Stochastic Structural Dynamics and Some Recent Developments Y. K. Lin Center for Applied Stochastics Research Florida Atlantic University Boca Raton, FL U.S.A.

More generally

Historical Development (1) Physicists – Brownian Motion Einstein (1905) W(t) = Gaussian white noise Ornstein – Uhlenbeck (1930) Wang – Uhlenbeck (1945) W(t) = vector of Gaussian white noises Gaussian Input Linear System Gaussian Output →→

(2)Electrical Engineers–Generalized Harmonic Analysis for Communication Systems Weiner (1930) Khintchine (1934) Rice (1944) vector of weakly stationary random processes Objective – obtain correlation functions (or spectral densities) of the response from those of excitations.

(3)Mechanical and Aerospace Engineers Turbulence Flight vehicles excited by turbulence, jet noise, rocket noise Rayleigh (1919) Pontryagin, Andronov, Vitt (1933) Taylor (1935) C. C. Lin (1944) Crandall Caughey Bolotin ….. (4)Civil Engineers – Winds, earthquakes, road roughness Housner (1941) …..

Solution Forms (1)Input Statistical Properties → Output Statistical Properties Possible if (a) system is linear, and (b) inputs are additive. (2)Input Probability Distribution → Output Probability Distribution Possible if (a) system is linear, and (b) inputs are additive and Gaussian.

Some exact solutions are obtainable, when (1)System is nonlinear (2)Some inputs are multiplicative Use mathematical theory of diffusive Markov processes.

Markov Random Process One-Step Memory Generalization to Multi-Dimensional Markov Process X (t) Transition probability distribution Transition probability density

Fokker-Planck-Kolmogorov (FPK) Equation for Markov Random Process Reduced FPK Equation for Stationary Markov Process The jth component of x drift coefficients diffusion coefficients

Exact Probability Solutions for Multi-Dimensional Nonlinear Systems (Restricted to Gaussian white noise excitations) Early Works (additive excitation only) (1)Nonlinear stiffness, linear damping (Pontryagin, et. al 1933). (2)Additional requirement for MDF systems – equipartition of kinetic energy. (3)Nonlinear damping (Caughey) – replacing constant damping coefficient by a function of total potential. Adding Multiplicative Excitations (1)First success (Dimentberg 1982) (2)Detailed balance (Yong-Lin 1987) (3)Generalized stationary potential (Lin-Cai 1988). (4)Removing the restriction of equipartition energy (Cai-Lin-Zhu).

A Single-Degree-of-Freedom System = damping term = stiffness term = Gaussian white noises = a constant (cross-spectral density of )

Fokker-Planck-Kolmogorov (F-P-K) Equation for

Additive Excitation Only Under the condition Equi-partition of kinetic energy

Method of Generalized Stationary Potential (Lin-Cai) Reduced F-P Equation Sufficient Conditions (a) (b)

Single DOF Reduced FPK equation

Split the Wong-Zakai correction term, if exists, into two parts The F-P-K equation can be re-arranged as follows: (B) (A)

Replacing the F-P-K equation by the sufficient conditions (A) (B) Solving for (A) where Restriction [from (B)]: (generalized stationary potential)

Detailed Balance - a special case - (Haken)

EXAMPLE 1: System in detailed balance

EXAMPLE 2: System not in detailed balance

A General Approximation Scheme

Method of Weighted Residual

Dissipation Energy Balancing

Reduction of Dimensionality (1)Averaging techniques (generalization of Krylov-Bogoliubov-Mitropolsky techniques for deterministic systems) (a)Stochastic averaging – linear or weakly nonlinear stiffness terms. (b)Quasi-conservative averaging – strongly nonlinear stiffness term. (c) Second-order averaging. (2) Slaving principle (Haken) Master – slow motion Slave – fast motion

Stochastic Averaging (Stratonovich, 1963; Khasminskii, 1966) An Example – A column excited by horizontal and vertical earthquakes Consider one dominant mode:

Transformation

For systems with strongly nonlinear stiffness: A is replaced by total energy U (or more generally Hamiltonian)

Quasi-Conservative Averaging For undamped free vibration Integrating Combining (a) and (b) Eq.(a) is now replaced by Eqs. (b) and (c) = quasi-period (a) (b) (c)

System Failures (i)First-Passage Failure (ii)Fatigue Failure (iii)Motion Instability

Stochastic Stability Concepts (1)Lyapunov Stability with Probability One (Sample Stability): (2) Stability in Probability (3) Stability in

For Linear System Stability in Probability Sample Stability Stability inSample Stability

Column under Fluctuating Axial Load

Mathiew-Hill Equation Strutt diagram

The Column Problem

Averaged A(t) – independent of the averaged, itself a scalar Markov process Condition for stability in probability

EXAMPLE – BRIDGE IN TURBULENT WIND TWO TORSIONAL AND TWO VERTICAL MODES

FIRST PASSAGE FAILURE

T = random time when the first-passage failure occurs

Reliability

Statistical Moments of First-Passage Time T Condition: a j, b jk independent of t 0

An example for first-passage failure

Average toppling time vs. base excitation level. Solid line: horizontal excitation only; dotted line: combined horizontal and vertical excitations.

Concluding Remarks (1)The present review is focused on analytical solutions. The important Monte Carlo simulation techniques are not covered, such as the works by Shinozuka, Schuëller, and Pradlwarter, etc. (2)Recent works by Arnold and his associates on dynamical systems are not covered. (3)Numerical solutions, such as those given by Naess and Johnson, Bergman and Spencer, Kloeden et al, etc. also are not covered.