Optimum absorber parameters for simple systems

In the classical problem a damped one degree-of-freedom absorber system is attached to a main system, which has one degree of freedom and is undamped. The optimum values of absorber stiffness and damping, which will minimize the resonant response of the main mass, are well known. In this paper the effect on these optimum conditions of light damping in the main system is studied. The authors show that optimum parameters for absorbers, which are attached to beams and plates, can be obtained simply and accurately from those for an equivalent one degree-of-freedom main system. This depends upon the concept of an effective mass for the elastic body and the representation of its response by the single relevant mode. It will be shown in a later paper that for more complex elastic bodies such as cylindrical shells, for which the natural frequencies are more closely spaced, these simple concepts do not predict accurately optimum absorber parameters.     

Optimum absorber parameters for minimizing vibration response

Optimum parameters are determined for absorbers, which, when attached to one mass of a main system with two degrees of freedom, minimize the harmonic response of that mass. Comparison is made with the absorber parameters that are determined by treating the main system as an equivalent one degree-of-freedom system and using classical results. Close agreement is obtained if the ratio of the two natural frequencies of the main system is reasonably large. This is in agreement with the author's recent work on optimum absorber parameters which minimize the response of elastic bodies. The extension of the method to multi degree-of-freedom main systems is outlined. The conditions for which different values of these parameters are predicted when the response is minimized over narrow and broad frequency bands are determined.     

Minimizing structural vibrations with absorbers

In a companion paper¹ the authors show that the parameters of an absorber which will minimize the resonant response of a simple elastic body can be determined from known results by treating the body as an equivalent single degree-of-freedom system. In this paper cylindrical shells are considered as examples of dynamically complex structures, for which the ratio of the natural frequencies of adjacent modes tends towards unity. It is shown that as dynamic complexity increases optimum absorber parameters for the reduction of resonant response deviate increasingly from those for an equivalent single degree-of-freedom system. Absorbers can be used also to reduce the random response of structures. Simple expressions for optimum parameters are given for an undamped main system, which has one degree of freedom and is subjected to white noise excitation. Optimum absorber parameters for beams, plates and cylindrical shells show similar qualitative behaviour for random and harmonic response with the concept of an equivalent single degree-of-freedom system being applicable only for the simpler structures.     

ENVELOPE OF GREEN'S FUNCTION FOR STRUCTURAL RESPONSE WITH SLIGHTLY DETUNED VIBRATION ABSORBERS

Explicit forms of the modal parameters and the envelope of the Green's function for transient response of main structures equipped with vibration absorbers are derived using a perturbation technique and assuming that the natural frequency of the absorber is slightly detuned from that of the main structure. Applying these perturbation solutions, the influence of the absorber parameters on the dynamic response of main structures is investigated and the ratio of the envelope for the main mass with absorber to that without absorber is constructed to provide insight to evaluate the effectiveness of the absorber to diminish the vibrations of main structures.     

Optimal weight absorber designs for vibrating structures exposed to random excitations

Optimal mass ratios that minimize the response of a laminated beam with an attached absorber are tabulated for various values of beam damping. The beam is treated as an equivalent one degree of freedom (1DOF) main system vibrating in the fundamental mode. The beam is subjected to Gaussian white noise force and Gaussian white noise base frame acceleration. Optimal absorber frequency ratios and absorber damping ratios have been tabulated by others; the results for the classical 1DOF main system with attached absorber suggest that the optimized non-dimensional response decreases monotonically as the mass ratio increases. However, to generalize this monotonic relation may lead to inappropriate conclusions. If we define a constraint such that an increase in absorber mass leads to a proportional decrease in available beam construction material, i.e. effectively the combined mass of the beam and absorber is minimized, then variations in the mass ratio will affect the beam's parameters such as mass, stiffness and damping. Since some of these parameters are used for non-dimensionalising the response, inspection of non-dimensional responses may in some cases lead to inappropriate conclusions. This paper shows the optimal mass ratios for minimizing the response of a structure exposed to earthquake or fluid flow type random excitations.     

Optimum Multiple Tuned Mass Dampers for base-excited undamped system

Optimum parameters of Multiple Tuned Mass Dampers (MTMD) for an undamped system to harmonic base excitation are investigated using a numerical searching technique. The criteria selected for the optimality is the minimization of steady-state displacement response of the main system. The explicit formulae for the optimum parameters of MTMD (i.e. damping ratio, bandwidth and tuning frequency) are then derived using curve-fitting scheme that can readily be used for engineering applications. The error in the proposed explicit expressions is investigated and found to be quite negligible. The optimum parameters of the MTMD system are obtained for different mass ratios and number of dampers. Copyright © 1999 John Wiley & Sons, Ltd.     

A note on distribution of amplitudes of peaks in structural response including uncertainties of the exciting ground motion and of the structural model

This note is an extension of earlier works that presented probability distribution functions for amplitudes of the peaks (the highest, the second highest … the m-th highest) in response of deterministic single degree-of-freedom (SDOF) and multi degree-of-freedom (MDOF) structures to ground motion, with deterministic Fourier spectrum and duration. It shows how these probability distribution functions can be evaluated if the Fourier spectrum and duration of the excitation are random variables specified via distribution functions. Two cases are considered: (l) when the structural model is deterministic, and (2) when the modal frequencies are random variables. The procedure presented here approximates the transfer function of the structural response by Dirac delta functions at the modal frequencies, and is applicable to multi-storey buildings with small modal damping, and with natural frequencies that are not too close. The resulting probability distribution functions are needed in seismic hazard calculations of peak response amplitudes of SDOF and MDOF structures that will not be exceeded with given confidence during the service time of the structure from any earthquake at all known faults within certain distance from the structure.     

Dynamic characteristics of multiple substructures with closely spaced frequencies

This paper considers a main structure supporting a large number of substructures. The substructures have closely spaced natural frequencies, and the combined main structure/multiple substructures system is subjected to harmonic or wide-band forces. The goal is to characterize the effects of the substructures on the response of the main structure. A special, fundamental case is studied in detail, where the substructures are oscillators with equal stiffnesses and equally spaced natural frequencies. The exact response expressions for the combined system are in terms of a complicated rational polynomial. However, by taking the limit where the number of substructures becomes large, the response expressions reduce to simple, physically meaningful results. It is found that the multiple substrutures are equivalent to a single viscous damping which is added to the damping of the main structure. An example illustrates how the results can be applied to passive vibration control of large structures.     

圆球减振装置对风力发电高塔的振动控制研究

本文提出了圆球减振装置对风力发电高塔振动控制的工作原理和计算方法,并对其控制效果进行了理论研究。首先利用拉格朗日方程推导得到圆球减振装置的自振频率及其对单自由度系统的被动控制力,并推广至多自由度系统。进而将风力发电高塔等效为集中质点模型,建立了风塔-减振装置体系的运动微分方程。用谐波叠加法模拟得到脉动风速时程,分析比较了风力发电高塔在无控及有控状态下的动力响应及疲劳寿命。计算结果表明,圆球减振装置是一种简单、经济和实用的减振装置,能够有效减小风塔的动力响应,延长其疲劳寿命。     

Vibrations of beams with an absorber consisting of a viscoelastic solid and a beam

This paper considers the response of a beam with a dynamic vibration absorber, which consists of a viscoelastic solid and a double-cantilever viscoelastic beam, attached to the centre of the main beam. The ends of the main beam are built in and excited sinusoidally by the base motion. The transfer matrix method is used in the analysis. The displacement transmissibility, i.e. the ratio of the displacement at the centre of the main beam to that of the base is investigated in the numerical example. It is shown that the present absorber is effective to suppress several resonances of the main beam simultaneously by only one compact device. Values of the optimum tuning design parameters are presented in the numerical example.     

Influence of lateral-load-resisting system on the earthquake response of structures—a system identification study

Analysis and comparison of the dynamic responses of three well instrumented (with accelerographs) high-rise buildings shaken during the 1984 Morgan Hill earthquake are presented. The buildings examined in the present work are (i) the Town Park Towers Apartment building, a 10-storey, concrete shear wall building; (ii) the Great Western Savings and Loan building, a 10-storey building with concrete frames and shear walls; and (iii) the Santa Clara County Office building, a 13-storey, moment-resistant steel frame building. The structures are located within 2 km of each other and, as may be confirmed by visual inspection of the recorded seismograms, experienced similar ground motions. One-dimensional and three-dimensional linear structural models are fitted to the observations using the modal minimization method' for structural identification, in order to determine optimal estimates of the parameters of the dominant modes of the buildings. The time-varying character of these parameters over the duration of the response is also investigated. Comparison of the recorded earthquake response of the structures reveals that the type of lateral-load-resisting system has an important effect on the dynamic behaviour of the structures because it controls the spacing of the characteristic modes on the frequency axis. The Santa Clara County Office building has closely spaced natural frequencies and exhibits strong torsional response and modal coupling. Its dynamic behaviour is contrasted with that of the Great Western Savings and Loan building which has well separated natural frequencies and exhibits small torsional response and no modal coupling. Strong modal coupling causes a beating-type phenomenon and makes earthquake response of structures different from that envisioned by codes.     

Probabilistic models for structures with bilinear demand-intensity relationships

An extension to the existing SAC/FEMA expressions to estimate mean annual frequency of exceedance (MAFE) for a given limit state is described. In specific, this study pertains to structural systems whose demand versus seismic intensity relationship cannot be reasonably represented by a linear fit in logspace, but rather a bilinear fit over the entire range of structural response. Using a predefined limiting intensity, the median demand is separated into two distinct zones of response. These expressions are derived using a second-order polynomial hazard model fit and can be considered a further extension of the closed-form expressions available in the literature. The steps in the derivation are described along with an example application of the proposed expressions. Comparing different models shows that the MAFE can be significantly misrepresented when using a linear demand-intensity model for systems whose behaviour deviates from this assumption in logspace. Similarly, a logarithmic function demand-intensity fit is examined and seen not to be suitable in the specific situations focused on here. Furthermore, significant underestimation or overestimation is observed when using local fits in the vicinity of the behaviour transition point, which highlights the need for such a bilinear model when assessing the structural performance at the transition point's vicinity. Adopting a bilinear model is shown to better represent structural systems with complex response characteristics, also allowing the use of a single demand model for the entire range of response. This is at the same time still compatible with the existing framework for performance-based seismic design and assessment.     

H_∞ drift control of time-delayed seismic structures

In this paper,an optimal H∞ control algorithm was applied to the design of an active tendon system installed at the first story of a multi-story building to reduce its interstory drift due to earthquake excitations.To achieve optimal control performance and to guarantee the stability of the control system,an optimum strategy to select control parameters γ and α was developed.Analytical expressions of the upper and the lower bounds of γ and α were obtained for a single degree-of-freedom system with state fee...     

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.     

Optimum multiple tuned mass dampers for structures under the ground acceleration based on the uniform distribution of system parameters

The five MTMD models, with natural frequencies being uniformly distributed around their mean frequency, have been recently presented by the first author. They are shown to have the near‐zero optimum average damping ratio (more precisely, for a given mass ratio there is an upper limit on the total number, beyond which the near‐zero optimum average damping ratio occurs). In this paper, the eight new MTMD models (i.e. the UM‐MTMD1～UM‐MTMD3, US‐MTMD1～US‐MTMD3, UD‐MTMD1 and UD‐MTMD2), with the system parameters (mass, stiffness and damping coefficient) being, respectively, uniformly distributed around their average values, have been, for the first time here, proposed to seek for the MTMD models without the near‐zero optimum average damping ratio. The structure is represented by the mode‐generalized system corresponding to the specific vibration mode that needs to be controlled. Through minimization of the minimum values of the maximum dynamic magnification factors (DMF) of the structure with the eight MTMD models (i.e. through the implementation of Min.Min.Max.DMF), the optimum parameters and values of Min.Min.Max.DMF for these eight MTMD models are investigated to evaluate and compare their control performance. The optimum parameters include the optimum mass spacing, stiffness spacing, damping coefficient spacing, frequency spacing, average damping ratio and tuning frequency ratio. The six MTMD models without the near‐zero optimum average damping ratio (i.e. the UM‐MTMD1～UM‐MTMD3, US‐MTMD1, US‐MTMD2 and UD‐MTMD2) are found through extensive numerical analyses. Likewise, the optimum UM‐MTMD3 offers the higher effectiveness and robustness and requires the smaller damping with respect to the rest of the MTMD models in reducing the responses of structures subjected to earthquakes. Additionally, it is interesting to note, by comparing the optimum UM‐MTMD3 with the optimum MTMD‐1 recently investigated by the first author, that the effectiveness and robustness for the optimum UM‐MTMD3 is almost identical to that for the optimum MTMD‐1 (without inclusion of the optimum MTMD‐1 with the near‐zero optimum average damping ratio). Recognizing these performance benefits, it is preferable to employ the optimum UM‐MTMD3 or the optimum MTMD‐1 without the near‐zero optimum average damping ratio, when installing the MTMD for the suppression of undesirable oscillations of structures under earthquakes. Copyright © 2003 John Wiley & Sons, Ltd.     

Periodic response of a sliding oscillator system to harmonic excitation

This paper deals with the periodic response of an oscillating system which is supported on a frictional interface. The base excitation is assumed harmonic and the frictional force is assumed to be of the Coulomb type. Though each segment of the motion of such a system is described by linear equations, its complete response is highly non-linear and varied. The most fundamental periodic solutions are derived analytically and numerically. The results indicate that such a system has several subharmonic resonant frequencies and that while the friction reduces the peak response of the system when it is excited at its ‘fixed-base’ natural frequency, ω_n, the sliding can induce considerably higher levels of response, when compared with those of a non-sliding, fixed-base system, for frequencies less than ω_n. The results obtained herein may find application in the area of vibration isolation.     

Optimum tuned-mass dampers for minimizing steady-state response of support-excited and damped systems

A numerical searching procedure to find the optimum tuning frequency and damping ratio of the tuned-mass damper which can reduce the steady-state response of damped main systems to a minimum level is developed and applied to the two different harmonic excitation sources, support motion of fixed-displacement amplitude and support motion of fixed-acceleration amplitude. The explicit formulae for these optimum parameters are then derived through a sequence of curve-fitting schemes. It has been found that, as the error of the explicit formulae is negligible, they provide a convenient tool to compute the optimum parameters in engineering applications. The numerical results show that the tuned-mass damper is less effective in reducing the system's response when there is a high level of damping incorporated into the system. It is also found that the optimum tuning frequency is strongly influenced by the damping level of a system, especially in regard to the fixed-acceleration support motion, but the optimum damping ratio of the tuned-mass damper is not sensitive to the damping level of a system. The response of the damped system using the undamped optimum value as the damping of the tuned-mass damper is not much different from the response using the damped optimum value.     

Fundamental characteristics of Multiple Tuned Mass Dampers for suppressing harmonically forced oscillations

Multiple Tuned Mass Dampers (MTMD's) consisting of many tuned mass dampers (TMD's) with distributed natural frequencies are considered for suppressing effectively the harmonically forced single mode response of structures. The fundamental characteristics of MTMD's are investigated analytically with the parameters of the covering frequency range of MTMD's, the damping ratio of each TMD and the total number of TMD's. The effectiveness and the robustness of MTMD's are also discussed in comparison with those of the usual single TMD. It is found that there exists an optimum MTMD for the given total number of TMD's with the optimum frequency range and the optimum damping ratio and that the optimum MTMD is more effective than the optimum single TMD. As for the robustness, it is also clarified that a MTMD can be much more robust than a single TMD while keeping the same level of effectiveness as the optimum single TMD.     

基于小波变换的补燃循环火箭发动机模态参数辨识

文中阐述基于小波变换的结构模态参数辨识方法的基本原理.首先介绍Heisenberg测不准原理,该原理限制了时频能量的同时集中.因小波变换的窗口可以随信号特征不同而变化,符合实际信号的要求.然后论述利用小波脊线的方法识别单自由度和多自由度系统的模态参数的理论.将该方法应用于某型补燃循环液体火箭发动机模态参数辨识中,得到了...     

Dynamic force control with hydraulic actuators using added compliance and displacement compensation

A new approach to dynamic force control of mechanical systems, applicable in particular to frame structures, over frequency ranges spanning their resonant frequencies is presented. This approach is implemented using added compliance and displacement compensation. Hydraulic actuators are inherently velocity sources, that is, an electrical signal regulates their velocity response. Such systems are therefore by nature high‐impedance (mechanically stiff) systems. In contrast, for force control, a force source is required. Such a system logically would have to be a low‐impedance (mechanically compliant) system. This is achieved by intentionally introducing a flexible mechanism between the actuator and the structure to be excited. In addition, in order to obtain force control over frequencies spanning the structure's resonant frequency, a displacement compensation feedback loop is needed. The actuator itself operates in closed‐loop displacement control. The theoretical motivation, as well as the laboratory implementation of the above approach is discussed along with experimental results. Having achieved a means of dynamic force control, it can be applied to various experimental seismic simulation techniques such as the effective force method and the real‐time dynamic hybrid testing method. Copyright © 2008 John Wiley & Sons, Ltd.