Calculation of selected ordinates of fourier spectra

A method is presented for calculating single or, by repeated use, selected ordinates of Fourier spectra rapidly and efficiently. The approach differs from standard methods in that any frequency consistent with the interval of digitization may be chosen and records of arbitrary length may be analysed. The method is based on the relation between Fourier spectra of an accelerogram and the final response of an undamped oscillator to the same accelerogram. This relation is reviewed and extended to aid the development of the computing method and to assist in the interpretation of Fourier spectrum techniques in earthquake engineering. Because only the terminal values of the response of an undamped oscillator are required to define the Fourier spectra, repeated application of superposition is used to form equivalent excitations with durations of only one-quarter of the period of the oscillator.     

Non-monotonic optimal damper placement via steepest direction search

An efficient and systematic procedure is proposed for finding the optimal damper positioning to minimize the dynamic compliance of a 3-D shear building model. The dynamic compliance is expressed in terms of the transfer function amplitudes of the local interstorey drifts evaluated at the undamped fundamental natural frequency. The dynamic compliance is minimized subject to a constraint on the sum of the damping coefficients of added dampers. Optimality criteria are derived and the optimal damper positioning is determined via an original steepest direction search algorithm. This algorithm enables one to find an optimal damper positioning sequentially for gradually increasing damper capacity levels. A non-monotonic design path with respect to the total damper capacity level often appears in the application of this algorithm. A new augmented algorithm via parameter switching is devised to find this non-monotonic design path. Copyright © 1999 John Wiley & Sons, Ltd.     

Quantification of ground-motion parameters and response spectra in the near-fault region

This study focuses on the characteristics of near-fault ground motions in the forward-direction and structural response associated with them. These ground motions are narrow-banded in nature and are characterized by a predominant period at which structures excited by them are severely affected. In this work, predominant period is defined as the undamped natural period of a single-degree-of-freedom (SDOF) oscillator at which its 5% damped linear elastic pseudo-spectral velocity (PSV) contains a clear and dominant peak. It is found that a linear relationship exists between predominant period and seismic moment. An empirical equation describing this relationship is presented by using a large set of accelerograms. Attenuation equations are developed to estimate peak ground velocity (PGV) as a function of earthquake magnitude and source-to-site distance. In addition, a predictive equation for spectral shapes of PSV (i.e., PSV normalized by PGV) is presented as a continuous function of the undamped natural period of SDOF oscillators. The model is independent of PGV, and can be used in conjunction with any available PGV attenuation relation applicable to near-fault ground motion exhibiting forward-directivity effects. Furthermore, viscous damping of the SDOF is included in the model as a continuous parameter, eliminating the use of so-called damping correction factors. Finally, simple equations relating force reduction factors and displacement ductility of elasto-plastic SDOF systems are presented.     

Solution of random structural system subject to nonstationary excitation: transforming the equation with random coefficients to one with deterministic coefficients and random initial conditions

This paper deals with the lower order (first four) nonstationary statistical moments of the response of linear systems with random stiffness and random damping properties subject to random nonstationary excitation modeled as white noise multiplied by an envelope function. The method of analysis is based on a Markov approach using stochastic differential equations (SDE). The linear SDE with random coefficients subject to random excitation with deterministic initial conditions are transformed to an equivalent nonlinear SDE with deterministic coefficients and random initial conditions subject to random excitation. In this procedure, new SDE with random initial conditions, deterministic coefficients and zero forcing functions are introduced to represent the random variables. The joint statistical moments of the response are determined by considering an augmented dynamic system with state variables made up of the displacement and velocity vectors and the random variables of the structural system. The zero time-lag joint statistical moment equations for the augmented state vector are derived from the Itô differential formula. The statistical moment equations are ordinary nonlinear differential equations where hierarchy of moments appear. The hierarchy is closed by the cumulant neglect closure method applied at the fourth order statistical moment level. General formulation is given for multi-degree-of-freedom (MDOF) systems and the performance of the method in problems with nonstationary excitations and large variabilities is illustrated for a single-degree-of-freedom (SDOF) oscillator.     

基于输出反馈的建筑结构闭开环次优控制

对传统的结构抗震闭开环控制算法进行改进。基于地面运动自回归模型,采用Kalman滤波利用可以量测到的地面加速度激励对未来时段即将发生的地面加速度激励进行预估,并在微分方程的求解中引入精确高效的精细积分算法。考虑到实际控制中量测全部状态变量的困难,改进算法仅需量测部分状态变量。数值仿真表明,基于输出反馈的闭开环次优控制策略能大大降低结构的地震响应。     

Non-linear inversion of resistivity profiling data for some regular geometrical bodies1

The inversion of resistivity profiling data involves estimation of the spatial distribution of resistivities and thicknesses of rock layers from the apparent resistivity data values measured in the field as a function of electrode separation. The drawbacks of using traditional curve-matching techniques to solve this inverse problem have been overcome by iterative linear techniques but these require good starting models even if the shape of the causative body is asssumed known. In spite of the recent developments in inversion techniques, no robust method exists for the inversion of resistivity profiling data for the simple model of dikes and spheres which are the classical models of geophysical prospecting. We apply three different non-linear inversion schemes to invert synthetic resistivity profiling data for the classical models embedded in a uniform matrix of contrasting resistivity. The three non-linear algorithms used are called the Metropolis simulated annealing (SA), very fast simulated annealing (VFSA) and a genetic algorithm (GA). We compare the performance of the three algorithms using synthetic data for an outcropping vertical dike model. Although all three methods were successful in obtaining optimal solutions for arbitrary starting models, VFSA proved to be computationally the most efficient.     

A note on the numerical evaluation of Duhamel's integral

General expressions are derived for the numerical evaluation of Duhamel's integral and its derivative. The work comprises an extension (to unequal time steps) and an application (to a piecewise linear forcing function) of the numerical integration approach adopted by Cronin.¹ The application is particularly relevant to the digital computation of response spectra from strong motion earthquake records.     

Study of a smart platform based on backstepping control method

A structural model is significant for the verification of structural control algorithms. However, for nonlinear behavior, experiments are mostly destructive tests that are costly, and conducting repetitive structural experiments is difficult. Therefore, a repetitive structural vibration model is important for structural vibration control. In this study, a smart platform to realize different structural behaviors is developed based on the backstepping control algorithm. Lyapunov functions are used to derive the control law. Simulations show that the designed model can track the structural responses of different arbitrary linear structures very well. In addition, the proposed platform can track responses of different piecewise linear structures and desired models with various hysteresis very well. Numerical results verify the effectiveness of the proposed tracking controller through the backstepping method for the established platform.     

On the optimality criterion in structural control

The classical performance index optimization control algorithm is considered in order to check the real optimality of the control procedure; the basic steps for the optimal algorithm are reviewed, and the equation for the optimal control force derived. It is shown that the optimality conditions cannot be met with regard to the performance index, unless one is concerned with simple free oscillations. It is proved that in this case on one side the optimal control turns out to be of the linear closed‐loop type, yielding explicit optimal control coefficients, and on the other side that no solution can exist of the optimal problem for a generic forcing function. It is concluded that one is forced to calibrate the control force for free oscillations, and that the reliability of the index procedure mainly rests on some implicit expectation that linear control can be calibrated in the absence of the external disturbance and that it works under forced oscillations as well. Furthermore, the problem of delayed active control, with reference to a s.d.o.f. system controlled by a closed‐loop linear algorithm and under the action of a dynamic forcing function is investigated. In particular, the effects produced on the response of the structure by the introduction in the control law of assessed critical values of time delay are analysed and the comparison is proposed between the numerical results that one gets by adopting two different procedures (on one hand the above‐mentioned optimal linear control law and on the other hand the constrained minimization of the structural response norm) to compensate for time lag occurring in the actuation of the active control servomechanisms. Copyright © 2000 John Wiley & Sons, Ltd.     

Ground-Motion Prediction Equations (GMPEs) for inelastic displacement and ductility demands of constant-strength SDOF systems

The objective of this paper is to present ground-motion prediction equations for ductility demand and inelastic spectral displacement of constant-strength perfectly elasto-plastic single-degree-of-freedom (SDOF) oscillators. Empirical equations have been developed to compute the ductility demand as a function of two earthquake parameters; moment magnitude, and source-to-site distance; one site parameter, the ground type; and three oscillator parameters, an undamped natural period, critical damping ratio, and the mass-normalized yield strength. In addition, a comparative study of the proposed model with selected previous studies and recommendations of Eurocode 8 is presented. Proposed equations can easily be incorporated in existing probabilistic seismic hazard analysis (PSHA) software packages with the introduction of an additional parameter. This leads to hazard curves for inelastic spectral displacement, which can provide better estimates of target displacement for nonlinear static procedures and an efficient intensity measure for probabilistic seismic demand analysis (PSDA). Proposed equations will be useful in performance evaluation of existing structures.     

On the interaction between a radiatively damped planetary wave and the zonally averaged circulation in the middle atmosphere

The interaction between a planetary wave damped by cooling to space and the zonally averaged circulation in the middle atmosphere is examined for a steady-state situation in middle latitudes. Quasi-geostrophic scaling of Type 2 is assumed (i.e. the space scales are planetary and the superrotation is small).A set of mean equations is derived for this scaling which is complementary to the set of perturbation equations previously studied. The mean equations show that a planetary wave induces a mean meridional circulation which is balanced by an eddy momentum forcing function and a mean diabatic heating which is balanced by an eddy heat flux forcing function. The vertical eddy fluxes enter the forcing at the same order as the horizontal eddy fluxes.An analytical wave solution is found for the case of an atmosphere in constant superrotation. The eddy fluxes and forcing functions are evaluated for this special case. It is found that they are very sensitive to the values of the radiative damping coefficient and the superrotation. Since the damping coefficient depends on the ozone concentration and the intensity of the solar ultraviolet flux, the results suggest that changes in these quantities can strongly modify the wave-mean flow interaction in the middle atmosphere. Possible implications for climate change are discussed.     

Energy-consistent integration method and its application to hybrid testing

The time-integration algorithm is an indispensable element to determine response of the boundary of the numerical as well as physical parts in a hybrid test. Instability of the time-integration algorithm may directly lead to failure of the test, so stability of an integration algorithm is particularly important for hybrid testing. The explicit algorithms are very popular in hybrid testing, because iteration is not needed. Many unconditionally stable explicit-algorithms have been proposed for hybrid testing. However, the stability analysis approaches used in all these methods are valid only for linear systems. In this paper, a uniform formulation for energy-consistent time integrations, which are unconditionally stable, is proposed for nonlinear systems. The solvability and accuracy are analyzed for typical energy-consistent algorithms. Some numerical examples and the results of a hybrid test are provided to validate the effectiveness of energy-consistent algorithms.     

Investigation of the elastic characteristics of a three-storey steel structure using system identification

In this report, three different models in increasing order of complexity have been used to identify the seismic behaviour of a three-storey steel structure subjected to arbitrary forcing functions, all of which excite responses within the elastic range. All of the models are constructed using system identification. In the first model, five parameters have been used to identify the frame. Treating the system as a shear building, we assign one stiffness coefficient to each floor and introduce Rayleigh-type damping with two additional parameters. The mass, assumed to be concentrated at a floor level, is kept constant throughout the study. The parameters are established using a modified Gauss-Newton algorithm. The match between measured and predicted quantities is satisfactory when these quantities are restricted to floor accelerations or displacements. To remove the constraint imposed by assuming that the frame deforms as a shear building, a second model with eight parameters is introduced, allowing rotations of the joints as independent degrees of freedom. Six of the eight parameters are related to the stiffness characteristics of the structural members while the remaining two are related to damping as before. In constructing the eight-parameter model, we learned that it is the effective lengths of the members that change during optimization. We also found that the independent response quantities, floor accelerations and joint rotations, must be used in the cost function for the optimization algorithm to converge. The match between measured and predicted quantities for the eight-parameter model is excellent. The set of parameters derived from the minimum squared error gives a model that shows very good correlation using information on the full duration of the pulse or only a portion of it. Also the same correlation exists between the coefficients obtained from different excitations. In an effort to explain the values of the parameters associated with the girders, an additional degree of freedom, namely, the pitching motion of the shaking table, is introduced as an additional degree of freedom. The paper presents, therefore, a five-, an eight- and, finally, a nine-parameter model.     

Earthquake response analysis considering non-proportional damping

Non-proportional damping may be defined as a form of linear viscous damping which introduces coupling between the undamped modal co-ordinate equations of motion. The standard mode superposition method of earthquake response analysis therefore cannot be employed with non-proportionally damped structures. In this paper, several methods for analysing the dynamic response of non-proportional damped structures are outlined. It is concluded that the most efficient procedure is to express the response in terms of a truncated set of undamped modal coordinates and to integrate directly the resulting coupled equations. The effectiveness of the method is demonstrated by a numerical example.     

Solution of dynamic response of SDOF system using piecewise Lagrange polynomial

As an extension of the procedure in which an arbitrary dynamic loading is approximated by piecewise linear segments, the second‐ and third‐degree piecewise Lagrangian interpolating polynomials are employed to approximate an arbitrary dynamic loading in the Duhamel integral for the solution of dynamic response of a SDOF system. The related formulae are derived. The proposed method offers computational advantage over the traditional step‐by‐step solution techniques for comparable accuracy, and far better accuracy than the piecewise linear approximation procedure for comparable time interval when the loading cannot be represented by straight‐line segments. Copyright © 2001 John Wiley & Sons, Ltd.     

Stochastic simulation of volcanic tremor from Ruapehu

The most common volcanic tremor produced by Ruapehu is a continuous signal with a dominant frequency of about 2 Hz. This signal has a sharply peaked spectrum, and an autocorrelation function with a high degree of coherence, even for lags of over 20 seconds. These characteristics strongly suggest that the cause of this tremor is a single resonator, probably a fluid-filled cavity resonating in an “organ-pipe” mode.The stochastic simulation of such a resonator uses the equation of motion of a Simple Harmonic Oscillator, which applies to an “organ-pipe” fundamental resonance, with either the characteristics of the oscillator, or the forcing function, containing a random element. A “white noise” forcing function, which would be appropriate for excitation of the cavity by a high pressure gas input, gave good agreement with the observed spectra and autocorrelation functions. Another possible model used an oscillator with a damping factor which varied randomly, and was sometimes negative, so oscillations built up, rather than decayed. This also gave a reasonable simulation of Ruapehu tremor.The third excitation model used a Poisson process, in which during each time interval there was a certain probability of applying a fixed impulse to the resonator. It was found that the impulses had to be frequent, i.e. several times a second, to match the characteristics of Ruapehu tremor.It has been suggested that tremor is composed of a succession of low-frequency (“B-type”) earthquakes. The results of this simulation show that at Ruapehu tremor could be produced by a resonator with positive feedback just sustaining oscillation, or by a resonator excited by external impulses. The most promising model for low-frequency earthquakes describes them as the result of a major external disturbance of the resonator.     

Subharmonic oscillations of a forced hydromagnetic cavity

Abstract

We study the propagation of nonlinear MHD waves in a highly magnetized plasma cavity. The cavity's moving boundaries generate Alfvén waves, which in turn drive and interact with slow magnetosonic waves. The interacting wave system is analyzed by a Galerkin and multiple-scale analyses leading to simple dynamical equations. When the frequency of the forcing provided by the moving boundaries and that of the fundamental Alfvén eigenmode are close, the cavity behaves like a Duffing oscillator. Application of the Melnikov function theory shows that the Alfvén wave's amplitude undergoes both flip and saddle-node bifurcations as the amplitude and the phase of the boundary forcing vary. Direct numerical integration confirms these results and provides an estimate of the amount of energy dissipated in the bifurcations.     

An algorithm to calculate the gravity anomaly of sedimentary basins with exponential density-depth relationships

We derive wavenumber domain expressions to calculate the gravity anomaly of a body with irregular bounding surfaces and an exponential density‐depth relationship. We apply the method to sedimentary basins, which commonly have this type of geometry and density distribution. The mathematical formulation also allows the exponential density‐depth relationship to be measured from an arbitrary irregular surface rather than the top surface. Using this arrangement, the gravity anomaly of exhumed sedimentary basins can be predicted if the amount of eroded section can be estimated. The corresponding inverse algorithms are also derived. Examples of the use of the forward algorithms, from the Galicia Interior Basin and the Central Irish Sea Basin, are used to illustrate these methods.     

Uncertainty estimates for surface runoff models

A lower bound for variance in surface runoff modelling estimates is advanced. The bound is derived using a linear unit hydrograph approach which utilizes a discretization of the catchment into an arbitrary number of subareas, a linear routing technique for channel flow effects, a variable effective rainfall distribution over the catchment, and calibration parameter distributions developed in correlating rainfall-runoff data by the model. The uncertainty bound reflects the dominating influence of the unknown rainfall distribution over the catchment and is expressed as a distribution function that can be reduced only by supplying additional rainfall-runoff data. It is recommended that this uncertainty distribution in modelling results be included in flood control design studies in order to incorporate a prescribed level of confidence in flood protection facilities.     

Stochastic partial differential equations in groundwater hydrology

Many problems in hydraulics and hydrology are described by linear, time dependent partial differential equations, linearity being, of course, an assumption based on necessity.Solutions to such equations have been obtained in the past based purely on deterministic consideration. The derivation of such a solution requires that the initial conditions, the boundary conditions, and the parameters contained within the equations be stipulated in exact terms. It is obvious that the solution so derived is a function of these specified, values.There are at least four ways in which randomness enters the problem. i) the random initial value problem; ii) the random boundary value problem; iii) the random forcing problem when the non-homogeneous part becomes random and iv) the random parameter problem.Such randomness is inherent in the environment surrounding the system, the environment being endowed with a large number of degrees of freedom.This paper considers the problem of groundwater flow in a phreatic aquifer fed by rainfall. The goveming equations are linear second order partial differential equations. Explicit form solutions to this randomly forced equation have been derived in well defined regular boundaries. The paper also provides a derivation of low order moment equations. It contains a discussion on the parameter estimation problem for stochastic partial differential equations.