Dynamic analogy between an electromagnetic shunt damper and a tuned mass damper

transcript

IOP PUBLISHING SMART MATERIALS AND STRUCTURES

Smart Mater. Struct. 22 (2013) 115018 (11pp) doi:10.1088/0964-1726/22/11/115018

Dynamic analogy between anelectromagnetic shunt damperand a tuned mass damper

Songye Zhu, Wenai Shen and Xin Qian

Department of Civil and Environmental Engineering, The Hong Kong Polytechnic University,Hung Hom, Kowloon, Hong Kong

E-mail: ceszhu@polyu.edu.hk

Received 7 April 2013, in final form 6 August 2013Published 17 October 2013Online at stacks.iop.org/SMS/22/115018

AbstractAn electromagnetic shunt damper (EMSD) is composed of an electromagnetic damperconnected to one or more RLC shunt circuits. Through a theoretical comparison, thispaper reveals the similarity and difference between an EMSD and a tuned mass damper(TMD), both of which are resonant-type vibration absorbers. The equivalent mass,stiffness and damping coefficient of the EMSD are derived based on the transfer functionsof structures with a TMD or an EMSD, and the functions of circuit capacitance andinductance are discussed accordingly. The optimal parameters of the RLC circuit in theEMSD are obtained through H∞ optimization. Despite their different optimal parameters,the EMSD and TMD exhibit comparable control performance with the same equivalentmass ratio. The dynamic analogy between these two types of dampers offers a newperspective for understanding novel EMSDs, given that TMDs have previously beenextensively studied. The potentials and constraints of the EMSD are further discussedthrough numerical case studies in which EMSD control performance is examined indifferent situations.

(Some figures may appear in colour only in the online journal)

Nomenclature

γ Ratio of the TMD’s frequency to structuralfrequency

γ Ratio of the EMSD’s frequency to structuralfrequency

γopt Optimal frequency ratio for the TMDγopt Optimal frequency ratio for the EMSDλ Ratio of excitation frequency to structural

frequencyµ Ratio of the TMD’s mass to structural massµ Ratio of the EMSD’s equivalent mass to

structural massξ2 Damping ratio of the TMDξ2 Damping ratio of the RLC circuitξ2,opt Optimal damping ratio for the TMDξ2,opt Optimal damping ratio for the EMSD

ω Frequency of harmonic excitation forceω1 Natural frequency of the primary structureω2 Natural frequency of the TMDω2 Natural frequency of the RLC shunt circuitψ Electromechanical coupling coefficientc1 Damping coefficient of the primary structurec2 Damping coefficient of the TMDc2 Equivalent damping coefficient of the EMSDf Excitation force acting on the primary structurek1 Stiffness of the primary structurek2 Stiffness of the TMDk2 Equivalent stiffness of the EMSDm1 Mass of the primary structurem2 Mass of the TMDm2 Equivalent mass of the EMSDq Electric charge transferred through the circuitx1 Displacement of the primary structure

Smart Mater. Struct. 22 (2013) 115018 S Zhu et al

x2 Displacement of the TMDC Total capacitance of the EMSD circuitCopt Optimal capacitance of the EMSD circuitGopt Peak dynamic amplification factor of the primary

structure with the TMDGopt Peak dynamic amplification factor of the primary

structure with the EMSDKem Machine constant of the electromagnetic damperL Total inductance of the EMSD circuitR Total resistance of the EMSD circuitRopt Optimal resistance of the EMSD circuit

1. Introduction

Electromagnetic devices can convert mechanical energy intoelectric energy through electromagnetic induction. Suchdevices are widely used in power generation or harvesting.Over the last two decades, growing interest has beendirected towards employing various electromagnetic devicesfor structural vibration control. For example, Palomera-Arias(2005), Palomera-Arias et al (2008) studied building vibrationcontrol using a passive electromagnetic damper connected to aresistive shunt circuit. Zhu et al (2012) proposed the use of anelectromagnetic damper for simultaneous vibration dampingand energy harvesting. Some researchers recently proposedand investigated an emerging electromagnetic dampingdevice called an electromagnetic shunt damper (EMSD) (deMarneffe 2007, Cheng and Oh 2009, Fleming and Moheimani2006, Niu et al 2009, Zhang et al 2012, Behrens et al 2005,Fleming 2004, Inoue et al 2008). Although different typesof shunt circuits were studied in the past, EMSD hereinafterrefers to an electromagnetic device connected to one ormore RLC shunt circuits that are commonly used in electricoscillators. If the natural frequency of the RLC circuit is tunedto be close to one of the structural vibration frequencies,an EMSD installed on a structure can effectively suppressstructural vibration through resonance and dissipation ofthe circuit. The concept of the EMSD was inspired by thepiezoelectric shunt damper (Hagood and Crawley 1991, Wu1999), and both of them essentially share a common principle.The promising features of the EMSD have motivated a seriesof investigations on this novel damping device. Behrenset al (2005) determined the optimal damping resistance ofthe shunt circuit of the EMSD and experimentally validatedit using a simple electromagnetic mass spring dampersystem. Fleming and Moheimani (2006) proposed a sensorlessactive shunt impedance for electromagnetic transducers usedas vibration absorbers. Their experimental results showedreductions of the two peaks in frequency response function(FRF) by 18.7 and 23.6 dB, respectively. de Marneffe (2007)derived the optimal parameters of an EMSD through rootlocus analysis and H∞ minimization. Inoue et al (2008)optimized the circuit parameters of an EMSD using theclassical Den Hartog’s method (Ormondroyd and Den Hartog1928, Den Hartog 1956), which is also known as the H∞optimization. Cheng and Oh (2009) studied multi-modevibration control of a cantilever beam using an EMSDconnected to multiple shunt circuits. Zhang et al (2012)

proposed an EMSD with adjustable frequency-independentdamping for multi-mode vibration control by using a negativeinductance and negative resistance technique.

Similar to a conventional tuned mass damper (TMD),an EMSD is a resonant-type damping device. Throughtheoretical comparisons, this paper reveals the similarityand difference in the dynamics of these two typesof damping devices. A single-degree-of-freedom (SDOF)primary structure equipped with an EMSD or a TMD isanalyzed. The dynamic analogy between the two dampingdevices is established based on the equivalent mass,stiffness and damping coefficient of the EMSD derivedfrom the transfer functions, which offers a new perspectiveinto understanding novel EMSDs, given that TMDs havepreviously been extensively studied. The optimal designparameters according to H∞ optimization in the frequencydomain are presented for both EMSDs and TMDs. Thecomparison indicates that comparable control performancecan be achieved by the optimally designed EMSD and TMD,despite their different optimal design laws. Compared with theTMD, the EMSD has the following salient features: (1) TheEMSD does not involve any moving mass, and (2) the EMSDcan imitate virtual mass using electronic capacitors, such thata substantial equivalent mass may be achieved by a smalldevice. Therefore, the EMSD is regarded as a promisingalternative to the conventional TMD in structural vibrationsuppression. The potentials and challenges of the EMSD invibration suppression are further discussed through numericalcase studies in which the control performance of the EMSD isexamined in different situations. The effects of some practicalconstraints on the optimization of the R,L, and C parametersare highlighted.

2. Tuned mass damper (TMD)

Figure 1(a) shows the mechanical model of a SDOF structurewith a TMD, and its equations of motion are given by[

[c1 + c2 −c2

−c2 c2

[k1 + k2 −k2

−k2 k2

where m1, c1, k1, and x1 are the mass, damping coefficient,stiffness and displacement of the primary structure, respec-tively; m2, c2, k2, and x2 are the mass, damping coefficient,stiffness and displacement of the TMD, respectively; and f isthe excitation force acting on the primary structure. ThroughLaplace transform, the transfer function relating the inputexcitation F(s) to the displacement output X1(s) is obtained

F(s)= {s2

+ 2ξ2ω2s+ ω22}

× {m1s4+ (2ξ2ω2m1 + c1 + c2)s

+ (ω22m1 + 2ξ2ω2c1 + k1 + k2)s

+ (ω22c1 + 2ξ2ω2k1)s+ ω

−1 (2)

Figure 1. A single-of-freedom (SDOF) primary structure attached to a TMD or an EMSD system. (a) SDOF structure with a TMD system.(b) SDOF structure with an EMSD.

where ω2 =√

k2/m2 is the natural frequency of the TMD,and ξ2 = c2/2

√k2m2 is the damping ratio of the TMD.

In theoretical studies, simple cases with c1 = 0 are oftenassumed. The displacement FRF of the system can beobtained by substituting s = jω into equation (2), which canbe further simplified to a dimensionless form:

H(λ) =X1(λ)

F(λ)/k1

=γ 2− λ2+ 2ξ2γ λj[

(1− λ2)(γ 2 − λ2)− µλ2γ 2]+ 2ξ2γ λ(1− λ2 − µλ2)j

where µ = m2/m1 is the ratio of the TMD mass to thestructural mass, γ = ω2/ω1 is the ratio of the TMD frequencyto the structural frequency, and λ = ω/ω1 is the ratio of theexcitation frequency to the structural frequency, also calleddimensionless excitation frequency. By disregarding thedamping of the primary structure (i.e. c1 = 0), Ormondroydand Den Hartog (1928) derived the optimal tuning of the TMDparameters through the H∞ optimization as follows

γopt =1

1+ µ(4a)

ξ2,opt =

√3µ

8(1+ µ)(4b)

where both the optimal frequency ratio and the optimaldamping ratio of the TMD are functions of the mass ratio.Corresponding to the optimal parameters, the two invariantpoints on the dimensionless FRF curve have equal amplitudesas follows

Gopt =

√2+ µµ

Equation (5) approximately represents the peak dynamicamplification factor ‖H‖∞, a common measure of vibrationcontrol performance. A large mass ratio µ always facilitatesa better control effect. The auxiliary mass of the TMDintroduces one more DOF, and thus the system has twovibration modes. Krenk (2005) proved that the optimalfrequency enables the two complex loci of natural frequenciesto meet at a bifurcation point. Moreover, the two vibration

modes have identical damping ratios if the damping is lowerthan the bifurcation point.

3. Electromagnetic shunt damper (EMSD)

3.1. Dynamic analog of EMSD

In the EMSD, an electromagnetic device can be connectedto more than one RLC shunt circuit (consisting ofresistors, inductors, and capacitors) with frequencies closeto the resonant frequencies of structures. This paper onlyinvestigates the EMSD for single-mode vibration suppression,that is, the electromagnetic device connected to one shuntcircuit. Figure 1(b) shows an SDOF primary structureequipped with an EMSD. The electromagnetic damper,comprising two major components, namely, permanentmagnets and coils, is capable of converting the mechanicalenergy of the structure into electrical energy. Structuralvibration generates an electromotive force (emf) proportionalto the motion velocity, and the emf further generates a currentflowing within the attached RLC circuit. In particular, whenthe vibration frequency is equal to the natural frequency ofthe circuit, the reactance of the shunt circuit will be equalto zero, which results in a maximum current in the circuitand a corresponding large damping force provided by thedamper. If the oscillating frequency of the RLC circuit istuned to be close to one of structural critical frequencies,the EMSD provides a resonant-type damping effect and isable to effectively suppress the vibration corresponding to thissingle mode. The dynamic equations of the system shown infigure 1(b) are given by[

[c1 Kem

−Kem R

where Kem is the machine constant of the electromagneticdevice; q is the electric charge transferred through the circuit;and L,R, and C are the total inductance, resistance, andcapacitance of the circuit, respectively, where R accounts

for the resistance of the coil and resistor as well as theequivalent series resistance of the inductor and capacitor, andL accounts for the inductance of the coil and the externalinductor. Through Laplace transform, equation (6) with zeroinitial conditions can be expressed as follows:[

m1s2 0

[c1s Kems

−Kems Rs

]. (7)

The transfer function relating the excitation F(s) to thedisplacement of the primary structure X1(s) can be obtainedby solving equation (7):

F(s)= [Ls2

+ Rs+ 1/C][m1Ls4+ (m1R+ c1L)s3

+ (m1/C + c1R+ k1L+ K2em)s

+ (c1/C + Rk1)s+ k1/C]−1

= [s2+ 2ξ2ω2s+ ω2

2][m1s4+ (m1R/L+ c1)s

+ (ω22m1 + c1R/L+ k1 + K2

em/L)s2

+ (ω22c1 + k1R/L)s+ ω2

2k1]−1 (8)

where ω2 and ξ2 are the natural frequency and damping ratioof the resonant shunt circuit, respectively

LC(9a)

ξ2 =R

L. (9b)

The similarity between equations (2) and (8) can be easilyobserved in both the numerators and denominators. If wedefine the equivalent mass, stiffness and damping coefficientof the EMSD as follows

m2 = K2emC (10a)

k2 = K2em/L (10b)

c2 = K2emRC/L (10c)

the transfer function in equation (8) can be converted to

F(s)= [s2

+ 2ξ2ω2s+ ω22][m1s4

+ (2ξ2ω2m1 + c1)s3

+ (ω22m1 + 2ξ2ω2c1 + k1 + k2)s

+ (ω22c1 + 2ξ2ω2k1)s+ ω

22k1]−1. (11)

Meanwhile, equations (9) can be rewritten as

m2(12a)

ξ2 =R

. (12b)

Similar to the structure–TMD system, if we assume anundamped primary structure, the dimensionless FRF can be

obtained by substituting s = jω into equation (11)

H(λ) =X1(λ)

F(λ)/k1

=γ 2− λ2

+ 2ξ2γ λj[(1− λ2)(γ 2 − λ2)− µλ2γ 2

]+ 2ξ2γ λ(1− λ2)j

where µ = m2/m1 is the ratio of the equivalent mass of theEMSD to the structural mass, and γ = ω2/ω1 is the ratioof the EMSD frequency to the structural frequency. Inoueet al (2008) defined the electromagnetic mechanical couplingcoefficient for the EMSD as ψ = K2

em/m1Lω21, which is

related to the equivalent mass ratio by

µ =m2

m1Lω21γ

γ 2 . (14)

3.2. Optimal tuning of the EMSD

Similar to the TMD system, the optimal tuning of the EMSDcan be derived based on the H∞ optimization by assumingzero damping of the primary structure. The derivation, asdetailed in the appendix, provides the optimal frequency ratioand damping ratio of the EMSD as follows

γopt =

2+ µ(15a)

ξ2,opt =

√3µ8. (15b)

Consequently, the optimal tuning of the electronic compo-nents, R,L and C can be obtained from equations (12), (14)and (15)

Copt =2m1

2k1L− K2em

Ropt = Kem

2m1. (16b)

Inoue et al (2008), de Marneffe (2007) also investigated theoptimal parameters of the EMSD. Compared with the optimalvalues derived by Inoue et al (2008), the optimal capacitancein equation (16a) is identical, whereas the optimum resistancein equation (16b) is slightly different because of the differentaveraging methods used. However, equations (16) differ fromthe results suggested by de Marneffe (2007).

Corresponding to equations (15), the two invariant pointson the dimensionless FRF curve have equal amplitudes asfollows

Gopt =

√2+ µµ

which is approximately equal to the peak dynamicamplification factor

∥∥H∥∥∞

of the displacement response ofthe primary structure with the EMSD.

Although derived for the SDOF primary structure, theoptimal design method can be applied to a multi-DOF

Figure 2. Optimal design parameters of the EMSD and TMD. (a) Optimal damping ratio versus mass ratio. (b) Optimal frequency ratioversus mass ratio.

(MDOF) structure by considering modal mass and modalstiffness. This method can be further extended to multi-mode vibration control of structures by connecting theelectromagnetic device with multiple RLC resonant circuits,provided that structural vibration modes are well separated.Cheng and Oh (2009) emphasized that the optimal frequencyratio and damping ratio in multi-mode vibration control areessentially the same as those in a single-mode control, as longas modal mass and modal stiffness are used.

4. Theoretical comparison of the TMD and EMSD

This section compares the dynamics of TMD and EMSDsystems as shown in figure 1. The transfer functions inequations (2) and (11) are almost the same, except forthe second terms in the denominators when the TMDhas the equivalent mass, stiffness, and damping defined inequations (10), which implies that the conventional TMDis a good dynamic analog to the EMSD. Given that theapplication of the TMD in structural vibration suppressionhas been extensively studied for decades, such an analogywould offer an insight into the mechanism of the EMSD andfacilitate understanding and acceptance of the novel EMSDby the mechanical or civil engineering society. However, thetwo transfer functions are identical only if the EMSD andthe equivalent TMD systems have zero damping, that is c2 =

c2 = 0, which represents a trivial case in real applications.As aforementioned, the damping of the TMD or EMSDshould be tuned to the optimal values in order to maximizetheir control effect. The similarity and difference between thecontrol performance of the EMSD and TMD are discussedtheoretically in this section.

Figure 2 shows a comparison of the optimal frequencyratios and optimal damping ratios between the EMSD andTMD, as defined by equations (4) and (15). The optimaldamping ratios of the EMSD and TMD are quite closewhen the (equivalent) mass ratio is small, but the differencebecomes evident with increasing µ (or µ). A more apparentdiscrepancy can be observed in the optimal frequency ratiosof the two systems, even when µ (or µ) is small. Equations (5)

Figure 3. Dimensionless frequency response functions of structuraldisplacement to external force (line: EMSD; dotted line: TMD; allwith optimum parameters).

and (17) predict the peak amplitudes of the dimensionless FRFcurves for the TMD and EMSD, respectively, which are oftenconsidered as a measure of control performance. Althoughthe EMSD and TMD have quite different optimal parameters,their optimal design can achieve an identical control effect interms of peak dynamic amplification factors. Figure 3 showsthe amplitude of the dimensionless FRFs H(λ) and H(λ)for the structures with the TMD and EMSD, respectively.Three (equivalent) mass ratios, namely 0.05, 0.2, and 1, arepresented in the figure, and the frequency ratios and dampingratios of the TMD and EMSD are tuned to the optimalvalues. The structure–TMD and structure–EMSD systemshave very similar FRF curves and identical amplitudes of thetwo peaks when their (equivalent) mass ratios µ and µ areequal. The infinite single peak of an uncontrolled structureis reduced to the two lower peaks with the amplitudes of16.1, 10.4, and 4.9 dB for mass ratios of 0.05, 0.2 and 1,respectively. Although the TMD causes the FRF curves toshift slightly towards the left compared with the EMSD,the difference between their overall control performancesshould be minimal. Furthermore, a larger equivalent mass

Figure 4. Change of structure–damper system’s dynamics with variation of (equivalent) mass ratio. (a) Root locus of open-loop poles.(b) System frequency versus mass ratio. (c) System damping ratio versus mass ratio.

ratio results in a lower peak amplitude of FRF and thus abetter control effect. Therefore, according to equations (10),a large capacitance C and a small inductance L are alwaysdesirable in the design of the EMSD in practice. Comparedwith conventional viscous dampers, the EMSD with a singleRLC branch offers damping only in a limited bandwidthnear the resonant frequency (as shown in figure 3). TheEMSD control is effective if structural vibration is dominatedby a specific vibration mode. Furthermore, such a limitedbandwidth of damping may be beneficial in base isolators,where high damping is required in the resonant range whilelow damping is desirable in a higher frequency range. Morediscussions on the pros and cons of the EMSD can be foundin Fleming et al (2005).

The transfer functions of the two systems (i.e. equa-tions (2) and (11)) have the same zeros but different poles.Installing the TMD or EMSD introduces an additional DOF tothe primary structure and yields two separate vibration modeswith significant damping. Figure 4(a) shows the root loci ofthe open-loop poles with variation of the (equivalent) massratio for the structure–TMD and structure–EMSD systems,in which both the frequency ratios and damping ratios ofthe dampers are tuned to the optimal values. With increasing(equivalent) mass ratio, the poles move from the imaginaryaxis towards the real axis. The four crosses on each linecorrespond to µ(or µ) = 0, 0.05, 0.2, and 1, respectively. Thedifference in the pole locations of the two systems is smallwhen µ and µ are low (e.g., less than 0.05) and becomesevident with the further increase in µ and µ. In general,increasing the (equivalent) mass ratio effectively enhancesthe damping level of the primary structure. As reported byKrenk (2005), the optimal tuning in equations (4) causesthe two vibration modes of the structure–TMD system to

have identical damping ratios. However, the equal-dampingfeature cannot be observed in the structure–EMSD system,as shown in figure 4(c). Figure 4(b) shows the change in theamplitude of the complex frequencies with the variation ofthe (equivalent) mass ratio. The two vibration frequencies ofboth systems are higher and lower than the original structuralfrequency, respectively. The upper and lower frequencies ofthe structure–EMSD system are always higher than those ofthe structure–TMD system.

Detuning of the parameters frequently occurs in realapplications. Figure 5 shows the impact of the detunedparameters of the EMSD on control performance. Similar tothe TMD, the control performance of the EMSD degradeswhen the EMSD parameters shift away from their optimalvalues. Control performance is quite sensitive to the detunedfrequency ratio when the equivalent mass ratio is small(as shown in figure 5(a)). Figure 5(b) indicates that thecontrol performance is insensitive to the slight over-dampingof the EMSD. However, significant over-damping (2ξ2,opt)

still degrades the control performance apparently. Notably,significant over-damping of the EMSD may occur in practicebecause of the constraints of circuit inherent resistance. Moredetails are discussed in section 5.

Both the EMSD and TMD are resonant-type dampingdevices. The similar dynamics and control performances ofthese two dampers imply that the extensive study on thelatter in the past may shed light on the understanding of theformer in vibration control applications. Compared with theconventional TMD, the EMSD does not involve a movingmass, and vibration control is realized using a single damper.Although the TMD has been successfully applied in manydifferent areas, a huge moving mass is often impractical, if notimpossible, in practical implementation. The mass ratios of

Figure 5. Sensitivity of control performance to detuned parameters. (a) Detuned frequency ratio. (b) Detuned damping ratio.

Figure 6. Effect of structural mass on the optimal RLC circuit parameters of the EMSD (structural frequency f1 = 6 Hz). (a) C versus m2.(b) L versus m2. (c) R versus m2.

the TMD in mechanical and civil engineering applications aretypically small. In the EMSD, however, the equivalent massis imitated by the capacitance (as shown by equations (10)),and a large mass ratio can be theoretically achieved byincreasing the machine constant and/or the capacitance, suchthat improved control performance may be achieved. Thepotentials and constraints of the EMSD are illustrated throughcase studies in section 5. Another advantage of the EMSDis that its virtual mass is immune to inertia force excitedby base motion, whereas the inertia force on auxiliary massalways reduces the TMD efficiency under base excitation.Thus, the EMSD may achieve better control performance thanthe conventional TMD under seismic ground motions.

5. Numerical case study of the EMSD

A prototype linear-motion electromagnetic device designedby Palomera-Arias (2005), Palomera-Arias et al (2008) isemployed in the case study. The moving part is made ofa N35 permanent magnet with a radius of 51 mm and a

length of 25 mm, the coil is made of AWG 15 wire with sixlayers and 195 turns, and the air gap is 1 mm. Consequently,the electromagnetic device has the following parameters:machine constant 17.35 V s m−1 or N/A, coil resistanceRcoil = 0.231 �, coil inductance Lcoil = 5.1 mH, and dampervolume 743 cm3. The electromagnetic device is optimallydesigned to achieve high damping density. A more detaileddesign procedure and other design parameters are elaboratedin Palomera-Arias (2005). The SDOF structure with a mass of43 kg and a natural frequency of 6 Hz is analyzed as a baselinecase, but its parameters are adjusted in the parametric studyin order to investigate the control performance of the EMSDin different situations. Considering that the peak amplitudeof FRF (i.e. ‖H‖∞) is directly related to the equivalent massratio µ in equation (17), the latter is employed as a measureof control performance in this section.

Figure 6 shows the variation of the RLC circuitparameters with increasing equivalent mass for three differentvalues of structural mass, m1 = 100, 43, and 30 kg. Thestructural frequency f1 is equal to 6 Hz in all three

Figure 7. Effect of structural frequency on the optimal RLC circuit parameters of the EMSD (structural mass m1 = 43 kg) (a) L versus m2.(b) R versus m2.

cases, whereas the parameters C,L and R are optimallydetermined according to equations (15) and (16). In general,a larger equivalent mass ratio, which facilitates bettercontrol performance, requires higher capacitance but lowerinductance and resistance. Figure 6(a) shows the relationshipbetween the capacitance and the equivalent mass describedby equation (10a). A small capacitance of 0.8F is analogousto a large equivalent mass of 240 kg. However, the equivalentmass cannot be increased arbitrarily because of the constraintsof the inductance and resistance in conventional RLC circuits.If the coil resistance and inductance are respectively setas the lower bounds of the L and R values (as shown infigures 6(b) and (c)), the achievable equivalent mass anddensity of the EMSD in the baseline case (m1 = 43 kg)are around 80 kg and 108 ton m−3, respectively; and thecorresponding equivalent mass ratio is around 1.9. Comparedwith the conventional TMD, the EMSD can achieve a veryhigh equivalent mass density and control efficiency. However,a larger structural mass (m1 = 100 kg) results in a muchsmaller achievable equivalent mass (19.7 kg) and density(26.5 ton m−3) of the EMSD, which is constrained bythe lower bound of circuit resistance; whereas a smallerstructural mass (m1 = 30 kg) results in a larger achievableequivalent mass (135 kg) of the EMSD, which is constrainedby the lower bound of circuit inductance. The prototypeelectromagnetic device has a relatively small resistance inthis case study. Considering a greater coil resistance as wellas the equivalent series resistance of circuit components,the typical lower bound of resistance may be higher thanthat of the baseline case. If the lower bound of R isassumed to be 0.5 �, the achievable equivalent mass of theEMSD, m2, will be 3.9, 9.9, and 16.2 kg for the structuralmass m1 = 100, 43, and 30 kg, respectively, where m2 isconstrained by the lower bound of R in all three cases. Itis noteworthy that the increased inherent resistance of thecircuit considerably reduces the equivalent mass and, thus, the

control effect. From the perspective of control performance,large capacitance and small inductance are always desirableto achieve a high equivalent mass ratio. However, the useof an extremely small L value requires a very low optimalresistance value according to equation (16b), which oftencannot be satisfied. Consequently, excessive resistance causesthe over-damping of the EMSD and considerably degradescontrol performance as observed in the experiments in Inoueet al (2008). Therefore, electromagnetic devices with minimalcoil resistance should be selected in practice. The additionof more resistors or insertion of any circuit with equivalentresistance in the RLC shunt circuit is not advisable.

Figure 7 shows the variation of the RLC circuitparameters for three different structural frequencies, f1 = 6,10, and 15 Hz. The structural mass m1 is equal to 43 kg inall three cases; and the parameters C,L, and R are optimallydetermined. The relationship between the capacitance andequivalent mass is the same as that shown in figure 6(a).The increasing frequencies require a smaller value of LCaccording to equation (9a). Given that the L value is oftenlimited by the lower bound that is dependent on the coilinductance or resistance, a smaller capacitance (i.e., a smallerequivalent mass) needs to be used for a higher frequency.For the structural frequencies f1 = 6, 10, and 15 Hz, theachievable equivalent masses of the EMSD m2 are 80.3, 18.1,and 7.2 kg, respectively; the corresponding equivalent massdensities are 108, 24.4, and 9.7 ton m−3, respectively; andthe corresponding equivalent mass ratios are 1.9, 0.42, and0.17, respectively. Accordingly, the control performance willdegrade rapidly with increasing structural frequency.

Figure 8 shows the effect of structural mass andfrequency on the control performance of the EMSD in a 3Dplot, where control performance is measured based on themaximum achievable equivalent mass ratio in considerationof the lower bounds of circuit inductance and resistance.The control performance of the EMSD deteriorates rapidly

Figure 8. Change of equivalent mass ratio with variation ofstructural mass and frequency.

with increasing structural mass and frequency. This conditionexplains the difficulty in using the EMSD for the optimalvibration control of large mass structures, high-frequencystructures, or higher vibration modes of MDOF structures thathave been observed in previous studies. For example, Chengand Oh (2009) utilized an EMSD to suppress the first twovibration modes of a cantilever beam, but the damping effectof the second mode is limited and the two dominant peaks inthe FRF cannot be observed around the second mode in eithernumerical or experimental results.

As a minimal resistance or inductance would benefit thecontrol performance of the EMSD, a potential solution tothe constraints of the internal coil is the negative impedancetechniques (Fleming et al 2005, Fleming and Moheimani2006, Niu et al 2009, Zhang et al 2012). Some sophisticatedelectric circuits with negative resistance or inductance can beused to offset too large inherent resistance and inductanceof the original circuit, such that the dotted lines in figures 6and 7 can be achieved. Consequently, the control performanceof the EMSD can be effectively enhanced, particularly insituations of high frequency or large structural mass. Fleminget al (2005) attached a negative inductor–resistor controllerto an EM damper to suppress structural vibration. Niuet al (2009) used a dSPACE system to implement negativeresistance in order to enhance the beam vibration controleffect using the EMSD. Zhang et al (2012) proposed the useof negative resistance and inductance circuits, implementedby using a dSPACE system as well, to cancel the inherentinductance and resistance of the EM damper. With the aidof a negative impedance circuit, the optimal inductance andresistance corresponding to extremely large capacitance canbe achieved. However, compared with conventional RLCresonant circuits such a strategy requires a power supply anda more complicated circuit or control system. In addition, thecurrent limit of the electromagnetic devices is another factorthat has to be considered in the actively controlled circuit.

Another approach to enhance the control performanceof the EMSD is to increase the machine constant Kem ofthe electromagnetic devices. The machine constants Kemof linear-motion electromagnetic devices are typically low.A potential solution is to use a linear-to-rotary conversionmechanism and a rotary electromagnetic device, in whichthe rotation of the electromagnetic device can be acceleratedby a gear box. A significantly higher Kem can be achievedusing this method with a given size of electromagnetic devices(Kem = ηngKem,rotary, where η relates the rotational and linearmotions, and ng is the ratio of the gear box). However, such amechanism may introduce more friction or parasitic dampingto the system, the effect of which needs to be carefullyexamined in future studies.

6. Discussions

(1) Both the EMSD and TMD are resonant-type dampingdevices that offer damping in a limited bandwidth close tothe resonant frequency. Through theoretical comparison,this study demonstrates the similarity and differencebetween the dynamics of the novel EMSD and theconventional TMD in terms of structural vibration control.Despite different optimal design laws, the EMSD hasa very similar vibration control effect and sensitivity todetuned parameters as the TMD.

(2) The dynamic analogy between the EMSD and TMDclearly reveals that maximum capacitance and minimuminductance can optimize the control performance of theEMSD. A large capacitance in the RLC shunt circuitof the EMSD is analogous to a large equivalent massin the TMD, which always benefits vibration controlperformance.

(3) Compared with the TMD, the EMSD does not involve anymoving mass and can theoretically achieve a substantialvirtual mass. A large equivalent mass ratio of the EMSDwould significantly improve control performance. It isnoteworthy that the installation of the EMSD and TMDare different in real applications, despite their similarcontrol dynamics. In a MDOF structure, the TMD canbe installed at any single point, while the EMSD has tobe installed between two points with relative motion. Thecontrol efficiency and optimal placement of the EMSD inMDOF structures warrant some future investigation.

(4) The optimization of RLC parameters is constrained by theminimum possible resistance and inductance of the entirecircuit. Consequently, the control performance decaysrapidly with increasing structural mass and frequency. Ingeneral, an electromagnetic device with a large machineconstant as well as low inherent resistance and inductanceshould be selected.

(5) Piezoelectric shunt dampers possess a dynamic principlesimilar to that of the EMSD. However, the high inherentresistance of piezoelectric materials may limit its controlefficiency.

(6) The optimal design law for the SDOF structure canbe extended to the multi-mode vibration control of

MDOF structures. However, the difficulty in controllinghigh-frequency modes has been envisioned.

(7) The negative impedance technique can effectivelyovercome the constraints of minimum resistance andinductance of the circuit and facilitate the use of a verylarge capacitance in the circuit. Although this techniqueconsiderably increases the equivalent mass ratio andimproves control performance, it requires a power supplyand a more complicated circuit or even an active controlsystem.

(8) Instead of a linear-motion electromagnetic device, acombination of a linear-to-rotary converter and a rotaryelectromagnetic device is recommended to enhance themachine constant Kem. However, such a mechanism mayintroduce more damping to structural systems, and its realeffect has to be evaluated in future studies.

Acknowledgments

The authors are grateful for the financial support from theResearch Grants Council of Hong Kong through a GRF grant(Project No. PolyU 5330/11E) and from The Hong KongPolytechnic University (Project No. G-YM29). Findings andopinions expressed here, however, are those of the authorsalone, not necessarily the views of the sponsors.

Appendix. Derivation of optimal EMSD parameters

The optimization method for the TMD presented in Cheung(2009) is adapted in this appendix for the H∞ optimizationof the EMSD installed on a SDOF primary structure. Theamplitude of the FRF in equation (13), also known as thedynamic amplification factor, is given by

|H(λ)| =

∣∣∣∣ X1(λ)

F(λ)/k1

∣∣∣∣=

√(γ 2 − λ2)2 + (2ξ2γ λ)2

[(1− λ2)(γ 2 − λ2)− µγ 2λ2]2 + [2ξ2γ λ(1− λ2)]2

√A+ Bξ2

C + Dξ22

A = (γ 2− λ2)2 (19a)

B = (2γ λ)2 (19b)

C = [(1− λ2)(γ 2− λ2)− µγ 2λ2

]2 (19c)

D = [2γ λ(1− λ2)]2. (19d)

The H∞ optimization aims to find parameters γ and ξ2such that they minimize the peak dynamic amplification factor

minimize∥∥H∥∥∞. (20)

Given specific γ and µ, there are two invariant points, denotedby points a and b, on the FRF curve of equation (18). Since the

two invariant points are independent of the EMSD dampingratio ξ2, the following relation should be satisfied

D(21a)

γ 2− λ2

(1− λ2)(γ 2 − λ2)− µλ2γ 2 = ±1

1− λ2 . (21b)

Taking a negative sign in equation (21b) yields

λ4−

(1+ γ 2

+µγ 2

)λ2+ γ 2

= 0. (22)

The two roots (denoted by λ2a and λ2

b) of equation (22)determine the locations of the two invariant points, and theircorresponding amplitudes of FRF are

|H(λa)| =

∣∣∣∣B

∣∣∣∣ = ∣∣∣∣ 1

1− λ2a

∣∣∣∣ (23a)

|H(λb)| =

∣∣∣∣B

∣∣∣∣ =∣∣∣∣∣ 1

1− λ2b

∣∣∣∣∣ . (23b)

The optimization procedure first makes the two invariantpoints a and b have equal amplitude by tuning the frequencyratio γ , and then attempts to make them be the two peak pointson the FRF curve. By equating equations (23a) and (23b), weobtain

1− λ2a= −

1− λ2b

. (24)

The optimal frequency ratio of EMSD can be determined bysolving equations (22) and (24)

γopt =

2+ µ. (15a)

The two roots λ2a and λ2

b in equation (22) can be obtained oncethe optimal frequency ratio is known. Substituting λ2

a or λ2b

into equation (24) yields the amplitude of the two points,

Gopt =

√2+ µµ

. (17)

The optimal damping ratio of the EMSD can be determinedby making the two invariant points be the two peaks of theFRF curve, i.e.

∂λ2

∣∣H(λ)∣∣2∣∣∣λ=λa= 0 (25a)

∂λ2

∣∣H(λ)∣∣2∣∣∣λ=λb= 0. (25b)

Let ∣∣H(λ)∣∣2 = p

q(26a)

p = (γ 2− λ2)2 + (2ξ2γ λ)

2 (26b)

q = [(1− λ2)(γ 2− λ2)

− µγ 2λ2]2+ [2ξ2γ λ(1− λ2)]2. (26c)

Equations (25) can then be converted to

∂λ2 −∂q

∂λ2 G2opt = 0. (27)

From equations (15a), (17) and (27), we can derive

ξ2,a =

√√√√ µ

)(28a)

ξ2,b =

√√√√ µ

). (28b)

Two different ξ2 values imply that equations (25a) and (25b)cannot be satisfied simultaneously. In practice, the optimaldamping ratio can be taken as the root-mean-square of ξ2,aand ξ2,b

ξ2,opt =

√ξ2

2,a + ξ22,b

√3µ8. (15b)

Equation (17) will be a good approximation of (but not exactlyequal to) the peak dynamic amplification factor

∥∥H∥∥∞

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