M.F. Dimentberg

Optimal bounded control of steady-state random vibrations

Abstract A SDOF system is considered, which is excited by a white-noise random force. The systems response is controlled by a force of bounded magnitude, with the aim of minimizing integral of the expected response energy over a given period of time. The integral to be minimized satisfies the Hamilton–Jacobi–Bellman (HJB) equation. An analytical solution of this PDE is obtained within a certain outer part of the phase plane. This solution is analyzed for large time intervals, which correspond to the limiting steady-state random vibration. The analysis shows the outer domain expanding onto the whole phase plane in the limit, implying that the simple dry-friction control law is the optimal one for steady-state response. The resulting value of the (unconditional) expected response energy, for the case of a stationary excitation, is also obtained. It matches with the corresponding result of energy balance analysis, as obtained by direct application of the SDE Calculus, as well as that of stochastic averaging for the case where the magnitude of dry friction force and intensity of excitation are both small. A general expression for mean absolute value of the response velocity is also obtained using the SDE calculus. Certain reliability predictions both for first-passage and fatigue-type failures are also derived for the optimally controlled system using the stochastic averaging method. These predictions are compared with their counterparts for the system with a linear velocity feedback and same r.m.s. response, thereby illustrating the price to be paid for the bounds on control force in terms of the reduced reliability of the system.

Modeling and random vibration analysis of SDOF systems with asymmetric hysteresis

This study focuses upon SDOF systems having non-linear, hysteretic stress-strain behavior which is asymmetric about the initial linear elastic relationship. Differences between tensile and compressive strength witnessed in various materials, particularly in the unique stress-strain properties of the shape-memory alloy Nitinol, motivate the need for mathematical models capable of simulating asymmetric response. Two models are presented, and the specific capabilities of each are explored. Both models follow the rate-type format of the Bouc-Wen model, and offer previously unavailable stress-strain relationships. Response statistics under Gaussian White Noise input are obtained using the method of equivalent linearization. For asymmetric systems, a need for non-zero-mean analysis is evident, even under zero-mean input.

Hybrid solution method for dynamic programming equations for MDOF stochastic systems

An optimal control problem is considered for a multi-degree-of-freedom (MDOF) system, excited by a white-noise random force. The problem is to minimize the expected response energy by a given time instantT by applying a vector control force with given bounds on magnitudes of its components. This problem is governed by the Hamilton-Jacobi-Bellman, or HJB, partial differential equation. This equation has been studied previously [1] for the case of a single-degree-of-freedom system by developing a “hybrid” solution. Specifically, an exact analitycal solution has been obtained within a certain outer domain of the phase plane, which provides necessary boundary conditions for numerical solution within a bounded in velocity inner domain, thereby alleviating problem of numerical analysis for an unbounded domain. This hybrid approach is extended here to MDOF systems using common transformation to modal coordinates. The multidimensional HJB equation is solved explicitly for the corresponding outer domain, thereby reducing the problem to a set of numerical solutions within bounded inner domains. Thus, the problem of bounded optimal control is solved completely as long as the necessary modal control forces can be implemented in the actuators. If, however, the control forces can be applied to the original generalized coordinates only, the resulting optimal control law may become unfeasible. The reason is the nonlinearity in maximization operation for modal control forces, which may lead to violation of some constraints after inverse transformation to original coordinates. A semioptimal control law is illustrated for this case, based on projecting boundary points of the domain of the admissible transformed control forces onto boundaries of the domain of the original control forces. Case of a single control force is considered also, and similar solution to the HJB equation is derived.

First-passage study and stationary response analysis of a BWB hysteresis model using quasi-conservative stochastic averaging method

The quasi-conservative stochastic averaging (QCSA) method is applied to a Bouc-Wen-Baber hysteretic system (BWB) under Gaussian white noise excitations. The stationary probability density of the systems response amplitude and energy is obtained for different excitation levels and different damping ratios. These results are compared with the studies presented by Cai and Lin in A New Solution Technique for Randomly Excited Hysteretic Structures, Technical Report on Grant No. NCEER-88-0012, State University of New York, Buffalo, NY, 1988. The first-passage time problem for the hysteretic system is also studied using the method of QCSA. The results for the expected time to failure as a function of the initial total energy are shown, for different excitation levels and for different threshold values of energy. The relationship between the expected time to failure when initial total energy is zero and excitation level is shown for a wide range of hysteresis shape parameters.

SPECTRAL DENSITY OF A NON-LINEAR SINGLE-DEGREE-OF-FREEDOM SYSTEM'S RESPONSE TO A WHITE-NOISE RANDOM EXCITATION: A UNIQUE CASE OF AN EXACT SOLUTION

Abstract A SDOF vibro-impact system with a one-sided rigid barrier is considered for the case of a perfectly elastic impact and stationary zero-mean Gaussian white-noise excitation. Special piece-wise-linear transformation of the systems state variables is applied, which effectively reduces the original problem to one without impacts. Moreover, for the special case, where position of the barrier coincides with that of the systems static equilibrium, the transformed equation of motion is found to be linear, implying Gaussian transition probability densities of the transformed state variables. Thus, the non-linear random vibration problem is reduced to one with “inertialess” non-linearity only. The above transformation is used to obtain an exact explicit expression for the response autocorrelation function, thus leading to quadrature expression for spectral density of the response.

A stationary model for periodic excitation with uncorrelated random disturbances

Abstract The paper presents a stationary model for periodic excitations with random amplitude and phase disturbances for linear and nonlinear random vibration analysis. The disturbances are modeled as uncorrelated stationary white noise processes. Application of the model is demonstrated by stationary moment response of a linear single-degree-of-freedom system subject to such excitations. To find moment responses, an equivalent augmented system subject to parametric white noise excitations under certain constraint conditions is studied. Numerical results for the second and fourth-order moment responses are presented. The probability density function of the response is calculated based on thhe cumulant-neglect closure method. NonGaussianity of the response is discussed in terms of the excess factor. The results show that the random amplitude disturbance can significantly increase system moment response. The random phase modulation may increase or reduce the system moment response, depending on the value of relative detuning berween the system natural frequency and the mean excitation frequency. The response may become Gaussian in the sense of up to the fourth-order moment for sufficiently large random phase or relative detuning.

Short-term Dynamic Instability of a System with Randomly Varying Damping

A single-degree-of-freedom system with temporal random variations of its total apparent damping is considered. It is demonstrated that the dynamic response of the system exhibits spontaneous transient outbreaks, which are induced by brief periods when the damping coefficient becomes negative. The analysis is based on a parabolic approximation for the random temporal variations of the damping coefficient during these excursions into the domain of dynamic instability, together with the asymptotic Krylov-Bogoliubov method of averaging over the response period. It results in an explicit relation for the response amplitude in terms of the peak value of the damping coefficient during the corresponding downcrossing of the zero level. In this way the reliability analysis for the system as based on a solution to the first-passage problem for the response or on evaluating the probability density (pdf) of the response peaks is reduced to the corresponding problems for the damping coefficient the accuracy of the derived expression for the above pdf is verified by direct Monte-Carlo simulation of the basic equation. Furthermore, the above analytical solution is used also to derive a simple identification procedure for the system from its measured (on-line) response. Specifically, the mean value of the damping coefficient can be estimated, as can its standard deviation and the mean frequency of its temporal variations.

A non-stationary stochastic model for periodic excitation with random phase modulation

Abstract The paper presents a non-stationary stochastic model for periodic excitation with random phase modulation, where the phase modulation is modeled as a modulated stationary. Gaussian process. Applications of the model are demonstrated by analysis of response of a single-degree-of-freedom (SDOF) system under such an excitation. The response is, in general, non-Gaussian. Cases of step, rectangular, and exponential envelopes are considered in the present study. The nonstationary second and fourth order moments are calculated by numerically solving the transient moment equations. Non-Gaussianity of the response is studied in terms of the non-stationary excess factor. Some numerical results are presented. The influences of system parameters, build-up and decay rates as well as duration of random phase modulation on the moment response of the SDOF system are discussed.

Transition from Planar to Whirling Oscillations in a Certain Nonlinear System

A single-mass two-degrees-of-freedom system is considered, witha radially oriented nonlinear restoring force. The latter is smooth andbecomes infinite at a certain value of a radial displacement. Stabilityanalysis is made for planar oscillation, or motion along a givendirection. As long as this motion is periodic, the nonlinearity in therestoring force provides a periodic parametric excitation in thetransverse direction. The linearized stability analysis is reduced tostudy of the Mathieu equation for the (infinitesimal) motions in thetransverse direction. For the case of free oscillations in the givendirection an exact solution is obtained, since a specific analyticalform is used for the (strongly nonlinear) restoring force, which permitsexplicit integration of the equation of motion. Stability of the planarmotion in this case is shown to be very sensitive to even slightdeviations from polar symmetry in the restoring force (as well as to theamplitude of oscillations in the given direction). Numerical integrationof the original equations of motion shows the resulting motion to be awhirling type indeed in case of the transversal instability. For thecase of a sinusoidal forcing in the given direction solution for the(periodic) response is obtained by Krylov–Bogoliubov averaging. Thisresults in the ‘transmitted’ Ince–Strutt chart – namely, stabilitychart for transverse direction on the amplitude-frequency plane of theexcitation in the original direction.

Random response of Duffing oscillator to periodic excitation with random disturbance

This paper discusses the random response of a non-linear Duffing oscillator subjected to a periodic excitation with random phase modulation. Effects of uncertainty in the periodic excitation and level of the system non-linearity on the response moments and non-Gaussian nature of the response caused by both the system non-linearity and the non-Gaussian loading are investigated. Results are presented in terms of the second- and the fourth-order moments as well as the excess factor of the response and some results are compared with those from the Monte Carlo simulation. An iterated linearisation technique is proposed to improve the accuracy of the numerical results for strongly non-linear systems.