Effects of Damping Device Nonlinearity on the Performance of Semiactive Tuned Mass Dampers

Transcription

1 Iowa State University From the SelectedWorks of Simon Laflamme 2010 Effects of Damping Device Nonlinearity on the Performance of Semiactive Tuned Mass Dampers C Lindh, Massachusetts Institute of Technology Simon Laflamme, Massachusetts Institute of Technology Jerome J. Connor, Massachusetts Institute of Technology Available at:

2 5 th World Conference on Structural Control and Monitoring 5WCSCM-274 5WCSCM-001 EFFECTS OF DAMPING DEVICE NONLINEARITY ON THE PERFORMANCE OF SEMIACTIVE TUNED MASS DAMPERS C. Lindh Massachusetts Institute of Technology, Cambridge, MA 02139, USA S. Laflamme Massachusetts Institute of Technology, Cambridge, MA 02139, USA J. J. Connor Massachusetts Institute of Technology, Cambridge, MA 02139, USA Abstract Studies have already demonstrated the successful use of linear semiactive damping devices, such as variable orifice (VO) dampers, for semiactive TMD systems. More recently, nonlinear semiactive damping devices, such as magnetorheological (MR) dampers, have also been shown to be effective for semiactive control of TMDs. Though semiactive dampers differ widely, with responses ranging from linear (VO) to nonlinear (MR), criteria for choosing an optimal semiacive device for a TMD have not been rigorously developed. This paper expands knowledge of semiactive TMD systems by assessing the effect of nonlinearity in the damping device on the effectiveness of a semiactive TMD. This is achieved by simulating a variable damping device (linear), and a variable friction device (nonlinear). The variable damping device consists of a VO damper, while the variable friction device consists of a new mechanically robust and reliable damping device with a dynamic resembling the MR damper. These simulations allow the influence of nonlinearity to be investigated and provide further insight into selecting an optimal semiactive damping device for improving the performance of a passive TMD. Introduction Recent decades have marked a trend towards the design and construction of very tall buildings. Advancements in analysis, coupled with an increased use of lighter building materials and a decrease in heavy claddings, have led to structures that are not only taller but also more flexible. Consequently, most modern towers are especially prone to oscillations under persistent winds, which can lead to swaying motions of several meters on the top floors (Chang, 1973, Miller, et al., 1988). In many cases, these large deflections may not threaten the integrity of the structure, but the steady rocking can cause considerable discomfort and even illness to building occupants. If persistent, the dynamic response under severe winds may render the top floors completely uninhabitable. Studies by Chang and Hansen investigated the effects of this motion on the human body, creating benchmarks for the perception of and physiological response to various increments of lateral acceleration (Chang, 1973, Hansen, et al., 1979). The maximum amplitudes of these responses are ultimately dictated by the ability of the structure to dissipate energy; the more significant the energy dissipation, the smaller the vibrations. All structures naturally release some energy through mechanisms such as internal stressing, rubbing, and plastic deformations. In large modern steel structures, however, the total damping may amount to as little as 1% of critical, making them very vulnerable to dynamic resonance effects (Housner, et al., 1997). As a result, additional measures are generally necessary to meet serviceability standards. Since eliminating the excitation source is impractical, this necessitates implementing a control scheme to enhance the effective damping of the structure. Lindh, Laflamme, and Connor 1

3 The most widely accepted control measure for mitigating the response of tall structures under wind loads is the implementation of a tuned mass damper (TMD). Consisting of a mass, a spring, and a damper, the natural frequency of the TMD is tuned to have a resonance very close to the fundamental mode of the primary structure, which allows a large amount of the structure vibrational energy to be transferred to the TMD and then dissipated by its damper. This system has been adapted in numerous tall structures, and it has proven to be an effective method for mitigating structural vibration under high wind loads. A major disadvantage of TMD systems is that their efficiency is limited to the ±15% range of the tuned frequency (Connor, 2003). In an effort to overcome this limitation and to improve performance, measures have been taken to incorporate active control techniques with passive TMD systems. The concept involves adding an external energy source to generate an additional force that complements the force generated by the TMD, usually through the use of an actuator. With the inclusion of sensors and a feedback loop, the actuator force can be adjusted nearly real-time to produce optimized results. However, active TMD systems require a significant power input to operate, and instabilities of the controller may lead to structural failure. To overcome those issues, semi-active TMDs (STMDs) have been proposed. Semi-active control systems typically require low power to operate, are inherently stable, and typically perform better than passive control systems (Laflamme & Connor, 2009). Literature on STMDs has focused both on earthquake loaded and wind-loaded structures, with a noticeable preponderance towards earthquake excited structures. The concept of a semi-active tuned mass damper was first proposed by Hrovat, who simulated results for a force-clipped active TMD that was prohibited from adding energy to the system (Hrovat, et al., 1983). The first practical STMD applications investigated used electrorheological (ER) dampers to supply the semi-active forces. Abè developed ER-STMD theory for controlling transient responses under impulse loading (Abé & Igusa, 1996), and Hidaka conducted experimental studies that coupled an ER-STMD with a three-story model building under ground excitation (Hidaka, et al. 1999). Both demonstrated improved performance. Magnetorheological (MR) fluid has been proposed as a replacement to ER fluid because of its enhanced robustness and low power requirement. Consequently, more recent research has focused on MR-based STMD systems. Koo assessed various groundhook-based control algorithms and demonstrated their effectiveness through an experimental study of a base-excited SDOF model structure coupled with an MR-STMD (Kuo, et al., 2004, Kuo et al., 2005). Ji further evaluated semiactive control algorithms for controlling a multi-degree-of-freedom (MDOF) structure subjected to earthquake loading when there are uncertainties about structural properties (Ji, et al., 2005). Performance benefits of an MR-STMD under earthquake loading were further documented by Loh through numerical simulations of the response of a 12-story building (Loh, et al., 2004). Other significant work has focused specifically on wind-loaded structures. Pinkaew was the first to demonstrate the steady-state efficacy of a damper-based STMD by simulating frequency-domain results due to harmonic excitation (Pinkaew & Fujino, 2001). Varadarajan proposed a novel device for a stiffness-based STMD capable of retuning to meet the demands of realistic wind excitations and changes in the stiffness properties of the primary structure (Varadarajan & Nagarajaiah, 2004). To the best of the authors knowledge, no semi-active TMD devices have been installed in an actual civil structure at this point, but theoretical work remains on-going due to the potential of semiactive devices. This paper expands knowledge of STMD systems by assessing the effect of nonlinearity in the damping device on the effectiveness of the STMD. Nonlinear damping devices, such as ER and MR dampers, exhibit variable stiffness-type behavior when current is applied, while linear devices, such as linear variable orifice (VO) dampers, explicitly exhibit a variable damping-like behavior. Thus, the effect of nonlinearity in the damping devices for STMD performance will be studied by simulating a variable damping-type device and a variable stiffness-type device. The variable damping device consists of a linear VO damper, while the variable friction device consists of a new mechanically robust and reliable damping device with a dynamic resembling the MR damper, termed a modified friction device (MFD). Lindh, Laflamme, and Connor 2

4 The paper is organized as follows. The next section describes both semiactive dampers utilized in STMD simulations. The subsequent section shows and discusses results of the simulations. The paper is then concluded. Semiactive Tuned Mass Dampers Recent work has shown the promise of STMD strategies to blend the performance benefits of active systems with the stability and energy advantages of passive systems. In this section, two STMDs are described: a linear STMD using of a linear VO as the damping device, and a nonlinear STMD utilizing an MFD. To contrast the differences between both strategies, those systems can be idealized as TMDs with either a variable damping element or a variable friction element, as schematized in Fig. 1, where the subscript denotes properties belonging to the TMD,,, and represent mass, damping, and stiffness, respectively, and is a coulomb friction element. The next subsections explain the theory describing the VO and MFD devices. m d m d k d c d (t) k d c d F c (t) M M k c k c Linear Variable Viscous (a) (b) Figure 1. diagram of a) VO-STMD; and b) MFD-STMD The variable orifice (VO) damper was the first semi-active damping device implemented in structural applications. It consists of a cylinder-piston system with a by-pass valve connected at both ends and behaves essentially like a conventional hydraulic fluid damper with adjustable resistance to fluid flow. By electromechanically controlling an orifice in this valve, it is possible to greatly vary the damping force in real-time. Consequently, VO dampers may be modeled mathematically as linear viscous dampers in which the damping coefficient,, now becomes a manipulable variable, ( ). A basic schematic of the VO damper can be seen in Figure 2. Figure 2. variable orifice damper schematic (Spencer & Sain, 1997) Lindh, Laflamme, and Connor 3

5 Matsunaga et al. (1998), Kurata et al. (1999), and Niwa et al. (2000) have each contributed to the development and classification of VO dampers capable of performing on a structural scale. These devices can produce maximum force outputs of 1 2 MN and require a power supply of only 70 W. The dynamic range of the dampers, which is determined by the minimum and maximum values for the damping coefficient, ( ), exceeds 200. These devices are relatively space efficient, with dimensions of 1.5 x 0.5 x 0.5 m, and masses of around 1300 kg. VO dampers have proven successful in several applications in civil structures. In 1994 Patten led installation of VO dampers on an I-35 bridge in Oklahoma to dissipate energy induced by vehicle traffic, marking the first full-scale implementation of structural control in the United States (Patten, 1994). Kobori implemented VO dampers to control stiffness elements in a semi-actively controlled building at the Kobori research complex (Housner, et al., 1997). Kurata was the first to make VO dampers the primary control system of a full-scale building, using several 1 MN models. Each of these advancements, among others, has led to acceptance of VO dampers as a viable means of improving control performance on a large scale. Modified Friction Device MR dampers have been proposed for control of civil structures, and aforementioned studies show that they are promising devices for vibration mitigation. Nevertheless, their application to civil structures is still in its infancy (Laflamme & Connor, 2009). Among factors that might impede their implementation, two key issues are related to their chemical robustness: robustness of the seal that is now guaranteed for vehicular applications (Carlson, 2006) but not yet documented over the very long term (50 years), and the sedimentation effect resulting from a limited use of the device for a long period of time (Avraam, 2009). Laflamme et al. (2010) proposed a mechanically robust damping device, termed the modified friction device (MFD), based on existing reliable technology that has a dynamic inspired by that of the MR damper. The authors preferred the MFD over the MR damper for the simulation to stimulate intellectual interest about the MFD. The MFD consists of a viscous element, a stiffness element, and a variable friction element installed in parallel. The variable friction element is a braking element inspired by vehicular braking. It consists of a drum with a self-energizing braking system. Laflamme et al. (2010) proposed a 200 kn capacity MFD actuated with a 12-volt power source. Figure 3 a) illustrates the dynamics, and Figure 3 b) shows its force diagram. (a) Figure 3. a) dynamics of the MFD; b) schematized force diagram of the MFD (Laflamme, et al., 2010) (b) Lindh, Laflamme, and Connor 4

6 The MFD is modeled using the LuGre model for the friction element. Its resisting force can be written: = + + (1) where is the stiffness of the friction element, is the damping coefficient of the viscous damper, and are the device displacement and velocity, respectively, and is a constant taken as =1 for a linear viscous damper. is the frictional force given as: = + + ( ) (2) with: = ( ) ( )= ( )+ ( ) ( ) ( )= (3) where,,,, and are constants, is an evolutionary variable, ( ) represents the Stribeck effect, is the coulomb friction force and depends on the voltage, is the increase in the friction force due to the Stribeck effect, and ( ) is the viscous friction. The parameters utilized for a 200 kn capacity are shown in Table 1. Table 1. Properties of the 200 kn MFD (Laflamme, et al., 2010) Simulation Numerical examples have been generated to compare how effectively VO dampers and MFD improve the performance of passive TMD systems. Because the primary function of TMDs in buildings has been to dampen fundamental mode vibrations under wind excitation, the simulations presented here focus on a structure modeled as an SDOF system subjected to an external load, ( ). Addition of a TMD with semiactive control forces results in a two degree of freedom system governed by the equation + + = + (4) Lindh, Laflamme, and Connor 5

7 in which = = 0 0 = + = + = ( ) 0 = 1 1 where is the displacement, and is the additional semi-active damping force provided by either the VO or the MFD. Dynamic Properties In general, the effectiveness of a passive TMD is limited by its mass, between 1% and 5% of the fundamental modal mass of the structure (Connor, 2003). Once has been established based on physical constraints, other TMD parameters may be selected using a variety of available optimization techniques (Warburton, 1982, Asami, 2002). In this study all passive TMD systems are tuned with the formulation presented by Asami (2002), which provides a series solution for optimized TMD parameters. The dynamic properties of the SDOF system and the TMD are: =5 N s /m =3.80 N s/m =1780 N/m = =2 = where is the mass ratio and where and are the optimum tuning and optimum damping, respectively, both of which come from optimization equations. Controller For simplicity and ease of comparison, a linear quadratic regulator (LQR) controller has been designed to investigate the performance of both semi-active devices. The importance weight matrix on structural motion,, and the weight on the control force,, are taken as: = 0 0 ( ) =10 (5) with: ( )= for = respectively, and maintained constant for both systems under study. To simplify comparison between devices, both the VO damper and the MFD are taken as ideal semi-active dampers. In other terms, if the required force from the LQR controller is within the set of accessible forces of the device ( ), the device is assumed to be capable of reaching that force. Moreover, if the Lindh, Laflamme, and Connor 6

8 required force is outside the range, the device will be set to its maximum voltage or minimum voltage, depending on the sign of the velocity: = if > and sign( )=sign otherwise (6) where is the control voltage, and is the control device velocity. Simulation Results In order to produce a general assessment of the dynamic behavior of a system, it is useful to work with a dynamic amplification function, ( ), defined as the frequency-dependent ratio between the system output and the excitation input. For SDOF structures, ( )= ( ) (7) where is the system stiffness and ( ) is the transfer function. Figure 4 demonstrates the dynamic responses of the systems under consideration for a TMD with µ=0.01. All results shown use the SDOF parameters given above; the VO and MFD were further constrained to have similar force accessibility, a concept described later in greater detail. Figure 4. Dynamic response of TMD systems for =. Both the VO-STMD and the MFD-STMD demonstrate improved performance as compared to the passive TMD, with the MFD system producing slightly superior suppression of peak dynamic amplifications. Simulations were also performed using systems with various mass ratios to document Lindh, Laflamme, and Connor 7

9 any dependence on. In each case, the STMD system was given optimal design parameters within realistic bounds, namely that the dynamic range of the VO not exceed 250 and that the maximum force of either system not exceed 3, where is the maximum force exerted by the viscous damper in the passive TMD system. Results were obtained for 5 discrete mass ratios, = For all systems considered, performance was compared in two ways. First, results are given in terms of the effective mass increase of the system, which demonstrates how much more mass would be necessary in a passive system to achieve the same mitigation results. Second, efficacy is presented in terms of effective damping,, the damping ratio necessary in an SDOF system to provide equivalent vibration suppression. The performance measure =max ( ) is utilized, which allows to be given by: = (8) Figures 5 and 6 present the results for all under consideration. In all cases, the MFD-STMD is seen to provide improved performance over the VO-STMD. The average effective mass increase was 69% for the VO system and 83% for the MFD system. Figure 5. Effective mass increase provided by semi-active device Lindh, Laflamme, and Connor 8

10 Figure 6. Effective damping of various systems Other than the constraints given previously, these results were obtained by designing each STMD solely for optimized performance. When assessing the feasibility of a mitigation system, however, other factors must be taken into account. Details of various factors can be seen in Table 2, where and refer to the effective mass ratio and effective damping as previously described. All other quantities presented are given in terms of their relationship to the equivalent passive TMD system:, gives the maximum STMD damper force as a function of the maximum force generated by the passive viscous damper;, gives the average power dissipated by the STMD damper in terms of the average power dissipated by the passive TMD; and, gives the maximum relative displacement between the STMD mass and the structure in terms of the maximum relative displacement of the passive TMD. The quantity is a metric characterizing how much control force is accessible by a given STMD system. Figure 7 portrays how is found for VO Table 2. Simulation results for the effect of the mass ratio Lindh, Laflamme, and Connor 9

11 (a) Figure 7. a) for the VO; b) for the MFD and MFD dampers, indicated by the shaded areas. In Table 2,, gives the accessible force of various VO and MFD systems in terms of an analogous passive value, =(1 2). As Table 2 indicates, the average power dissipated and the maximum relative displacements are fairly comparable for both the VO- and the MFD-STMD systems, both of which see slight increases over their passive TMD counterparts. More significant variations are present in, and,, which is to be expected given the marked difference in accessible forces between the linear VO and the nonlinear MFD. The nonlinear MFD-STMD attains superior results with a much lower requisite force. Figure 8 explicitly shows the influence of the maximum damper force for STMD systems with a mass ratio of Results are given as a percent of the maximum mitigation attained by either system and demonstrate significantly better results for the MFD-STMD. It is noted, however, that for an equivalent maximum force, there is a much greater range of accessible forces available to the nonlinear device than to the linear device. It is therefore of interest to investigate the influence of, on the overall effectiveness of the STMD system. As demonstrated in Figure 9, the MFD- STMD indeed lags behind the mitigation of the VO-STMD for low,, but even for realistically attainable values of, the MFD-STMD again supersedes the performance efficiency of the VO-STMD. (b) Figure 8. Influence of maximum damper force for =. Lindh, Laflamme, and Connor 10

12 Figure 9. Influence of accessible force range for =. Discussion As the above results have demonstrated, both the VO and the MFD are capable of improving the control performance of the STMD. More specifically, performance results have shown the effectiveness of the STMD systems considered to be equivalent to passive systems with 60% - 90% more mass. This is significant because one possible design objective in choosing a semi-active system could be to decrease the required mass. Particularly in retrofit situations, space or weight constraints may present difficult design issues that could be ameliorated through the addition of a semi-active damper. Furthermore, while both the VO and MFD offered improved performance, the nonlinear MFD demonstrated superior vibration mitigation for all mass ratios considered in this study. Because this could be accomplished with both less maximum damper force and less overall accessible force, it is readily apparent that the presence of nonlinearity in the damping device has a direct bearing on the effectiveness of the STMD system. Consequently, use of a suitable nonlinear semiactive device is seen to be paramount to the optimization of an STMD. Due to its robustness and mechanical simplicity, the MFD shows potential to accomplish this goal. Conclusion The analysis presented here has compared the potential of linear and nonlinear semiactive damping devices to improve passive tuned mass damper performance. Representative simulations involving a linear variable orifice (VO) damper and a nonlinear modified friction device (MFD) have demonstrated the nonlinear device to be superior at improving vibration mitigation. In addition to improved performance, the MFD-STMD exhibits the added benefit of requiring smaller maximum damper forces than the VO-STMD. Both the VO-STMD and the MFD-STMD, however, have demonstrated the ability to add to the effective mass of the TMD system. This result gives particular promise to STMD implementation in both retrofit situations in which space constraints may be significant and in design situations where extraordinarily large masses would otherwise be required. Lindh, Laflamme, and Connor 11

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