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Experimental and analytical study on vibration control effects of eddy-current tuned mass dampers under seismic excitations
Journal of Sound and Vibration 421 (2018) 153e165
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Experimental and analytical study on vibration control effects of eddy-current tuned mass dampers under seismic excitations Zheng Lu a, b, Biao Huang b, Qi Zhang b, Xilin Lu a, b, * a b
State Key Laboratory of Disaster Reduction in Civil Engineering, Tongji University, Shanghai, 200092, China Research Institute of Structural Engineering and Disaster Reduction, Tongji University, Shanghai, 200092, China
a r t i c l e i n f o
a b s t r a c t
Article history: Received 18 August 2017 Received in revised form 20 October 2017 Accepted 23 October 2017
Eddy-current tuned mass dampers (EC-TMDs) are non-contacting passive control devices and are developed on the basis of conventional tuned mass dampers. They comprise a solid mass, a stiffness element, and a damping element, wherein the damping mechanism originates from eddy currents. By relative motion between a non-magnetic conductive metal and a permanent magnet in a dynamic system, a time-varying magnetic field is induced in the conductor, thereby generating eddy currents. The eddy currents induce a magnetic field with opposite polarity, causing repulsive forces, i.e., damping forces. This technology can overcome the drawbacks of conventional tuned mass dampers, such as limited service life, deterioration of mechanical properties, and undesired additional stiffness. The experimental and analytical study of this system installed on a multi-degreeof-freedom structure is presented in this paper. A series of shaking table tests were conducted on a five-story steel-frame model with/without an EC-TMD to evaluate the effectiveness and performance of the EC-TMD in suppressing the vibration of the model under seismic excitations. The experimental results show that the EC-TMD can effectively reduce the displacement response, acceleration response, interstory drift ratio, and maximum strain of the columns under different earthquake excitations. Moreover, an analytical method was proposed on the basis of electromagnetic and structural dynamic theories. A comparison between the test and simulation results shows that the simulation method can be used to estimate the response of structures with an EC-TMD under earthquake excitations with acceptable accuracy. © 2018 Elsevier Ltd. All rights reserved.
Keywords: Eddy-current tuned mass damper Tuned mass damper Passive control Shaking table test Analytical method Structural control
1. Introduction Since Yao introduced the concept of vibration control in civil engineering in 1972 [1], relevant structural control theories and methods have been significantly developed and have been proven to be economical and efficient in practice. The various control strategies, proposed by several researchers worldwide, can be classified into active control, passive control, hybrid control, and semi-active control, among which passive control is the prevalent strategy owing to its simplicity and lack of reliance on additional energy input [2]. Among numerous passive control devices, the tuned mass damper (TMD), wherein
* Corresponding author. State Key Laboratory of Disaster Reduction in Civil Engineering, Tongji University, Shanghai, 200092, China. E-mail addresses: [email protected] (Z. Lu), [email protected] (B. Huang), [email protected] (Q. Zhang), [email protected] (X. Lu). https://doi.org/10.1016/j.jsv.2017.10.035 0022-460X/© 2018 Elsevier Ltd. All rights reserved.
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the input energy is dissipated by tuning the frequency, is widely used owing to its simple characteristics, convenient installation, low cost, and favorable control effects at specific tuning frequencies [3]. The damping component of a TMD is mostly provided by conventional fluid viscous dampers. However, in practice, several issues arise in viscous dampers. In particular, these dampers may leak over time, thus limiting the service life. The physical properties may degrade in high-temperature environments, resulting in failure [4,5]. The stiffness of the overall system increases because of the viscous dampers, which is unexpected for some structures. Moreover, it is difficult or rather expensive to change the damping ratio of the TMD once the viscous dampers are installed. Consequently, some innovative damping mechanisms have been developed to address the shortcomings of conventional TMD technology. For example, some researchers replaced viscous damping with particle damping. However, the damping mechanism of particle dampers has not been thoroughly understood because of their high nonlinearity [6e13]. Another simple and effective solution involves using eddy current damping. Eddy current damping is a non-contacting damping mechanism which is suitable for solving the aforementioned problems. Because the device in eddy current damping is usually made of metal, it hardly degrades in its life cycle, making it useful for high-temperature applications. Owing to the non-contacting property, eddy current damping does not lead to an increase in the overall stiffness of the structures; therefore, the stiffness characteristics of the structures remain unaffected. Furthermore, the damping ratio can be easily adjusted by varying the air gap between the permanent magnet and the conductor. The damping mechanism via eddy currents has been reported by many researchers [14e18]. By the relative motion between a non-magnetic conductive metal and a permanent magnet or by changing the strength of the field magnets, a timevarying magnetic field is induced in the conductor, thereby generating eddy currents (shown in Fig. 1). The eddy currents induce another magnetic field with opposite polarity, thereby causing repulsive forces, i.e., damping forces. Because of the electrical resistance of the conductor, the induced currents dissipate in the form of heat at the rate of I2R, where I and R represent the current intensity and electrical resistance of the conductor, respectively. Therefore, the vibration energy of the main structures is transferred into the conductor and is dissipated in the form of heat. Eddy current damping is largely employed in the field of magnetic backing, and another application involves suppressing the lateral vibration in rotor shafts [20], which was reviewed by Henry and Bae [16]. In the field of structural vibration control, Matsuzaki et al. presented a vibration control system wherein the motion of a beam with an eddy current damper (ECD) could be suppressed. They derived a theoretical solution for the beam and demonstrated the viability of the concept [21]. In another study, an experiment was conducted to demonstrate the effectiveness of the ECD. Their results show that the electromagnetic force is capable of damping the first few modes of vibration of the beam [22]. In another study, Zheng et al. analyzed the effectiveness of an ECD in suppressing the free vibration response of a beam by numerical simulation and found that the damping effect of the ECD varies with the amplitude of the beam. The larger the amplitude, the greater is the effective damping [23]. Kwak et al. developed a new type of ECD to suppress the vibration of a beam. The ECD comprised a copper plate, which was rigidly fixed onto the end of the beam, and two permanent magnets. The ECD was constructed and tested to analyze the efficiency [15]. Later, Bae at el. developed the theoretical model of the ECD proposed by Kwak et al. and investigated the damping performance of a new ECD model. The experimental and simulation results demonstrate the potential of the ECD in controlling the vibration of a cantilever beam [14]. Sodano et al. proposed a new damper configuration providing more damping to the beam structure and developed an improved theoretical model with enhanced accuracy [24]. The
Fig. 1. Eddy current induced by the relative motion between a non-magnetic conductive metal and a permanent magnet [19].
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aforementioned studies primarily concentrated on the evaluation and analysis of ECDs in suppressing the vibration of cantilever beams with small-scale models. Kienholz et al. combined the advantages of the ECD and tuned mass damper and developed an eddy-current tuned mass damper (EC-TMD) to suppress the vibration of a solar sail array. They found that the magnetic damping techniques can achieve high damping forces [25]. In another study, Larose et al. investigated the effectiveness of the EC-TMD in suppressing the vibration of a small-scale bridge model by conducting wind tunnel tests. With the EC-TMD properly tuned, the global vibration of the bridge model could be damped [26]. Bae et al. confirmed the effective damping of an EC-TMD in suppressing the vibration of a beam model through both simulations and experiments [27]. In recent years, studies have been conducted on large-scale models with EC-TMDs. Wang et al. studied the feasibility of a large-scale EC-TMD and found that the eddy current damping was the dominant damping source. In addition, they demonstrated that the damping ratio could be easily adjusted by varying the gap between the permanent magnets and the conductive plates [28]. Bae et al. proposed a simple yet effective method for suppressing the vibration of a large beam structure using an EC-TMD without substantially increasing the overall weight [29]. However, to improve the practical application of ECDs to engineering structures such as high-rise buildings, it is necessary to analyze the damping performance in structural models by conducting large-scale tests. In this study, the damping performance of an EC-TMD installed on top of a five-story steel-frame model was evaluated by conducting an experimental study and determining an analytical solution. Shaking table tests conducted on the large-scale five-story frame structure are first introduced. An analytical solution for the damping performance of the EC-TMD, derived on the basis of electromagnetic and structural dynamic theories, is then discussed. Finally, a computational method is presented, simulated, and validated.
2. Experimental design 2.1. Model design The test model is a five-story steel-frame structure with a height of 5.48 m, comprising planes and columns, as shown in Fig. 2(a). The columns were made of Q690 steel whereas the planes were made of Q345 steel. The dimensions of the columns are 15 mm 180 mm 1060 mm and those of the planes are 30 mm 2000 mm 2000 mm. The total mass of the test model is 6000 kg. The seismic excitations were inputted to the structure along the weak axis direction in the shaking table tests. The natural vibration periods of the first five modes along the weak axis were 1.00, 0.35, 0.22, 0.17, and 0.15 s, respectively. Two acceleration sensors and two displacement sensors were set up in each floor. The acceleration and displacement sensors are numbered from 4 to 13 and from 76 to 85, respectively, as shown in Fig. 2(b). Four strain sensors were set up at the bottom of the columns to monitor the maximum strain of the columns; the strain sensors were numbered from 101 to 104.
Fig. 2. Model configuration. (a) Image of the test model, and (b) sensor arrangement.
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2.2. Damper design The EC-TMD system, manufactured by the Shanghai Research Institute of Materials, includes a supporting frame, four cables, a tuning mass, a permanent magnet, and a copper plane, as shown in Fig. 3. The supporting frame was fixed on top of the model. The lateral stiffness of the supporting frame was significantly greater than that of the model, thus ensuring that the supporting frame moves along with the top of the model. The tuning mass with a permanent magnet was suspended from the top of the supporting frame using four cables. Between the tuning mass and the steel frame model, the copper plane was fixed using four small columns. To improve the vibration-reduction effects, the frequency of the EC-TMD should be designed as close as possible to the first mode frequency of the primary structure. Hence, the length L of the suspended cables was decided by the classical equation for a simple pendulum: L ¼ ðT=2pÞ2 $g ¼ 0:248 m. The EC-TMD is primarily used for adjusting the damping properties of the auxiliary mass; however, it is not providing stiffness tuning, which is provided by the coupling structural member. In engineering practice, the mass of a tall building is considerable. Therefore, the mass ratio of the damper to the primary structure should be small. For example, a TMD with a mass ratio of 1% was attached to the Shanghai Center Tower in China [30], and a tuned liquid damper (TLD) with a mass ratio of 1% was attached to the Shin Yokohama Prince hotel in Japan [31]. On the basis of these practical cases, the mass ratio of the damper to the model structure in the tests was chosen to be approximately 1%. 2.3. Excitations in shaking table tests Fig. 4 shows the time histories and the corresponding power spectral density curves of three earthquake waves, including the El Centro, Wenchuan, and Japan 311 waves. The earthquake waves were incorporated in the shaking table tests to evaluate the damping performance of the EC-TMD system under different seismic conditions. The seismic excitations were inputted in only the weak axis direction of the test model, with peak ground accelerations of 0.05, 0.10, and 0.15 g, respectively (g is the acceleration due to gravity). 3. Response analysis To evaluate the vibration control performance of the damper, the vibration damping effect is defined as follows.
vibration damping effect ¼
the pðRMSÞ of structure without dampers the pðRMSÞ of structure with an EC TMD the pðRMSÞ of structure without dampers 100%
Here, the p and RMS represent the peak and root-mean-square values of the displacement/acceleration response at the top of the model, respectively. 3.1. Displacement response The RMS value of the displacement/acceleration is an indicator of the vibration energy and is usually employed to indicate the random variable energy in random vibration. The peak value of the displacement/acceleration is of concern in engineering
Fig. 3. Damper configuration.
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Fig. 4. Time histories of (a) El Centro, (b) Wenchuan, and (c) Japan 311 waves; Power spectral densities of (d) El Centro, (e) Wenchuan, and (f) Japan 311 waves.
design, because it is related to the safety of buildings and the comfort of the occupants. Table 1 lists the peak and RMS values of the displacement responses at the roof of the test model in different cases. With regard to the vibration control effects in reducing the displacement responses, the results show that (i) The response of the controlled model with an EC-TMD is lower than that of the uncontrolled model, thus implying that the EC-TMD can dissipate vibration energy and provide effective damping for the test model; (ii) The vibration-reduction effect in terms of the RMS values of the displacement responses is significantly better than that in terms of the peak values, wherein the range of the former is 18.6e69.4% and that of the latter is 10.4e34.9%; (iii) The vibration-reduction effect is different under different seismic inputs. The reduction in vibration is favorable in the case of the El Centro wave; however, the damping effects are relatively poorer in the cases of the Wenchuan and Japan 311 waves. This may be because of the frequency characterization of the input earthquake excitations. Fig. 4 shows the excitations of the El Centro, Wenchuan, and Japan 311 waves in the time and frequency domains. The frequency of the El Centro wave is largely in the range of 0.5e2 Hz, which is near the natural frequency of the primary structure, whereas the differences between the dominant frequencies of the Wenchuan and Japan 311 waves and the fundamental frequency of the structure are considerable. This frequency characterization makes the displacement response of the model in the case of the El Centro wave to be significantly greater than that in the cases of the other inputs. Accordingly, in the case of the El Centro wave, the EC-TMD dissipates more energy via mass tuning and eddy currents, thereby exhibiting a more favorable damping effect.
3.2. Acceleration response Not only can the displacement be reduced, the acceleration can also be attenuated in most cases. Table 2 lists the peak and RMS values of the acceleration responses at the roof of the test model in different cases. In all cases of the RMS values and most cases of the peak values, the acceleration responses of the controlled model are lower than those of the uncontrolled model. This shows that the EC-TMD can effectively dissipate the vibration energy of the test model. However, the vibrationreduction performance of the model with the EC-TMD in the case of the Japan 311 wave is the worst, particularly for peak ground accelerations of 0.05 and 0.15 g where the peak acceleration slightly increases. This is because although the EC-TMD is a modified version of the conventional TMD and therefore is expected to exhibit improved performance, the EC-TMD is still a linear damper, the performance of which is influenced by the frequency of the excitation, similar to the conventional TMD. This is observed in both Table 1 (Section 3.1) and Table 2 (Section 3.2). In fact, the key advantage of the EC-TMD is the non-contacting damping mechanism. Moreover, from Tables 1 and 2, the vibration damping effect in terms of the RMS values is better than that in terms of the peak values in most cases. This is clarified by analyzing the time history of the displacement/acceleration response in the following section.
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Table 1 Peak and RMS values of the displacement response at the roof of the test model. Seismic input
El Centro
Wenchuan
Japan 311
Peak input value (g)
0.05 0.1 0.15 0.05 0.1 0.15 0.05 0.1 0.15
Without dampers
With an EC-TMD
p (mm)
RMS (mm)
p (mm)
RMS (mm)
Reduction effect p (%)
RMS (%)
34.008 79.213 118.925 8.924 18.537 27.305 4.475 10.842 22.438
13.125 29.810 43.056 1.806 4.082 6.262 1.415 4.089 8.273
30.456 68.882 102.521 7.048 14.738 22.082 3.449 8.797 14.602
4.121 9.124 13.662 1.470 3.219 4.788 0.996 2.517 4.250
10.4 13.0 13.8 21.0 20.5 19.1 22.9 18.9 34.9
68.6 69.4 68.3 18.6 21.1 23.5 29.6 38.5 48.6
3.3. Time histories and power spectral densities The entire time history and power spectral density of the displacement/acceleration response are necessary for evaluating the damage inflicted to the structures and can be regarded as indicators of the intensity of the earthquake ground motion to some extent. Fig. 5 shows the displacementetime history and power spectral density curves at the roof level of the test model, wherein the solid and dotted lines represent the controlled and uncontrolled models, respectively. From the time history curves, it can be found that the vibration control effect is poor at the very beginning, and the controlled curves begin to decay quickly thereafter. If the peak values are observed at the very beginning, the damping performance in terms of the peak values will be poor; this is one of the reasons why the peak values of the responses is high in some cases and the vibration-reduction effects in terms of the RMS values are better than those in terms of the peak values listed in Tables 1 and 2. At the beginning, it takes some time for the EC-TMD to start moving and dissipate the vibration energy. Fig. 5(a) and (b) show that the peak values of the test model in the cases of the El Centro and Wenchuan wave excitations are 118.9 and 27.3 mm, respectively. Therefore, it was easier for the EC-TMD to create the damping effect in the case of the El Centro wave. Fig. 5(c) and (d) show the corresponding power spectral density curves in the cases of the El Centro and Wenchuan waves, respectively. Fig. 5(c) shows that the vibration energy is significantly high at the fundamental frequency of the test model. Moreover, the peak value of the model with the EC-TMD is significantly lower than that without the dampers, thus indicating that the EC-TMD has favorable damping effects at the fundamental frequency. Fig. 5(d) shows that the vibration energy is significantly high at the first and second mode frequencies of the test model. Because of the additional mass of the EC-TMD, the structural frequencies of the controlled model are slightly lower than those of the uncontrolled model. The peak values of the model with the EC-TMD are significantly lower than those of the model without the dampers. This is because the EC-TMD provides effective damping for the test model and suppresses the vibration. In most cases, the displacement responses of the model with the EC-TMD are lower than those of the model without the dampers. This is observed not only in the time history curves but also in the power spectral density curves.
3.4. Interstory drift ratio The maximum interstory drift ratio of a building is an important index for structural design and should not exceed specified ultimate values in the relevant design codes. Fig. 6 shows the maximum interstory drift ratio of the test model in the case of the El Centro wave with different peak ground accelerations (0.05, 0.1, and 0.15 g). Although the amplitudes of the excitations are different, the maximum interstory drift ratios with respect to the five stories of the test model are reduced using the EC-TMD in all cases. The damping performance of the EC-TMD for a peak ground acceleration of 0.15 g is better than those for peak ground accelerations of 0.05 and 0.1 g. This is because the vibration response of the test model in the former case is greater than those in the other cases. Therefore, the EC-TMD moves more violently and dissipates more vibration energy.
3.5. Maximum strain of the columns The maximum strain of the columns is a parameter used to check whether the structure is in the elastic or elastoplastic phase, in relation to the damage phenomena. Table 3 lists the maximum strains of the columns of the test model with/without the EC-TMD for all cases. It can be seen that: (i) The maximum strain of the model in the case of the El Centro wave is the highest, and the maximum strain in the case of the Japan 311 wave is the lowest, which coincides with the results of the displacement and acceleration responses, given in Sections 3.1 and 3.2, respectively; (ii) The maximum strains of the columns in all the cases do not exceed the yield strain of the Q690 steel columns, indicating that the test model remains in an elastic phase during the shaking table tests; (iii) The maximum strain of the columns can be reduced using the EC-TMD.
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Table 2 Peak and RMS values of the acceleration response at the roof of the test model. Seismic input
El Centro
Wenchuan
Japan 311
Peak input value (g)
0.05 0.1 0.15 0.05 0.1 0.15 0.05 0.1 0.15
Without dampers
With an EC-TMD
p (g)
RMS (g)
p (g)
RMS (g)
Reduction effect p (%)
RMS (%)
0.219 0.508 0.757 0.201 0.450 0.665 0.108 0.271 0.458
0.059 0.134 0.193 0.047 0.108 0.161 0.034 0.080 0.141
0.142 0.328 0.501 0.186 0.407 0.631 0.115 0.261 0.490
0.021 0.051 0.089 0.033 0.071 0.113 0.022 0.053 0.097
35.1 35.3 33.8 7.4 9.4 5.2 6.7 3.9 6.8
63.6 61.7 53.8 30.2 34.2 29.9 36.2 33.3 31.6
Fig. 5. Time history curves of the displacement responses of the model in the cases of (a) El Centro, and (b) Wenchuan waves; Power spectral density curves of the displacement responses of the model in the cases of (c) El Centro, and (d) Wenchuan waves.
The series of shaking table tests conducted on the model with/without the EC-TMD, along with the analysis of the experimental results, prove that the EC-TMD performs effectively in suppressing the vibration of a large-scale primary structure. In the following section, a corresponding analytical/simulation solution is proposed to estimate the response of the structures with the EC-TMD, and the simulation results are analyzed by comparing the experimental results. 4. Analytical/simulation solution 4.1. Analytical method The magnetic field distribution with respect to the rectangular permanent magnet is first introduced, and subsequently, the electromagnetic force is analyzed. The effects of the electromagnetic force can be reflected in the form of the eddy-current damping coefficient, which is a function of the relative distance between the permanent magnet and the conductor. The eddy-current damping coefficient can be then incorporated into the governing equations of the primary system.
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Fig. 6. Maximum interstory drift ratios of the test model in the case of the El Centro wave with peak ground accelerations of (a) 0.05, (b) 0.1, and (c) 0.15 g.
Table 3 Maximum strains of the columns in the test model. Seismic input
Peak input value (g)
Uncontrolled (mε)
Controlled (mε)
Reduction effect (%)
El Centro
0.05 0.1 0.15 0.05 0.1 0.15 0.05 0.1 0.15
338 776 1163 193 443 676 75 174 323
279 633 958 139 307 473 67 157 296
17.46 18.43 17.63 27.98 30.70 30.03 10.67 9.77 8.36
Wenchuan
Japan 311
Gou et al. [32] derived analytical expressions to determine the magnetic field distribution of a rectangular permanent magnet. To complete this analytical method, some key equations to calculate the magnetic field distribution are given and
Fig. 7. Coordinate system in a rectangular permanent magnet.
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derived herein. The dimensions of the permanent magnet are assumed to be a mm (length) b mm (width) c mm (height), and the direction of polarity of the magnet is along the z axis, as shown in Fig. 7. The components of the magnetic field distribution along the three axes can be derived on the basis of the molecular circulation model and the BioteSavart law.
Zh Bx ¼
N dBx ¼ ½Gða x; y; zÞ þ Gða x; b y; zÞ Gðx; y; zÞ Gðx; b y; zÞ 2
(1)
N dBy ¼ ½Gðb y; x; zÞ þ Gðb y; a x; zÞ Gðy; x; zÞ Gðy; a x; zÞ 2
(2)
0
Zh By ¼ 0
Zh Bz ¼
dBz 0
¼ N½Jðx; b y; zÞ þ Jðy; a x; zÞ þ Jða x; b y; zÞ þ Jðb y; a x; zÞ þ Jðb y; x; zÞ þ Jðy; x; zÞ þ Jða x; y; zÞ þ Jðx; y; zÞ
(3)
The auxiliary functions are defined as follows.
N¼
m0 J s 4p
(4)
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi g21 þ g22 þ ðg3 z0 Þ2 g2 z0 ¼ h Gðg1 ; g2 ; g3 Þ ¼ ln qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi z ¼0 g21 þ g22 þ ðg3 z0 Þ2 þ g2 0 3 j3 z0 6j1 7 z 0 ¼ h Jðj1 ; j2 ; j3 Þ ¼ arctan4 $qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi5 j2 j2 þ j2 þ ðj z Þ2 z0 ¼ 0 0 3 1 2
(5)
2
(6)
where m0 ð¼ 4p 107 Þ and Js ð¼ 9:55 105 A=m2 Þ represent the magnetic permeability in the air and the equivalent current density, respectively. The magnetic field distribution of the rectangular permanent magnet can be calculated using the aforementioned equations. Wang et al. determined the magnetic field distribution along the z direction at the height of 5.5 cm and found that the magnetic field is largely distributed on the projected area of the permanent magnet and decayed rapidly outside this area [28]. Compared to the magnetic field distribution in the projected area, the magnetic field outside this area is insignificant and therefore can be neglected, thus simplifying the analytical method and improving the efficiency of the calculation. When a conductive metal passes through a magnetic field, eddy current forces are created in the conductor. It is assumed that the EC-TMD tunes along the y axis. The velocity of the EC-TMD is expressed as follows.
vTMD ¼ 0i þ vy j þ 0k:
(7)
The current density J can be calculated as follows.
2
i J ¼ sðv BÞ ¼ s4 0 Bx
j vy By
3 k 0 5 ¼ svy ðBz i Bx kÞ Bz
(8)
where s is the conductivity of the conductor in the magnetic field, and B is the magnetic field strength at the location of the conductor. Once the eddy currents are generated, they induce a magnetic field with opposite polarity as the primary field, causing a repulsive electromagnetic force.
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Z
J BdV ¼ svy
F¼ ¼ svy
Z h
Z
2
i
6 4 Bz
j
k
3
7 Bx 5dV
0
(9)
Bx By Bz i Bx By i B2x þ B2z j þ By Bz k dV
The electromagnetic force in the motion direction of the EC-TMD is Fy .
Fy ¼ svy
Z B2x þ B2z dV
(10)
According to the properties of the magnetic field distribution, the magnetic field is largely distributed in the projected area of the permanent magnet, and the magnitudes of the magnetic fields are the same in this area. Therefore, Eq. (10) can be simplified as follows.
Fy ¼ sdS B2x þ B2z vy
(11)
where d and S are the thickness and projective area of the permanent magnet, respectively. Bx and Bz are the components of the magnetic field strength at the position of the conductor and are decided by the relative distance between the conductor and the permanent magnet. From Eq. (11), it can be found that the electromagnetic force is proportional to the relative velocity between the permanent magnet and the conductor; this force is the same as the linear damping force. Hence, the vibration control effect of the ECTMD can be considered equivalent to that of the linear damping coefficient, which is a function of the relative position between the conductor and the permanent magnet.
Ce ¼ sdS B2x þ B2z
(12)
When an EC-TMD is present in a multi-degree-of-freedom system, the effect of the EC-TMD is equivalent to that of a system with one spring and two dampers, as shown in Fig. 8, where Ct is the damping provided by the conventional TMD, and Ce is the damping component due to the eddy currents, which depends on the relative position between the system and the EC-TMD. The governing equations for a multi-degree-of-freedom system with an EC-TMD can be written as follows.
MX€ þ C X_ þ KX ¼ Ex€g M ¼ diag½M1 M2 …Mn Mt 2 C1 þ C2 C2 6 C C 6 2 2 þ C3 C3 6 6 C¼6 6 4 2
K1 þ K2 6 K 6 2 6 K¼6 6 6 4
3 1
Cn
Cn
Cn þ Ct þ Ce Ct Ce
K2 K2 þ K3
K3 1
Kn
Kn
Kn þ Kt Kt
X ¼ ½X1 X2 /Xn Xt
7 7 7 7 7 7 Ct Ce 5 Ct þ Ce 3
7 7 7 7 7 7 Kt 5 Kt
T
E ¼ ½ M1 M2 / Mn Mt T where M, C, and K are the mass, damping, and stiffness matrices of the system with the EC-TMD, respectively, and Ex€g is the inertial force matrix with respect to the ground acceleration. Xi is the relative displacement of the ith floor with respect to the ground, and Mi ; Ci ; and Ki are the mass, damping, and stiffness values of the ith floor, respectively (i ¼ 1,2 …,n). Xt is the relative displacement of the EC-TMD with respect to the ground; Mt ; Ct ; and Kt are the mass, damping, and stiffness values of the conventional TMD; and Ce is the time-varying damping coefficient of the EC-TMD with respect to the eddy currents.
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Fig. 8. Simplified diagram of a system with an EC-TMD.
Fig. 9. Calculated and experimental displacementetime histories of the model in the cases of (a) El Centro, (b) Wenchuan waves (0.15 g), (c) zoom-in view of (a), and (d) zoom-in view of (b).
Table 4 Comparison of the simulation and experimental results for the peak values of the displacement at the roof of the model with the EC-TMD. Seismic input
Peak input value (g)
Simulation (mm)
Experiment (mm)
Error (%)
El Centro
0.05 0.1 0.15 0.05 0.1 0.15 0.05 0.1 0.15
32.158 69.524 106.682 7.207 16.109 22.954 3.740 8.972 13.612
30.456 68.882 102.521 7.048 14.738 22.082 3.449 8.797 14.602
5.3 0.9 3.9 2.2 8.5 3.8 7.8 1.9 7.3
Wenchuan
Japan 311
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4.2. Computational results In Section 2, the shaking table tests conducted on the five-story steel-frame structure were discussed. The corresponding computational simulations are presented in this section. The mass and stiffness matrices of the five-story model are indicated by M s and K s , respectively.
2 6 6 6 Ms ¼ 6 6 6 4
3
M1 M3 M4
3
1100
52 104 52
K2 K2 þ K3 K3
1100 1100 3
K3 K3 þ K4 K4
K4 K4 þ K5 3
52 104 52
7 7 7 7 kg 7 7 5
1100
M5
K1 þ K2 6 6 K2 6 Ks ¼ 6 6 6 4 104 6 52 6 6 ¼6 6 6 4
7 6 7 6 7 6 7¼6 7 6 7 6 5 4
M2
2
2
2
K5
1227
7 7 7 7 7 7 K5 5 K5
7 7 7 7 104 N=m 52 7 7 104 52 5 52 52
The damping ratios of the test structure in the first two structural modes are x1 (¼0.85%) and x2 (¼0.25%), respectively. For the EC-TMD, the mass (Mt ) is 35 kg. Because the frequencies of the test model and the EC-TMD are equal, the stiffness between the EC-TMD and the test model can be calculated as Kt ¼ 1381 ðN=mÞ. The damping coefficient with respect to the eddy currents varies with the change in the relative distance between the conductor and the permanent magnet. Therefore, it was necessary to calculate the damping coefficient of the EC-TMD for every step using the equations given in Section 4.1. Fig. 9 shows the calculated and experimental displacementetime histories for the roof of the test model with the EC-TMD in the cases of the El Centro and Wenchuan waves for a peak ground acceleration of 0.15 g. The calculated displacementetime histories coincide with the experimental ones in most cases. Table 4 lists the comparison of the simulation results and the corresponding experimental results for the peak displacement at the roof of the test model with the EC-TMD. Fig. 9 and Table 4 show that the simulation results are in good agreement with the experimental ones, thus demonstrating that the proposed analytical method can be used to estimate the response of structures with an EC-TMD under earthquake excitations. 5. Conclusion Compared to conventional TMDs, the EC-TMD with a non-contacting damping mechanism shows excellent performance in several ways: 1) it does not degrade in its life cycle; 2) the damping performance remains largely unaffected even in hightemperature environment; 3) the stiffness of the overall structure remains unaffected; 4) the damping ratio can be easily adjusted by varying the air gap between the permanent magnet and the conductor. A series of shaking table tests were conducted on a five-story steel-frame model. From the experimental results, the ECTMD was proven both feasible and effective in suppressing the vibration of large-scale structures. In terms of the displacement and acceleration responses, the damping effects of the EC-TMD were found to be excellent, particularly in terms of the RMS responses. Although the frequency characteristics of the El Centro, Wenchuan, and Japan 311 waves are significantly different, the EC-TMD can be used to reduce the structural vibrations due to the seismic waves, demonstrating the effective vibration-control performance of the EC-TMD. In addition, the damper helps in reducing the interstory drift ratio and the maximum strain of the test model. The relationship between the permanent magnet and the conductor is derived on the basis of the electromagnetic theory, and the effect of the electromagnetic force is simplified to a time-varying damping coefficient. An analytical method is proposed to evaluate the response of the structures with the EC-TMD. The analytical/simulation method was validated via the shaking table test results, and the simulation accuracy was found to be reasonable. The results of the experimental and analytical investigations demonstrate the effectiveness and excellent performance of the EC-TMD in suppressing the vibration of a large-scale model. Therefore, this technology has a good potential for future applications to engineering projects.
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Acknowledgement Financial support from the National Natural Science Foundation of China (Grant No. 51478361), Science and Technology Commission of Shanghai Municipality (STCSM, Grant No. 13231201600) are highly appreciated. Shanghai Research Institute of Materials are acknowledged as well. References [1] J.T.P. Yao, Concept of structural control, J. Struct. Div. ASCE 98 (1972) 1567e1574. [2] F. Casciati, L. Faravelli, A passive control device with SMA components: from the prototype to the model, Struct. Contr. Health Monit. 16 (2009) 751e765. [3] A.J. Roffel, S. Narasimhan, Results from a full-scale study on the condition assessment of pendulum tuned mass dampers, J. Struct. Eng. 142 (2016), https://doi.org/10.1061/(asce)st.1943-541x.0001339. [4] Z. Lu, X. Chen, D. Zhang, K. Dai, Experimental and analytical study on the performance of particle tuned mass dampers under seismic excitation, Earthq. Eng. Struct. Dynam. 46 (2017) 697e714. [5] Z. Lu, X. Lu, W. Lu, S.F. 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Ikeda, Vibration control system using electromagnetic forces, J. Intell. Mater. Syst. Struct. 8 (1997) 751e756. [22] Y. Matsuzaki, T. Ikeda, A. Nae, T. Sasaki, Electromagnetic forces for a new vibration control system: experimental verification, Smart Mater. Struct. 9 (2000) 127e131. [23] X.J. Zheng, Y.H. Zhou, K. Miya, An analysis of variable magnetic damping of a cantilever beam-plate with end coils in transverse magnetic fields, Fusion Eng. Des. 55 (2001) 457e465. [24] H.A. Sodano, J.-S. Bae, D.J. Inman, W.K. Belvin, Improved concept and model of eddy current damper, J. Vib. .Acoust. Transac. ASME 128 (2006) 294e302. [25] D.A. Kienholz, S.C. Pendleton, K.E. Richards Jr., D.R. Morgenthaler, Demonstration of solar array vibration suppression, in: 1994 North American Conference on Smart Structures and Materials, International Society for Optics and Photonics, 1994, pp. 59e72. [26] G.L. Larose, A. Larsen, E. 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Journal of Sound and Vibration journal homepage: www.elsevier.com/locate/jsvi
Experimental and analytical study on vibration control effects of eddy-current tuned mass dampers under seismic excitations Zheng Lu a, b, Biao Huang b, Qi Zhang b, Xilin Lu a, b, * a b
State Key Laboratory of Disaster Reduction in Civil Engineering, Tongji University, Shanghai, 200092, China Research Institute of Structural Engineering and Disaster Reduction, Tongji University, Shanghai, 200092, China
a r t i c l e i n f o
a b s t r a c t
Article history: Received 18 August 2017 Received in revised form 20 October 2017 Accepted 23 October 2017
Eddy-current tuned mass dampers (EC-TMDs) are non-contacting passive control devices and are developed on the basis of conventional tuned mass dampers. They comprise a solid mass, a stiffness element, and a damping element, wherein the damping mechanism originates from eddy currents. By relative motion between a non-magnetic conductive metal and a permanent magnet in a dynamic system, a time-varying magnetic field is induced in the conductor, thereby generating eddy currents. The eddy currents induce a magnetic field with opposite polarity, causing repulsive forces, i.e., damping forces. This technology can overcome the drawbacks of conventional tuned mass dampers, such as limited service life, deterioration of mechanical properties, and undesired additional stiffness. The experimental and analytical study of this system installed on a multi-degreeof-freedom structure is presented in this paper. A series of shaking table tests were conducted on a five-story steel-frame model with/without an EC-TMD to evaluate the effectiveness and performance of the EC-TMD in suppressing the vibration of the model under seismic excitations. The experimental results show that the EC-TMD can effectively reduce the displacement response, acceleration response, interstory drift ratio, and maximum strain of the columns under different earthquake excitations. Moreover, an analytical method was proposed on the basis of electromagnetic and structural dynamic theories. A comparison between the test and simulation results shows that the simulation method can be used to estimate the response of structures with an EC-TMD under earthquake excitations with acceptable accuracy. © 2018 Elsevier Ltd. All rights reserved.
Keywords: Eddy-current tuned mass damper Tuned mass damper Passive control Shaking table test Analytical method Structural control
1. Introduction Since Yao introduced the concept of vibration control in civil engineering in 1972 [1], relevant structural control theories and methods have been significantly developed and have been proven to be economical and efficient in practice. The various control strategies, proposed by several researchers worldwide, can be classified into active control, passive control, hybrid control, and semi-active control, among which passive control is the prevalent strategy owing to its simplicity and lack of reliance on additional energy input [2]. Among numerous passive control devices, the tuned mass damper (TMD), wherein
* Corresponding author. State Key Laboratory of Disaster Reduction in Civil Engineering, Tongji University, Shanghai, 200092, China. E-mail addresses: [email protected] (Z. Lu), [email protected] (B. Huang), [email protected] (Q. Zhang), [email protected] (X. Lu). https://doi.org/10.1016/j.jsv.2017.10.035 0022-460X/© 2018 Elsevier Ltd. All rights reserved.
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the input energy is dissipated by tuning the frequency, is widely used owing to its simple characteristics, convenient installation, low cost, and favorable control effects at specific tuning frequencies [3]. The damping component of a TMD is mostly provided by conventional fluid viscous dampers. However, in practice, several issues arise in viscous dampers. In particular, these dampers may leak over time, thus limiting the service life. The physical properties may degrade in high-temperature environments, resulting in failure [4,5]. The stiffness of the overall system increases because of the viscous dampers, which is unexpected for some structures. Moreover, it is difficult or rather expensive to change the damping ratio of the TMD once the viscous dampers are installed. Consequently, some innovative damping mechanisms have been developed to address the shortcomings of conventional TMD technology. For example, some researchers replaced viscous damping with particle damping. However, the damping mechanism of particle dampers has not been thoroughly understood because of their high nonlinearity [6e13]. Another simple and effective solution involves using eddy current damping. Eddy current damping is a non-contacting damping mechanism which is suitable for solving the aforementioned problems. Because the device in eddy current damping is usually made of metal, it hardly degrades in its life cycle, making it useful for high-temperature applications. Owing to the non-contacting property, eddy current damping does not lead to an increase in the overall stiffness of the structures; therefore, the stiffness characteristics of the structures remain unaffected. Furthermore, the damping ratio can be easily adjusted by varying the air gap between the permanent magnet and the conductor. The damping mechanism via eddy currents has been reported by many researchers [14e18]. By the relative motion between a non-magnetic conductive metal and a permanent magnet or by changing the strength of the field magnets, a timevarying magnetic field is induced in the conductor, thereby generating eddy currents (shown in Fig. 1). The eddy currents induce another magnetic field with opposite polarity, thereby causing repulsive forces, i.e., damping forces. Because of the electrical resistance of the conductor, the induced currents dissipate in the form of heat at the rate of I2R, where I and R represent the current intensity and electrical resistance of the conductor, respectively. Therefore, the vibration energy of the main structures is transferred into the conductor and is dissipated in the form of heat. Eddy current damping is largely employed in the field of magnetic backing, and another application involves suppressing the lateral vibration in rotor shafts [20], which was reviewed by Henry and Bae [16]. In the field of structural vibration control, Matsuzaki et al. presented a vibration control system wherein the motion of a beam with an eddy current damper (ECD) could be suppressed. They derived a theoretical solution for the beam and demonstrated the viability of the concept [21]. In another study, an experiment was conducted to demonstrate the effectiveness of the ECD. Their results show that the electromagnetic force is capable of damping the first few modes of vibration of the beam [22]. In another study, Zheng et al. analyzed the effectiveness of an ECD in suppressing the free vibration response of a beam by numerical simulation and found that the damping effect of the ECD varies with the amplitude of the beam. The larger the amplitude, the greater is the effective damping [23]. Kwak et al. developed a new type of ECD to suppress the vibration of a beam. The ECD comprised a copper plate, which was rigidly fixed onto the end of the beam, and two permanent magnets. The ECD was constructed and tested to analyze the efficiency [15]. Later, Bae at el. developed the theoretical model of the ECD proposed by Kwak et al. and investigated the damping performance of a new ECD model. The experimental and simulation results demonstrate the potential of the ECD in controlling the vibration of a cantilever beam [14]. Sodano et al. proposed a new damper configuration providing more damping to the beam structure and developed an improved theoretical model with enhanced accuracy [24]. The
Fig. 1. Eddy current induced by the relative motion between a non-magnetic conductive metal and a permanent magnet [19].
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aforementioned studies primarily concentrated on the evaluation and analysis of ECDs in suppressing the vibration of cantilever beams with small-scale models. Kienholz et al. combined the advantages of the ECD and tuned mass damper and developed an eddy-current tuned mass damper (EC-TMD) to suppress the vibration of a solar sail array. They found that the magnetic damping techniques can achieve high damping forces [25]. In another study, Larose et al. investigated the effectiveness of the EC-TMD in suppressing the vibration of a small-scale bridge model by conducting wind tunnel tests. With the EC-TMD properly tuned, the global vibration of the bridge model could be damped [26]. Bae et al. confirmed the effective damping of an EC-TMD in suppressing the vibration of a beam model through both simulations and experiments [27]. In recent years, studies have been conducted on large-scale models with EC-TMDs. Wang et al. studied the feasibility of a large-scale EC-TMD and found that the eddy current damping was the dominant damping source. In addition, they demonstrated that the damping ratio could be easily adjusted by varying the gap between the permanent magnets and the conductive plates [28]. Bae et al. proposed a simple yet effective method for suppressing the vibration of a large beam structure using an EC-TMD without substantially increasing the overall weight [29]. However, to improve the practical application of ECDs to engineering structures such as high-rise buildings, it is necessary to analyze the damping performance in structural models by conducting large-scale tests. In this study, the damping performance of an EC-TMD installed on top of a five-story steel-frame model was evaluated by conducting an experimental study and determining an analytical solution. Shaking table tests conducted on the large-scale five-story frame structure are first introduced. An analytical solution for the damping performance of the EC-TMD, derived on the basis of electromagnetic and structural dynamic theories, is then discussed. Finally, a computational method is presented, simulated, and validated.
2. Experimental design 2.1. Model design The test model is a five-story steel-frame structure with a height of 5.48 m, comprising planes and columns, as shown in Fig. 2(a). The columns were made of Q690 steel whereas the planes were made of Q345 steel. The dimensions of the columns are 15 mm 180 mm 1060 mm and those of the planes are 30 mm 2000 mm 2000 mm. The total mass of the test model is 6000 kg. The seismic excitations were inputted to the structure along the weak axis direction in the shaking table tests. The natural vibration periods of the first five modes along the weak axis were 1.00, 0.35, 0.22, 0.17, and 0.15 s, respectively. Two acceleration sensors and two displacement sensors were set up in each floor. The acceleration and displacement sensors are numbered from 4 to 13 and from 76 to 85, respectively, as shown in Fig. 2(b). Four strain sensors were set up at the bottom of the columns to monitor the maximum strain of the columns; the strain sensors were numbered from 101 to 104.
Fig. 2. Model configuration. (a) Image of the test model, and (b) sensor arrangement.
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2.2. Damper design The EC-TMD system, manufactured by the Shanghai Research Institute of Materials, includes a supporting frame, four cables, a tuning mass, a permanent magnet, and a copper plane, as shown in Fig. 3. The supporting frame was fixed on top of the model. The lateral stiffness of the supporting frame was significantly greater than that of the model, thus ensuring that the supporting frame moves along with the top of the model. The tuning mass with a permanent magnet was suspended from the top of the supporting frame using four cables. Between the tuning mass and the steel frame model, the copper plane was fixed using four small columns. To improve the vibration-reduction effects, the frequency of the EC-TMD should be designed as close as possible to the first mode frequency of the primary structure. Hence, the length L of the suspended cables was decided by the classical equation for a simple pendulum: L ¼ ðT=2pÞ2 $g ¼ 0:248 m. The EC-TMD is primarily used for adjusting the damping properties of the auxiliary mass; however, it is not providing stiffness tuning, which is provided by the coupling structural member. In engineering practice, the mass of a tall building is considerable. Therefore, the mass ratio of the damper to the primary structure should be small. For example, a TMD with a mass ratio of 1% was attached to the Shanghai Center Tower in China [30], and a tuned liquid damper (TLD) with a mass ratio of 1% was attached to the Shin Yokohama Prince hotel in Japan [31]. On the basis of these practical cases, the mass ratio of the damper to the model structure in the tests was chosen to be approximately 1%. 2.3. Excitations in shaking table tests Fig. 4 shows the time histories and the corresponding power spectral density curves of three earthquake waves, including the El Centro, Wenchuan, and Japan 311 waves. The earthquake waves were incorporated in the shaking table tests to evaluate the damping performance of the EC-TMD system under different seismic conditions. The seismic excitations were inputted in only the weak axis direction of the test model, with peak ground accelerations of 0.05, 0.10, and 0.15 g, respectively (g is the acceleration due to gravity). 3. Response analysis To evaluate the vibration control performance of the damper, the vibration damping effect is defined as follows.
vibration damping effect ¼
the pðRMSÞ of structure without dampers the pðRMSÞ of structure with an EC TMD the pðRMSÞ of structure without dampers 100%
Here, the p and RMS represent the peak and root-mean-square values of the displacement/acceleration response at the top of the model, respectively. 3.1. Displacement response The RMS value of the displacement/acceleration is an indicator of the vibration energy and is usually employed to indicate the random variable energy in random vibration. The peak value of the displacement/acceleration is of concern in engineering
Fig. 3. Damper configuration.
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Fig. 4. Time histories of (a) El Centro, (b) Wenchuan, and (c) Japan 311 waves; Power spectral densities of (d) El Centro, (e) Wenchuan, and (f) Japan 311 waves.
design, because it is related to the safety of buildings and the comfort of the occupants. Table 1 lists the peak and RMS values of the displacement responses at the roof of the test model in different cases. With regard to the vibration control effects in reducing the displacement responses, the results show that (i) The response of the controlled model with an EC-TMD is lower than that of the uncontrolled model, thus implying that the EC-TMD can dissipate vibration energy and provide effective damping for the test model; (ii) The vibration-reduction effect in terms of the RMS values of the displacement responses is significantly better than that in terms of the peak values, wherein the range of the former is 18.6e69.4% and that of the latter is 10.4e34.9%; (iii) The vibration-reduction effect is different under different seismic inputs. The reduction in vibration is favorable in the case of the El Centro wave; however, the damping effects are relatively poorer in the cases of the Wenchuan and Japan 311 waves. This may be because of the frequency characterization of the input earthquake excitations. Fig. 4 shows the excitations of the El Centro, Wenchuan, and Japan 311 waves in the time and frequency domains. The frequency of the El Centro wave is largely in the range of 0.5e2 Hz, which is near the natural frequency of the primary structure, whereas the differences between the dominant frequencies of the Wenchuan and Japan 311 waves and the fundamental frequency of the structure are considerable. This frequency characterization makes the displacement response of the model in the case of the El Centro wave to be significantly greater than that in the cases of the other inputs. Accordingly, in the case of the El Centro wave, the EC-TMD dissipates more energy via mass tuning and eddy currents, thereby exhibiting a more favorable damping effect.
3.2. Acceleration response Not only can the displacement be reduced, the acceleration can also be attenuated in most cases. Table 2 lists the peak and RMS values of the acceleration responses at the roof of the test model in different cases. In all cases of the RMS values and most cases of the peak values, the acceleration responses of the controlled model are lower than those of the uncontrolled model. This shows that the EC-TMD can effectively dissipate the vibration energy of the test model. However, the vibrationreduction performance of the model with the EC-TMD in the case of the Japan 311 wave is the worst, particularly for peak ground accelerations of 0.05 and 0.15 g where the peak acceleration slightly increases. This is because although the EC-TMD is a modified version of the conventional TMD and therefore is expected to exhibit improved performance, the EC-TMD is still a linear damper, the performance of which is influenced by the frequency of the excitation, similar to the conventional TMD. This is observed in both Table 1 (Section 3.1) and Table 2 (Section 3.2). In fact, the key advantage of the EC-TMD is the non-contacting damping mechanism. Moreover, from Tables 1 and 2, the vibration damping effect in terms of the RMS values is better than that in terms of the peak values in most cases. This is clarified by analyzing the time history of the displacement/acceleration response in the following section.
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Table 1 Peak and RMS values of the displacement response at the roof of the test model. Seismic input
El Centro
Wenchuan
Japan 311
Peak input value (g)
0.05 0.1 0.15 0.05 0.1 0.15 0.05 0.1 0.15
Without dampers
With an EC-TMD
p (mm)
RMS (mm)
p (mm)
RMS (mm)
Reduction effect p (%)
RMS (%)
34.008 79.213 118.925 8.924 18.537 27.305 4.475 10.842 22.438
13.125 29.810 43.056 1.806 4.082 6.262 1.415 4.089 8.273
30.456 68.882 102.521 7.048 14.738 22.082 3.449 8.797 14.602
4.121 9.124 13.662 1.470 3.219 4.788 0.996 2.517 4.250
10.4 13.0 13.8 21.0 20.5 19.1 22.9 18.9 34.9
68.6 69.4 68.3 18.6 21.1 23.5 29.6 38.5 48.6
3.3. Time histories and power spectral densities The entire time history and power spectral density of the displacement/acceleration response are necessary for evaluating the damage inflicted to the structures and can be regarded as indicators of the intensity of the earthquake ground motion to some extent. Fig. 5 shows the displacementetime history and power spectral density curves at the roof level of the test model, wherein the solid and dotted lines represent the controlled and uncontrolled models, respectively. From the time history curves, it can be found that the vibration control effect is poor at the very beginning, and the controlled curves begin to decay quickly thereafter. If the peak values are observed at the very beginning, the damping performance in terms of the peak values will be poor; this is one of the reasons why the peak values of the responses is high in some cases and the vibration-reduction effects in terms of the RMS values are better than those in terms of the peak values listed in Tables 1 and 2. At the beginning, it takes some time for the EC-TMD to start moving and dissipate the vibration energy. Fig. 5(a) and (b) show that the peak values of the test model in the cases of the El Centro and Wenchuan wave excitations are 118.9 and 27.3 mm, respectively. Therefore, it was easier for the EC-TMD to create the damping effect in the case of the El Centro wave. Fig. 5(c) and (d) show the corresponding power spectral density curves in the cases of the El Centro and Wenchuan waves, respectively. Fig. 5(c) shows that the vibration energy is significantly high at the fundamental frequency of the test model. Moreover, the peak value of the model with the EC-TMD is significantly lower than that without the dampers, thus indicating that the EC-TMD has favorable damping effects at the fundamental frequency. Fig. 5(d) shows that the vibration energy is significantly high at the first and second mode frequencies of the test model. Because of the additional mass of the EC-TMD, the structural frequencies of the controlled model are slightly lower than those of the uncontrolled model. The peak values of the model with the EC-TMD are significantly lower than those of the model without the dampers. This is because the EC-TMD provides effective damping for the test model and suppresses the vibration. In most cases, the displacement responses of the model with the EC-TMD are lower than those of the model without the dampers. This is observed not only in the time history curves but also in the power spectral density curves.
3.4. Interstory drift ratio The maximum interstory drift ratio of a building is an important index for structural design and should not exceed specified ultimate values in the relevant design codes. Fig. 6 shows the maximum interstory drift ratio of the test model in the case of the El Centro wave with different peak ground accelerations (0.05, 0.1, and 0.15 g). Although the amplitudes of the excitations are different, the maximum interstory drift ratios with respect to the five stories of the test model are reduced using the EC-TMD in all cases. The damping performance of the EC-TMD for a peak ground acceleration of 0.15 g is better than those for peak ground accelerations of 0.05 and 0.1 g. This is because the vibration response of the test model in the former case is greater than those in the other cases. Therefore, the EC-TMD moves more violently and dissipates more vibration energy.
3.5. Maximum strain of the columns The maximum strain of the columns is a parameter used to check whether the structure is in the elastic or elastoplastic phase, in relation to the damage phenomena. Table 3 lists the maximum strains of the columns of the test model with/without the EC-TMD for all cases. It can be seen that: (i) The maximum strain of the model in the case of the El Centro wave is the highest, and the maximum strain in the case of the Japan 311 wave is the lowest, which coincides with the results of the displacement and acceleration responses, given in Sections 3.1 and 3.2, respectively; (ii) The maximum strains of the columns in all the cases do not exceed the yield strain of the Q690 steel columns, indicating that the test model remains in an elastic phase during the shaking table tests; (iii) The maximum strain of the columns can be reduced using the EC-TMD.
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Table 2 Peak and RMS values of the acceleration response at the roof of the test model. Seismic input
El Centro
Wenchuan
Japan 311
Peak input value (g)
0.05 0.1 0.15 0.05 0.1 0.15 0.05 0.1 0.15
Without dampers
With an EC-TMD
p (g)
RMS (g)
p (g)
RMS (g)
Reduction effect p (%)
RMS (%)
0.219 0.508 0.757 0.201 0.450 0.665 0.108 0.271 0.458
0.059 0.134 0.193 0.047 0.108 0.161 0.034 0.080 0.141
0.142 0.328 0.501 0.186 0.407 0.631 0.115 0.261 0.490
0.021 0.051 0.089 0.033 0.071 0.113 0.022 0.053 0.097
35.1 35.3 33.8 7.4 9.4 5.2 6.7 3.9 6.8
63.6 61.7 53.8 30.2 34.2 29.9 36.2 33.3 31.6
Fig. 5. Time history curves of the displacement responses of the model in the cases of (a) El Centro, and (b) Wenchuan waves; Power spectral density curves of the displacement responses of the model in the cases of (c) El Centro, and (d) Wenchuan waves.
The series of shaking table tests conducted on the model with/without the EC-TMD, along with the analysis of the experimental results, prove that the EC-TMD performs effectively in suppressing the vibration of a large-scale primary structure. In the following section, a corresponding analytical/simulation solution is proposed to estimate the response of the structures with the EC-TMD, and the simulation results are analyzed by comparing the experimental results. 4. Analytical/simulation solution 4.1. Analytical method The magnetic field distribution with respect to the rectangular permanent magnet is first introduced, and subsequently, the electromagnetic force is analyzed. The effects of the electromagnetic force can be reflected in the form of the eddy-current damping coefficient, which is a function of the relative distance between the permanent magnet and the conductor. The eddy-current damping coefficient can be then incorporated into the governing equations of the primary system.
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Fig. 6. Maximum interstory drift ratios of the test model in the case of the El Centro wave with peak ground accelerations of (a) 0.05, (b) 0.1, and (c) 0.15 g.
Table 3 Maximum strains of the columns in the test model. Seismic input
Peak input value (g)
Uncontrolled (mε)
Controlled (mε)
Reduction effect (%)
El Centro
0.05 0.1 0.15 0.05 0.1 0.15 0.05 0.1 0.15
338 776 1163 193 443 676 75 174 323
279 633 958 139 307 473 67 157 296
17.46 18.43 17.63 27.98 30.70 30.03 10.67 9.77 8.36
Wenchuan
Japan 311
Gou et al. [32] derived analytical expressions to determine the magnetic field distribution of a rectangular permanent magnet. To complete this analytical method, some key equations to calculate the magnetic field distribution are given and
Fig. 7. Coordinate system in a rectangular permanent magnet.
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derived herein. The dimensions of the permanent magnet are assumed to be a mm (length) b mm (width) c mm (height), and the direction of polarity of the magnet is along the z axis, as shown in Fig. 7. The components of the magnetic field distribution along the three axes can be derived on the basis of the molecular circulation model and the BioteSavart law.
Zh Bx ¼
N dBx ¼ ½Gða x; y; zÞ þ Gða x; b y; zÞ Gðx; y; zÞ Gðx; b y; zÞ 2
(1)
N dBy ¼ ½Gðb y; x; zÞ þ Gðb y; a x; zÞ Gðy; x; zÞ Gðy; a x; zÞ 2
(2)
0
Zh By ¼ 0
Zh Bz ¼
dBz 0
¼ N½Jðx; b y; zÞ þ Jðy; a x; zÞ þ Jða x; b y; zÞ þ Jðb y; a x; zÞ þ Jðb y; x; zÞ þ Jðy; x; zÞ þ Jða x; y; zÞ þ Jðx; y; zÞ
(3)
The auxiliary functions are defined as follows.
N¼
m0 J s 4p
(4)
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi g21 þ g22 þ ðg3 z0 Þ2 g2 z0 ¼ h Gðg1 ; g2 ; g3 Þ ¼ ln qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi z ¼0 g21 þ g22 þ ðg3 z0 Þ2 þ g2 0 3 j3 z0 6j1 7 z 0 ¼ h Jðj1 ; j2 ; j3 Þ ¼ arctan4 $qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi5 j2 j2 þ j2 þ ðj z Þ2 z0 ¼ 0 0 3 1 2
(5)
2
(6)
where m0 ð¼ 4p 107 Þ and Js ð¼ 9:55 105 A=m2 Þ represent the magnetic permeability in the air and the equivalent current density, respectively. The magnetic field distribution of the rectangular permanent magnet can be calculated using the aforementioned equations. Wang et al. determined the magnetic field distribution along the z direction at the height of 5.5 cm and found that the magnetic field is largely distributed on the projected area of the permanent magnet and decayed rapidly outside this area [28]. Compared to the magnetic field distribution in the projected area, the magnetic field outside this area is insignificant and therefore can be neglected, thus simplifying the analytical method and improving the efficiency of the calculation. When a conductive metal passes through a magnetic field, eddy current forces are created in the conductor. It is assumed that the EC-TMD tunes along the y axis. The velocity of the EC-TMD is expressed as follows.
vTMD ¼ 0i þ vy j þ 0k:
(7)
The current density J can be calculated as follows.
2
i J ¼ sðv BÞ ¼ s4 0 Bx
j vy By
3 k 0 5 ¼ svy ðBz i Bx kÞ Bz
(8)
where s is the conductivity of the conductor in the magnetic field, and B is the magnetic field strength at the location of the conductor. Once the eddy currents are generated, they induce a magnetic field with opposite polarity as the primary field, causing a repulsive electromagnetic force.
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Z
J BdV ¼ svy
F¼ ¼ svy
Z h
Z
2
i
6 4 Bz
j
k
3
7 Bx 5dV
0
(9)
Bx By Bz i Bx By i B2x þ B2z j þ By Bz k dV
The electromagnetic force in the motion direction of the EC-TMD is Fy .
Fy ¼ svy
Z B2x þ B2z dV
(10)
According to the properties of the magnetic field distribution, the magnetic field is largely distributed in the projected area of the permanent magnet, and the magnitudes of the magnetic fields are the same in this area. Therefore, Eq. (10) can be simplified as follows.
Fy ¼ sdS B2x þ B2z vy
(11)
where d and S are the thickness and projective area of the permanent magnet, respectively. Bx and Bz are the components of the magnetic field strength at the position of the conductor and are decided by the relative distance between the conductor and the permanent magnet. From Eq. (11), it can be found that the electromagnetic force is proportional to the relative velocity between the permanent magnet and the conductor; this force is the same as the linear damping force. Hence, the vibration control effect of the ECTMD can be considered equivalent to that of the linear damping coefficient, which is a function of the relative position between the conductor and the permanent magnet.
Ce ¼ sdS B2x þ B2z
(12)
When an EC-TMD is present in a multi-degree-of-freedom system, the effect of the EC-TMD is equivalent to that of a system with one spring and two dampers, as shown in Fig. 8, where Ct is the damping provided by the conventional TMD, and Ce is the damping component due to the eddy currents, which depends on the relative position between the system and the EC-TMD. The governing equations for a multi-degree-of-freedom system with an EC-TMD can be written as follows.
MX€ þ C X_ þ KX ¼ Ex€g M ¼ diag½M1 M2 …Mn Mt 2 C1 þ C2 C2 6 C C 6 2 2 þ C3 C3 6 6 C¼6 6 4 2
K1 þ K2 6 K 6 2 6 K¼6 6 6 4
3 1
Cn
Cn
Cn þ Ct þ Ce Ct Ce
K2 K2 þ K3
K3 1
Kn
Kn
Kn þ Kt Kt
X ¼ ½X1 X2 /Xn Xt
7 7 7 7 7 7 Ct Ce 5 Ct þ Ce 3
7 7 7 7 7 7 Kt 5 Kt
T
E ¼ ½ M1 M2 / Mn Mt T where M, C, and K are the mass, damping, and stiffness matrices of the system with the EC-TMD, respectively, and Ex€g is the inertial force matrix with respect to the ground acceleration. Xi is the relative displacement of the ith floor with respect to the ground, and Mi ; Ci ; and Ki are the mass, damping, and stiffness values of the ith floor, respectively (i ¼ 1,2 …,n). Xt is the relative displacement of the EC-TMD with respect to the ground; Mt ; Ct ; and Kt are the mass, damping, and stiffness values of the conventional TMD; and Ce is the time-varying damping coefficient of the EC-TMD with respect to the eddy currents.
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Fig. 8. Simplified diagram of a system with an EC-TMD.
Fig. 9. Calculated and experimental displacementetime histories of the model in the cases of (a) El Centro, (b) Wenchuan waves (0.15 g), (c) zoom-in view of (a), and (d) zoom-in view of (b).
Table 4 Comparison of the simulation and experimental results for the peak values of the displacement at the roof of the model with the EC-TMD. Seismic input
Peak input value (g)
Simulation (mm)
Experiment (mm)
Error (%)
El Centro
0.05 0.1 0.15 0.05 0.1 0.15 0.05 0.1 0.15
32.158 69.524 106.682 7.207 16.109 22.954 3.740 8.972 13.612
30.456 68.882 102.521 7.048 14.738 22.082 3.449 8.797 14.602
5.3 0.9 3.9 2.2 8.5 3.8 7.8 1.9 7.3
Wenchuan
Japan 311
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4.2. Computational results In Section 2, the shaking table tests conducted on the five-story steel-frame structure were discussed. The corresponding computational simulations are presented in this section. The mass and stiffness matrices of the five-story model are indicated by M s and K s , respectively.
2 6 6 6 Ms ¼ 6 6 6 4
3
M1 M3 M4
3
1100
52 104 52
K2 K2 þ K3 K3
1100 1100 3
K3 K3 þ K4 K4
K4 K4 þ K5 3
52 104 52
7 7 7 7 kg 7 7 5
1100
M5
K1 þ K2 6 6 K2 6 Ks ¼ 6 6 6 4 104 6 52 6 6 ¼6 6 6 4
7 6 7 6 7 6 7¼6 7 6 7 6 5 4
M2
2
2
2
K5
1227
7 7 7 7 7 7 K5 5 K5
7 7 7 7 104 N=m 52 7 7 104 52 5 52 52
The damping ratios of the test structure in the first two structural modes are x1 (¼0.85%) and x2 (¼0.25%), respectively. For the EC-TMD, the mass (Mt ) is 35 kg. Because the frequencies of the test model and the EC-TMD are equal, the stiffness between the EC-TMD and the test model can be calculated as Kt ¼ 1381 ðN=mÞ. The damping coefficient with respect to the eddy currents varies with the change in the relative distance between the conductor and the permanent magnet. Therefore, it was necessary to calculate the damping coefficient of the EC-TMD for every step using the equations given in Section 4.1. Fig. 9 shows the calculated and experimental displacementetime histories for the roof of the test model with the EC-TMD in the cases of the El Centro and Wenchuan waves for a peak ground acceleration of 0.15 g. The calculated displacementetime histories coincide with the experimental ones in most cases. Table 4 lists the comparison of the simulation results and the corresponding experimental results for the peak displacement at the roof of the test model with the EC-TMD. Fig. 9 and Table 4 show that the simulation results are in good agreement with the experimental ones, thus demonstrating that the proposed analytical method can be used to estimate the response of structures with an EC-TMD under earthquake excitations. 5. Conclusion Compared to conventional TMDs, the EC-TMD with a non-contacting damping mechanism shows excellent performance in several ways: 1) it does not degrade in its life cycle; 2) the damping performance remains largely unaffected even in hightemperature environment; 3) the stiffness of the overall structure remains unaffected; 4) the damping ratio can be easily adjusted by varying the air gap between the permanent magnet and the conductor. A series of shaking table tests were conducted on a five-story steel-frame model. From the experimental results, the ECTMD was proven both feasible and effective in suppressing the vibration of large-scale structures. In terms of the displacement and acceleration responses, the damping effects of the EC-TMD were found to be excellent, particularly in terms of the RMS responses. Although the frequency characteristics of the El Centro, Wenchuan, and Japan 311 waves are significantly different, the EC-TMD can be used to reduce the structural vibrations due to the seismic waves, demonstrating the effective vibration-control performance of the EC-TMD. In addition, the damper helps in reducing the interstory drift ratio and the maximum strain of the test model. The relationship between the permanent magnet and the conductor is derived on the basis of the electromagnetic theory, and the effect of the electromagnetic force is simplified to a time-varying damping coefficient. An analytical method is proposed to evaluate the response of the structures with the EC-TMD. The analytical/simulation method was validated via the shaking table test results, and the simulation accuracy was found to be reasonable. The results of the experimental and analytical investigations demonstrate the effectiveness and excellent performance of the EC-TMD in suppressing the vibration of a large-scale model. Therefore, this technology has a good potential for future applications to engineering projects.
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