No-Jerk Skyhook Control Methods for effectively reduce vibration transmission across the suspensionand control the suspended body ~or sprung mass!, as compared to

Downloaded FSemiactive Suspensions

Mehdi AhmadianProfessor, Fellow ASME,Center for Vehicle Systems and Safety,Department of Mechanical Engineering,Virginia Tech, Blacksburg, VA 24061

Xubin SongMem. ASME,Visteon Corporation, 6100 Mercury Dr.,Dearborn, MI 48126

Steve C. SouthwardMem. ASME,Lord Corporation, 100 Lord Drive, Cary, NC

This paper presents two alternative implementations of skyhookcontrol, named skyhook function and no-jerk skyhook, forreducing the dynamic jerk that is often experienced with conven-tional skyhook control in semiactive suspension systems. Ananalysis of the relationship between the absolute velocity of thesprung mass and the relative velocity across the suspension areused to show the damping-force discontinuities that result fromthe conventional implementation of skyhook control. This analysisshows that at zero crossings of the relative velocity, conventionalskyhook introduces a sharp increase (jump) in damping force,which, in turn, causes a jump in sprung-mass acceleration. Thisacceleration jump, or jerk, causes a significant reduction in iso-lation benefits that can be offered by skyhook suspensions. Thealternative implementations of skyhook control included in thisstudy offer modifications to the formulation of conventional sky-hook control such that the damping force jumps are eliminated.The alternative policies are compared to the conventional skyhookcontrol in the laboratory, using a base-excited semiactive systemthat includes a heavy-truck seat suspension. An evaluation of thedamping force, seat acceleration, and the electrical currents sup-plied to a magnetorheological damper, which is used for thisstudy, shows that the alternative implementations of skyhook con-trol can entirely eliminate the damping-force discontinuities andthe resulting dynamic jerks caused by conventional skyhook con-trol. @DOI: 10.1115/1.1805001#

IntroductionThe idea of skyhook control has been in existence for more

than three decades. Introduced by Crosby and Karnopp @1,2#, sky-hook control has been studied most often for vehicle-suspensionapplications. Suspension systems with skyhook control draw asmall amount of energy to operate a valve that adjusts the damp-ing force; therefore, they are referred to semiactive suspensionsor dampers. The damper valve can be a mechanical element ~as isthe case for mechanically adjustable dampers! or a fluid valve thattakes advantage of the rheological changes of the damper fluid,such as a magnetorheological ~MR! damper @3,4#.

The virtues of semiactive suspensions versus conventional, pas-sive suspensions have been addressed in several past studies@511#. Using various analytical methods, these studies have con-cluded that in nearly all cases of semiactive suspensions can more

Contributed by the Technical Committee on Vibration and Sound for publicationin the JOURNAL OF VIBRATION AND ACOUSTICS. Manuscript received November2003; final revision, February 2004. Associate Editor: Lawrence A. Bergman.

580 Vol. 126, OCTOBER 2004 Copyright rom: http://vibrationacoustics.asmedigitalcollection.asme.org/ on 04/28/2passive suspensions. Some of these studies have led to differentcontrol methods for semiactive suspensions, including severalvariations to skyhook control, which have been reviewed by Iversand Miller @12#.

Although there exists a large number of studies that provide acomprehensive analysis of the skyhook control method and itsvariations, very few studies have provided an analysis of the dy-namic jerk that is caused by the change in damping force in sky-hook systems. This study will provide a comprehensive analysisof the dynamic jerk that is induced by skyhook control. After abrief overview of skyhook control, we will provide two alternativeformulations of skyhook control for eliminating dynamic jerk. Ad-ditionally, we will provide the results of a series of dynamic teststhat compare the alternative methods to conventional skyhookcontrol for a test setup that includes a single-degree-of-freedom~dof! base-excited system with a magnetorheological damper.

BackgroundFor a semiactive suspension, the equations governing skyhook

control can be described by

H V1V12.0 Fsa5GsV1V1V12,0 Fsa50 (1)where

V15absolute velocity of the sprung massV125relative velocity across the suspensionFsa5skyhook damping forceGs5a gain factor that is commonly used such that the full

damping range of the skyhook damper can be usedSkyhook control, which is illustrated in Fig. 1, indicates that

when the absolute velocity of the sprung mass and the relativevelocity across the suspension have the same sign, the damper isresisting the motion of the sprung mass and a damping force pro-portional to the sprung mass velocity is desired. Otherwise, thedamping force does not resist the motion of the sprung mass andmust be minimized. This logic emulates the ideal configuration ofa passive damper hooked between the sprung mass and animaginary sky, hence, the name skyhook.

Problem StatementIn order to better highlight the dynamic jerk problem due to the

skyhook control, we will evaluate a three-dimensional surface plotof Eq. ~1!, as is shown in Fig. 2. The figure clearly indicates thatin the second and fourth quardrants, when the relative and abso-lute velocities have opposite signs, the damper force is zero. Oth-erwise it is proportional to the absolute velocity. Figure 2 alsoshows that the damping force jumps sharply at the transitionsbetween the quardrants corresponding to the relative velocity zerocrossings, which is further depicted in Fig. 3. This force disconti-nuity causes acceleration jumps or jerks in various physical sys-tems, such as the single degree-of-freedom system shown in Fig.4, which reptresents a heavy-truck seat suspension.

Skyhook Function. The continuous equation

Csa5Is1e f2e2 f

Ione f1Io f fe2 f(2)

is proposed to represent the damping coefficient of the semiativesuspension, which can be adjusted in real time. The constants Isand Ion are such that they are small as compared to Io f f , i.e.,

Is!Io f fand

Ion!Io f f

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Downloaded FThe exponential power f represents the switching function of thesemiactive suspension. In the case of skyhook control, it is a func-tion of relative and absolute velocity, i.e.,

f 5Kv1v12 (3)where K is a positive constant. In practice, the selection of theconstants in Eqs. ~2! and ~3! would be according to the forcecapabilities of the adjustible damper that is used for the semiactivesuspension. Such selections, which must be done on a case-by-case basis, will have little effect on the comparison of the damperforces that will be dicussed later and, as such, will not be furtherelaborated on here.

Figure 5 shows a surface plot of the damping force with respectto the absolute and relative velocity. In Fig. 5, it can be observedthat smooth transition of the damping force from one quardrant toanother is achieved, as compared to Fig. 2. A similar differencecan be observed between Figs. 6 and 3. Both Figs. 5 and 6 indi-cate that the skyhook function control would cause substantiallylower dynamic jerk.

Based on Figs. 2 and 3, the approaches for reducing or elimi-nating the dynamic jerks could include:

Fig. 1 Skyhook damping force illustration

Fig. 2 Surface plot of damper force for conventional imple-mentation of skyhook control using a rate filter

Journal of Vibration and Acousticsrom: http://vibrationacoustics.asmedigitalcollection.asme.org/ on 04/28/21. Using an analytic continuous function instead of the sky-hook control, shown in Eq. ~1!, in order to avoid the damp-ing force discontinuity

2. Modifing the skyhook control policy to enable smooth tran-sition of damper current, even when the damper is not at itstrue relative velocity zero crossing

No-Jerk Skyhook. An alternative formulation to Eq. ~1! forreducing dynamic jerk is

Csa5H KuV1V12u V1V12>00 V1V12,0 (4)where K is a constant gain. This formulation is quite similar to theskyhook control shown in Eq. ~1!, except that the damping coef-

Fig. 3 Damping force discontinuities resulting from conven-tional implementation of skyhook control using a rate filter

Fig. 4 A single-dof base-excited system: a schematic model,b physical system in the form of a heavy-truck seat suspen-sion

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Downloaded Fficient is a function of both absolute and relative velocity. As isshown in Figs. 7 and 8, including the relative velocity in skyhookcontrol formulation eliminates damping force discontinuity anddynamic jerk. Compared with Fig. 2, Fig. 7 shows that the damp-ing force has a smooth transition between the four quadrants. Thesame phenomenon is observed in comparing Figs. 3 and 8. Ofcourse, reducing the force discontinuity reduces dynamic jerks, aswill be shown in our laboratory test results, which are included inthe next sections.

Test SetupThe truck-seat suspension with a magnetorheological damper,

shown in Fig. 4, is used for laboratory evaluation of the methodsthat were suggested earlier for reducing jerk. The test setup thatwas used for this study is shown schematically in Fig. 9, andfurther documented by Song @13# and Reichert @14#. The seat isexcited at the base by a hydraulic actuator with a force capacity of2000 lb. The sprung mass is supported on a suspension that con-sists of an airspring and an adjustable magnetorheological ~MR!damper. The airspring has a linear spring constant within the rangeof the relative displacement that was used in our tests.

The MR damper force is directly proportional to the currentsupplied to the damper. An MR damper was used instead of a

Fig. 5 Surface plot of damper force for implementation of sky-hook control using skyhook function

Fig. 6 Elimination of damping force discontinuties due to us-ing skyhook function

582 Vol. 126, OCTOBER 2004rom: http://vibrationacoustics.asmedigitalcollection.asme.org/ on 04/28/2mechanically adjustable damper because of its availability, ease ofuse, and the growing trend of using such dampers for suspensionapplications.

A standard input profile known as ISO Class 2 Excitation ~alsoknown as ISO2 Excitation!, documented in @15#, was used forexciting the system. The absolute velocity needed for skyhookcontrol was derived from an accelerometer on the sprung mass.The relative velocity across the suspension was estimated fromthe relative displacements across the suspension, which was mea-sured by a rotary sensor that was mounted at the cross-pivot of thescissor suspension, for the truck-seat suspension in Fig. 3.

Test ResultsThe test results are shown in Figs. 1013. These figures show

the results of three different implementation of the skyhook con-trol that are described below, using the terminology that we haveselected for this study:

1. Rate Skyhook, is the implementation of the skyhook con-trol in Eq. ~1!, using a rate filter that is able to estimatevelocity across the suspension from relative displacement

Fig. 7 Surface plot of no-jerk skyhook control damper force

Fig. 8 Elimination of damping-force discontinuities due to us-ing no-jerk skyhook formulation

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Downloaded Fmeasurement, as described earlier. This is the implementa-tion of skyhook control frequently adopted for vehicle sus-pensions and similar systems.

2. Skyhook Function, described by Eqs. ~2! and ~3!, is oneof the two alternative implementations of skyhook controlsuggested in this study.

3. No-jerk Skyhook, is the other suggested alternativeimplementation of skyhook control, described by Eq. ~4!.

First, we investigate the currents to the MR damper as shown inFigs. 1113. It is of no surprise that the skyhook function is quitesimilar to the no-jerk skyhook. Both control policies have asmooth transition of the current to the MR damper from the off tothe on states because the new modified skyhook control policiestake advantage of the fact that at least one velocity changessmoothly from the on to the off state ~or vice versa!. They gener-ate a smooth dynamic response, as shown in Figs. 10 and 12.These figures show that both the no-jerk skyhook and skyhookfunction-control policies cause a relatively smooth dynamic re-sponse because they eliminate the damping-force discontinuity.

Fig. 9 Implementation of skyhook control on a physical sys-tem representing a single-dof base-excited system with a semi-active suspension

Fig. 10 Sprung mass accelerations caused by different sky-hook control formulations in response to a 1.45 Hz pure-tonebase excitation

Journal of Vibration and Acousticsrom: http://vibrationacoustics.asmedigitalcollection.asme.org/ on 04/28/2ConclusionsTwo different implementations of skyhook control, named

skyhook function and no-jerk skyhook, were suggested forreducing the dynamic jerk that is often experienced with conven-tional skyhook control in suspension systems. Through an analysisof the relationship between the absolute velocity of the sprungmass and the relative velocity across the suspension, we showedthe source of force discontinuity in conventional implementationof skyhook control, called rate skyhook, for the purpose of thisstudy. This analysis showed that at the zero crossings of the rela-tive velocity, rate skyhook introduces a jump in the damping forcethat causes a jump in sprung mass acceleration. This accelerationjump, or dynamic jerk, causes a significant reduction in isolationbenefits that can be offered by skyhook suspensions. Two alterna-tive implementations of skyhook control included in the studyoffer modifications to the formulation of conventional skyhookcontrol such that the damping-force jumps are eliminated. Onealternative policy introduced a continuous function that can beused in place of the conventional formulation of skyhook control,which is described as a bistate function in terms of the products ofthe absolute velocity of the sprung mass and the suspension rela-

Fig. 11 Comparison of electrical currents supplied to a mag-netorheological damper in a semiactive suspension in re-sponse to a 1.45 Hz pure-tone base excitation

Fig. 12 Sprung mass accelerations caused by different sky-hook control formulations in response to a broadband ISO2base excitation

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tive velocity. Another alternative policy maintains the bistate for-mulation of the conventional skyhook control, but introduces therelative velocity of the suspension in the force-switching formu-lation such that the force discontinuities are eliminated.

The two alternative methods were compared with rate skyhookcontrol, in terms of the relationship between the damper force andthe relative and absolute velocities of the system in order to showthe elimination of the force discontinuity at the zero crossings ofthe relative velocity across the suspension. Additionally, the alter-

References@1# Karnopp, D. C., and Cosby, M. J., 1974, System for Controlling the Trans-

mission of Energy Between Spaced Members, U.S. Patent 3,807,678.@2# Crosby, M. J., and Karnopp, D. C., 1973, The Active Damper, Shock Vib.

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@8# Chalasani, R. M., 1986, Ride Performance Potential of Active SuspensionSystemsPart I: Simplified Analysis Based on a Quarter-Car Model, Pro-ceedings of the 1986 ASME Winter Annual Meeting, Los Angeles, ASME, NewYork.

@9# Ahmadian, M., 1999, On the Isolation Properties of Semiactive Dampers, J.Vib. Control, 5~2!, pp. 217232.

@10# Krasnicki, E. J., 1980, Comparison of Analytical and Experimental Resultsfor a Semiactive Vibration Isolator, Shock Vib. Bull., 50.

@11# Miller, L. R., 1988, An approach to Semiactive Control of Multiple-Degree-of-Freedom Systems, Ph.D. thesis, North Carolina State University, Raleigh,NC.

@12# Ivers, D. E., and Miller, L. R., 1991, Semiactive Suspension Technology: AnEvolutionary View, ASME Advanced Automotive Technologies, DE-40, BookNo. H00719-1991, ASME, New York, pp. 327346.

Fig. 13 Comparison of electrical currents supplied to a mag-netorheological damper in a semiactive suspension in re-sponse to a broadband ISO2 base excitation

Downloaded Fnative policies were compared with conventional skyhook controlusing a laboratory setup that included a heavy-truck seat suspen-sion representing a base-excited single-dof system. An evaluationof the seat acceleration, as well as the electrical current suppliedto the magnetorheological damper that was used for this study,clearly showed the elimination of jerk with the alternative sky-hook controls.

584 Vol. 126, OCTOBER 2004rom: http://vibrationacoustics.asmedigitalcollection.asme.org/ on 04/28/2@13# Song, X., 1999, Design of Adaptive Vibration Control Systems With Appli-cation to Magneto-Rheological Dampers, Ph.D. Dissertation, Virginia Poly-technic Institute and State University, Blacksburg, VA.

@14# Reichert, B. A., 1997, Application of Magneto Rheological Dampers forVehicle Seat Suspensions, M.S. thesis, Virginia Polytechnic Institute andState University, Blacksburg, VA.

@15# International Standards Organization, 1994, Earth-Moving MachineryLaboratory Evaluation of Operator Seat Vibration, Standard 7096, ISO.

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