简谐荷载作用下伴生自由振动的研究

本文基于结构动力学的基本理论，推导了简谐荷载作用下，瞬态自由振动反应的理论解。与受迫振动的理论解比较，详细研究了伴生自由振动的各种动力反应（位移、加速度、内力）的振动规律。研究结果表明，在一定的参数范围内，伴生自由振动不能忽略，有时甚至是动力反应中的主要成分。     

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.     

A new friction law for sliding rigid blocks under cyclic loading

A rigid block sliding down an inclined plane under the action of gravity was monitored with accelerometers and an LVDT to investigate how the transition from static to kinetic friction develops. Once the transition patterns were identified, experiments were carried out to study the response of a dynamically excited rigid block sliding down the inclined plane of a shaking table. Harmonic time series were used as input motions. The laboratory results allowed the definition of a continuous friction law to model the continuous variation of the friction from its static to kinetic condition. This law was implemented into the commercial 3D distinct element code 3DEC to numerically reproduce the experiments carried out, thus validating the friction law. Afterwards, the friction law was used to evaluate the sliding due to the kinetics of the block. Three cases were analyzed: constant friction coefficient, Coulomb friction law and the proposed friction law. These computations were compared to laboratory measurements. It is found that permanent displacements computed by considering the new law are in better agreement with laboratory measurements.     

First passage of a sliding rigid structure on a frictional foundation

The first passage problem for a sliding rigid block on a frictional foundation, subjected to Gaussian white noise through the foundation, is studied. A well-posed boundary value problem is formulated from Markov process theory and solved for the ordinary moments of time to first passage by the finite element method. A range of parameters is studied representing a simple seismic base isolation system. Results of Monte Carlo simulation are given for comparison.     

摩擦滑动起始的动力学特征

摩擦界面的失效方式对于我们从根本上认识工程学至地震学领域中的失效过程非常关键.摩擦运动起始于将两个剪切体分离的薄界面内传播的破裂前沿.通过测定沿界面的剪切应力和法向应力,结合随后的实际接触面积动态学特性,我们发现剪切应力与法向应力之比在局部可远远超过没有发生突然滑动的静摩擦系数.而且,系统所选的不同破裂模式与局部应力比...     

Vibration characteristics of rigid body placed on sand ground

This study aims to investigate experimentally the vibration properties of rigid body placed on sand ground surface. The rigid body models with circular or rectangular base with variable mass, inertial moment and base sizes, were prepared, and the vibration behavior was observed in some series of free vibration tests and forced vibration tests. The observed behavior was analyzed and the vibration mode, vibration period and damping ratio were examined. It was found that the natural vibration period depends not only on the mechanical properties of rigid body and ground, but also on the magnitude of vibration amplitude. This suggested the notable effect of nonlinear strain dependent stiffness of ground material. A physical model with distributed spring–dashpot element was used to model the interactive mechanical behavior between rigid body and ground. The stiffness of the spring–dashpot element was evaluated through the modal analysis of the observed vibration behaviors. The effects of base shape, base size and base pressure on the stiffness of spring–dashpot element are discussed. The spring–dashpot model was verified with the behavior observed in forced vibration tests.     

Steady-state harmonic response of a rigid plate bearing on a liquid-saturated poroelastic halfspace

Soil–structure interaction problems are typically modelled by assuming subgrade behaviour to be either elastic or viscoelastic. Herein, compliance functions that may be used to solve soil–structure interaction problems are evaluated by treating the subgrade as a liquid-saturated poroelastic material whose behaviour is governed by Biot's theory. The compliances are evaluated for the harmonic rocking and vertical motions of rigid permeable and impermeable plates bearing on a poroelastic halfspace. Comparisons are made with elastic solutions which assume the subgrade to be either completely drained or undrained. Also, solid and fluid contact stresses are reported for the poroelastic case and compared to the solid contact stresses for the elastic cases.     

水热条件下富层状硅酸盐矿物糜棱岩的摩擦特性实验研究

为探索断层岩石摩擦特性对于断层力学性质的影响,我们采集了龙门山汶茂断裂韧性剪切带中的富含层状硅酸盐矿物的糜棱岩样品进行了水热条件下摩擦滑动实验研究.实验在三轴压机之上完成,实验温度为100~600℃,有效正压力100 MPa,孔隙水压分别为30 MPa和130 MPa.为获得摩擦滑动的稳定性参数（a-b）,剪切滑移速率在1.22 μm·s^-1,0.244 μm·s^-1和0.0488 μm·s^-1之间切换.实验发现在200~500℃的温度范围内,摩擦系数随着温度的增加而显著增大（约0.56~0.72）.在200~300℃范围内,随温度的升高糜棱岩的摩擦滑动表现出由稳定的速度强化向不稳定速度弱化转变的趋势.在有效正压力不变的情况下,孔隙水压的增大会促进糜棱岩的摩擦滑动在500~600℃温度范围内由不稳定的速度弱化向稳定的速度强化的转变.实验给出的断层在原地深度处的脆性和塑性变形机制的转变,有助于理解断层深部的地震成核机制以及成核的温压条件.

     

Periodic response of a rigid block resting on a footing subjected to harmonic excitation

This study presents the dynamic behaviour of a rigid block which rests on a footing supported by a spring and a dashpot on a rigid base. The response of the rigid body is examined carefully when the base is excited by a harmonic force. It is found that a periodic motion appears in three different modes: stick-stick, stick-slip and slip-slip. The conditions that initiate the stick-stick and slip-slip modes are derived in explicit forms and the maximum sliding displacement is also obtained analytically. Useful dimensionless parameters are proposed for the presentation of the dynamic behaviour. The accuracy of results is confirmed by the response history computed by the Nigam-Jennings method.     

Linearization and first‐order expansion of the rocking motion of rigid blocks stepping on viscoelastic foundation

In structural mechanics there are several occasions where a linearized formulation of the original non‐linear problem reduces considerably the computational effort for the response analysis. In a broader sense, a linearized formulation can be viewed as a first‐order expansion of the dynamic equilibrium of the system about a ‘static’ configuration; yet caution should be exercised when identifying the ‘correct’ static configuration. This paper uses as a case study the rocking response of a rigid block stepping on viscoelastic supports, whose non‐linear dynamics is the subject of the companion paper, and elaborates on the challenge of identifying the most appropriate static configuration around which a first‐order expansion will produce the most dependable results in each regime of motion. For the regime when the heel of the block separates, a revised set of linearized equations is presented, which is an improvement to the unconservative equations published previously in the literature. The associated eigenvalues demonstrate that the characteristics of the foundation do not affect the rocking motion of the block once the heel separates. Copyright © 2008 John Wiley & Sons, Ltd.     

Response of rigid body assemblies to dynamic excitation

Shaking table tests were conducted to investigate the response of rectangular wooden blocks and block assemblies of various sizes and slenderness to harmonic and earthquake base excitation. The shaking tests were followed by an analytical and a numerical study of response of single blocks and block assemblies. The analytical study was aimed at establishing criteria for the initiation of rocking and of overturning in response to harmonic base motion and consisted of solving numerically the differential equations of motion of a rigid block on a rigid foundation. The numerical study, in the course of which the response of both single blocks and block assemblies was examined, was implemented by means of the Distinct Element Method (DEM). Prior to the DE simulation of actual shaking tests, preliminary analyses were conducted to confirm numerical stability and to evaluate material and damping parameters. Comparing the recorded time histories with those given by the analytical study and the DE simulation, good agreement was found. The distinct element model in use appeared to follow the highly non-linear motion of rigid body assemblies faithfully to reality. On the basis of the results, provided that the necessary parameters are carefully estimated, the employed DE model can be regarded as an appropriate tool to simulate response of rigid body assemblies to dynamic base excitation.     

The dynamics and thermodynamics of precipitation loading

A one-dimensional, time-dependent numerical cloud model is used to analyze the factors in the dynamic and thermodynamic equations which lead to a steady-state or nonsteady-state solution for the cloud vertical motion, buoyancy, precipitation, and cloud water fields. ‘Bulk water’ microphysical techniques are used for the cloud, rain, and hail variables. An atmospheric sounding from a severe storm situation is used as initial and environmental conditions, yielding model updrafts of 40 m sec^?1 maximum and more than 10 m sec^?1 over the entire cloud region. ‘Early conversion’ of the cloud water to rain leads to loading of lower portions of the updraft by rain, the formation of appreciable amounts of hail by freezing of the supercooled rain, and subsequent loading of the middle and upper portions of the updraft so that the updraft erodes throughout the cloud depth and the cloud dissipates, yielding a vigorous rain shower. A delay in the conversion of the cloud water to rain results in a steady-state solution, no rain or hail falling through the updraft. A two-dimensional cloud simulation of this same case shows rain and hail in the upper cloud regions recycled in the two-dimensional flow into the updraft near cloud base and a breakdown of the updraft with resultant rainout (negligible hail reaching the ground). The breakdown of the updraft has profound effects on the temperature field within the cloud, causing the lapse rate to deviate from the steady-state condition and approach the initial environmental conditions. The results emphasize the fact that the local change in temperature (and other dependent variables as well) is not independent of the vertical velocity, in general. This has implications for the interpretation of measurements made within clouds.     

Effects of turbidity caused by nutrient loading on benthic dynamics

The benthic response to nutrient loading of the water column was determined in the transition from early to mid-summer in a semi-enclosed, shallow estuarine ecosystem of the Baltic Sea. The procedure involved the inoculation of an in situ mesocosm with nitrogen and phosphorus to induce hypertrophic conditions, and comparison of this treated compartment with an untreated control. Sediment activity was assessed as oxygen turn-over, and measured in the laboratory with a simulated in situ polarographic flow-through method. Nutrient addition induced the development of pelagic organisms, particularly phytoplankton and protozoa, which resulted in turbidity within the water column. The latter appeared to depress benthic primary productivity. On the other hand, the greater organic matter load, upon sedimenting, caused increased biological respiration of the sediment. Chemical oxygen demand was unaffected by the increased organic matter input.     

The dynamics of a rigid foundation on the surface of an elastic half-space

The response of a rigid rectangular foundation block resting on an elastic half-space has been determined by considering first the displacement functions for any position on the surface of an unloaded half-space due to a harmonic point force. The influence of the foundation has been taken into account by assuming a relaxed condition at the interface, i.e. a uniform displacement under the foundation and that the sum of the point forces must equal the total applied force. The three motions of vertical, horizontal and rocking have been considered and numerical values for the in-phase and the quadrature components of the displacement functions are presented for a Poisson's ratio of 0.25. The effect of the mass and inertia of the foundation can be allowed for by an impedance matching technique. Response curves and non-dimensional resonant frequency curves are given for a square and a rectangular foundation for different mass and inertia ratios and for several values of Poisson's ratio. These curves are for design purposes and are an addition to similar published curves for circular and infinitely long rectangular foundations. Some of the calculated results have been verified by a laboratory experiment.     

Analytical models for the rigid body motions of skew bridges

The rigid body motions of the bridge deck, along with the impact between the bridge deck and the abutments, were the source of extensive damage on skew highway bridges during the 1971 San Fernando earthquake. In this paper, a model for the rigid body motions of skew bridges is presented and analysed. The focus of the model is the appropriate representation of the impact between the bridge deck and the abutments and the explanation of the inducement of in-plane rotational vibrations of the bridge deck as the result of this impact. A simplified model is briefly described first, and the kinematic mechanism of the problem is explained. Then, the analysis of a more detailed and realistic model follows. This model is applied on a short skew bridge located in Riverside, California. The response of this bridge to several earthquake shakings revealed that the planar rigid body rotations of the deck are induced primarily as a result of the skewness of the deck and the impact between the deck and the abutments.     

热水条件下黑云母断层泥的摩擦强度与稳定性

黑云母是自然界常见的层状硅酸盐矿物,其摩擦系数不高且化学稳定性好,对其摩擦性质的关注可能会对弱断层的研究有所帮助.本次工作选取的实验温度条件对应于典型地壳强度模型中脆塑性转化带的范围,为300 ℃和400 ℃.有效正应力为200 MPa,孔隙水压包括10 MPa和30 MPa,在此条件下对黑云母模拟断层泥进行摩擦实验研究.实验得出黑云母的摩擦系数平均在0.36左右.速度依赖性随温度升高速度弱化的程度增强,表现为300 ℃为十分微弱的速度弱化,而在400 ℃出现了黏滑行为,代表了更强的速度弱化.显微结构中同时出现了脆性剪切变形和塑性扭折变形,但决定宏观力学性质的显然是脆性剪切变形.在黑云母存在的情况下,本研究的实验结果有助于理解大陆地壳脆塑性转化带中地震的可能性和弱断层深部的变形机制、宏观力学行为以及地震活动.     

Spherical harmonic analysis of the Navier-Stokes equation in magnetofluid dynamics

The equation of motion (Navier-Stokes equation) for a uniformly rotating, compressible, magnetic, viscous fluid is analyzed in terms of infinite series of spherical surface harmonics. Differential equations are obtained for the radial functions of the poloidal and toroidal harmonics of the velocity, corresponding to those obtained by Bullard and Gellman for the magnetic field from the electromagnetic induction equation. This new analysis opens the way for the dynamical problem of electromagnetic induction in the earth's core to be considered by the spherical harmonic method.     

Propagation of normal modes due to impulsive loading to 3-D medium on a rigid basement

Generally, waves in layered media include several surface wave modes showing dispersive properties. These properties often complicate the understanding of the propagation characteristics of elastic waves. In this paper, not only a method for decomposing transient elastic waves into normal modes is presented but also investigations are applied to decomposed normal modes. This method is developed by means of DWFE and the modal superposition method. In the formulation process, transfer functions describing the relationship between the amplitude of normal modes and that of force density are defined. It was found that the propagation properties of the normal modes were well explained by the dispersion curves and transfer functions. From investigations of the normal mode propagation, properties of each normal mode can be discussed.     

On free nutation of the rotation vector of a rigid body

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Semiactive control of rigid blocks under earthquake excitation

This paper investigates the usefulness of a semiactive control to reduce the overturning vulnerability of a rigid block on a rigid plane under earthquake excitation. The proposed feedback law is used to set the stiffness of restraints placed at the 2 lower corners of the block. The performance of the semiactive control is numerically validated by subjecting the block to 100 recorded accelerograms. Specific simulations are performed to study the effect of different anchorage design parameters on the utility of the control. Finally, the robustness of the proposed control is addressed with respect to typical issues of the real‐world implementation.