Stationary Random Excitation Research Articles

Influence of the supporting braces on the dynamic control of buildings with added viscous dampers

The aim of this paper is to suggest a procedure to analyse the supporting brace stiffness influence on the damping optimization of a linear-elastic shear-type model. This analysis appropriately extends the Takewaki's technique based on the minimization of an objective function, which represents the sum of mean-square inter-storey drifts to stationary random excitation. The new procedure is a two-phase optimization technique in which an optimal value of the supporting brace stiffness is also involved. In order to demonstrate the applicability of the proposed procedure, the results of the numerical analysis for an 8-dof shear-type model subjected to a base excitation are presented. Moreover, response in the time domain is worked out for the analysed model. The procedure presented in this paper is readily available for practical applications. Copyright © 2008 John Wiley & Sons, Ltd.

Power delivered to mechanical systems by random vibrations

This paper develops deformational response power descriptions of multiple degree-of-freedom systems due to stationary random vibration excitation. Two new concepts are developed. The deformational response power density (DRPD) can be computed when a structure's natural frequencies and modal masses are available. The DRPD shows the spectral content of the deformational power delivered to a specific structure by the stationary, random excitation. This function can be found through a weighted windowing of the power spectrum of the input acceleration excitation. Deformational response input power spectra (DRIPS), similar to the input energy spectrum and shock response spectrum, give the power delivered to single-degree-of-freedom systems as a function of natural frequency. It is shown that the DRIPS is simply a smoothed version of the power spectrum of the input acceleration excitation. The DRIPS gives rise to a useful power-based data smoothing operation.

Power Delivered to Mechanical Systems by Random Vibrations

This paper develops deformational response power descriptions of multiple degree-of-freedom systems due to stationary random vibration excitation. Two new concepts are developed. The deformational response power density (DRPD) can be computed when a structure's natural frequencies and modal masses are available. The DRPD shows the spectral content of the deformational power delivered to a specific structure by the stationary, random excitation. This function can be found through a weighted windowing of the power spectrum of the input acceleration excitation. Deformational response input power spectra (DRIPS), similar to the input energy spectrum and shock response spectrum, give the power delivered to single-degree-of-freedom systems as a function of natural frequency. It is shown that the DRIPS is simply a smoothed version of the power spectrum of the input acceleration excitation. The DRIPS gives rise to a useful power-based data smoothing operation.

Effects of Overlapping and Windowing on Frequency Response Function Estimates of Systems With Random Inputs

The use of block overlapping and time windowing in the estimation of the frequency response functions (FRFs) of linear systems is considered. The systems are subjected to stationary white random excitation, and there is no external noise. The sources of errors for the bias and variance of the frequency response are identified to be due to the Fourier transforms of the transient terms or end effects that arise due to nonperiodic signals. Using a pure time delay as an example for the system, it is shown that there is a limit to the improvement achievable in reducing the variance as the block overlap is increased.

Isotropic layered soil–structure interaction caused by stationary random excitations

Soil–structure interaction emanating from seismic stationary random excitations is studied using the pseudo-excitation method in combination with the precise integration method. The soil considered is a viscoelastic, transversely isotropic and layered half space and the structure which it supports is modelled by the finite element method. The excitation sources are random field ones that are stationary in the time domain and are located in the soil. The pseudo-excitation method is used to transform this stationary random soil–structure interaction problem into a series of deterministic harmonic response analyses and the precise integration method is used to integrate the ordinary differential equations in the frequency–wavenumber domain. The power spectral densities of the soil–structure interaction responses caused by the stationary random excitations are investigated.

A top-down approach to calibration, validation, uncertainty quantification and predictive accuracy assessment

This paper describes a “top-down” uncertainty quantification (UQ) approach for calibration, validation and predictive accuracy assessment of the SNL Validation Workshop Structural Dynamics Challenge Problem. The top-down UQ approach differs from the more conventional (“bottom-up”) approach in that correlated statistical analysis is performed directly with the modal characteristics (frequencies, mode shapes and damping ratios) rather than using the modal characteristics to derive the statistics of physical model parameters (springs, masses and viscous damping elements in the present application). In this application, a stochastic subsystem model is coupled with a deterministic subsystem model to analyze stochastic system response to stochastic forcing functions. The weak nonlinearity of the stochastic subsystem was characterized by testing it at three different input levels, low, medium and high. The calibrated subsystem models were validated with additional test data using published NASA and Air Force validation criteria. The validated subsystem models were first installed in the accreditation test bed where system response simulations involving stochastic shock-type force inputs were conducted. The validated stochastic subsystem model was then installed in the target application and simulations involving limited duration segments of stationary random vibration excitation were conducted.

Optimal active random vibration control for smart truss structures based on reliability

The current paper presents the reliability-based optimization of the location and feedback gains of active bar in a closed-loop control system for random smart truss structures under stationary random excitation. The mathematical model with reliability constraints on the root mean square value of the structural random dynamic displacement and stress response is developed based on maximization of dissipation energy due to the control action. The randomness of the structural physical materials, geometry, and damping are included in the analysis, and the applied forces are considered as stationary random excitation. The numerical characteristic corresponding to mean values and standard deviations of the stationary random responses of random smart structures is developed by the random factor method. Numerical examples of truss structures are presented to demonstrate the rationality and validity of the active control model.

On dry friction modelling and simulation in kinematically excited oscillatory systems

This paper deals with the analysis and simulation of a general single degree of freedom (sdof) oscillatory system with idealised linear viscous damper and dry friction. For dry friction modelling the phenomenological macro-slip approach is employed, described in mathematical form either by the signum function approach or by the physically correct stick–slip approach assuming switching phenomena on a short time scale. Both approaches are illustrated first using a steady-state harmonic acceleration excitation with constant amplitude and then a stationary random acceleration excitation, corresponding to a field-measured excitation in a vehicle. The differences in the two approaches are highlighted, indicating that the physically correct stick–slip approach describes the friction phenomenon better than the standard signum approach. The signum approach is prone to false numerical oscillations completely distorting the acceleration response signal in comparison to measured suspension system response. The acceleration transmissibility response is analysed in respect to the dry friction force magnitude, employing stationary random excitation. A sdof oscillatory system without viscous damping, subjected to both stationary random acceleration and harmonic acceleration is analysed, too. It is shown that such a system can be used without serious practical problems; however, no implications on its performance from the analysis under harmonic constant amplitude acceleration excitation can be made.

Damage Detection Based on the Cross Correlation Function Amplitude Vector and its Application to the ASCE Benchmark Structure

A new method for structural damage detection based on the Cross Correlation Function Amplitude Vector (CorV) of the measured vibration responses is presented. Under a stationary random excitation with a specific frequency spectrum, the CorV of the structure only depends on the frequency response function matrix of the structure, so the normalized CorV has a specific shape. Thus the damage can be detected and located with the correlativity and the relative difference between CorVs of the intact and damaged structures. With the benchmark problem sponsored by ASCE Task Group on Structural Health Monitoring, the CorV is proved an effective approach to detecting the damage in structures subject to random excitations.

Optimal control of random vibration in plate and shell structures with distributed piezoelectric components

The state covariance assignment (SCA) method of Skelton and associates has been extended to the optimal random vibration control of large-scale complicated shell structures with distributed piezoelectric components and under nonstationary random excitations. It provides a direct approach for achieving performance goals stated in terms of the root-mean-square (RMS) values which are common in many engineering system designs. The large-scale shell structures with distributed piezoelectric components of complicated geometrical configurations are approximated by the hybrid strain or mixed formulation based lower order triangular shell finite elements developed in the present investigation. Each of these shell finite elements has three nodes. Every node has seven degrees of freedom (dof) which include three translational, three rotational and one electric potential dof. These elements are better alternatives to those based on the displacement formulation and that hinged on the truly hybrid strain formulation in that they eliminate locking problems commonly associated with displacement based lower order shell finite elements. Computed results applying the proposed approach for a plate and a cylindrical shell structures with distributed piezoelectric components and under stationary random excitations are included to demonstrate its simplicity of use and efficiency of implementing the proposed approach.

Random seismic response analysis of truss structures with uncertain parameters

This paper presents an approach called random factor method (RFM) for the random vibration analysis of seismic response of random truss structures. An antenna and a truss bridge with random material parameters and geometric dimensions are considered to illustrate the application of the RFM in the seismic response analysis of random structures under stationary or non-stationary random earthquake excitation. Using the RFM, the structural stiffness and mass matrices can respectively be described by the product of two parts corresponding to the random factors of structural parameters and the deterministic matrix. From the expressions of structural seismic random response in the frequency domain, computational expressions for the mean value and standard deviation of the mean square value of the seismic random displacement and stress response are developed by means of the random variable’s functional moment method and the algebra synthesis method. The influences of the randomness of the structural parameters on the structural seismic responses are investigated.

Studies of the Performance of Multi-Unit Impact Dampers Under Stochastic Excitation

The performance of particle dampers whose behavior under broadband excitations involves internal friction and momentum transfer is a highly complex nonlinear process that is not amenable to exact analytical solutions. While numerous analytical and experimental studies have been conducted over many years to develop strategies for modeling and controlling the behavior of this class of vibration dampers, no guidelines currently exist for determining optimum strategies for maximizing the performance of particle dampers, whether in a single unit or in arrays of dampers, under random excitation. This paper focuses on the development and evaluation of practical design strategies for maximizing the damping efficiency of multi-unit particle dampers under random excitation, both the stationary and nonstationary type. High-fidelity simulation studies are conducted with a variable number of multi-unit dampers ranging from 1 to 100, with the magnitude of the “dead-space” nonlinearity being a random variable with a prescribed probability distribution spanning a feasible range of parameters. Results of the computational studies are calibrated with carefully conducted experiments with single-unit/single-particle, single-unit/multi-particle, and multiple-unit/multi-particle dampers. It is shown that a wide latitude exists in the trade-off between high vibration attenuation over a narrow range of damper gap size versus slightly reduced attenuation over a much broader range. The optimum configuration can be achieved through the use of multiple particle dampers designed in accordance with the procedure presented in the paper. A semi-active algorithm is introduced to improve the rms level reduction, as well as the peak response reduction. The utility of the approach is demonstrated through numerical simulation studies involving broadband stationary random excitations, as well as highly nonstationary excitations resembling typical earthquake ground motions.

Stochastically optimal active control of a smart truss structure under stationary random excitation

Optimization of the placement and feedback gains of an active bar in a closed-loop control system for random intelligent truss structures under stationary random excitation are studied in this paper. Based on maximization of dissipation energy due to the control action, a mathematical model with reliability constraints on the mean square value of the structural dynamic displacement and stress response is developed. The randomness of the physical parameters corresponding to the structural materials, geometric dimensions and structural damping are included in the analysis, and the applied forces are considered as stationary random excitation. The numerical characteristic of the stationary random responses of a stochastic smart structure is developed. Numerical examples of stochastic truss structures are presented to demonstrate the rationality and validity of the active control model, and some useful conclusions are obtained.

Stationary random waves propagation in 3D viscoelastic stratified solid

Propagation of stationary random waves in viscoelastic stratified transverse isotropic materials is investigated. The solid was considered multi-layered and located above the bedrock, which was assumed to be much stiffer than the soil, and the power spectrum density of the stationary random excitation was given at the bedrock. The governing differential equations are derived in frequency and wave-number domains and only a set of ordinary differential equations (ODEs) must be solved. The precise integration algorithm of two-point boundary value problem was applied to solve the ODEs. Thereafter, the recently developed pseudo-excitation method for structural random vibration is extended to the solution of the stratified solid responses.

Responses of discretized systems under narrow band nonstationary random excitations. Part 1: linear problems

The extended stochastic central difference (ESCD) method is proposed as a viable alternative for computing linear responses of discretized multi-degrees-of-freedom (mdof) systems under narrow band stationary and nonstationary random disturbances. The method provides a means of controlling the center frequencies and bandwidths of narrow band stationary and nonstationary random excitation processes. It is suitable for larger-scale random response analysis of complicated structures idealized by the finite element method. Its additional important feature is that application of normal mode or complex normal mode analysis or direct numerical integration algorithms such as the fourth-order Runge–Kutta scheme is not required. Examples, including one of flow-induced vibration of a pipe containing a moving fluid are included to demonstrate: (1) the capability of the proposed method and difference between responses of discretized systems under narrow band and wide band random excitations, and (2) its accuracy and efficiency by way of comparison to the Monte Carlo simulation data. Generalization of the ESCD method for computation of responses of nonlinear mdof systems is presented in a companion paper.

Responses of discretized systems under narrow band nonstationary random excitations. Part 2: nonlinear problems

In a companion paper, the extended stochastic central difference (ESCD) method was presented for computation of responses of linear multi-degrees-of-freedom (mdof) systems under narrow band stationary and nonstationary random excitations. The present paper is concerned with the generalization of the ESCD method for computation of nonlinear mdof systems. The generalization is based on a combination of the ESCD method with statistical linearization (SL) technique, modified adaptive time scheme (ATS), and time coordinate transformation (TCT). Unlike the conventional SL technique, the generalized ESCD method is applicable to mdof systems with large nonlinearities. Comparison is made of the computed results applying the generalized ESCD method to those obtained by the Monte Carlo simulation (MCS). Excellent agreements were obtained. It was observed that the proposed method is very efficient compared with the MCS.

Experiment study of structural random loading identification by the inverse pseudo excitation method

The inverse pseudo excitation method is used in the identification of random loadings. For structures subjected to stationary random excitations, the power spectral density matrices of such loadings are identified experimentally. The identification is based on the measured acceleration responses and the structural frequency response functions. Numerical simulation is used in the optimal selection of sensor locations. The proposed method has been successfully applied to the loading identification experiments of three structural models, two uniform steel cantilever beams and a four-story plastic glass frame, subjected to uncorrelated or partially correlated random excitations. The identified loadings agree quite well with actual excitations. It is proved that the proposed method is quite accurate and efficient in addition to its ability to alleviate the ill conditioning of the structural frequency response functions.

Dynamic response analysis of linear stochastic truss structures under stationary random excitation

This paper presents a new method for the dynamic response analysis of linear stochastic truss structures under stationary random excitation. Considering the randomness of the structural physical parameters and geometric dimensions, the computational expressions of the mean value, variance and variation coefficient of the mean square value of the structural displacement and stress response under the stationary random excitation are developed by means of the random variable's functional moment method and the algebra synthesis method from the expressions of structural stationary random response of the frequency domain. The influences of the randomness of the structural physical parameters and geometric dimensions on the randomness of the mean square value of the structural displacement and stress response are inspected by the engineering examples.

A generally applicable 3D truck ride simulation with coupled rigid bodies and finite element models

The aim of this paper is to present a three dimensional ride simulation for a medium weight truck equipped with two axles. The dynamics of both a rigid body and a flexible frame structure are considered in the vehicle model. The road surface irregularities are modelled as stationary Gaussian random excitations taking into account the cross correlation between the vehicle axles and tracks. The free-free modal properties of the frame structure have been calculated using the FEM (finite-element method) considering two different configurations for the vehicle frame. The results from the FE models are incorporated with the general vehicle model to study their effect on the vehicle ride behaviour. The equations of motion of the general vehicle model have been formulated applying the Lagrangian approach and the modal superposition theory. Random responses, including power spectral densities and root mean square values, are calculated and analysed. The results covered 23 response variables including driver, cab and body accelerations and the induced dynamic forces in the cab and main suspension systems and tyres. The results showed that the structure dynamics of the frame strongly affects several response variables. Moreover, it has been found that the present simulation technique is practical, powerful, and economic, and can be commercially used for the assessment of truck dynamics. Conclusions and recommendations for further work are given.

Random loading identification of a plastic glass cantilever beam

The loading power spectral density matrices of a cantilever beam of plastic glass subjected to stationary random excitations are identified by utilizing inverse pseudo excitation method using response measurements and structural frequency response functions. Numerical simulation is conducted for the purpose of optimal selection of sensor locations. Advantages of this method include its computational efficiency compared with other approaches and the benefit of alleviating the ill conditioning of frequency response function near some resonant frequencies. The identified loadings agree with actual ones perfectly well and this method is robust to random noise.