Abstract
In Chap. 3, the Wiener path integral formulation developed in Chap. 2 for first-order stochastic differential equations is extended to treat second-order stochastic differential equations modeling the dynamics of stochastically excited linear multi-degree-of-freedom structural systems. Specifically, considering Gaussian white noise excitation, it is shown that higher than second variations in the path integral functional expansion vanish. Thus, retaining only the first term (most probable path approximation) yields the exact response joint transition Gaussian probability density function. Two illustrative examples pertaining to a stochastically excited 2-degree-of-freedom linear oscillator and to a bending beam with Young’s modulus modeled as a stochastic field are considered to demonstrate the exact nature of the closed-form solutions.
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References
Caughey, T. K., and Stumpf, H. J. (1961). Transient response of a dynamic system under random excitation. Journal of Applied Mechanics, 28, 563–566.
Chaichian, M., and Demichev, A. (2001). Path integrals in physics, Vol. I: Stochastic processes and quantum mechanics. Institute of Physics Publishing.
Colet, P., Wio, H. S., and San Miguel, M. (1989). Colored noise: A perspective from a path-integral formalism. Physical Review A, 39, 6094.
Conte, J. P., and Peng, B.-F. (1996). An explicit closed-form solution for linear systems subjected to nonstationary random excitation. Probabilistic Engineering Mechanics, 11, 37–50.
Donoso, J. M., Salgado, J. J., and Soler, M. (1999). Short-time propagators for nonlinear Fokker-Planck equations. Journal of Physics A: Mathematical and General, 32, 3681.
Drozdov, A. N., and Talkner, P. (1998). Path integrals for Fokker–Planck dynamics with singular diffusion: Accurate factorization for the time evolution operator. The Journal of Chemical Physics, 109, 2080–2091.
Einchcomb, S. J. B., and McKane, A. J. (1995). Use of Hamiltonian mechanics in systems driven by colored noise. Physical Review E, 51, 2974.
Ewing, G. M. (1985). Calculus of variations with applications. Dover Publications.
Gelfand, I. M., and Fomin, S. V. (1963). Calculus of variations. Prentice Hall.
Grigoriu, M. (1997). Local solutions of laplace, heat, and other equations by Ito processes. Journal of Engineering Mechanics, 123, 823–829.
Grigoriu, M., and Papoulia, K. D. (2005). Effective conductivity by a probability-based local method. Journal of Applied Physics, 98, 033706.
Hänggi, P. (1989). Path integral solutions for non-Markovian processes. Zeitschrift für Physik B Condensed Matter, 75, 275–281.
Kougioumtzoglou, I. A. (2017). A Wiener path integral solution treatment and effective material properties of a class of one-dimensional stochastic mechanics problems. Journal of Engineering Mechanics, 143, 04017014.
Lu, T.-T., and Shiou, S.-H. (2002). Inverses of 2 \(\times \) 2 block matrices. Computers and Mathematics with Applications, 43, 119–129.
Machlup, S., and Onsager, L. (1953). Fluctuations and irreversible process. II. Systems with kinetic energy. Physical Review, 91, 1512.
McKane, A. J., Luckock, H. C., and Bray, A. J. (1990). Path integrals and non-Markov processes. I. General formalism. Physical Review A, 41, 644.
Newman, T. J., Bray, A. J., and McKane, A. J. (1990). Inertial effects on the escape rate of a particle driven by colored noise: An instanton approach. Journal of Statistical Physics, 59, 357–369.
Psaros, A. F., Petromichelakis, I., and Kougioumtzoglou, I. A. (2019). Wiener path integrals and multi-dimensional global bases for non-stationary stochastic response determination of structural systems. Mechanical Systems and Signal Processing, 128, 551–571.
Psaros, A. F., Zhao, Y., and Kougioumtzoglou, I. A. (2020). An exact closed-form solution for linear multi-degree-of-freedom systems under Gaussian white noise via the Wiener path integral technique. Probabilistic Engineering Mechanics 60, 103040.
Risken, H. (1984). The Fokker-Planck equation: Methods of solution and applications. Springer.
Roberts, J. B., and Spanos, P. D. (1990, 2003). Random vibration and statistical linearization. Wiley (1990); and Dover Publications (2003).
Seber, G. A. (2008). A matrix handbook for statisticians (Vol. 15). Wiley.
Shinozuka, M. (1987). Structural response variability. Journal of Engineering Mechanics, 113, 825–842.
Silvester, J. R. (2000). Determinants of block matrices. The Mathematical Gazette, 84, 460–467.
Wio, H. S. (2013). Path integrals for stochastic processes: An introduction. World Scientific.
Wio, H. S., Colet, P., San Miguel, M., Pesquera, L., and Rodriguez, M. A. (1989). Path-integral formulation for stochastic processes driven by colored noise. Physical Review A, 40, 7312.
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Kougioumtzoglou, I.A., Psaros, A.F., Spanos, P.D. (2024). Linear Systems Under Gaussian White Noise Excitation: Exact Closed-Form Solutions. In: Path Integrals in Stochastic Engineering Dynamics. Springer, Cham. https://doi.org/10.1007/978-3-031-57863-2_3
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DOI: https://doi.org/10.1007/978-3-031-57863-2_3
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