Impact interactions between two vibration systems under random excitation

The exact stationary solutions of the random response of two special vibration systems with impact interactions are formulated in this paper. Between the two systems, the Hertz contact law is used to model the contact process during vibration. A clearance is also introduced. The excitation is assumed to be a stationary white Gaussian process with zero mean and acting on the two systems independently. By assuming the ratio of the excitation and damping parameter of each system to be the same, the exact solutions can be found through solving the time-independent Fokker-Planck equation. The effects of contact stiffness, clearance and the system properties on the response are discussed probabilistically. From this study, it is found that, under some cases, the contact phenomena still play an important role on the response even when the clearance is larger than three times the root mean square response of the corresponding linear systems.     

Fluidization and representative wave transformation on muddy beds

Irregular wave-induced mud fluidization and wave spectrum transformation on muddy profiles are studied through representative wave technique. The constitutive equations of visco-elastic model are adopted for the rheological behavior of fluid mud, while the behavior of stationary mud is assumed to be elastic. A set of representative waves are employed to investigate wave–mud interaction. The results are verified using real field data. Comparing the performance of common representative waves, it is concluded that the phenomena can be better predicted by root mean square wave.     

A response spectrum method for random vibration analysis of mdf systems

A response spectrum method for stationary random vibration analysis of linear, multi-degree-of-freedom systems is developed. The method is based on the assumption that the input excitation is a wide-band, stationary Gaussian process and the response is stationary. However, it can also be used as a good approximation for the response to a transient stationary Gaussian input with a duration several times longer than the fundamental period of the system. Various response quantities, including the mean-squares of the response and its time derivative, the response mean frequency, and the cumulative distribution and the mean and variance of the peak response are obtained in terms of the ordinates of the mean response spectrum of the input excitation and the modal properties of the system. The formulation includes the cross-correlation between modal responses, which is shown to be significant for modes with closely spaced natural frequencies. The proposed procedure is demonstrated for an example structure that is subjected to an ensemble of earthquake-induced base excitations. Computed results based on the response spectrum method are in close agreement with simulation results obtained from time-history dynamic analysis. The significance of closely spaced modes and the error associated with a conventional method that neglects the modal correlations are also demonstrated.     

Particle swarm optimization of TMD by non‐stationary base excitation during earthquake

There are many traditional methods to find the optimum parameters of a tuned mass damper (TMD) subject to stationary base excitations. It is very difficult to obtain the optimum parameters of a TMD subject to non‐stationary base excitations using these traditional optimization techniques. In this paper, by applying particle swarm optimization (PSO) algorithm as a novel evolutionary algorithm, the optimum parameters including the optimum mass ratio, damper damping and tuning frequency of the TMD system attached to a viscously damped single‐degree‐of‐freedom main system subject to non‐stationary excitation can be obtained when taking either the displacement or the acceleration mean square response, as well as their combination, as the cost function. For simplicity of presentation, the non‐stationary excitation is modeled by an evolutionary stationary process in the paper. By means of three numerical examples for different types of non‐stationary ground acceleration models, the results indicate that PSO can be used to find the optimum mass ratio, damper damping and tuning frequency of the non‐stationary TMD system, and it is quite easy to be programmed for practical engineering applications. Copyright © 2008 John Wiley & Sons, Ltd.     

Stochastic analysis of the linear equivalent response of bridge piers with aseismic devices

The dynamic response of bridge piers with aseismic devices to earthquake excitation is evaluated by the stochastic equivalent linearization technique. The seismic acceleration is schematized through a Gaussian stationary random process. The pier is considered linear elastic, the span is idealized as a rigid mass, the restoring force of the device is represented through a non-linear differential model. The study of the complex modes of the linearized system gives an interpretation of the mechanical behaviour, leads to a formally elementary solution and highlights some phenomena which are typical of the hysteretic systems, particularly of those marked by weak hardening. Even though the solution is limited to the stationary field, it brings out several noteworthy considerations about the effective non stationary behaviour of the structure. Copyright © 1999 John Wiley & Sons, Ltd.     

Modal decomposition method for stationary response of non-classically damped systems

The stationary response of multi-degree-of-freedom non-classically damped linear systems subjected to stationary input excitation is studied. A modal decomposition procedure based on the complex eigenvectors and eigenvalues of the system is used to derive general expressions for the spectral moments of response. These expressions are in terms of cross-modal spectral moments and explicitly account for the correlation between modal responses; thus, they are applicable to structures characterized with significant non-classical damping as well as structures with closely spaced frequencies. Closed form solutions are presented for the important case of response to white-noise input. Various quantities of response of general engineering interest can be obtained in terms of these spectral moments. These include mean zero-crossing rate and mean, variance and distribution of peak response over a specified duration. Examples point out several instances where non-classical damping effects become significant and illustrate the marked improvement of the results of this study over conventional analysis based on classical damping approximations.     

A robust delay-compensation technique based on memory

This paper addresses the effects of time delay on actively isolated structures subjected to support excitation. A force proportional to the absolute velocity of the base of the isolated structure applied at the base of the structure (co-located active damping) is the control scheme considered. The actuating mechanisms have some non-zero time response. Consequently, if no compensation is provided, the performance of the system is worse than that of the ideal delay-free controller. The dynamics of the controller-actuator system is modelled by a delay operator on the feedback signal. Time delays producing instability of the controlled structure are investigated, and the effect of time delay on the mean square acceleration of the structure subjected to stationary random excitation is assessed. A delay-compensation technique that requires memory of past control actions is proposed and compared with a commonly used compensation technique. By using simple numerical examples, it is shown that the proposed delay-compensation strategy improves the performance of the system significantly. The technique is extended to other models of the actuator dynamics and modified to provide robustness to small parameter uncertainties.     

Non-stationary random vibrations of a shear beam under high frequency seismic effects

Band-limited, non-stationary random vibrations of a shear beam are studied in order to investigate high frequency seismic effects on building structures. A solution for the evolutionary spectral density of the shear beam response to a time segment of band-limited white noise is given in a closed form. The root mean square (rms) and peak response of the shear beam are studied for two characteristic frequency bands: the conventional 1–4 Hz and higher frequency 4–16 Hz, characteristic for rockburst ground motion. Applying the criterion of equal excitation intensity with constant rms velocity, both responses are analyzed in detail and compared. The “switching off” fundamental mode for high frequency excitations results in characteristic overshoot of the stationary response level by the non-stationary rms response and an amplification of the response in the upper part of the shear beam.     

THE THEORY OF RESPONSE ANALYSIS OF FUZZY STOCHASTIC DYNAMICAL SYSTEMS WITH A SINGLE DEGREE OF FREEDOM

Most real-life structural/mechanical systems have complex geometrical and material properties and operate under complex fuzzy environmental conditions. These systems are certainly subjected to fuzzy random excitations induced by the environment. For an analytical treatment of such a system subjected to fuzzy random excitations, it becomes necessary to establish the general theory of dynamic response of a system to fuzzy random excitations. In this paper, the theory of response, fuzzy mean response and fuzzy covariance response of a single-degree-of-freedom (sdf) system to fuzzy random excitations in the time domain and frequency domain is put forward. The theory of response analysis of an sdf system to both stationary and non-stationary fuzzy random excitations in the time domain and frequency domain is established. Two examples are considered in order to demonstrate the rationality and validity of the theory, and the models of stationary filtered white noise and non-stationary filtered white noise fuzzy stochastic processes of the earthquake ground motion are set up. Methods of analysis for fuzzy random seismic response of sdf systems are put forward using the principles of response analysis of an sdf fuzzy random dynamic system.     

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.     

Simplified analysis of dynamic response of rigid foundations with arbitrary geometries

A simple, efficient numerical procedure is presented for the analysis of the steady-state dynamic response of rigid foundations of arbitrary shape on a homogeneous, isotropic elastic half-space. The contact area of the foundation and soil medium is discretized into square subregions. The influence of the rigid square subregion is approximated by that of an equivalent rigid circular base with the same contact area. This approximation is a better representation of a rigid square subregion than a uniformly stressed subregion used in earlier approaches. Simple expressions are used to determine the influence of the subregion on itself and on the rest of the subregions, without the expense of numerical integration procedures. The present method is demonstrated to converge rapidly. The accuracy of this method is examined by comparison with available published theoretical and experimental results.     

Efficiency of base isolation systems in structural seismic protection and energetic assessment

This paper concerns the seismic response of structures isolated at the base by means of High Damping Rubber Bearings (HDRB). The analysis is performed by using a stochastic approach, and a Gaussian zero mean filtered non‐stationary stochastic process is used in order to model the seismic acceleration acting at the base of the structure. More precisely, the generalized Kanai–Tajimi model is adopted to describe the non‐stationary amplitude and frequency characteristics of the seismic motion. The hysteretic differential Bouc–Wen model (BWM) is adopted in order to take into account the non‐linear constitutive behaviour both of the base isolation device and of the structure. Moreover, the stochastic linearization method in the time domain is adopted to estimate the statistical moments of the non‐linear system response in the state space. The non‐linear differential equation of the response covariance matrix is then solved by using an iterative procedure which updates the coefficients of the equivalent linear system at each step and searches for the solution of the response covariance matrix equation. After the system response variance is estimated, a sensitivity analysis is carried out. The final aim of the research is to assess the real capacity of base isolation devices in order to protect the structures from seismic actions, by avoiding a non‐linear response, with associated large plastic displacements and, therefore, by limiting related damage phenomena in structural and non‐structural elements. In order to attain this objective the stochastic response of a non‐linear n‐dof shear‐type base‐isolated building is analysed; the constitutive law both of the structure and of the base devices is described, as previously reported, by adopting the BWM and by using appropriate parameters for this model, able to suitably characterize an ordinary building and the base isolators considered in the study. The protection level offered to the structure by the base isolators is then assessed by evaluating the reduction both of the displacement response and the hysteretic dissipated energy. Copyright © 2003 John Wiley & Sons, Ltd.     

确定土层地震最大反应的简化方法

本文将水平土层简化为剪切型多自由度线性体系,根据对土层地震反应的振型迭加法所得结果的统计分析,确定体系的传递函数。参考概率理论中有关求极值的办法,给出与功率谱密度相应的反应谱。同时,根据振型均方根的办法,给出土层的加速度、相对位移、剪应变和剪应力的最大反应值沿深度的分布。分析结果表明,文中所提供的方法与精确法所得的结果有很好的吻合。      

穿越逆冲断层的埋地管道非线性反应分析

穿越逆冲断层的埋地管道在地震作用下,容易发生局部屈曲或整体失稳等形式的破坏,研究逆冲断层作用下的埋地管道地震反应规律,对管道抗震设计及施工等具有重要的意义。本文将埋地管线及周围土体从半无限地球介质中取出,分别以空间薄壳单元和实体单元进行离散,采用非线性接触力学方法模拟管、土之间的滑移、分离及闭合现象;采用线性位移加载模拟断层的错动,考虑了系统初始应力状态的影响,对土体未开裂前的管土相互作用系统进行了拟静力数值分析;分析了位错量、土体刚度、埋设深度、径厚比及跨越角度对埋地管道反应的影响,得出了一些有益的结论。     

Stochastic seismic sliding of rigid mass supported through non-symmetric friction

The sliding behaviour of a rigid mass supported on a randomly vibrating foundation through a non-symmetric Coulomb-friction contact is studied both analytically and by numerical simulation. The analysis is based on a stationary solution of the associated Fokker-Planck equation, and makes use of equivalent linearization and of a suitable decomposition of the non-zero mean non-stationary sliding process. It is shown that the analytical results yield several exact asymptotic expansions for both small and large values of time. An extensive Monte Carlo type numerical simulation study produces non-stationary response statistics which are in very good accord with the analytical results. Furthermore, it is found that Gumbel's Extreme Value Distribution reproduces with remarkable accuracy the observed cumulative frequency of maximum slip displacement. The results of this paper may find application in seismic design of embankment dams, earth retaining walls and base ‘isolation’ systems.     

Stochastic analysis of structures and piping systems subjected to stationary multiple support excitations

A stochastic method has been developed for seismic analysis of structures and piping systems subjected to multiple support excitations. In either the time or the frequency domain, mean and extreme values of structural and piping system response can be found, including the effects of cross-correlations of modal response and cross-correlations of multiple support excitations. Stationary white noise and stationary filtered white noise ground excitations are used. A computer program has been developed to carry out the stochastic seismic analysis. Results for a realistic nuclear power plant structure and piping system with and without modal cross-correlations and support excitation cross-correlations are compared. From these results, it is concluded that neglecting cross-correlations can lead to large errors. The stochastic method reported is shown to be more accurate than the response spectrum method and more economical than the time-history method; therefore, it is recommended for seismic analysis of nuclear power plants.     

Response of SDOF system to non‐stationary earthquake excitation

Earthquake excitation is often modelled by non‐stationary random process (i.e. uniformly modulated broad‐band excitation) for analysis of structural safety subjected to seismic hazards. In this paper, the non‐stationary response of a single‐degree‐of‐freedom (SDOF) system to non‐stationary earthquake motion is investigated for different shapes of modulating functions. The evolutionary power‐spectral density function (PSDF) of the displacement of the SDOF system is obtained using the time‐varying frequency response function and the PSDF of the earthquake excitation. The close form expressions for time‐varying frequency response function are derived for different shapes of the modulating functions. In order to study the effects of the shape of the modulating function, a comparison of the non‐stationary earthquake response of the SDOF system is also made for different modulating functions having the same energy content. Copyright © 2004 John Wiley & Sons, Ltd.     

Analysis of non-stationary structural systems by using a band-variable filter

One of the main tools used to study the dynamic response of structural systems is certainly the Fourier Transform. This tool is very useful and reliable to investigating the response of a stationary system, i.e. a generic system that does not changes its characteristics over time. Conversely, the Fourier Transform is no longer reliable if the main goal is to study the evolution of the dynamic response of a system whose features rapidly vary with time. To this regard, several mathematical tools were developed to analyze time-variable dynamic responses. Soil and buildings, subject to transient forcing such as an earthquake, may change their characteristics over time with the initiation of nonlinear phenomena. This paper proposes a new methodology to approach the study of non-stationary response of soil and buildings: a band-variable filter based on S-Transform. In fact, with the possibility of changing the bandwidth of each filtering window over time, it becomes possible to extract from a generic record only the response of the system focusing on the variation of individual modes of vibration. Practically, it is possible to extract from a generic non-stationary signal only the phase of interest. The paper starts from examples and applications on synthetic signals, then examines possible applications to the study of the non-stationary response of soil and buildings. The last application focuses on the possibility to evaluate the mode shapes over time for both numerical and scaled model subjected to strong motion inputs.     

平稳随机激励下随机结构非线性动力响应分析

为了在结构强震灾变过程模拟中正确反映结构参数的随机性并准确界定结构失效模式,提出一种平稳随机激励下随机结构非线性动力响应分析的简化方法.首先运用点估计法将复合非线性随机系统转化为一系列平稳随机激励下的确定性非线性结构,然后再应用成熟的等效线性化方法计算出平稳随机激励下非线性结构动力响应的各阶矩.文末进行了算例分析与验证...     

Seismic response of segmental buildings

This paper proposes an aseismic design concept in which the superstructure of a base-isolated building is divided into several segments. Each segment may comprise a few storeys and is interconnected by additional vibrational isolation systems. The dynamic characteristics of the segmental buildings are investigated. The optimum parameters of the vibration isolation systems are determined by minimizing the mean square acceleration response. The seismic response of a typical segmental building subjected to the N—S component of the 1940 El Centro earthquake input is evaluated and compared with the responses of the corresponding fixed-base and conventional base-isolated buildings. The comparisons show that, when the superstructure is segmented, while the acceleration response in the superstructure remains as small as that in the conventional base-isolated building, the displacement across the base isolation system at foundation level is substantially reduced.