Abstract
The vibroimpact systems with bilateral barriers are often encountered in practice. However, the dynamics of the vibroimpact system with bilateral barriers is full of challenges. Few closed-form solutions were obtained. In this paper, we propose a novel method for random vibration analysis of single-degree-of-freedom (SDOF) vibroimpact systems with bilateral barriers under Gaussian white noise excitations. A periodic approximate transformation is employed to convert the equations of the motion to a continuous form. The probabilistic description of the system is subsequently defined through the corresponding Fokker-Planck-Kolmogorov (FPK) equation. The closed-form stationary probability density function (PDF) of the response is obtained by solving the reduced FPK equation and using the proposed iterative method of weighted residue together with the concepts of the circulatory probability flow and the potential probability flow. Finally, the versatility of the proposed approach is demonstrated by its application to two typical examples. Note that the solution obtained by using the proposed method can be used as the benchmark to examine the accuracy of approximate solutions obtained by other methods.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
XU, W., FENG, J., and RONG, H. Melnikov’s method for a general nonlinear vibro-impact oscillator. Nonlinear Analysis: Theory, Methods & Applications, 71(1), 418–426 (2009)
DIMENTBERG, M. F. Statistical Dynamics of Nonlinear and Time-Varying Systems, Research Studies Press, Taunton (1988)
BROGLIATO, B. Nonsmooth Impact Mechanics: Models, Dynamics and Control, Springer-Verlag, London (1996)
BABISTKY, V. Theory of Vibro-Impact Systems and Applications, Springer-Verlag, Berlin (1998)
IBRHIM, R. A., CHALHOUB, N. G., and FALZARANO, J. Interaction of ships and ocean structures with ice loads and stochastic ocean waves. Applied Mechanics Reviews, 60(5), 246–289 (2007)
ALBERT LUO, C. J. and GUO, Y. Vibro-Impact Dynamics, John Wiley & Sons, New York (2012)
DIMENTBERG, M. F. and IOURTCHENKO, D. V. Random vibrations with impacts: a review. Nonlinear Dynamics, 36(2), 229–254 (2004)
IBRAHIM, R. A. Vibro-Impact Dynamics: Modeling, Mapping and Applications, Springer-Verlag, Berlin (2009)
JIN, X. L., HUANG, Z. L., and LEUNG, Y. T. Nonstationary probability densities of system response of strongly nonlinear single-degree-of-freedom system subject to modulated white noise excitation. Applied Mathematics and Mechanics (English Edition), 32(11), 1389–1398 (2011) https://doi.org/10.1007/s10483-011-1509-7
LIU, Q., XU, Y., XU, C., and KURTHS, J. The sliding mode control for an airfoil system driven by harmonic and colored Gaussian noise excitations. Applied Mathematical Modelling, 64, 249–264 (2018)
LIU, Q., XU, Y., and KURTHS, J. Active vibration suppression of a novel airfoil model with fractional order viscoelastic constitutive relationship. Journal of Sound and Vibration, 432, 50–64 (2018)
JIANG, W. A., SUN, P., ZHAO, G. L., and CHEN, L. Q. Path integral solution of vibratory energy harvesting systems. Applied Mathematics and Mechanics (English Edition), 40(4), 579–590 (2019) https://doi.org/10.1007/s10483-019-2467-8
DIMENTBERG, M. F., IOURTCHENKO, D. V., and VAN EWIJK, O. Subharmonic response of a quasi-isochronous vibroimpact system to a randomly disordered periodic excitation. Nonlinear Dynamics, 17(2), 173–186 (1998)
NAMACHCHIVAYA, N. S. and PARK, J. H. Stochastic dynamics of impact oscillators. Journal of Applied Mechanics, 72(6), 862–870 (2004)
RONG, H. W., WANG, X. D., XU, W., and FANG, T. Resonant response of a non-linear vibro-impact system to combined deterministic harmonic and random excitations. International Journal of Non-Linear Mechanics, 45(5), 474–481 (2010)
RONG, H. W., WANG, X. D., LUO, Q. Z., XU, W., and FANG, T. Subharmonic response of single-degree-of-freedom linear vibroimpact system to narrow-band random excitation. Applied Mathematics and Mechanics (English Edition), 32(9), 1159–1168 (2011) https://doi.org/10.1007/s10483-011-1489-x
LI, C., XU, W., FENG J. Q., and WANG, L. Response probability density functions of Duffing-Van der Pol vibro-impact system under correlated gaussian white noise excitations. Physica A: Statistical Mechanics and its Applications, 392(6), 1269–1279 (2013)
YANG, G. D., XU, W., GU, X. D., and HUANG, D. M. Response analysis for a vibroimpact Duffing system with bilateral barriers under external and parametric gaussian white noises. Chaos, Solitons & Fractals, 87(S), 125–135 (2016)
XIE, X., LI, J., LIU, D., and GUO, R. Transient response of nonlinear vibro-impact system under Gaussian white noise excitation through complex fractional moments. Acta Mechanica, 228(3), 1153–1163 (2017)
ZHAO, X. R., XU, W., YANG, Y. G., and WANG, X. Y. Stochastic responses of a viscoelastic-impact system under additive and multiplicative random excitations. Communications in Nonlinear Science and Numerical Simulation, 35, 166–176 (2016)
YURCHENKO, D., BURLON, A., PAOLA, M. D., and PIRROTTA, A. Approximate analytical mean-square response of an impacting stochastic system oscillator with fractional damping. ASCE-ASME Journal of Risk and Uncertainty in Engineering Systems, Part B: Mechanical Engineering, 3(3), 030903 (2017)
LIU, L., XU, W., YUE, X. L., and HAN, Q. Stochastic response of Duffing-Van der Pol vibroimpact system with viscoelastic term under wide-band excitation. Chaos, Solitons & Fractals, 104, 748–757 (2017)
SUN, J. Q. and HSU, C. S. First-passage time probability of non-linear stochastic systems by generalized cell mapping method. Journal of Sound and Vibration, 124(2), 233–248 (1988)
SUN, J. Q. and HSU, C. S. A statistical study of generalized cell mapping. Journal of Applied Mechanics, 55(3), 694–701 (1988)
SUN, J. Q. and HSU, C. S. The generalized cell mapping method in nonlinear random vibration based upon short-time Gaussian approximation. Journal of Applied Mechanics, 57(4), 1018–1025 (1990)
HAN, Q., XU, W., and YUE, X. L. Stochastic response analysis of noisy system with non-negative real-power restoring force by generalized cell mapping method. Applied Mathematics and Mechanics (English Edition), 36(3), 329–336 (2015) https://doi.org/10.1007/s10483-015-1918-6
WANG, L., MA, S., JIA, W. T., and XU, W. The stochastic response of a class of impact systems calculated by a new strategy based on generalized cell mapping method. Journal of Applied Mechanics, 85(5), 054502 (2018)
IOURTCHENKO, D. V. and SONG, L. L. Numerical investigation of a response probability density function of stochastic vibroimpact systems with inelastic impacts. International Journal of Non-Linear Mechanics, 41(3), 447–455 (2006)
ER, G. K. An improved closure method for analysis of nonlinear stochastic systems. Nonlinear Dynamics, 17(3), 285–297 (1998)
ZHU, H. T. Stochastic response of a parametrically excited vibro-impact system with a nonzero offset constraint. International Journal of Dynamics and Control, 4(2), 180–194 (2016)
ZHU, H. T. Stochastic response of a vibro-impact Duffing system under external poisson impulses. Nonlinear Dynamics, 82(1), 1001–1013 (2015)
DIMENTBERG, M. F., GAIDAI, O., and NAESS, A. Random vibrations with strongly inelastic impacts: response PDF by the path integration method. International Journal of Non-Linear Mechanics, 44(7), 791–796 (2009)
ZHURAVLEV, V. F. A method for analyzing vibration-impact systems by means of special functions. Mechanics of Solids, 11, 23–27 (1976)
KUMER, P., NARAYANAN, S., and GUPTA, S. Stochastic bifurcations in a vibro-impact Duffing-Van der Pol oscillator. Nonlinear Dynamics, 85(1), 439–452 (2016)
KUMER, P., NARAYANAN, S., and GUPTA, S. Bifurcation analysis of a stochastically excited vibro-impact Duffing-Van der Pol oscillator with bilateral rigid barriers. International Journal of Mechanical Sciences, 127(S), 103–117 (2017)
CHEN, L. C., QIAN, J. M., ZHU, H. S., and SUN, J. Q. The closed-form stationary probability distribution of the stochastically excited vibro-impact oscillators. Journal of Sound and Vibration, 439, 260–270 (2019)
CHEN, L. C., LIU, J., and SUN, J. Q. Stationary response probability distribution of SDOF nonlinear stochastic systems. Journal of Applied Mechanics, 84(5), 051006 (2017)
PAOLA, M. D. and SOFI, A. Approximate solution of the Fokker-Planck-Kolmogorov equation. Probabilistic Engineering Mechanics, 17(4), 369–384 (2002)
CHEN, L. C. and SUN, J. Q. The closed-form solution of the reduced Fokker-Planck-Kolmogorov equation for nonlinear systems. Communications in Nonlinear Science and Numerical Simulation, 41(12), 1–10 (2016)
Author information
Authors and Affiliations
Corresponding author
Additional information
* Citation: CHEN, L. C., ZHU, H. S., and SUN, J. Q. Novel method for random vibration analysis of single-degree-of-freedom vibroimpact systems with bilateral barriers. Applied Mathematics and Mechanics (English Edition), 40(12), 1759–1776 (2019) https://doi.org/10.1007/s10483-019-2543-5
Project supported by the National Natural Science Foundation of China (Nos. 11672111, 11332008, 11572215, and 11602089), the Program for New Century Excellent Talents in Fujian Province University, the Natural Science Foundation of Fujian Province of China (No. 2019J01049), and the Promotion Program for Young and Middle-Aged Teacher in Science and Technology Research of Huaqiao University (Nos. ZQNYX307 and ZQNYX505)
Rights and permissions
About this article
Cite this article
Chen, L., Zhu, H. & Sun, J.Q. Novel method for random vibration analysis of single-degree-of-freedom vibroimpact systems with bilateral barriers. Appl. Math. Mech.-Engl. Ed. 40, 1759–1776 (2019). https://doi.org/10.1007/s10483-019-2543-5
Received:
Revised:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10483-019-2543-5