Extreme events and their optimal mitigation in nonlinear structural systems excited by stochastic loads: Application to ocean engineering systems

We develop an efficient numerical method for the probabilistic quantification of the response statistics of nonlinear multi-degree-of-freedom structural systems under extreme forcing events, emphasizing accurate heavy-tail statistics. The response is decomposed to a statistically stationary part and an intermittent component. The stationary part is quantified using a statistical linearization method while the intermittent part, associated with extreme transient responses, is quantified through i) either a few carefully selected simulations or ii) through the use of effective measures (effective stiffness and damping). The developed approach is able to accurately capture the extreme response statistics orders of magnitude faster compared with direct methods. The scheme is applied to the design and optimization of small attachments that can mitigate and suppress extreme forcing events delivered to a primary structural system. Specifically, we consider the problem of suppression of extreme responses in two prototype ocean engineering systems. First, we consider linear and cubic springs and perform parametric optimization by minimizing the forth-order moments of the response. We then consider a more generic, possibly asymmetric, piecewise linear spring and optimize its nonlinear characteristics. The resulting asymmetric spring design far outperforms the optimal cubic energy sink and the linear tuned mass dampers.

[2]  C. Garrett.  Rogue waves , 2012 .

[3]  R. A. Shenoi,et al.  High speed marine craft motion mitigation using flexible hull design , 2012 .

[4]  A. Pirrotta,et al.  Probabilistic characterization of nonlinear systems under Poisson white noise via complex fractional moments , 2014 .

[5]  Søren Nielsen,et al.  Response and Reliability of Poisson-Driven Systems by Path Integration , 1995 .

[6]  Mark J. Beran,et al.  Statistical Continuum Theories , 1965 .

[7]  Alexander F. Vakakis,et al.  Shock Mitigation by Means of Low- to High-Frequency Nonlinear Targeted Energy Transfers in a Large-Scale Structure , 2016 .

[8]  Themistoklis P. Sapsis,et al.  Probabilistic response and rare events in Mathieu׳s equation under correlated parametric excitation , 2016, 1706.00109.

[9]  W. Zhu.  Stochastic Averaging Methods in Random Vibration , 1988 .

[10]  Themistoklis P. Sapsis,et al.  A moment-equation-copula-closure method for nonlinear vibrational systems subjected to correlated noise , 2015, 1502.01208.

[11]  Andrew J. Majda,et al.  Lessons in uncertainty quantification for turbulent dynamical systems , 2012 .

[12]  T. T. Soong,et al.  Random Vibration of Mechanical and Structural Systems , 1992 .

[13]  Y. K. Lin,et al.  Stochastic stability of wind-excited long-span bridges , 1996 .

[14]  P. Spanos,et al.  Random vibration and statistical linearization , 1990 .

[15]  Dongbin Xiu,et al.  The Wiener-Askey Polynomial Chaos for Stochastic Differential Equations , 2002, SIAM J. Sci. Comput..

[16]  Michael R Riley,et al.  Ride Severity Index - A New Approach to Quantifying the Comparison of Acceleration Responses of High-Speed Craft , 2011 .

[17]  Alexander F. Vakakis,et al.  Effective Stiffening and Damping Enhancement of Structures With Strongly Nonlinear Local Attachments study the stiffening and damping effects that local essentially nonlinear attachments , 2012 .

[18]  K. Sobczyk.  Stochastic Differential Equations: With Applications to Physics and Engineering , 1991 .

[19]  Karl Garme,et al.  Prediction and evaluation of working conditions on high-speed craft using suspension seat modelling , 2015 .

[20]  Themistoklis P. Sapsis,et al.  Probabilistic Description of Extreme Events in Intermittently Unstable Dynamical Systems Excited by Correlated Stochastic Processes , 2014, SIAM/ASA J. Uncertain. Quantification.

[21]  Oleg Gendelman,et al.  Dynamics of linear discrete systems connected to local, essentially non-linear attachments , 2003 .

[22]  Gerassimos A. Athanassoulis,et al.  Beyond the Markovian assumption: response–excitation probabilistic solution to random nonlinear differential equations in the long time , 2015, Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences.

[23]  R. A. Shenoi,et al.  A simplified 3-D human body–seat interaction model and its applications to the vibration isolation design of high-speed marine craft , 2009 .

[24]  G. Kerschen,et al.  Nonlinear Targeted Energy Transfer in Mechanical and Structural Systems , 2008 .

[25]  Y. K. Lin.  Application of Nonstationary Shot Noise in the Study of System Response to a Class of Nonstationary Excitations , 1963 .

[26]  Seymour M.J. Spence,et al.  Large scale reliability-based design optimization of wind excited tall buildings , 2012 .

[27]  G. Karniadakis,et al.  A computable evolution equation for the joint response-excitation probability density function of stochastic dynamical systems , 2012, Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences.

[28]  Alexander F. Vakakis,et al.  Large-scale experimental evaluation and numerical simulation of a system of nonlinear energy sinks for seismic mitigation , 2014 .

[29]  Gerassimos A. Athanassoulis,et al.  New partial differential equations governing the joint, response–excitation, probability distributions of nonlinear systems, under general stochastic excitation , 2008 .

[30]  Themistoklis P. Sapsis,et al.  A probabilistic decomposition-synthesis method for the quantification of rare events due to internal instabilities , 2015, J. Comput. Phys..

[31]  James T. P. Yao,et al.  Mathematical modelling of structural behaviour during earthquakes , 1988 .

[32]  Alexander F. Vakakis,et al.  Numerical and experimental investigation of a highly effective single-sided vibro-impact non-linear energy sink for shock mitigation , 2013 .

[33]  Alexander F. Vakakis,et al.  Irreversible Passive Energy Transfer in Coupled Oscillators with Essential Nonlinearity , 2005, SIAM J. Appl. Math..

[34]  R. Iwankiewicz,et al.  Analytical vs. Simulation Solution Techniques for Pulse Problems in Non-linear Stochastic Dynamics , 1997 .

[35]  V L Belenky,et al.  Stability and Safety of Ships: Risk of Capsizing , 2007 .

[36]  L. Bergman,et al.  Solution of the Four Dimensional Fokker-Planck Equation: Still a Challenge , 2005 .

[37]  W. F. Wu,et al.  CUMULANT-NEGLECT CLOSURE FOR NON-LINEAR OSCILLATORS UNDER RANDOM PARAMETRIC AND EXTERNAL EXCITATIONS , 1984 .

[38]  Alexander F. Vakakis,et al.  Inducing Passive Nonlinear Energy Sinks in Vibrating Systems , 2001 .

[39]  Paul C. Liu.  A chronology of freauqe wave encounters , 2007 .

[40]  L. Isserlis.  ON A FORMULA FOR THE PRODUCT-MOMENT COEFFICIENT OF ANY ORDER OF A NORMAL FREQUENCY DISTRIBUTION IN ANY NUMBER OF VARIABLES , 1918 .

[41]  Jie Luo,et al.  Equivalent modal damping, stiffening and energy exchanges in multi-degree-of-freedom systems with strongly nonlinear attachments , 2012 .

[42]  Edwin Kreuzer,et al.  The effect of sea irregularities on ship rolling , 2006, Computing in Science & Engineering.