for stochastic Duffing oscillators. J. Sound Vib. 186, 846–855, 1995.

    Article  MATH  Google Scholar 

  52. Legras, J.: Méthodes et Techniques de l’analyse numérique. Dudon, Paris, 1971.

    Google Scholar 

  53. Lyon, R., Heckl, M., and Hazlegrove, C.: Narrow band excitation of the hard-spring oscillator. J. Acoust. Soc. Am. 33, 1404–1411, 1961.

    Article  Google Scholar 

  54. Masri, S., Smyth, A., and Traina, M.: Probabilistic representation and transmission of nonstationary processes in multi-degree-of-freedom systems. Trans. ASME J. Appl. Mech. 65, 398–409, 1998.

    Article  Google Scholar 

  55. Mel’ts, I., Pukhova, T., and Uskov, G.: Multidimensional statistical linearization of functions containing factors of the power, exponential and trigonometric types and also δ functions. Avtomatika i Telemekhanika 28, 1871–1880, 1967.

    Google Scholar 

  56. Morosanov, J.: Practical methods for the calculation of the linearization factors for arbitrary nonlinearities. Avtomatika i Telemekhanika 29, 81–86, 1968.

    Google Scholar 

  57. Naess, A., Galeazzi, F., and Dogliani, M.: Extreme response predictions of nonlinear compliant offshore structures by stochastic linearization. Appl. Ocean Res. 14, 71–81, 1992.

    Article  Google Scholar 

  58. Naumov, B.: The Theory of Nonlinear Automatic Control Systems. Frequency Methods. Nauka, Moscow, 1972 (in Russian).

    Google Scholar 

  59. Orabi, I. and Ahmadi, G.: A functional series expansion method for response analysis of nonlinear random systems subjected to random excitations. Int. J. Nonlinear Mech. 22, 451–465, 1987.

    Article  MathSciNet  MATH  Google Scholar 

  60. Orabi, I. and Ahmadi, G.: An iterative method for nonstationary response analysis of nonlinear random systems. J. Sound Vib. 119, 145–157, 1987.

    Article  MathSciNet  Google Scholar 

  61. Orabi, I. and Ahmadi, G.: Nonstationary response analysis of a Duffing oscillator by the Wiener-Hermite expansion method. Trans. ASME J. Appl. Mech. 54, 434–440, 1987.

    MathSciNet  MATH  Google Scholar 

  62. Pervozwanskii, A.: Random Processes in Nonlinear Automatic Control Systems. Fizmatgiz, Moskwa, 1962 (in Russian).

    Google Scholar 

  63. Pradlwarter, H.: Nonlinear stochastic response distributions by local statistical linearization. Int. J. Non-Linear Mech. 36, 1135–1151, 2001.

    Article  MathSciNet  MATH  Google Scholar 

  64. Priestly, M.: Spectral Analysis and Time Series, 5th edn. Academic Press, New York, 1981.

    Google Scholar 

  65. Pugachev, V. and Sinitsyn, I.: Stochastic Differential Systems. Wiley, Chichester, 1987.

    MATH  Google Scholar 

  66. Pugachev, V.: Theory of Random Functions. Gostkhizdat, Moskwa, 1957 (in Russian).

    Google Scholar 

  67. Ricciardi, G.: A non-Gaussian stochastic linearization method. Probab. Eng. Mech. 22, 1–11, 2007.

    Article  Google Scholar 

  68. Roberts, J. and Spanos, P.: Random Vibration and Statistical Linearization. John Wiley and Sons, Chichester, 1990.

    MATH  Google Scholar 

  69. Romanov, V. and Khuberyan, B. K.: Approximate moment dynamics method using statistical linearization. Awtomatika i Telemechanika 37, 29–35, 1978.

    Google Scholar 

  70. Sakata, M. and Kimura, K.: Calculation of the non-stationary mean square response of a non-linear system subjected to non-white excitation. J. Sound Vib. 73, 333–343, 1980.

    Article  MATH  Google Scholar 

  71. Samorodnitsky, G. and Taqqu, M.: Stable non-Gaussian Random Processes, Stochastic Models with Infinite Variance. Chapman and Hall, London, 1994.

    Google Scholar 

  72. Sawargai, Y., Sugai, N., and Sunahara, Y.: Statistical Studies of Nonlinear Control Systems. Nippon Printing and Publishing Co., Osaka, 1962.

    Google Scholar 

  73. Sinitsyn, I.: Methods of statistical linearization (survey). Awtomatika i Telemechanika 35, 765–776, 1974.

    MathSciNet  Google Scholar 

  74. Smyth, A. and Masri, S.: Nonstationary response of nonlinear systems using equivalent linearization with a compact analytical form of the excitation process. Probab. Eng. Mech. 17, 97–108, 2002.

    Article  Google Scholar 

  75. Smyth, A. and Masri, S.: The robustness of an efficient probabilistic data-based tool for simulating the nonstationary response of nonlinear systems. Int. J. Non-Linear Mech. 39, 1453–1461, 2004.

    Article  MATH  Google Scholar 

  76. Sobiechowski, C.: Statistical linearization of dynamical systems under parametric delta-correlated excitation. Z. Angew. Math. Mech. 79, 315–316, 1999 S2.

    Google Scholar 

  77. Sobiechowski, C. and Socha, L.: Statistical linearization of the Duffing oscillator under non-Gaussian external excitation. In: P. D. Spanos (ed.), Proc. Int. Conf. on Computational Stochastic Mechanics, pp. 125–133. Balkema, Rotterdam, 1998.

    Google Scholar 

  78. Sobiechowski, C. and Socha, L.: Statistical linearization of the Duffing oscillator under non-Gaussian external excitation. J. Sound Vib. 231, 19–35, 2000.

    Article  MathSciNet  Google Scholar 

  79. Socha, L.: Equivalent linearization for dynamical systems excited by colored noise. Z. Angew. Math. Mech. 70, 738–740, 1990.

    MathSciNet  Google Scholar 

  80. Socha, L.: Moment Equations in Stochastic Dynamic Systems. PWN, Warszawa, 1993 (in Polish).

    Google Scholar 

  81. Socha, L.: Statistical and equivalent linearization methods with probability density criteria. J. Theor. Appl. Mech. 37, 369–382, 1999.

    MATH  Google Scholar 

  82. Socha, L.: Probability density statistical and equivalent linearization methods. Int. J. Syst. Sci. 33, 107–127, 2002.

    Article  MathSciNet  MATH  Google Scholar 

  83. Spanos, P.: Numerical simulations of Van der Pol oscillator. Comput. Math. Appl. 6, 135–145, 1980.

    Article  MathSciNet  MATH  Google Scholar 

  84. Steinwolf, A., Ferguson, N., and White, R.: Variations in steepness of probability density function of beam random vibration. Eur. J. Mech. A/Solids 19, 319–341, 2000.

    Article  MATH  Google Scholar 

  85. Steinwolf, A. and Ibrahim, R.: Numerical and experimental studies of linear systems subjected to non-Gaussian random excitations. Probab. Eng. Mech. 14, 289–299, 1999.

    Article  Google Scholar 

  86. Traina, M., R.K., M., and Masri, S.: Orthogonal decomposition and transmission of nonstationary random process. Probab. Eng. Mech. 1, 136–149, 1986.

    Google Scholar 

  87. Tylikowski, A. and Marowski, W.: Vibration of a non-linear single degree of freedom system due to Poissonian impulse excitation. Int. J. Non-Linear Mech. 21, 229–238, 1986.

    Article  MathSciNet  MATH  Google Scholar 

  88. Valsakumar, M., Murthy, K.P., and Ananthakrishna, G.: On the linearization of nonlinear langevin-type stochastic differential equations. J. Stat. Phys. 30, 617–631, 1983.

    Article  MATH  Google Scholar 

  89. Wang, G. and Dai, M.: 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, 947–955, 2001 (English edn).

    MATH  Google Scholar 

  90. Wicher, J.: On the coefficients of statistical linearization in E. D. Zaydenberg’s method. Nonlinear Vib. Probl. 12, 147–154, 1971.

    MathSciNet  MATH  Google Scholar 

  91. Zaydenberg, E. D.: A third mehod for the statistical linearization of a class of nonlinear differential equations. Avtomatika i Telemekhanika 25, 195–200, 1964.

    Google Scholar 

  92. Zhang, R.: Work/energy-based stochastic equivalent linearization with optimized power. J. Sound Vib. 230, 468–475, 2000.

    Article  Google Scholar 

  93. Zhang, R., Elishakoff, I., and Shinozuka, M.: Analysis of nonlinear sliding structures by modified stochastic linearization methods. Nonlinear Dyn. 5, 299–312, 1994.

    Article  Google Scholar 

  94. Zhang, X., Elishakoff, I., and Zhang, R.: A new stochastic linearization method based on minimum mean-square deviation of potential energies. In: Y. Lin and I. Elishakoff (eds.), Stochastic Structural Dynamics – New Theoretical Developments, pp. 327–338. Springer Verlag, Berlin, 1991.

    Google Scholar 

  95. Zhang, X. and Zhang, R.: Energy-based stochastic equivalent linearization with optimized power. In: B. Spencer and E. Johnson (eds.), Stochastic Structural Dynamics, pp. 113–117. Balkema, Rotterdam, 1999.

    Google Scholar 

  96. Zhang, X., Zhang, R., and Xu, Y.: Analysis on control of flow-induced vibration by tuned liquid damper with crossed tube-like containers. J. Wind Eng. Ind. Aerodyn. 50, 351–360, 1993.

    Article  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Faculty of Mathematics and Natural Sciences College of Sciences, Cardinal Stefan Wyszyński University in Warsaw, ul.Dewajtis 5, 01-815Warsaw, Poland

    Leslaw Socha

  2. Department of Mathematics, Physics and Chemistr, University of Silesia, ul.Akademicka 4, 40-007 Katowice, Poland

    Leslaw Socha

Authors
  1. Leslaw Socha
    View author publications

    You can also search for this author in PubMed Google Scholar

Rights and permissions

Reprints and permissions

Copyright information

© 2008 Springer-Verlag Berlin Heidelberg

About this chapter

Cite this chapter

Socha, L. (2008). Statistical Linearization of Stochastic Dynamic Systems Under External Excitations. In: Linearization Methods for Stochastic Dynamic Systems. Lecture Notes in Physics, vol 730. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-540-72997-6_5

Download citation

Publish with us

Policies and ethics

Search

Navigation