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Magnetic negative stiffness dampers

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2015 Smart Mater. Struct. 24 072002

(http://iopscience.iop.org/0964-1726/24/7/072002)

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Fast Track Communication

Magnetic negative stiffness dampers

Xiang Shi and Songye Zhu

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

E-mail: ceszhu@polyu.edu.hk

Received 4 March 2015, revised 13 May 2015Accepted for publication 29 May 2015Published 17 June 2015

AbstractThis communication presents the design principle and experimental validation of two novelcongurations of magnetic negative stiffness dampers (MNSDs), both of which are composed ofseveral permanent magnets arranged in a conductive pipe. The MNSD, as a passive device,efciently integrates negative stiffness and eddy-current damping in a simple and compactdesign, in which the negative stiffness behavior depends on the different arrangements of thepermanent magnets. When applied to structural vibration control, passive MNSD may achieve aperformance comparable with semi-active or active control in some applications. Laboratoryexperiments of small-scale prototypes successfully veried the proposed MNSD design concept.

Keywords: magnetic negative stiffness damper, passive negative stiffness mechanism, eddycurrent damping, vibration control

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

Introduction

Structural vibration control refers to the protection of primarystructural systems against excessive vibrations induced bydynamic loads. Various dampers in passive, semi-active, oractive modes have been proposed to alleviate structuralvibrations, including some emerging self-powered self-sen-sing active and semi-active dampers (Chen and Liao 2012).Active and semi-active control techniques usually achievebetter control performance than passive dampers. The linearquadratic regulator algorithm, a commonly adopted optimalcontrol theory for active dampers, may produce the hysteresis(i.e., forcedeformation) relationship of a damper with anapparent negative-stiffness feature in some situations thatbenets control effect (Iemura and Pradono 2009). Thisobservation has led to the exploration of a passive negative-stiffness damper that produces similar hysteresis and achievescontrol performance comparable to those of active dampers.

Some researchers attempted to implement such a nega-tive-stiffness mechanism using semi-active dampers (e.g.,Iemura et al 2006, Hgsberg 2011, Weber et al 2011, Weberand Boston 2011). Iemura et al (2006) introduced a controllaw to produce negative-stiffness hysteresis using variable-orice oil dampers. Hgsberg (2011) produced negative

stiffness by applying linearly or cosine decaying voltage tomagnetorheological (MR) dampers. Weber and Boston(2011) optimized MR dampers with a hysteresis of clippedviscous damping and negative stiffness. Weber et al (2011)further combined a tuned mass damper with an MR damper toachieve controllable stiffness. Although more attractive thanactive control in many aspects, semi-active control stillrequires a feedback system involving sensors and controllers.Therefore, past studies have explored several truly passivenegative-stiffness systems, such as a negative stiffness springbased on snap-through behavior of a pre-buckled beam(Dijkstra et al 1988, Wang and Lakes 2004, Lee et al 2007,Kalathur and Lakes 2013), a friction pendulum sliding iso-lator with convex friction interface (Iemura and Pra-dono 2009), a negative stiffness system composed of a pre-compressed spring (Pasala et al 2012), and so on. Meanwhile,the negative-stiffness mechanism has seen potential applica-tions in vibration suppression of civil and mechanical struc-tures. For example, Iemura and Pradono (2009) veried theeffectiveness of semi-active negative-stiffness dampers in theseismic protection of buildings and cable-stayed bridges.Hgsberg (2011) found that a semi-active MR damper thatproduces negative stiffness considerably reduced the overallresponse of a 10-story frame with slight amplication of local

Smart Materials and Structures

Smart Mater. Struct. 24 (2015) 072002 (7pp) doi:10.1088/0964-1726/24/7/072002

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response at the damper position. Pasala et al (2012) demon-strated that an appropriate combination of negative stiffnessand viscous damping can effectively reduce the seismicresponse of a building, such as deformations, accelerations,and base shear. Growing interest in negative stiffness systemshas also been seen in vehicle seat suspensions, vibrationisolation table and vibration transducers (Lee et al 2007, Leand Ahn 2011, Zhu et al 2012). Le and Ahn (2011) developeda seat isolation system using pre-stressed springs to achieveexcellent vibration reduction. Passive negative-stiffnessvibration-isolation systems have recently become an eco-nomical option for nanotechnology applications (Platus andFerry 2007). However, some inherent limitations in theexisting designs may hinder the widespread application ofthese negative-stiffness systems. For example, a pre-com-pressed spring exhibits negative-stiffness feature only whendisplaced beyond a certain limit. A pre-buckled beam isassociated with asymmetrical negative stiffness about itsinitial position. A sliding isolator with convex friction inter-face is only effective in the horizontal direction with thepresence of gravity.

This communication proposes two novel designs ofnegative-stiffness dampers, termed as magnetic negativestiffness dampers (MNSD) in this paper. The proposedMNSDs are based on a magnetism principle, which is com-pletely distinct from existing negative stiffness congura-tions. The MNSD is a fully passive damper that efcientlyintegrates the negative stiffness mechanism and eddy-currentdamping in a compact and simple conguration. In thisconguration, the negative stiffness is contributed by theinteraction forces among several sequentially placed magnets.The proof-of-concept experiments were conducted in thelaboratory through cyclic tests of small-scale prototypes. Thenonlinearity in negative stiffness, either hardening or soft-ening, was observed in the experimental results.

Design concept

Figure 1 shows the two design congurations of the proposedMNSDs, both of which consist of static and moving magnets,sliding bearings, xing spacers, a conductive pipe, and ashaft. Some basic dimensions are also shown in their sectionviews. The moving magnets are mounted on the moving shaftby the xing spacers, and the static magnets are xed on thestatic conductive pipe. The shaft connected with the movingmagnet is aligned by the sliding bearings at the two ends ofthe damper. Some soft stops (e.g. rubber spacers) should beadded in both congurations to avoid collision or directcontact when the moving magnet gets close to the staticmagnets or bearings. Design A is composed of three perma-nent magnets aligned in the longitudinal direction(gure 1(a)), and design B is composed of two concentricpermanent magnets (gure 1(b)).

The principle of design A is presented in gure 2. Indesign A, three permanent magnets, which include a movingmagnet and two static magnets, are aligned along the long-itudinal axis with the same pole orientations. The interaction

force between two magnets nonlinearly increases withdecreasing separation distance. When the moving magnetstays at the zero position that is equidistant to the two staticmagnets, the net force acting on the moving magnet is equalto zero. When the moving magnet in the middle movesupwards or downwards, a force F in the opposite direction ofthe displacement X is required to keep the moving magnet inequilibrium (gure 2(a)). This phenomenon represents anegative-stiffness forcedisplacement relationship. Design Binvolves a magnet ring and a magnet cylinder with the samepole orientation. The two permanent magnets are concentricwhen the moving magnet is at the zero position (gure 3(a)).When the inner moving magnet moves away from the zeroposition, the repelling force between the two magnets shouldbe counterbalanced by an external force F in the oppositedirection of the displacement X (gure 3(a)). Thus, design Balso exhibits a negative-stiffness behavior.

When a magnet is placed inside a non-ferromagneticconductive pipe, the movement of the magnet leads to thevariation in the magnetic ux and produces eddy currentwithin the pipe. Consequently, the eddy current exerts anelectromagnetic force, which is equivalent to a passive vis-cous damping force, on the moving magnet (gures 2(b)or 3(b)).

Finally, designs A and B efciently combine the nega-tive-stiffness feature contributed by the magnetic interactionforce and the passive damping induced by eddy current into asingle device (gures 2(c) and 3(c)). Compared with theexisting design of negative-stiffness devices or systems, theproposed congurations represent more efcient and compactdesigns associated with favorable energy dissipation cap-ability and symmetrical negative stiffness about the equili-brium position. These arrangements are advantageous tovibration control of many mechanical and civil structures.Larger damper force may be achieved by increasing either thestrength and dimensions of magnets or the number of units.However, there often exists an upper limit on the strength anddimensions of strong magnets in practical manufacture, andthis fact may limits the force of a single MNSD. A morepractical solution to a larger damper force is repeating the unitshown in gures 1(a) or (b) in the longitudinal direction. Toavoid any interactions, adjacent units should be spaced withsufcient separation distance or be magnetically isolated. Theeddy-current damping in the proposed congurations can beconveniently replaced by conventional viscous uid dampingif the latter is desirable. The permanent magnets can also bereplaced by electromagnets so that an adjustable damperbehavior can be achieved.

Experimental verication

The prototypes of the two MNSD congurations were fabri-cated and tested in the laboratory to verify the design concept.The neodymium permanent magnets (also known as NdFeBmagnets) with grades of N48 and N35 are used in designs Aand B, respectively. Three magnetic rings with the outerdiameter d ,out inner diameter din and thickness hm equal to 48,

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10 and 20 mm, respectively, are employed in design A. Twoconcentric magnetic rings with the same thickness ofhm= 20 mm are used in design B, where the outer magnet hasthe outer and inner diameters of d 76 mmout,2 = andd 50 mm,in,2 = respectively, and the inner magnet has theouter and inner diameters of d 40 mmout,1 = andd 10 mm,in,1 = respectively. The notations of the dimensionsare shown in gure 1. All materials, except the magnets in theMNSDs, are non-ferromagnetic. The conductive pipes andshaft are made of brass and stainless steel, respectively. Thediameter of shaft is d 8 mm,s = which is rigid enough to holdthe moving magnet. The brass pipes with various wallthicknesses ( )t 15 and 25 mmp = are tested. The static andmoving magnets are mounted on the conductive pipe andmoving shaft, respectively. Sliding bearings were installedbetween the shaft and end plates of the pipe to guide the

movement and minimize the friction. The MNSD prototypes(both design A and design B) were cyclically tested underdifferent loading frequencies, including a quasi-static loadingrate. Figure 4 shows the photo of design A on an MTS uni-versal testing machine. The cyclic tests were conducted with adisplacement control and the damper force was measured by aload cell shown in gure 4. The eddy-current damping andnegative stiffness were experimentally characterized indivi-dually rst: the eddy-current damping was characterizedthrough dynamic testes of the congurations without staticmagnets (as shown in gure 2(b)); the negative stiffness wascharacterized through quasi-static tests of the complete con-gurations in which the damping effect became ignorable.Finally, the eddy-current damping and negative stiffnesswere experimentally characterized simultaneously throughdynamic tests of the complete conguration.

Figure 1. Two congurations of MNSD.

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Negative stiffness

Figure 5 shows the forcedisplacement relationship at aquasi-static loading rate, in which the damping force isminimal so that the negative-stiffness mechanism can beclearly observed. The opposite directions of the force anddisplacement validate the concept of magnetic negative-

stiffness mechanism presented in gures 2(a) and 3(a). Asymmetric but nonlinear negative stiffness can be observed inboth MNSD congurations. The regression curves for theforcedisplacement relationships are also obtained and shownin gure 5. The tangent stiffness can be subsequently esti-mated based on the regression curves. Figure 6 shows thedisplacement-dependent stiffness variation for both designs.

Figure 2. Principle of design A of MNSD.

Figure 3. Principle of design B of MNSD.

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Designs A and B exhibit the hardening and softening patternsof negative stiffness, respectively. As shown in gure 6(a),design A shows the minimum negative stiffness at the zeroposition where the moving magnet is equidistant from the twostatic magnets, and the magnitude of negative stiffnessincreases nonlinearly with the displacement. The stroke ofdesign A is limited by the gap distance between the movingand static magnets. By contrast, design B provides the max-imum magnitude of negative stiffness at the zero position(gure 6(b)), and its initial negative stiffness is considerablygreater than that of design A. However, the tangent stiffnessdrops quickly with increasing displacement. The stiffnessdecreases to zero when the magnetic force reaches the max-imum points. Then, the tangent stiffness becomes positive,and the MNSD essentially loses the negative-stiffness feature.Although the movement of the magnet in design B is notrestrained in the axial direction, the displacement range withnegative stiffness is from 10 to +10 mm. Thus the effectivestroke of design B is approximately equal to the thickness ofthe magnets in design B.

Eddy-current damping

Figure 7 presents the experimental results corresponding tothe conguration shown in gure 2(b), in which a singlemagnet is moving inside a brass pipe with a wall thickness of25 mm. The moving magnet is excited with harmonic dis-placement, and the eddy-current damping force is character-ized at different excitation frequencies with diverse pipethickness. The eddy-current damping force is proportional to

the vibration frequency, and the damping force correspondingto 2 Hz excitation is approximately twice as much as that of1 Hz (gure 7(a)). More tests with various brass pipes indi-cates that the eddy-current damping force increases with thethickness of the pipe wall, because a larger thickness of theconductive pipe is associated with smaller resistance andhigher eddy current inside the pipe wall.

MNSD

Figure 8 shows the experimental result of design A thatcombines negative-stiffness and eddy-current damping beha-vior. The cyclic tests were conducted under different loadingfrequencies. The eddy-current damping force of the MNSDincreases with the vibration frequency, whereas the stiffnessforce remains nearly the same. The experimental resultssuccessfully prove the design concept shown in gure 2(c).The proposed design of MNSD can efciently integrate thenegative-stiffness mechanism and eddy-current damping in asimple and compact conguration.

Figure 4. MSND of design A under cyclic testing.

Figure 5. Negative-stiffness behavior of MNSDs.

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Conclusion

Negative-stiffness dampers for vibration suppression ofmechanical and civil structures have recently attractedgrowing interest. Two novel designs of negative-stiffnessdampers based on magnetism are proposed in this study. Thetwo MNSD designs can efciently integrate the negativestiffness and eddy-current damping in a simple and compactdesign. The proof-of-concept experiments were conductedthrough the cyclic loading of scaled MNSD prototypes on anMTS machine. The nonlinearity in negative stiffness isobserved in both congurations. Designs A and B exhibit thehardening and softening patterns of negative stiffness,respectively, with increasing displacement. Compared withthe existing designs of negative-stiffness systems, the salient

features of the proposed MNSD include the following: (1)symmetrical negative-stiffness behavior; (2) integrateddamping characteristic; and (3) a compact design that can beinstalled in any direction. The proposed MNSDs have a greatpotential to replace semi-active or active dampers in diversevibration suppression or isolation applications.

Acknowledgments

The authors are grateful for the nancial support from theResearch Grants Council of Hong Kong through a GRF grant(Project No. PolyU 152222/14E) and from the Innovation andTechnology Commission of Hong Kong through an ITF grant(Project No. ITS/344/14).

References

Chen C and Liao W H 2012 A self-sensing magnetorheologicaldamper with power generation Smart Mater. Struct. 21 025014

Dijkstra K, Videc B P and Huizinga J 1988 Mechanical springhaving negative spring stiffness useful in an electroacoustictransducer J. Acoust. Soc. Am. 84 8045

Hgsberg J 2011 The role of negative stiffness in semi-active controlof magnetorheological dampers Struct. Control HealthMonit. 18 289304

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Iemura H and Pradono M H 2009 Advances in the development ofpseudonegativestiffness dampers for seismic response controlStruct. Control Health Monit. 16 78499

Kalathur H and Lakes R S 2013 Column dampers with negativestiffness: high damping at small amplitude Smart Mater.Struct. 22 084013

Le T D and Ahn K K 2011 A vibration isolation system in lowfrequency excitation region using negative stiffness structurefor vehicle seat J. Sound Vib. 330 631135

Lee C M, Goverdovskiy V N and Temnikov A I 2007 Design ofsprings with negative stiffness to improve vehicle drivervibration isolation J. Sound Vib. 302 86574

Figure 6. Nonlinear negative stiffness of the tested MNSDs.

Figure 7. Eddy-current damping behavior of MNSD.

Figure 8. Forcedisplacement behavior of MNSD (Design A).

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Pasala D T R, Sarlis A A, Nagarajaiah S, Reinhorn A M,Constantinou M C and Taylor D 2012 Adaptive negativestiffness: new structural modication approach for seismicprotection J. Struct. Eng. 139 111223

Platus D L and Ferry D K 2007 Negative-stiffness vibration isolationimproves reliability of nanoinstrumentation Laser Focus World43 107

Weber F and Boston C 2011 Clipped viscous damping with negativestiffness for semi-active cable damping Smart Mater. Struct. 20045007

Weber F, Boston C and Malanka M 2011 An adaptive tuned massdamper based on the emulation of positive and negativestiffness with an MR damper Smart Mater. Struct. 20 015012

Wang Y C and Lakes R S 2004 Stable extremely-high-dampingdiscrete viscoelastic systems due to negative stiffness elementsAppl. Phys. Lett. 84 44513

Zhu Y, Li Q, Xu D, Hu C and Zhang M 2012 Modeling andanalysis of a negative stiffness magnetic suspension vibrationisolator with experimental investigations Rev. Sci. Instrum.83 095108

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IntroductionDesign conceptExperimental verificationNegative stiffnessEddy-current dampingMNSD

ConclusionAcknowledgmentsReferences