PDF solution of nonlinear oscillators subject to multiplicative Poisson pulse excitation on displacement. (English) Zbl 1170.70379
Summary: A solution procedure for the stationary probability density function (PDF) of the responses of nonlinear oscillators subjected to Poisson white noises is formulated with exponential-polynomial closure (EPC) method. The effectiveness of the solution procedure is investigated with nonlinear oscillators subjected to both external and multiplicative Poisson white noises at different levels of system nonlinearity, excitation intensity, and impulse arrival rates. Numerical results show that the PDFs obtained with the EPC procedure are in good agreement with those from Monte Carlo simulation.
MSC:
| 70L05 | Random vibrations in mechanics of particles and systems |
Keywords:
nonlinear oscillator; probability density function; Poisson white noise; stochastic processesReferences:
| [1] | Grigoriu, M.: Dynamic systems with Poisson white noise. Non-Linear Dyn. 36, 255–266 (2004) · Zbl 1091.70013 · doi:10.1023/B:NODY.0000045518.13177.3c |
| [2] | Di Paola, M., Pirrotta, A.: Direct derivation of corrective terms in SDE through nonlinear transformation on Fokker–Planck equation. Non-Linear Dyn. 36, 349–360 (2004) · Zbl 1090.70016 · doi:10.1023/B:NODY.0000045511.89550.57 |
| [3] | Pirrotta, A.: Multiplicative cases from additive cases: Extension of Kolmogorov–Feller equation to parametric Poisson white noise processes. Probab. Eng. Mech. 22, 127–135 (2007) · doi:10.1016/j.probengmech.2006.08.005 |
| [4] | Hu, S.L.J.: Responses of dynamic systems excited by non-Gaussian pulse processes. ASCE J. Eng. Mech. 119, 1818–1827 (1993) · doi:10.1061/(ASCE)0733-9399(1993)119:9(1818) |
| [5] | Hu, S.L.J.: Closure on discussion by Di Paola, M. and Falsone, G. on response of dynamic system excited by non-Gaussian pulse processes. ASCE J. Eng. Mech. 120, 2473–2474 (1994) · doi:10.1061/(ASCE)0733-9399(1994)120:11(2473) |
| [6] | Grigoriu, M.: The Itô and Stratonovich integrals for stochastic differential equations with Poisson white noise. Probab. Eng. Mech. 13, 175–182 (1998) · doi:10.1016/S0266-8920(97)00032-5 |
| [7] | Roberts, J.B.: System response to random impulses. J. Sound Vib. 24, 23–34 (1972) · Zbl 0266.70018 · doi:10.1016/0022-460X(72)90119-8 |
| [8] | Cai, G.Q., Lin, Y.K.: Response distribution of non-linear systems excited by non-Gaussian impulsive noise. Int. J. Non-Linear Mech. 27, 955–967 (1992) · Zbl 0765.70017 · doi:10.1016/0020-7462(92)90048-C |
| [9] | Vasta, M.: Exact stationary solution for a class of non-linear systems driven by a non-normal delta-correlated process. Int. J. Non-Linear Mech. 30, 407–418 (1995) · Zbl 0844.60034 · doi:10.1016/0020-7462(95)00009-D |
| [10] | Proppe, C.: The Wong–Zakai theorem for dynamical systems with parametric Poisson white noise excitation. Int. J. Eng. Sci. 40, 1165–1178 (2002) · Zbl 1211.45012 · doi:10.1016/S0020-7225(01)00087-8 |
| [11] | Proppe, C.: Exact stationary probability density functions for non-linear systems under Poisson white noise excitation. Int. J. Non-Linear Mech. 38, 557–564 (2003) · Zbl 1346.60110 · doi:10.1016/S0020-7462(01)00084-1 |
| [12] | Köylüoǧlu, H.U., Nielsen, S.R.K., Iwankiewicz, R.: Reliability of non-linear oscillators subject to Poisson driven impulses. J. Sound Vib. 176, 19–33 (1994) · Zbl 0945.70542 · doi:10.1006/jsvi.1994.1356 |
| [13] | Köylüoǧlu, H.U., Nielsen, S.R.K., Iwankiewicz, R.: Response and reliability of Poisson-driven systems by path integration. ASCE J. Eng. Mech. 121, 117–130 (1995) · doi:10.1061/(ASCE)0733-9399(1995)121:1(117) |
| [14] | Köylüoǧlu, H.U., Nielsen, S.R.K., Çakmak, A.Ş.: Fast cell-to-cell mapping (path integration) for nonlinear white noise and Poisson driven systems. Struct. Saf. 17, 151–165 (1995) · doi:10.1016/0167-4730(95)00006-P |
| [15] | Wojtkiewicz, S.F., Johnson, E.A., Bergman, L.A., Spencer, B.F. Jr., Grigoriu, M.: Stochastic response to additive Gaussian and Poisson white noises. In: Spencer, B.F., Johnson, E.A. (eds.) Stochastic Structural Dynamics, 4th Int. Conf. on Stochastic Structural Dynamics, Notre Dame, August, 1998, pp. 53–60. Balkema, Rotterdam (1999) |
| [16] | Wojtkiewicz, S.F., Johnson, E.A., Bergman, L.A., Grigoriu, M., Spencer, B.F. Jr.: Response of stochastic dynamical systems driven by additive Gaussian and Poisson white noise: Solution of a forward generalized Kolmogorov equation by a spectral finite difference method. Comput. Methods Appl. Mech. Eng. 168, 73–89 (1999) · Zbl 0956.70003 · doi:10.1016/S0045-7825(98)00098-X |
| [17] | Iwankiewicz, R., Nielsen, S.R.K.: Solution techniques for pulse problems in non-linear stochastic dynamics. Probab. Eng. Mech. 15, 25–36 (2000) · doi:10.1016/S0266-8920(99)00006-5 |
| [18] | Pirrotta, A.: Non-linear systems under parametric white noise input: Digital simulation and response. Int. J. Non-Linear Mech. 40, 1088–1101 (2005) · Zbl 1349.74060 · doi:10.1016/j.ijnonlinmec.2005.04.001 |
| [19] | Tylikowski, A., 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) · Zbl 0589.70023 · doi:10.1016/0020-7462(86)90006-5 |
| [20] | Grigoriu, M.: Equivalent linearization for Poisson white noise input. Probab. Eng. Mech. 10, 45–51 (1995) · doi:10.1016/0266-8920(94)00007-8 |
| [21] | Sobiechowski, C., Socha, L.: Statistical linearization of the Duffing oscillator under non-Gaussian external excitation. J. Sound Vib. 231, 19–35 (2000) · Zbl 1237.70122 · doi:10.1006/jsvi.1999.2668 |
| [22] | Proppe, C.: Equivalent linearization of MDOF systems under external Poisson white noise excitation. Probab. Eng. Mech. 17, 393–399 (2002) · doi:10.1016/S0266-8920(02)00036-X |
| [23] | Proppe, C.: Stochastic linearization of dynamical systems under parametric Poisson white noise excitation. Int. J. Non-Linear Mech. 38, 543–555 (2003) · Zbl 1346.74091 · doi:10.1016/S0020-7462(01)00083-X |
| [24] | Iwankiewicz, R., Nielsen, S.R.K., Thoft-Christensen, P.: Dynamic response of non-linear systems to Poisson-distributed pulse trains: Markov approach. Struct. Saf. 8, 223–238 (1990) · doi:10.1016/0167-4730(90)90042-N |
| [25] | Iwankiewicz, R., Nielsen, S.R.K.: Dynamic response of non-linear systems to Poisson-distributed random impulses. J. Sound Vib. 156, 407–423 (1992) · doi:10.1016/0022-460X(92)90736-H |
| [26] | Di Paola, M., Falsone, G.: Non-linear oscillators under parametric and external Poisson pulses. Nonlinear Dyn. 5, 337–352 (1994) · doi:10.1007/BF00045341 |
| [27] | Er, G.K.: A new non-Gaussian closure method for the PDF solution of nonlinear random vibrations. In: Murakami, H., Luco, J.E. (eds.) Engineering Mechanics: A Force for the 21st Century, 12th Engrg. Mech. Conf., San Diego, May, pp. 1403–1406. ASCE, Reston (1998) |
| [28] | Er, G.K.: An improved closure method for analysis of nonlinear stochastic systems. Nonlinear Dyn. 17, 285–297 (1998) · Zbl 0923.70019 · doi:10.1023/A:1008346204836 |
| [29] | Er, G.K., Iu, V.P.: Probabilistic solutions to nonlinear random ship roll motion. ASCE J. Eng. Mech. 125, 570–574 (1999) · doi:10.1061/(ASCE)0733-9399(1999)125:5(570) |
| [30] | Er, G.K., Iu, V.P.: Stochastic response of base-excited Coulomb oscillator. J. Sound Vib. 233, 81–92 (2000) · doi:10.1006/jsvi.1999.2792 |
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