Steady state rocking response of rigid blocks part 1: Analysis

The result of a theoretical study on the rocking response of rigid blocks subjected to sinusoidal base motion is presented. The study indicates that, for a given excitation amplitude and frequency, a rigid block can respond in several different ways. Based on analysis, the regions of different classes of steady state symmetric response solutions are mapped on the excitation amplitude-frequency parameter space. The steady state response solutions (both harmonic and subharmonic) are classified into two classes, out-of-phase and in-phase with respect to the excitation. Only out-of-phase solutions are found to be stable. A parametric study shows that steady rocking response amplitude is highly sensitive to the size of the block and the excitation frequency in the low frequency range. It is relatively insensitive to the excitation amplitude and the system's coefficient of restitution of impact. For two blocks of the same aspect ratio and coefficient of restitution subjected to the same excitation, the larger block always responds in smaller amplitude than the smaller block. Computer simulation is carried out to study the stability of the symmetric steady state response solutions obtained from analysis. It is found that as the excitation frequency is decreased beyond the boundary of stable symmetric response, the response becomes unsymmetric where the mean amplitude of oscillation is non-zero. Further decrease in excitation frequency beyond the stable unsymmetric response boundary causes instability in the form of overturning.     

The many steady state responses of a rigid block under harmonic forcing

Recent mathematical advances in the study of the response of a rigid block to horizontal simple harmonic forcing conclude that (i) all types of subharmonic response are possible, (ii) several types can occur at one point in parameter space, (iii) exact expressions are available for stability boundaries in parameter space, (iv) asymmetric solutions exist just outside the upper boundaries of symmetric solutions, (v) period- and impact-doubling cascades occur as parameter values are varied even further outside the boundaries, (vi) aperiodic (or chaotic) responses are possible, (vii) periodic responses can occur which appear to violate West's formula and (viii) steady state responses of the forced system can be so large as to produce toppling of the block if the system were unforced. These results are used to explain and extend recent computational work. Also we find quantitative agreement between our theoretical work and some recent experimental work of Tso and Wong.     

Modal pushover‐based scaling of earthquake records for nonlinear analysis of single‐story unsymmetric‐plan buildings

The modal pushover‐based scaling (MPS) procedure, currently restricted to symmetric‐plan buildings, is extended herein to unsymmetric‐plan buildings. The accuracy of the extended MPS procedure was evaluated for a large set of three‐degree‐of‐freedom unsymmetric‐plan structures with variable stiffness and strength. The structures were subjected to nonlinear response history analysis considering sets of seven records scaled according to the MPS procedure. Structural responses were compared against the benchmark values, defined as the median values of the engineering demand parameters due to 30 unscaled records. This evaluation of the MPS procedure has led to the following conclusions: (i) the MPS procedure provided accurate estimates of median engineering demand parameter values and reduced record‐to‐record variability of the responses; and (2) the MPS procedure is found to be much superior compared to the ASCE/SEI 7‐10 scaling procedure for three‐dimensional analysis of unsymmetric‐plan buildings. Copyright © 2013 John Wiley & Sons, Ltd.     

DYNAMIC RESPONSE OF EQUIPMENT IN STRUCTURES WITH SLIDING SUPPORT

The dynamic behaviour of a single degree-of-freedom (DOF) equipment mounted on a sliding primary structures subjected to harmonic and earthquake ground motions is studied numerically. To deal with the discontinuity nature of sliding structural systems, in this work the fictitious spring model is adopted. With the problem formulated in a state space form, an incremental numerical scheme capable of dealing with multi-DOF sliding structural systems is proposed for solving the time history responses. Numerical examples excited by harmonic and real earthquake ground motions are considered in order to study the following three effects: (1)the variation of the frictional coefficient of the sliding support, (2)subharmonic resonance and (3)effect of tuning (i.e. when the frequency of the equipment is coincident with or close to the fundamental frequency of the primary structure) on the mounted equipment. The dynamic characteristics of the mounted equipment are highlighted in the analysis of the numerical examples. © 1997 by John Wiley & Sons, Ltd.     

Development of basic technique to improve seismic response accuracy of tributary area-based lumped-mass stick models

Although a detailed finite element(FE) model provides more precise results, a lumped-mass stick(LMS) model is preferred because of its simplicity and rapid computational time. However, the reliability of LMS models has been questioned especially for structures dominated by higher modes and seismic inputs. Normally, the natural frequencies and dynamic responses of a LMS model based on tributary area mass consideration are different from the results of the FE model. This study proposes a basic updating technique to overcome these discrepancies; the technique employs the identical modal response, D(t), to the detailed FE model. The parameter D(t) is a time variable function in the dynamic response composition and it depends on frequency and damping ratio for each mode, independent of the structure's mode shapes. The identical response D(t) for each mode is obtained from the frequency adaptive LMS model; the adaptive LMS model which can provide identical modal frequencies as the detailed FE model. Theoretical backgrounds and formulations of the updating technique are proposed. To validate the updating technique, two types of structures(a symmetric straight column and an unsymmetric T-shaped structure) are considered. From the seismic response results including base shear and base moment, the updating technique considerably improves the seismic response accuracy of the tributary area-based LMS model.     

Lanczos method for dynamic analysis of damped structural systems

In this paper we extend the Lanczos algorithm for the dynamic analysis of structures⁷ to systems with general matrix coefficients. The equations of dynamic equilibrium are first transformed to a system of first order differential equations. Then the unsymmetric Lanczos method is used to generate two sets of vectors. These vectors are used in a method of weighted residuals to reduce the equations of motion to a small unsymmetric tridiagonal system. The algorithm is further simplified for systems of equations with symmetric matrices. By appropriate choice of the starting vectors we obtain an implementation of the Lanczos method that is remarkably close to that in Reference 7, but generalized to the case with indefinite matrix coefficients. This simplification eliminates one of the sets of vectors generated by the unsymmetric Lanczos method and results in a symmetric tridiagonal, but indefinite, system. We identify the difficulties that may arise when this implementation is applied to problems with symmetric indefinite matrices such as vibration of structures with velocity feedback control forces which lead to symmetric damping matrices. This approach is used to evaluate the vibration response of a damped beam problem and a space mast structure with symmetric damping matrix arising from velocity feedback control forces. In both problems, accurate solutions were obtained with as few as 20 Lanczos vectors.     

Non-linear steady state vibration and dynamic snap through of shallow arch beams

The non-linear steady state vibration of shallow arch beams is studied by a finite element method based on the principle of virtual work. Both the free and forced periodic vibrations are considered. The axial and flexural deformations are coupled by the induced axial force along the beam element. The spatial discretization is achieved by the usual finite element method and the steady state nodal displacements are expanded into a Fourier series. The harmonic balance method gives a set of non-linear algebraic equations in terms of the vibrating frequency and the Fourier coefficients of nodal displacements. The non-linear algebraic equations are solved by the Newtonian algorithm iteratively. The combined algorithm is called the incremental harmonic balance method. The importance of the conditions of completeness and balanceability is presented. Since the non-linearity is essentially softening, different orders of internal resonances between two modes can occur repeatedly. Isolated response curves are possible and are connected to the bifurcation of a particular excited mode.     

A modal pushover analysis procedure to estimate seismic demands for unsymmetric‐plan buildings

An Erratum has been published for this article in Earthquake Engng. Struct. Dyn. 2004; 33:1429. Based on structural dynamics theory, the modal pushover analysis (MPA) procedure retains the conceptual simplicity of current procedures with invariant force distribution, now common in structural engineering practice. The MPA procedure for estimating seismic demands is extended to unsymmetric‐plan buildings. In the MPA procedure, the seismic demand due to individual terms in the modal expansion of the effective earthquake forces is determined by non‐linear static analysis using the inertia force distribution for each mode, which for unsymmetric buildings includes two lateral forces and torque at each floor level. These ‘modal’ demands due to the first few terms of the modal expansion are then combined by the CQC rule to obtain an estimate of the total seismic demand for inelastic systems. When applied to elastic systems, the MPA procedure is equivalent to standard response spectrum analysis (RSA). The MPA estimates of seismic demand for torsionally‐stiff and torsionally‐flexible unsymmetric systems are shown to be similarly accurate as they are for the symmetric building; however, the results deteriorate for a torsionally‐similarly‐stiff unsymmetric‐plan system and the ground motion considered because (a) elastic modes are strongly coupled, and (b) roof displacement is underestimated by the CQC modal combination rule (which would also limit accuracy of RSA for linearly elastic systems). Copyright © 2004 John Wiley & Sons, Ltd.     

Effective elimination of subharmonic ghost events from vibroseis data

Harmonic or subharmonic noise is often present in vibroseis data as reverberation‐like, laterally coherent bands occurring parallel to and before or after, the main events. Such periodic noise is typically generated during the standard correlation process when the actual source signal travelling through the subsurface is, for whatever reason, different from the desired source signal, i.e., the pilot‐sweep controlling the baseplate and used for correlation. A typical cause can be that harmonic or subharmonic frequency partials are generated in addition to the vibroseis sweep's desired fundamental frequencies. These harmonics produce strong ‘ghost events’ during correlation of the geophone trace with the pilot‐sweep, originating from additional correlations between the fundamental and harmonic frequencies. Especially subharmonic ‘ghosts’ will overlap with ‘good’ fundamental signals, since for typically used up‐sweeps they are folded to later traveltimes, where the signal/noise‐ratio is already lower, thus aggravating or preventing a reliable interpretation of possible later reflections. Here, a method is introduced to remove these unwanted noise trains (with only negligible impact on the fundamental signal) by transforming the seismogram traces into a so‐called ‘(sub)harmonic domain’. In this domain, the respective harmonic noise portions are focused and separated from the fundamental signals, enabling easier detection and appropriate suppression. After back‐transformation to the x‐T domain, the records are free from the corresponding harmonic contamination and can then be processed as usual. The method operates in a data‐driven fashion, i.e., the traces are not uniformly processed but are processed depending upon their actual (sub)harmonic content. The decontamination procedure can be applied universally, i.e., to uncorrelated/correlated and/or vertically unstacked/stacked data either in a manual, semiautomated or fully automated manner. The method works perfectly for synthetic vibroseis traces with or without harmonic/subharmonic portions. The application to real, crustal‐scale vibroseis records that were acquired in 2006 in the Dead Sea region, Israel and that were severely contaminated by subharmonic ground‐roll ghosts covering reflectivity from the basement to the Moho, shows the robustness and success of the presented method.     

Hilbert spectrum and intrinsic oscillation mode of dynamic response of a bilinear SDOF system： influence of harmonic excitation amplitude

Under harmonic wave excitation, the dynamic response of a bilinear SDOF system can be expressed by the Hilbert spectrum. The Hilbert spectrum can be formulated by (1) the inter-wave combination mechanism between the steady response and the transient response when the system behaves linearly, or (2) the intra-wave modulation mechanism embedded in one intrinsic mode function (IMF) component when the system behaves nonlinearly. The temporal variation of the instantaneous frequency of the IMF component is consistent with the system nonlinear behavior of yielding and unloading. As a thorough study of this fundamental structural dynamics problem, this article investigates the influence of the amplitude of the harmonic wave excitation on the Hilbert spectrum and the intrinsic oscillatory mode of the dynamic response of a bilinear SDOF system.     

Simplified mechanical subgrade model for dynamic response analysis of shallow foundations

Baranov proposed an approach for the treatment of the subgrade in the dynamic response analysis of a rigid foundation which is partially embedded. The approach is based on a Winkler approximation together with wave equation solutions developed for simplified conditions. The paper extends and modifies Baranov's approach so as to be applicable to surface foundation problems, by introducing an internal coupling mechanism. Closed form expressions are presented for the steady state harmonic responses of a massless rigid footing and an elastic beam on the surface of the presented subgrade model. These expressions can accommodate the inhomogeneity of the subgrade in a straightforward manner. The model is found to perform well in reproducing the continuous subgrade medium behaviour. The dynamic responses of structures are often highly dependent on the subgrade modelling and therefore great caution is needed in modelling the subgrade.     

Parameter identification of torsionally coupled shear buildings from earthquake response records

This paper presents an efficient procedure to determine the natural frequencies, modal damping ratios and mode shapes for torsionally coupled shear buildings using earthquake response records. It is shown that the responses recorded at the top and first floor levels are sufficient to identify the dominant modal properties of a multistoried torsionally coupled shear building with uniform mass and constant eccentricity even when the input excitation is not known. The procedure applies eigenrealization algorithm to generate the state‐space model of the structure using the cross‐correlations among the measured responses. The dynamic characteristics of the structure are determined from the state‐space realization matrices. Since the mode shapes are obtained only at the instrumented floor (top and first floors) levels, a new mode shape interpolation technique has been proposed to estimate the mode shape coefficients at the remaining floor levels. The application of the procedure has been demonstrated through a numerical experiment on an eight‐storied torsionally coupled shear building subjected to earthquake base excitation. The results show that the proposed parameter identification technique is capable of identifying dominant modal parameters and responses even with significant noise contamination of the response records. Copyright © 2008 John Wiley & Sons, Ltd.     

Experimental study of the response of a gravel streambed to increased sediment supply

If increased sediment supply to a river channel exceeds its transport capacity, deposition necessarily occurs as the bed adjusts to accommodate the increased supply. Both the mean and spatial patterns in bed elevation and grain size may change and an ability to understand their relative importance is needed to predict bed response. We report on an experiment in a field‐scale flume in which sediment supply is increased to a gravel bed with alternate bars. Sediment was recirculated in the experiments, but augmented in two steps, after which the bed was allowed to reach a new steady state. The transport rate at the end of the experiment was three times larger than at the start. High‐resolution sediment flux and topographic measurements, grain size derived from photographs, and hydrodynamic modeling allow us to document the topographic and textural response of the bed to increased sediment supply. The spatial patterns of bed topography and texture were forced by the flume setup and the initial and final steady states included long stationary alternate bars with associated grain size sorting. The transient bed contained several scales of shorter wavelength migrating bedforms superimposed on, and temporarily replacing the stationary alternate bars. Bed topography and textural patterns adjusted to increased sediment supply over different timescales. Bed slope and mean stress increased directly with sediment supply rate to produce a new transport steady state in a time about 2.5 times the minimum needed to deposit the required sediment wedge, indicating a trap efficiency of about 40% for the aggrading wedge. Adjustments in local topography and sorting, primarily in the form of smaller, migrating bars, continued for a period approximately equal to that required to initially reach transport steady state. Copyright © 2013 John Wiley & Sons, Ltd.     

Theory of internal gravity wave saturation

Gravity wave saturation is an important process affecting the transport and deposition of momentum, heat, and constituents in the earth's atmosphere. This paper informally discusses several saturation mechanisms and their effects, including convection, Kelvin-Helmholtz instability, vortical mode instability, parametric subharmonic instability, and mean flow interaction. Convective saturation is emphasized. The parameterization of convective adjustment is discussed and a few remarks are made concerning the effects of turbulence localization on the convective saturation process. Several outstanding problems in saturation theory are identified that could be addressed with observational, numerical, and laboratory studies.     

Optimum viscous damper for connecting adjacent SDOF structures for harmonic and stationary white‐noise random excitations

The dynamic behaviour of two adjacent single‐degree‐of‐freedom (SDOF) structures connected with a viscous damper is studied under base acceleration. The base acceleration is modelled as harmonic excitation as well as stationary white‐noise random process. The governing equations of motion of the connected system are derived and solved for relative displacement and absolute acceleration responses of connected structures. The response of structures is found to be reduced by connecting with a viscous damper having appropriate damping. For undamped SDOF structures, the closed‐form expressions for optimum damping of viscous damper for minimum steady state as well as minimum mean square relative displacement and absolute acceleration of either of the connected SDOF structures are derived. The optimum damper damping is found to be functions of mass and frequency ratio of two connected structures. Further, numerical results had indicated that the damping of the connected structures does not have noticeable effects on the optimum damper damping and the corresponding optimized response. This implies that the derived closed‐form expressions for optimum damper damping of undamped structures can also be used in practical applications for damped structures. Copyright © 2006 John Wiley & Sons, Ltd.     

Dynamic soil–structure interaction of a single-span bridge

Experimental and analytical studies were conducted to determine dynamic soil–structure interaction characteristics of a single-span, prestressed-concrete bridge with monolithic abutments supported by spread footings. The experimental programme, consisting of harmonic forced vibration excitation of the bridge in the transverse and longitudinal directions, revealed the presence of four modes in the frequency band, 0 to 11 Hz, and the onset of a fifth mode at 14 Hz, the highest frequency attained during the tests. The fundamental mode at 4.7 Hz was the primary longitudinal bending mode of the deck and had a relatively low damping ratio (ζ₁), that was approximately 0.025 of critical. The second and third modes at 6.4 Hz and 8.2 Hz were the primary twisting modes of the deck which involved substantial transverse rocking, transverse translation and torsion of the footings. As expected, the damping ratios associated with these two modes, ζ₂ = 0.035 and ζ₃ = 0.15, were directly related to the relative amounts of deck and footing motion. The fourth mode at 10.6 Hz was the second twisting mode of the deck and involved relatively little motion of the footings and abutment walls, which was consistent with the low damping, ζ₄ = 0.02, observed in this mode. The response data at 14 Hz suggested that the fifth mode beyond this frequency was the second longitudinal bending mode of the deck involving longitudinal translation and bending of the abutment walls. A three-dimensional finite element model of the bridge, with Winkler springs attached to the footings and abutment walls to represent the soil–structure interaction, was able to reproduce the experimental data (natural frequencies, mode shapes and bridge response) reasonably well. Although the stiffnesses assigned to the Winkler springs were based largely on the application of a form of Rayleigh's principle to the experimental data, these stiffnesses were similar to theoretical foundation stiffnesses of the same size footings on a linearly elastic half space and theoretical lateral stiffnesses of a rigid retaining wall against a linearly elastic backfill.     

利用GCITEM-IGGCAS模拟DE2潮汐Hough波模对电离层的影响

本文利用GCITEM-IGGCAS模式,从电动力学耦合作用和直接上传两种作用方式,详细模拟研究了DE2潮汐4种Hough波模分量对电离层的影响.我们将不同种类的Hough波模分别输入到模式当中作为底层边界条件,驱动模式模拟得到电离层的电子密度变化,从中分离两种作用机制的响应进行分析.模拟结果发现电离层对DE2的4种Hough波模的响应都表现为半年变化,波峰出现在春季和秋季,波谷则出现在冬季和夏季.一天的变化特性上,赤道对称波模的响应出现明显的4个峰值和谷值,其他3种波模响应主要表现为一个峰值和谷值.4种波模当中赤道对称波模对电离层的作用最为明显,占据主导地位,对电离层的影响表现为波动效应,其中3波分量的响应最强,主要由电动力学作用控制.其他3种波模对电离层则是削弱作用.本研究可以帮助我们更深刻的理解非迁移潮汐对电离层的作用方式和效果.     

Some properties of flows in unsaturated soils with an exponential dependence of the hydraulic conductivity upon the pressure head

A systematic development of the consequences of an exponential dependence of the hydraulic conductivity upon the pressure head is presented. Alternative expressions for the flux are discussed in detail. For steady flows, partial differential equations in terms of the matric flux potential, the pressure head, and the total head are derived. For steady, plane and axially symmetric flows, partial differential equations for the stream function are given. A theoretical basis for the construction of viscous flow analogs for steady, plane and axially symmetric flows is also presented.     

Linearization of rigid body dynamics on frictional interfaces under harmonic loading

The non-linear dynamic response of the rigid block is linearized by means of a friction model that implicitly considers block response through an experimental parameter obtained from shaking table experiments. In view of the great difficulty in carrying out shaking table experiments, in this technical note some recommendations to estimate friction model parameters are given. The selection of parameters considers the sliding response mode of the block–plane-excitation system: stick–slip or continuous sliding. Once all the friction parameters and block response mode were estimated, a methodology was proposed to compute rigid block dynamic response. The numerical results were then compared to actual experimental data for a rigid block sliding on a geotextile–wood interface, along an inclined plane subjected to base harmonic acceleration. Experiments were carried out for both the stick–slip and the continuous sliding modes. Computed and measured responses for both cases showed good agreement, thus indicating that the methodology developed in this research is adequate to capture the physics (of non-linear nature) of rigid blocks sliding on frictional interfaces subjected to complex harmonic loading. The findings encourage the extension of the linearization technique to the more general seismic loading case.     

Analytical solutions to nonlinear conservative oscillator with fifth-order nonlinearity

This paper describes analytical and numerical methods to analyze the steady state periodic response of an oscillator with symmetric elastic and inertia nonlinearity. A new implementation of the homotopy perturbation method (HPM) and an ancient Chinese method called the max-min approach are presented to obtain an approximate solution. The major concern is to assess the accuracy of these approximate methods in predicting the system response within a certain range of system parameters by examining their ability to establish an actual (numerical) solution. Therefore, the analytical results are compared with the numerical results to illustrate the effectiveness and convenience of the proposed methods.