Frequency band-wise passive control of linear time invariant structural systems with H∞ optimization

Probabilistic assessment of steel buildings installed with passive control devices under multi-hazard scenario of earthquake and wind

The Tuned Mass Damper (TMD) is generally as a passive vibration control device consisting of added auxiliary mass with functioning spring and damping elements. TMD is basically designed to be tuned to the dominant frequency of a structure which the excitation of frequency will resonate the structural motion out of phase to reduce unwanted vibration. However, a single unit TMD is only capable of suppressing the fundamental structural mode. In order to control multimode vibrations and to cater wide band seismic frequency, more than one TMD is required to improve the effectiveness of a control mechanism. For the purpose of this study, a 3-storey benchmark reinforced structural building subjected to El Centro seismic ground motion is modelled as uncontrolled Primary Structure (PS) by considering appropriate structural properties such as stiffness and damping. Mathematical modelling of uncontrolled PS is developed and further evaluated numerically by assuming the PS as an equivalent lumped system. For the case of controlled PS which the passive mechanism is included to the system, optimum parameters of both TMD and Multiple TMD (MTMD) are designed to be tuned to the dedicated structural modes where the performance is dependent on specified parameters such as auxiliary mass ratio, optimum damping ratio, and optimum frequency ratio. The eigen value analysis is carried out by assuming that the structure is a linear time-invariant system. The input and output components of structural system arrangements are then characterized in the transfer function manner and then converted into state space function. To enhance structural control effectiveness, the adaptive system is incorporated by the attachment of Magneto-Rheological (MR) damper to both single TMD and MTMD passive system. The response analysis of the control system arrangements is executed using both time history and frequency response analysis. The main objectives of the design are to minimize both structural peak and Root Mean Square (RMS) displacements. From the analysis, the designed control mechanisms are concluded as highly effective in reducing all structural floor displacements for the semi-active cases with 99% displacement reduction for the third and second floors, and 98% for the first floor, compared to the uncontrolled case. It is concluded that the MR damper significantly contributed to the enhancement of the passive system to mitigate structural seismic vibration.

Computations were performed to investigate the utility of three passive devices for controlling obliqueshock-wave/boundary-layer interactions on a flat plate. All three devices involve a cavity with two slots one upstream of the incident shock where blowing takes place and another downstream of it where bleeding takes place. The blowing and bleeding rates were determined passively by the pressure difference across the shock. Results are presented which show the nature of the flow field induced by passive blowing and bleeding. This study is based on the ensemble-averaged conservation equations of mass, momentum (compressible Navier-Stokes), and energy closed by a low Reynolds number k-oo model of turbulence. Solutions were obtained by using a cell-centered finite-volume method based on flux-difference splitting of Roe with slope limiter of Chakravarthy and Osher and a diagonalized alternating-direction implicit scheme with multigrid on a patched/ overlapped grid system. * Graduate student. ** Graduate student. Now, research staff at Sterling Software/NASA Ames Research Center. Professor. Senior Member AIAA. Research Engineer. Member AIAA. Copyright © 1997 by the American Institute of Aeronautics and Astronautics, Inc. No copyright is asserted in the United States under Title 17, U.S. Code. The U.S. Government has a royaltyfree license to exercise all rights under the copyright claimed herein for government purposes. All other rights are reserved by the copyright owner. INTRODUCTION Effective control of shock-wave/boundary-layer interactions to prevent flow separation and minimize flow distortions is important in a number of applications. These include airframes of supersonic aircraft, external and mixed-compression inlets, and wind tunnels. For some of these applications, shock-wave induced boundary-layer separations are not just annoyances because of reductions in efficiency but rather a very serious operational hazard. As an example, shock-wave induced separations in mixed-compression inlets can lead to the unstart condition requiring the entire propulsion system to undergo a restart sequence during flight. A number of techniques have been developed to control the detrimental effects of incident shock waves on boundary layers. Of these, bleeding and blowing have been found to be highly effective in eliminating both flow separation and improving boundary-layer profile. In terms of implementation, blowing and bleeding have been operated mostly in an active mode. By active, it is meant that a source of high or low pressure must be maintained in order to ensure blowing or bleeding. Since maintaining such a source requires additional energy which also reduces efficiency, a passive mode of operation is desirable. In 1979, Bushnell and Whitcomb (see Ref. 11) proposed a passive control device made up of a porous surface over a cavity or plenum. It was suggested that placing this device over a shock-wave/boundary-layer interaction region will produce a flow through the cavity from downstream to upstream of the shock because of the static pressure rise across the shock. Since that Copyright© 1997, American Institute of Aeronautics and Astronautics, Inc. time, a number of investigators have studied passive devices for controlling shock-wave/boundary-layer interactions. They found success in reducing shock drag over transonic airfoils (see review in Ref. 11). Recently, passive control was also found to be successful in controlling normal shock/boundary-layer interactions about an external compression inlet. So far, passive blowing and bleeding have been applied only to flows with low supersonic speeds (Mach numbers less than 1.4) so that the resulting shocks have been relatively weak. Also, no studies computational or experimental have been reported that show the nature of the multidimensional flow field about passive control devices. The objective of this study is to investigate computationally the utility of three passive devices for controlling oblique-shock-wave / boundary-layer interactions on a flat plate in which the freestream Mach number is much higher than 1.4 and the incident shock wave can induce boundary-layer separation in the absence of control. The focus is on understanding the nature of the flow and the usefulness of passive bleed in controlling strong shocks. In the next section, the problem studied along with the three passive control devices investigated are described. Afterwards, the formulation, numerical method of solution, and results obtained are presented. DESCRIPTION OF PROBLEM Figure 1 shows a schematic diagram of the passively controlled oblique-shock-wave/boundary-layer interaction problem being studied. It involves a supersonic turbulent boundary layer flowing past a flat plate with an incident oblique shock wave and blowing/bleeding from a passive device. Figures 2 to 4 show details of the three passive devices analyzed. The reasons for selecting these three configurations are given in the results section. The physical dimensions in Figs. 1 to 4 are as follows: LI = 57.34 cm, H = 25.4 cm, L = 7.65 cm, al = 1.14 cm, a2 = 0.76cm, a3 =0.38 cm, bl = 1.62 cm, b2 = 1.08 cm, b3 = 0.54 cm, cl = 8.6 cm, c2 = 7.65 cm, cl = 6.06 cm, c2' = 5.11 cm, h = 0.0381 cm, and s = 0.635 cm. Othei dimensions are given later in this section. Fig. 1. Schematic of problem studied. Fig. 2. Passive device 1: flap. Fig. 3. Passive device 2: big ellipse. ff{\_ JL-' — i l IBlcalin Fig. 4. Passive device 3: small ellipse. For the problem shown in Fig. 1, the computational domain is the region bounded by the dashed lines. The fluid that enters the domain above the flat plate is air Copyright© 1997, American Institute of Aeronautics and Astronautics, Inc. with a specific-heats ratio (y) of 1.4. The freestream Mach number (Moo), static pressure (Poo), and static temperature (T») are 2.46, 10.736 KPa, and 131.93 K, respectively. This supersonic flow has a turbulent, boundary layer next to the flat plate. At the inflow boundary, the thickness of that boundary layer (5) is 3.244 cm. The displacement thickness (8*) is 0.196 cm. A shock-wave generator characterized by L2 and a causes an oblique shock wave to strike the turbulent, boundary layer on the flat plate, a was set at 8° which produced a shock wave that was strong enough to induce flow separation on the flat plate in the absence of bleed or blowing. L2 was chosen so that the shock would incident on the flat plate at a distance of 34.4 cm measured from the inflow boundary under inviscid flow conditions (i.e., zero boundary-layer thickness). Aside from studying the three passive control devices shown in Figs. 2 to 4, the following were investigated: two locations of the incident shock wave relative to the passive control device (one just upstream of the bleed slot as shown in Figs. 2 to 4 and another just downstream of the blowing slot as shown in Figs. 5 and 6) and two boundary-layer heights at the inflow boundary (one finite as shown in Figs. 2 to 5 and the other zero as shown in Fig. 6). Note that when the shock impingement location change, the passive control device moved not the flat plate (i.e., L2 remains constant). Also, note when the boundary-layer height is zero at the inflow, the wall was kept inviscid until reaching the passive device. All cases studied are summarized in Table 1. Table 1 Summary of Cases Studied*

This research article investigates the attenuation characteristics of passive control devices, viz., particle tuned mass damper (PTMD) and tuned mass damper (TMD) are used for a single degree of freedom (SDOF) and multidegree of freedom (MDOF) structure. Numerous parameters, like filling ratio, auxiliary mass ratio, particle density, and so on are pretending attenuation characteristics of particle damper. Recent research in the field of dynamics has explored enhanced characteristics of particle dampers toward attenuation of time history responses using both tuning of natural frequency and energy dissipation generated due to impact between particles and between particles and the container. TMD works in many cases as a simple pendulum and has not been found to attenuate the time history response of different earthquakes concentrating over contrasting frequencies. It works well when the earthquake concentrates its’ frequency near the damper frequency. However, PTMD works well during earthquakes, focusing on varying frequencies. An SDOF and MDOF structure generates a time history response subjected to standard earthquakes and sinusoidal waves using the shake table test results under damping from passive control devices. Also, a simulation model using SAP2000 has been established to discover the time history responses generated by TMD and PTMD. PTMD was more effective than TMD for SDOF and MDOF systems of structure in all cases of waveforms. However, TMD was more effective for the sine wave input waveform in the case of MDOF, as seen from the experimental and simulation studies.

Summary Structures today may be equipped with passive structural control devices to achieve some performance criteria. The optimal design of these passive control devices, whether via a formal optimization algorithm or a response surface parameter study, requires multiple solutions of the dynamic response of that structure, incurring a significant computational cost for complex structures. These passive control elements are typically point-located, introducing a local change (possibly nonlinear, possibly uncertain) that affects the global behavior of the rest of the structure. When the structure, other than these localized devices, is linear and deterministic, conventional solvers (e.g., Runge–Kutta, MATLAB's ode45, etc.) ignore the localized nature of the passive control elements. The methodology applied in this paper exploits the locality of the uncertain and/or nonlinear passive control element(s) by exactly converting the form of the dynamics of the high-order structural model to a low-dimensional Volterra integral equation. Design optimization for parameters and placement of linear and nonlinear passive dampers, tuned mass dampers, and their combination, as well as their design-under-uncertainty for a benchmark cable-stayed bridge, is performed using this approach. For the examples considered herein, the proposed method achieves a two-orders-of-magnitude gain in computational efficiency compared with a conventional method of comparable accuracy. Copyright © 2016 John Wiley & Sons, Ltd.

The tuned mass damper (TMD) is a widely used passive control device which is attached to a main system to suppress undesired vibration. In this paper, a non-traditional form of TMD system is investigated. Unlike the traditional TMD configuration, the considered TMD system has a linear viscous damper connecting the absorber mass directly to the ground instead of the main mass. There have been some studies on the optimization design of the non-traditional TMD (NT-TMD) for undamped main structures. Those studies have indicated that the NT-TMD provides better performance than the traditional TMD does. When there is a frequency shifting in the structural frequency or tuning frequency of TMD, to the best knowledge of the authors, there has been no study on the performance of the NT-TMD. The main idea of the study is to investigate the effect of frequency detuning on the control performance of the NT-TMD. The optimum parameters of the NT-TMD system and corresponding effectiveness are obtained for different mass ratios of the NT-TMD system. The numerical results indicate that the NT-TMD with high mass ratio provides better robustness to the changes in the target frequency ratio than the traditional TMD.

Linear time-invariant (LTI) systems are systems that are both linear and time-invariant. A system is considered linear if the output of the system is scaled by the same amount as the input given to the system. Moreover, this system follows the superposition principle, which implies that the sum of all the inputs will be the sum of the outputs of the individual inputs. Time-invariant systems are systems in which the output caused by a particular input does not change with time and only depends on when that input was applied. This class of systems is very important in the control field where many mature tools and methods exist. Before discussing nonlinear systems, the results of LTI systems must be reviewed to allow for the easy introduction of repetitive control (RC, or repetitive controller, which is also designated as RC) for nonlinear systems. On the other hand, any RC methods applicable to nonlinear systems should be first applicable to LTI systems. Therefore, it is necessary to apply these methods to LTI systems first and then move onto nonlinear systems. Moreover, some RC methods for nonlinear systems are an extended form or a combination of the methods in LTI systems. Therefore, it is important and necessary to first be aware of the RC methods for LTI systems. LTI systems often employ two types of models, i.e., transfer function models and state-space models. This chapter aims to answer the following question: How do you design a repetitive controller for LTI systems based on transfer function models and state-space models?

— Linear time invariant (LTI) systems are widely used for modeling of dynamics systems in science and engineering problems. Harmonic oscillation of LTI systems is an outstanding case of LTI system behavior and it is employed for modeling of many periodic physical phenomenon in nature. This study investigates sufficient conditions to obtain harmonic oscillation by using high-order LTI systems. A design procedure for controlling harmonic oscillation of single-input single-output high-order LTI systems is presented. LTI system coefficients are calculated by solving equation sets, which imposes a stable sinusoidal oscillation solution for the characteristic polynomials of LTI systems. Moreover, these analyses are extended to fractional order LTI systems. Simulation examples are demonstrated for high-order LTI systems and the control of harmonic oscillations are discussed by using Hilbert transform and spectrogram of oscillation signals

In this paper, we formulate continuous time linear fractional shift invariant (LFSI) systems that generalize the well-known linear time invariant (LTI) systems by means of an angle parameter /spl phi/. LTI systems are obtained as a special case of LFSI systems for /spl phi/ = 0. LFSI systems belong to the large class of time-varying systems. Whereas LTI systems commute with time shifts, LFSI systems commute with fractional shifts defined on the time-frequency plane. Just as the conventional Fourier transform (FT) diagonalizes LTI systems, an LFSI system associated with angle /spl phi/ is diagonalized by the fractional Fourier transform (FrFT) defined at the perpendicular angle /spl phi/ + (/spl phi//2). We show that the eigen-functions of an LFSI system at angle /spl phi/ are linear FM (chirp) signals with a sweep rate of tan /spl phi/. Finally, we demonstrate via a simulation example that, in certain cases, LFSI systems can outperform LTI systems.

Floating offshore wind turbines (FOWT) are subjected to strong loads, mainly due to wind and waves. These disturbances cause undesirable vibrations that affect the structure of these devices, increasing the fatigue and reducing its energy efficiency. Among others, a possible way to enhance the performance of these wind energy devices installed in deep waters is to combine them with other marine energy systems, which may, in addition, improve its stability. The purpose of this work is to analyze the effects that installing some devices on the platform of a barge-type wind turbine have on the vibrations of the structure. To do so, two passive control devices, TMD (Tuned Mass Damper), have been installed on the platform of the floating device, with different positions and orientations. TMDs are usually installed in the nacelle or in the tower, which imposes space, weight, and size hard constraints. An analysis has been carried out, using the FAST software model of the NREL-5MW FOWT. The results of the suppression rate of the tower top displacement and the platform pitch have been obtained for different locations of the structural control devices. They have been compared with the system without TMD. As a conclusion, it is possible to say that these passive devices can improve the stability of the FOWT and reduce the vibrations of the marine turbine. However, it is indispensable to carry out a previous analysis to find the optimal orientation and position of the TMDs on the platform.

Research in vibration response control deals not only with prevention of catastrophic failures of structures during natural or accidental/manmade hazards but also ensures the comfort of occupants through serviceability. Therefore, the focus of this book is on the theory of dynamic response control of structures by using different kinds of passive vibration control devices. The strategies used for controlling displacement, velocity, and acceleration response of structures such as buildings, bridges, and liquid storage tanks under the action of dynamic loads emanating from earthquake, wind, wave, and so forth are detailed. The book: Explains fundamentals of vibration response control devices and their practical applications in response mitigation of structures exposed to earthquake, wind, and wave loading Offers a comprehensive overview of each passive damper, its functioning, and mathematical modeling in a dynamical system Covers practical aspects of employing the passive control devices to some of the benchmark problems that are developed from existing buildings and bridges in different countries worldwide Includes MATLAB® codes for determining the dynamic response of single degree of freedom (SDOF) and multi-degree of freedom (MDOF) systems along with computational models of the passive control devices This book is aimed at senior undergraduate students, graduate students, and researchers in civil, earthquake, aerospace, automotive, mechanical engineering, engineering dynamics, and vibration control, including structural engineers, architects, designers, manufacturers, and other professionals.

The fractional Fourier transform (FRFT) has been shown to be a powerful analyzing tool in signal processing. Many properties of this transform have been currently derived as counterparts to the corresponding properties of the Fourier transform (FT), including the theory of the linear fractional shift invariant (LFSI) systems. However, the LFSI systems, which are derived as extensions of the linear time invariant (LTI) systems, available in the literature do not generalize very nicely the LTI systems, which state that the output of a LTI system does not depend on the particular time the input is applied, and the FT of the output is the product of the transfer function and the FT of the input in the Fourier domain. In this paper, we propose a new LFSI system structure associated with the FRFT, and the LTI systems are noted as special cases. The eigenfunctions and eigenvalues of the new LFSI systems are presented. Moreover, the conditions for distortionless transmission in the fractional Fourier domain and basic properties of the ideal fractional filter are derived. Some applications of the achieved results are also discussed.

This paper presents a recursive system identification method for multi-degree-of-freedom (MDoF) structures with tuned mass dampers (TMDs) considering abrupt stiffness changes in case of sudden events, such as earthquakes. Due to supplementary non-classical damping of the TMDs, the system identification of MDoF+TMD systems disposes a challenge, in particular, in case of sudden events. This identification methods may be helpful for structural health monitoring of MDoF structures controlled by TMDs. A new adaptation formulation of the unscented Kalman filter allows the identification method to track abrupt stiffness changes. The paper, firstly, describes the theoretical background of the proposed system identification method and afterwards presents three parametric studies regarding the performance of the method. The first study shows the augmented state identification by the presented system identification method applied on a MDoF+TMD system. In this study, the abrupt stiffness changes of the system are successfully detected and localized under earthquake, impulse and white noise excitations. The second study investigates the effects of the state covariance and its relevance for the system identification of MDoF+TMD systems. The results of this study show the necessity of an adaptive definition of the state covariance as applied in the proposed method. The third study investigates the effects of modeling on the performance of the identification method. Mathematical models with discretization of different orders of convergence and system noise levels are studied. The results show that, in particular, MDoF+TMD systems require higher order mathematical models for an accurate identification of abrupt changes.

The seismic response of structures is often enhanced by introducing passive control devices that can operate through the dissipation of the input energy or by modifying the dynamic characteristics of the main structure. The inherent non-linearities in the constitutive laws of some of them lead to computation difficulties and have limited the large-scale use and design of these devices. In this study, a procedure for the optimal design of multi passive control devices is proposed. The general case of linear Multi-Degree-Of-Freedom (MDOF) not-classically-damped structural systems controlled by Fluid Viscous Dampers (FVD) are investigated in a stochastic framework. The procedure consists of evaluation of the device optimal pattern by minimizing an objective function related to the dampers cost and subjected to a constraint on the structural behaviour. For each configuration, the complete probabilistic characterization of the response is achieved by employing random vibration theory, Stochastic Linearisation (SL) techniques and a novel analytic model which provides closed-form PSD functions of earthquakes accelerations coherent to response spectra suggested by seismic codes. Exploiting this model, a procedure to speed up the Stochastic Linearisation technique by avoiding any numerical integration is proposed. Applications on MDOF building structures have been carried out to validate the proposed approach in terms of accuracy and reduction of the computational effort and to obtain optimal pattern of the passive control device coherently with the provisions of seismic building codes.

The effectiveness of passive vibration control devices used to retrofit multi-storied steel buildings during their design life is investigated under the dynamic forces induced by earthquake and wind. The passive vibration control devices include steel bracing, viscous and viscoelastic dampers. The buildings without and with the retrofitting devices are modelled as multi-degree of freedom (M-DOF) systems, with inertial masses lumped at each floor level. The governing differential equations of motion for the uncontrolled and controlled buildings are solved using Newmark’s time marching scheme. The obtained dynamic responses for the buildings exposed to the earthquake- and wind-induced forces are subsequently compared. It is concluded that upon retrofitting, the modified dynamic properties, such as modal frequencies and damping ratio play an active role to attract forces during the two hazards, which in turn influences the response reduction achieved. It may be worth noting that the buildings retrofitted for earthquake tend to attract more forces under wind load and vice versa. Therefore, a retrofit strategy providing beneficial effects against a particular hazard may prove to be catastrophic for the other, which underlines the need for careful selection of the retrofit solution and design for a structure considering such multi-hazard scenario.