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Shake table investigation of a structure isolated by recycled rubber devices and magnetorheological dampers

2016; Wiley; Volume: 24; Issue: 5 Linguagem: Inglês

10.1002/stc.1906

1545-2263

Giuseppe Maddaloni, Nicola Caterino, Antonio Occhiuzzi,

Structural Engineering and Vibration Analysis

Structural Control and Health MonitoringVolume 24, Issue 5 e1906 Research ArticleOpen Access Shake table investigation of a structure isolated by recycled rubber devices and magnetorheological dampers Giuseppe Maddaloni, Corresponding Author Giuseppe Maddaloni giuseppe.maddaloni@unisannio.it Department of Engineering, University of Benevento ‘Sannio’, Benevento, 82100 Italy Construction Technologies Institute, Italian National Research Council (ITC-CNR), San Giuliano Milanese (MI), Italy Correspondence to: Giuseppe Maddaloni, Department of Engineering, University of Benevento ‘Sannio’, Palazzo ex-INPS - Piazza Roma, 21, 82100 Benevento, Italy. E-mail: giuseppe.maddaloni@unisannio.itSearch for more papers by this authorNicola Caterino, Nicola Caterino Department of Engineering, University of Naples ‘Parthenope’, Naples, 80143 Italy Construction Technologies Institute, Italian National Research Council (ITC-CNR), San Giuliano Milanese (MI), ItalySearch for more papers by this authorAntonio Occhiuzzi, Antonio Occhiuzzi Department of Engineering, University of Naples ‘Parthenope’, Naples, 80143 Italy Construction Technologies Institute, Italian National Research Council (ITC-CNR), San Giuliano Milanese (MI), ItalySearch for more papers by this author Giuseppe Maddaloni, Corresponding Author Giuseppe Maddaloni giuseppe.maddaloni@unisannio.it Department of Engineering, University of Benevento ‘Sannio’, Benevento, 82100 Italy Construction Technologies Institute, Italian National Research Council (ITC-CNR), San Giuliano Milanese (MI), Italy Correspondence to: Giuseppe Maddaloni, Department of Engineering, University of Benevento ‘Sannio’, Palazzo ex-INPS - Piazza Roma, 21, 82100 Benevento, Italy. E-mail: giuseppe.maddaloni@unisannio.itSearch for more papers by this authorNicola Caterino, Nicola Caterino Department of Engineering, University of Naples ‘Parthenope’, Naples, 80143 Italy Construction Technologies Institute, Italian National Research Council (ITC-CNR), San Giuliano Milanese (MI), ItalySearch for more papers by this authorAntonio Occhiuzzi, Antonio Occhiuzzi Department of Engineering, University of Naples ‘Parthenope’, Naples, 80143 Italy Construction Technologies Institute, Italian National Research Council (ITC-CNR), San Giuliano Milanese (MI), ItalySearch for more papers by this author First published: 26 July 2016 https://doi.org/10.1002/stc.1906Citations: 18AboutSectionsPDF ToolsRequest permissionExport citationAdd to favoritesTrack citation ShareShare Give accessShare full text accessShare full-text accessPlease review our Terms and Conditions of Use and check box below to share full-text version of article.I have read and accept the Wiley Online Library Terms and Conditions of UseShareable LinkUse the link below to share a full-text version of this article with your friends and colleagues. Learn more.Copy URL Share a linkShare onFacebookTwitterLinkedInRedditWechat Summary This paper describes the main results obtained from shaking table tests performed on a 1-storey steel frame isolated by innovative rubber bearings and magnetorheological dampers (hybrid isolation). The proposed base isolation technology is based on the use of bearings made of a low-cost recycled elastomer and reinforced with fibre sheets. Bounding the strain demand for such isolators because of severe earthquakes within acceptable values is a key point of this research, crucial for the effective application of this low-cost technology to real cases. A specific semi-active controller has been designed and adopted to achieve this goal. A set of natural earthquake records is applied to the base of the said structure using a shaking table system. The experimental results in terms of base displacement, roof acceleration and interstory drift are compared with the response of the same structure base isolated in the absence of magnetorheological dampers. It is found that when the semi-active control is suitably designed and implemented, the seismic performance of the structure can be significantly improved. Even if referred to the case study, the conclusions encourage further investigating the application of the proposed low-cost technology for seismic protection of buildings. © 2016 The Authors. Structural Control and Health Monitoring Published by John Wiley & Sons, Ltd. 1 Introduction Seismic base isolation (BI) is one of the most successful techniques used to protect structures against earthquakes. Many different devices can be adopted to implement such control strategy 1. However, observations from recent earthquakes have demonstrated that an isolation system alone in some cases cannot provide a full protection against structural damage given that a base-isolated building may experience large and unexpected displacements, especially because of near-field earthquakes 2. Such excessive displacements are generally undesirable for many reasons, not least the possible collision with adjacent constructions 3. In recent years, several researchers have investigated the possibility of using supplemental dampers to reduce displacement demand for base-isolated structures. This is the concept of the hybrid BI system. In a BI system, the use of passive dampers as supplemental devices is not always agreed upon. Actually, in such cases, the base displacement demand is generally reduced, even if interstory drifts and floor accelerations may increase 4. The use of adaptive ‘smart’ dampers is proposed herein to enhance the performance of a BI system, avoiding the above undesired. A pioneering study on hybrid isolation through the use of variable dampers was proposed by Makris in 1997 5. In particular, in this paper, the problem of protecting base-isolated structures from rapid, long-period motions is addressed. He shows that in certain cases, the presence of high-friction forces in the isolation system of a structure is responsible for the presence of permanent displacements. Electrorheological dampers can eliminate this problem because the friction forces can be removed during shaking, thus allowing the superstructure to be recentred once the seismic action is ended. Subsequently, a design approach for the use of semi-active (SA) devices in base-isolated buildings was proposed by Li and Ou 6. The effectiveness of the SA control of base-isolated structures was tested in several experiments. In Madden et al. 7, the ability of an adaptive seismic isolation system to protect structures subjected to earthquake ground motions was investigated. The isolation system consists of sliding isolation bearings in combination with an adaptive hydraulic damper. The damping capacity of the hydraulic damper can be modified in real time to respond to the effects that the earthquake ground motion has on the structure. An experimental laboratory implementation of the adaptive isolation system within a scale-model building structure was described. The experimental results demonstrated that for both near-field and far-field earthquake ground motions, an adaptive sliding BI system is capable of reducing the interstory drift response of structures while simultaneously limiting the displacement response of the isolation system. Other experimental studies have been conducted on hybrid BI systems by pairing rubber bearings with magnetorheological (MR) dampers 8, 9. In particular, in 8, a series of large-scale experimental tests was conducted on a mass equipped with a BI system that consists of high-damping rubber bearings and an MR damper. The specimen was subjected to various intensities of near- and far-fault earthquakes. The results showed that high-damping rubber bearings can reduce the absolute acceleration of the mass but that they also allow large displacements. Therefore, an MR damper is needed to control displacement. In 9, the performance of an isolation system for a base-isolated, two-degree-of-freedom structural model employing MR fluid dampers was investigated. The efficacy of this smart system in reducing structural responses for a wide range of loading conditions was demonstrated in a series of experiments. The results for the smart solution were compared with tests wherein an MR damper was operated in a passive mode (i.e. with a constant current being applied to the MR damper). A comparison between numerical and experimental results obtained using isolated structures controlled with MR dampers is shown in 10 and 11. Researchers' efforts on hybrid isolation systems realised using MR dampers have been focused on the development of new control algorithms for reducing the response of base-isolated structures 12-14. In 14, two control algorithms were developed to monitor the voltage input to an MR damper so that the desirable performance of the structural system could be achieved. The novelty of the models is that they consider, for the first time, the effect of the supplied voltage on the commanded voltage dynamics of the MR damper. The efficiency of the proposed techniques has been shown and compared using an example of a base-isolated three-storey building subject to a set of seismic excitations. An interesting hybrid BI system composed of linear elastomeric bearings, friction-pendulum bearings, shape-memory alloy wires and magnetorheological dampers was proposed for the mitigation of seismic motions by Shook et al. 15. To compare different isolation systems, such as rubber bearings, friction pendulums and roller systems, a wide experimental campaign was recently conducted in Naples (Italy) under the framework of the Joint Experimental Testing on Base Isolation Systems (JETBIS) program, which is financed by the Italian Department of Civil Protection. In particular, the seismic performance of a steel frame that is isolated with an innovative type of rubber bearing was evaluated by shaking table tests. The novel BI technology consists of bearings made of a low-cost recycled elastomer and reinforced with fibre sheets 16. For sites requiring large design displacements (e.g. in some epicentral areas), the rubber bearings cannot be used in the absence of dampers that are used in parallel. Therefore, the behaviour of the same base-isolated structure described earlier, but also equipped with MR dampers (hybrid isolation), is experimentally investigated using the shaking table. The present paper describes the main results obtained comparing the seismic response of the base-isolated structure with that of the same structure equipped with the hybrid control system. 2 Description of Experimental Tests In the following, the main features of the JETBIS structural mock up (test setup and instrumentation) are introduced. 2.1 Test setup configuration The shake table tests, which are performed to investigate the seismic behaviour of the hybrid BI system, were conducted at the laboratory of the Department of Structures for Engineering and Architecture of the University of Naples Federico II. The tests were performed using an earthquake simulator system, which consists of two 3 × 3 m square shake tables. Each table is characterised by two degrees of freedom in the two horizontal directions. The maximum payload of each shake table is 200 kN with a frequency range of 0–50 Hz, peak acceleration equal to 1 g at the maximum payload, peak velocity equal to 1 m/s and total displacement equal to 500 mm (±250 mm). Only one shake table was used in this experimental campaign. The test setup is shown in Figure 1(a) and (b). It was adapted from one previously used for experimental tests at the same laboratory 17, 18. The test frame has a height of 2900 mm and plan dimensions of 2150 × 2650 mm. The frame is constructed using welded square hollow columns (150 × 150 × 15 mm) of C45 steel material and rolled square hollow beams (120 × 120 × 12.5 mm) of S275 steel material; the beam–column connections are bolted. A 3.3-tons reinforced concrete slab is placed on the roof of the structure. Its plan dimensions are 2150 × 2650 mm with a thickness equal to 250 mm. The concrete slab is connected to the test frame by prestressed bolts to guarantee sufficient friction strength at the steel-beam-to-slab interface for the transfer of inertia forces. Figure 1Open in figure viewerPowerPoint Setup configuration: sketch (a) and general view (b) of the test frame at the Department of Structures for Engineering and Architecture laboratory; sketch (c) and general view (d) of the magnetorheological (MR) device. A total mass of 3.6 tons is added at the base of the frame using concrete blocks. The mass is restrained on the shaking table by a rigid horizontal frame (in red) fixed at the vertical frame, as shown in Figure 1(b). The total mass of the setup (vertical steel frame + top concrete slab + concrete blocks + horizontal steel frame) is approximately 7.8 tons. Four safety frames are added to the setup configuration to prevent the overturning due to excessive displacement (Figure 1(a)). Two MR dampers, placed on the East and West sides of the setup (Figure 2), were used in this test campaign. They are connected to the frame and shaking table by specially designed steel parts (Figures 1(c) and (d)). Figure 2Open in figure viewerPowerPoint Overview of the instrumentation positioning. Four rubber bearings placed at the corners of the test frame complete the setup in the test configuration (Figures 1(a) and (c)). The steel frame adopted in the tests simulates a 1:3-scale 1-storey structure. The tests were performed by moving the shaking table in one direction, as indicated in Figures 1(a) and (b). 2.2 Instrumentation A total of 10 transducers were used to measure the response of the structure during the hybrid tests (Figure 2). Specifically, four laser displacement sensors were placed on the South side of the test setup (two at the top and two at the base of the vertical steel frame) to measure the absolute displacements in the direction of the applied motions. The laser displacement sensors are connected to an external reference steel frame. Accelerations are measured by a total of six triaxial accelerometers. Three accelerometers are located on the concrete slab, two are located at the base of the vertical steel frame in the South-West and South-East corner, and one is located on the shaking table platform (Figure 2). The sampling frequency of the measured quantities is 200 Hz; during data processing, a 30-Hz low-pass filter was applied to all recorded signals. Finally, piezoresistive load cells mounted at the end of the MR dampers are used to measure the force of the dissipating devices (Figures 1(c) and (d)). Table I summarises the description of all response parameters measured during the tests as well as the adopted sensors, including their position in height and in plan. Table I. Response parameters measured: description, position in height and in plan of the instruments. 3 Rubber Bearings and Magnetorheological Dampers In the following, the adopted rubber bearings and MR dampers are briefly described. Furthermore, the electronic equipment used by the SA control system is introduced, also allowing to highlight specific aspects of this type of experiment that are unusual compared with other types of structural testing. 3.1 Description The rubber bearings used in the tests are named recycled rubber–fibre-reinforced bearings (RR-FRBs). They are made of a low-cost recycled elastomer and are reinforced with fibre sheets. Devices with dimensions of 70 × 70 mm in plane and a thickness of 63 mm were adopted in the tests. Each bearing is manufactured by bonding 12 layers of recycled rubber and 11 bidirectional carbon fibre sheets with a polyurethane adhesive (Figure 3). Figure 3Open in figure viewerPowerPoint Picture of a rubber bearing. The design vertical pressure on the bearings is approximately 3.85 MPa. At a design displacement of 30 mm, the horizontal and vertical stiffness of the bearing are approximately 80 kN/m and 7000 kN/m, respectively. The manufacture of the bearings is very simple. The entire process took only few hours for the curing of the binder. Each bearing is shaped by a table cutting machine that divides a long pad of large dimensions. Currently, only a few prototypes have been produced, and an accurate estimate of the cost of production is not easy. However, it is reasonable to assume that the large-scale production of these bearings for real applications could bring the price up to a hundred euro per isolator. More details can be found in Spizzuoco et al. 19. The MR devices adopted for the tests are two full-scale prototypes (Figure 4) designed and manufactured by the German company Maurer Söhne. The overall dimension of each device is 675 mm (length). A maximum force of 30 kN can be developed along the longitudinal axis, whereas the presence of special spherical pin joints at both ends prevents the development of bending, shear and torsional moments in the piston rod. The dampers have a stroke of ±25 mm. A magnetic circuit composed of coils in series with a total resistance of 3.34 Ω is used to generate the magnetic field in the device. The current in the circuit is provided in the range of 0 ÷ 3 A. Detail about mechanical response and dissipative capability of such prototype devices can be found in 20 and 21 respectively. Figure 4Open in figure viewerPowerPoint One of the two prototype magnetorheological (MR) dampers used in the tests. 3.2 Extra equipment for structural control Additional special electronic equipment was utilised to drive the MR devices in hybrid tests (Figure 5). The equipment includes the following components: (1) a set of two operational power supplies (model BOP 50-4 M from Kepco Inc., New York, USA) featuring an output range of ±50 V, ±4 A (i.e. power source-power sink capabilities of 200 W), which can provide the current needed to feed the circuitry inside the MR device; (2) a 1.0-μF capacitor mounted in parallel with the output for stabilising the current loop operating with inductive loads and measuring both the current and voltage inside the MR damper in real time; (3) the Labview Professional Development System (release 8.5), that is the environment in which the software that acquires and generates all analogue signals involved in the experimental tests was written; (4) a National Instruments (NI) SCC-A10 10-to-1 voltage attenuator for scaling the ±50 V output signal from the power supply to be measured by the acquisition board; (5) an NI PXI 1042 chassis; (6) an NI PXI-8196 RT embedded real-time controller; (7) two NI PXI-6259 data acquisition boards each with 16 analogue inputs and four analogue outputs (±10 V voltage signals, 16-bit resolution and 2800 kHz maximum sampling rate); (8) an NI PXI-4065 digital multimeter for measuring the current in the damper's circuit; and (9) two NI BNC-2110 connector blocks. Figure 5Open in figure viewerPowerPoint Electronic equipment for acquisition and control for the Joint Experimental Testing on Base Isolation Systems experimental tests: power supplies (1), 1.0 μF capacitor (2), Labview software (3), voltage attenuator (4), National Instruments chassis (5), real-time National Instruments CPU (6), data acquisition board (7), digital multimeter (8) and connector block (9). The MR dampers have two parallel and independent configurations of sensors, cables, control algorithms and power supplies. For on-line control, the sampling frequency of relevant quantities was set to 1 kHz. Signals have been low-pass processed using a second-order Butterworth filter and a 30 Hz cutting frequency. Because of structural symmetry, the data recorded for each damper are practically identical; therefore, in the following, we will refer to either of the two sides. 4 Input and Testing Protocol A set of three acceleration time histories was used for the tests. The earthquake accelerograms were derived from the experimental campaign on isolated structures performed by Calabrese et al. in 2014 16. In particular, spectrum-matching accelerograms were used for the tests by selecting a set of seven waveforms compatible with the Italian Seismic Code 22 according to the code provision described in 23, 24. Only three more severe earthquakes (Bingol, Campano-Lucano, Montenegro) out of the seven selected in Calabrese et al. 16 were used in the present experimental tests. Three is the minimum number of events to be considered, according to many standards, when time history analyses are adopted to characterise the seismic response of a given structure. Time constraints did not allow implementing additional shaking table analyses, each requiring an intense preliminary activity to ensure being able to perform the test in a safe way, taking into account the unusual behaviour of the adopted low-cost bearings. Given a geometry scale factor of the test frame of 1/3, to satisfy the dynamic similitude requirements, the selected earthquakes have been compressed in time by the same scale factor. The magnitudes of the signals are also scaled (see SF in Table II) to obtain a peak ground acceleration equal to 2.55 m/s2 for all time histories. The maximum value of the acceleration scale factors is 1.01, the minimum is 0.80 and the mean is 0.89. The acceleration time-history inputs for the shaking table tests are shown in Figure 6. Table II lists the main parameters of these ground motions and demonstrates that the values of peak ground acceleration for all of the events are equal (2.55 m/s2). Table II. Selected ground motion, specifics and significant parameters. Earthquake name Waveform ID Station ID Date [dd/mm/yyyy] SF PGA [m/s2] PGV [cm/s] PGD [cm] Bingol (BIN) 7142ya ST539 01/05/2003 0.87 2.55 18.29 3.25 Campano-Lucano (CAM) 290ya ST96 23/11/1980 0.80 2.55 44.10 16.20 Montenegro (MON) 200ya ST68 15/04/1979 1.01 2.55 12.87 9.60 PGV, peak ground velocity; PGD, peak ground displacement. Figure 6Open in figure viewerPowerPoint Acceleration time-history inputs for shaking table tests. The experimental activity included 12 tests. Table III shows their characteristics. In tests #1, #2 and #3, the frame is isolated by rubber bearings only (BI) and subject to the Bingol, Campano-Lucano and Montenegro earthquakes, respectively, without supplemental dampers. These tests form the basis of a comparison for evaluating the effectiveness of the proposed hybrid BI system. Subsequently, MR dampers are added to the initial configuration to reduce the base displacements. In tests #4, #5 and #6, the MR dampers are not fed with a current (passive OFF configuration), thus corresponding to a moderate level of supplemental damping. A high level of supplemental damping was adopted for tests #7 to #9, where the current fed to the dampers is 0.4 A (passive ON configuration). Tests #10, #11 and #12 refer to devices operated in a semi-active manner according to a properly designed algorithm (SA configuration). Table III. Experimental activity. MR, magnetorheological; BIN, Bingol; MON, Montenegro; CAM, Campano-Lucano. 5 Semi-active Control Algorithm. Evaluation of Additional Damping The previously described structural mock up can be modelled as a 2-DOF system. A preliminary set of identification tests allowed characterising the tested structure as follows: m2 = 4625 kg, m1 = 3208 kg, k2 = 3051000 N/m, k1 = 606000 N/m, c2 = 1176 Ns/m, c1 = 12250 Ns/m. With reference to the preceding values, mass, damping and stiffness matrices are organised as follows: (1) The equations of free motion of the bare frame, that is, without the influence of MR dampers, can be written as usual: (2)where x = [x1 x2]T is the two-component vector of the system displacements relative to the base. Each component is a scalar function of time; however, the time dependence will be explicitly written only when strictly needed. In the typical hypothesis of neglecting damping, the main modal parameters are modal periods T1 = 0.74 s; T2 = 0.15 s modal shapes Modal shapes are normalised so that the second row component (corresponding to DOF n. 2) is set to 1. The first eigenvector clearly shows that the isolation effect is partial and with a significant structural strain. An alternative representation of the modal shapes (modulus and phase) is shown for comparison with what follows. In this case, the 180° phase delay of the second component of the second mode corresponds to the minus sign in the typical representation of mode shapes in the time domain. Modal periods Ti fit the experimentally evaluated counterparts reasonably well. The linear system of Eq. 2 (two time-invariant second-order differential equations) can also be written as a set of four linear, time-invariant, first-order differential equations in the state space representation: (3)where (4)and (5)is the system state. By explicitly considering damping (inherent structural damping and that associated to the isolation devices), the main modal parameters computed in the state space domain are modal periods T1 = 0.74 s; T2 = 0.15 s modal damping ratios ζ1 = 8,0%; ζ2 = 3,7% modal shapes Compared with the undamped analysis, the BI structure without supplemental dampers includes an insignificant phase between the components of the first eigenvector and a reduction in the phase between the components of the second eigenvector, whereas absolute values are comparable. The addition of a significant amount of damping should correspond to a ‘degradation’ of modal shapes, that is, to higher values of phase between the components of each mode. Specifically, benefits associated with the increase in damping would be reduced or vanish because of the modal modifications induced by damping. By considering the force f of the dampers, Eq. 2 becomes (6)or (7) Accordingly, the equation of free vibrations in the state space becomes (8)where (9) If the force f is provided by a linear viscous damper with constant c, that is, (10)where (11) Eq. 7 could be written as (12)and Eq. 8 would become (13)where (14) The effect of the additional dissipation provided by MR dampers is analysed in terms of equivalent viscous damping 25. When currents of 0 and 0.4 A are applied, each MR damper corresponds to a linear equivalent damper with a viscous constant that is approximately equal to c0 = 30 kN s/m and c0.4 = 60 kN s/m in the cases of Bingol and Montenegro base accelerations, respectively, where the maximum velocities of the base with respect to the ground are approximately 0.09 m/s. For the Campano-Lucano accelerogram, where the maximum recorded velocities are close to 0.19 m/s, the equivalent linear viscous coefficients are approximately c0 = 20 kN s/m and c0.4 = 40 kN s/m. The complex modal analysis corresponding to the passive configurations of MR dampers (current set to 0 and 0.40 A) yields the results shown in Table IV, where data corresponding to the BI structure without dampers are also presented for comparison. Table IV. Modal parameters of passive configurations. BI Passive off c0 = 20 kNs/m Passive off c0 = 30 kNs/m Passive on c0.4 = 40 kNs/m Passive on c0.4 = 60 kNs/m First mode Second mode First mode Second mode First mode Second mode First mode Second mode First mode Second mode Period 0.74 s 0.15 s 0.73 s 0.15 s 0.71 s 0.16 s 0.69 s 0.16 s 0.64 s 0.18 s Damping ratio 8.0% 3.7% 35% 13% 49% 18% 64% 23% 99% 31% Modal shape 0.89∠1° 1.62∠175° 0.91∠5° 1.58∠156° 0.94∠6° 1.53∠147° 0.98∠7° 1.47∠137° 1.15∠6° 1.27∠117° 1.00∠0° 1.00∠0° 1.00∠0° 1.00∠0° 1.00∠0° 1.00∠0° 1.00∠0° 1.00∠0° 1.00∠0° 1.00∠0° As shown in the next section, the addition of a high level of damping to the base of the structure corresponds, as expected, to a significant reduction in base displacements at the cost of a general increase in both interstory drift and peak accelerations. In an attempt to overcome the undesirable side effects of the additional damping, in a further series of tests, the current in the MR devices was continuously varied according to a SA control scheme. Generally speaking, in a base-isolated n-DOF structure with m independent actuators, the equations of free motion can be represented as (15)where M, C and K are the mass, damping and stiffness matrices, respectively, and P is the n × m allocation matrix of the control forces fdi collected in the control vector fd. If is assumed that the SA forces in the dampers ‘fd’ have specific values ‘fu’, for example, functions of the system's state through a gain matrix G (16)then the state-space representation of the closed feedback loop n-DOF structural system under free vibration becomes (17) Eigenvalues and eigenvectors of ACL (summarising the properties of the closed-loop controlled system) are different from those of A; in other words, frequencies, damping ratios and modal shapes of the controlled system are different from those of the uncontrolled system. Therefore, the question arises if modal parameters can be modified in a more favourable way. An answer to this question was most likely first introduced by Moore 26 and subsequently explored by many scholars, but the authors are unaware of any application to the particular case of SA-controlled BI structures. In this latter case, it would be desirable to have an initial modal shape that resembles the motion of a rigid body above isolators with a high-damping ratio, whereas higher modes should have participation factors of close to 0. Let us assume that a matrix G does exist such that sd,i and ψd,i are the desired eigenvalue and eigenvector of the ith mode of the closed-loop (CL) system. When the CL system vibrates according to that mode, the system's state vari