Vibration analysis for the rotationalmagnetorheological damper

Yousef IskandaraniDepartment of Engineering

University of AgderGrimstad, Norway

Email: yousef.iskandarani@uia.no

Hamid Reza KarimiDepartment of Engineering

University of AgderGrimstad, Norway

Email: Hamid.r.karimi@uia.no

Abstract—Comfort, reliability, functionality performancewhich provide a longer life cycle requires thourogh understandingand analysis of the vibrations, this is a general rule for most ofthe static and dynamic functionality applications. Vibrations isan extremely important issue to consider when designing varioussystems.

The hysteresis in the dampers is very important issue whencharacterizing a damper, it is a very complex phenomena butvery important to consider.The hysteresis equations of Bouc-Wen, Lugre, and Dahl have been modeled and simulated inMatlab/Simulink. We have manipulated the different parametersin the models and analyzed their effects on the outcome. Thehysteresis models of Bouc-Wen, Dahl and LuGre have beenanalyzed and compared analytically to really show the differencein the models. At last the Bouc-Wen model was implementedtogether with the SAS(Semi Active Suspension) system in thelaboratory. The model parameters were tuned manually to tryto fit the response of the system.

In this paper a predefined methodology has been applied fordetermining the hysteresis loop parameters using the data col-lected for vibration analysis under predefined test specifications.The following data has been used later to regenerate the vibrationsignal, so on get as closer to the real signal.

In the coming work, advanced method will be used to deter-mine the exact parameters for the hysteresis loop as well as usingthe inverse hysteresis to improve the performace of the vibrationsuspension in the Semi Active Suspension system.

The behavior of MR dampers can be presented with differentmathematical models. The Bouc-Wen model was found to bemodel to both illustrate the MR damper and recreate thebehavior of the SAS system.

Keywords: Magnetorheological damper, Bouc-Wen, Lugre,Dahl, SemiActive Suspension

I. VIBRATION ANALYSIS

The hysteresis identification can be a complex process, un-derstanding the static and the dynamic vibration in the systemenable facilitating the process of the hysteresis identification,In this work different vibration responses of the MR damper.When using fluid dampers the exerted force, or torque, willrespond differently to vibrations with respect to the system’snatural frequency and the exerted vibration on the damper.We will discuss the static response compared to the dynamicresponse using the models discussed in III.

A. Static vibration

Static vibration on the MR damper is interpreted as whenthe damper is working without the mechanical dynamics ofthe system. The damper’s exerted force, or torque, is thena function fully described by the current and velocity. Toillustrate the effects of the vibration, the Matlab Simulinkmodels are used to simulate the maximum torque of thedamper with respect to frequency when the displacement isforced in a sine motion. The result using the Bouc Wen modelis shown in Figure 1

Fig. 1. Vibration response with respect to frequency using the Bouc Wenmodel. Displacement is forced in a sine motion creating a frequency dependenttorque.

When using a dynamic model, here Bouc Wen, someonecan observe high torque even at low frequencies.

B. Dynamic vibration

Considering the dynamics in a system, it is important tohave knowledge of its vibrational response. The system willhave a natural frequency and applying a harmonic excitationnear the system’s natural frequency can create a highly unsta-ble system. We use the semi-active damper to control and mutethese critical vibrations. To analyze, the mechanical system isbuilt up using

T = Iθ = −(ksθ + Tmr θ + Tin

)(1)

The model is shown in Figure 2 with fictitious values. Thesystem has a natural frequency of

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Fig. 2. The physical system with the MR damper

ω0 =√ks/m (2)

To make a resonance plot, it is required to give the inputtorque in Figure 2 a frequency, measure the maximum dis-placement and redo the process with a new frequency. Theresult is shown in Figure 3.

Fig. 3. Vibration analysis of the Dynamic system using the Bouc Wen modelwith current [0, 1

3, 23, 1]A

Note how the peak is shifting towards higher frequencyratios as the current is increased. For frequency ratios higherthan 2.2, the best result may come with lower or no currentapplied.

In Figure 4 the actual result retrieved from the SAS systemis shown. The frequency increases linearly from low (wheelslowly turning) to high. At around 30 seconds, the systemis near the its natural frequency (∼ 1.6Hz) resulting inmaximum displacement change. Passing this critical frequencywill again calm the system.

Fig. 4. Vibration Response of SAS system with no current applied

II. TYPES OF FRICTION

The MR damper can be modeled with different types ofmathematical models. Each model describes different aspectsof friction and/or dynamic properties of the MR damper. Inthis section, some friction aspects together with the modelswill be later simulated, compared and analyzed.

A. Hysteresis

Hysteresis is a dynamic friction phenomena whichrepresents the history dependence of physical systems.[1]We get a model for the systems nonlinear behavior at lowvelocities. Figure 5 show an example of hysteresis. Theforce versus velocity curve is not coincide for increasing anddecreasing velocities.

Fig. 5. Example of hysteric behavior[2]

B. Coloumb friction

Fig. 6. Columb friction model[3]

The Columb friction model represents the static relationshipbetween friction forces and velocities. This model cannotreproduce friction characteristics that depend on time.[4]

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Fig. 7. Viscous friction[3]

C. Viscous frictionViscous fricton is static and lineary dependent to the sliding

velocity.

D. StictionIf someone combine static, Coulomb and the viscous friction

model, someone will get a model of stiction. A stiction modelincludes the threshold force that is needed to start a movementbetween two bodies, the constant Coulomb friction and thevelocity dependent viscous friction.

E. Viscous friction

Fig. 8. Viscous friction[3]

Viscous friction is static and linearly dependent to thesliding velocity.

F. StictionIf someone combine static, Coulomb and the viscous friction

model, someone will get a model of stiction. A stiction modelincludes the threshold force that is needed to start a movementbetween two bodies, the constant Coulomb friction and thevelocity dependent viscous friction.

G. Stribeck effectThe Stribeck effect appears when the friction is decreasing

and later is increasing with increasing velocities starting atzero velocity. This effect is seen in for example bearings. Theoil thickness is building up in the beginning, causing the dropin the friction force.

Fig. 9. Stribeck effect[3]

H. Stick-slip motion

Fig. 10. Model of stick slip motion[5]

If a mass m is pulled with a spring at a constant speed vpalong a surface; the mass will have a periodic motion wherethe mass varies between sticks and slips.

I. Zero-slip motion

m · a = Fd − F (3)

The applied force Fd is smaller than the stiction force.This occurs when a masses is dragged along a surface with alow constant velocity while the friction keeps the mass fromslipping.

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Fig. 11. Zero-slip motion[5]

III. MODELS

A. The Bingham model

Fig. 12. Bingham mechanical model[2]

Fmr = Fc · sign(x) + c0x+ F0 (4)

x : Piston velocityFc : Frictional forcec0 : Damping constantF0 : Offset value (Constant force)

Fig. 13. Response of the Bingham model

From equation 4 it can assumed that the shape of theBingham force profile will be equal to the coloumb plusviscous friction as seen in Figure 13. The friction force Fc

equals the starting point of the graph on the y-axis. The

damping constant c0 equals the linear relationship ∆F∆x which

appears when the velocity x is not zero.The Bingham model is clearly linear and since the MR

damper is highly nonlinear, this model will not be an areaof focus and detailed study.

B. Dahl friction model

The Dahl’s-model was made for the purpose of simulatingwith friction. It’s a simple dynamic model which capturesmany properties such as the hysteresis and zero slip displace-ment, but does not capture the Stribeck effect or stiction[6]. The friction represented by this model depends onlyon displacement. This relatively simple model has a typicalhysteresis curve as shown in Figure 14

dF

dx= σ0

(1− F

Fcsign(v)

)(5)

σ0 : StiffnessFc : Columb frictionv : Velocity

Fig. 14. Basic Dahl hysteresis model

Figure 14 shows the force versus displacement hysteresisof the basic Dahl model. We see that the graph is differentwhen the displacement is increasing compared to when itsdecreasing.

The Dahl model has been modified to show some of theproperties of the MR damper[7]:

fmr = kx · x+ [kwa + kwb · v]w (6)w = |ρ|(x− |x|w) (7)

x : Velocityv : Control voltagew : Hysteric variablekx, kwa, kwb and ρ : Parameters

C. LuGre friction model

dz

dt= v − σ0 ·

|v|g(v)

· z = v − h(v) · z (8)

F = σ0 · z + σ1 · z + f(v) (9)

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For constant velocity, the steady state force Fss is [4]:

Fss = g(v)sgn(v) + f(v) (10)

g(v) = Fc + (Fs + Fc)−(v/v0)2 (11)

v : Velocity between the surfaces in contactz : Internal friction state (average bristle deflection)f(v) : The viscous friction (velocity dependent)g(v) : Approximation of the Stribeck effect and Coloumb frictionFs : Stiction force (static friction)Fc : Columb frictionσ0 : Stiffnessσ1 : Microdampingvs, α : Constants

The LuGre model is an extension of the Dahl model.[4] vsis a constant which determine how quickly g(v)approachesthe friction force Fc. The Lugre model is very advanced andcaptures many aspects of friction.

D. The Bouc-Wen model

Fig. 15. Bouc-Wen mechanical model[2]

fmr = c0 · x+ k0(x− x0) + α · z (12)z = −ϕ · |x| · z · |z|n−1 − β · x · |z|n + κ · x (13)

c0 : Damping constantx : Pistion velocityk0 : Spring constantα, β, ϕ, κ : Parametersz : Bouc-Wen hysteresis constant

The Bouc-Wen model is the most commonly used model tocapture the behavior of non-linear hysteric systems such as theMR damper.[8] This model can produce a variety of hystericpatterns.[2]

IV. TESTING HYSTERESIS-MODELS ON THE SAS SYSTEM

In this section, the zooming and investigating if the Dahland the bouc-wen can fit the MR damper in the suspensionsystem in the lab. We have not been focusing on the Lugremodel because this is just an extension to the Dahl model andmuch harder to tune because of the many parameters. The SASsystem is simplified and modeled in matlab/simulink.

Fig. 16. Model of system

V. SAS MODEL

Since a static vibration analysis with no motion of thewheel, in this situation, it is assumed the wheel to be stiffand with no damping.

The spring represents the physical spring on the model. Thedamper represents the constant damping in the MR damperplus other external things as friction and so on. The last partis representing the force from the nonlinear damping in theMR damper. θ1 and θ(θ = θ2 − θ1) is measured trough aangle sensor on the model and plotted on the computer.

VI. EQUATIONS FOR SYSTEM WITH BOUC-WEN ANDDAHL

Model-equations with Bouc-Wen:

T = Jθ = −kθ − c(i)θ − α(i)z (14)

J is redundant:

θ = −k1θ − c(i)θ − α(i)z (15)

α(i) and c(i) are defined in the following:

c(i) = c1 + c2 · i (16)α(i) = α1 + α2 · i (17)

The Bouc-Wen non linearity is a function of z.Z can be calculated from the differential equation:

z = −γ|θ|z|z|n − βθ|z|n + δθ (18)

This equation was modeled in matlab/simulink and thenmerged together with the rest of the system.

Fig. 17. Total system with Bouc-Wen

On the left side, the bouc-wen model(green system) forcalculating det parameter z. The yellow system is the con-tribution from the active damping, dependent on the current.

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The white blocks illustrates the physical system with a linearspring and damper. The purple block multiplied by z is thepassive nonlinear damping.

Model-equations with Dahl:

T = Jθ = −kθ − kx · θ + (kwa + kwb · v) · w (19)

J is redundant:

θ = −kθ − kx · θ + (kwa + kwb · v) · w (20)w = ρ · (θ − |θ|w) (21)

Fig. 18. Dahl and SAS system in matlab/simulink

The orange blocks represents the dahl-model for MRdamper and the white blocks represents the SAS-model.

VII. EXPERIMENTAL TESTING

The experimental tests done using the SAS test setup asshown in figure(19) when pressing the ”arm” down to an initialθ(0) = θ0 = −5deg and then drop. The arm would thenoscillate around equilibrium.

Fig. 19. The mechatronics setup for the SAS

The results with 0A and 1A on the MR damper gave usthese curves:

First, trying to fit the dahl and bouc-wen into the passivesystem with no current.

Bouc-Wen(passive) Dahl(passive)n = 0.099 ρ = 10γ = 1.0 kwa = 0.6β = 873 k1 = 109δ = 700 c1 = 0.7c1 = 0.7 kx = 0.15α1 = 1k1 = 109

Fig. 20. Results with 1 and 0 amp.

Fig. 21. Left: Bouc-Wen. Right: Dahl

After trying to fit the model by changing the parameters forthe hysteresis models, it is shown that the Bouc-Wen actuallyis a better fit than Dahl. Fitting the Dahl model was moredifficult. The problem was that the nonlinear damping leads toa varying frequency, which the Dahl has a bigger problem withthan Bouc-Wen. The Bouc-Wen is more adjustable because ofthe nonlinear terms with the exponential n-parameter.

Next, the same test with 1 amp into the MR damper isperformed.

Fig. 22. Top: Bouc-Wen. Bottom: Dahl

Bouc-Wen(active) Dahl(active)c2 = 1.5 kwb = 2.8α2 = 1

When looking at the Figure(22), it is shown that the Bouc-Wen model has a much better fit than the Dahl.

Seen from Figures(23) and (24) the hysteresis of Dahl is abit larger than the Bouc-Wen hysteresis(Figure(23)). When thecurrent is set to one, the Changes are increasing dramatically.The Bouc When which is the most correct model for this MR

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Fig. 23. comparing hysteresis with 0A (Bouc-Wen:[green],Dahl:[blue])

Fig. 24. comparing hysteresis with 1A (Bouc-Wen:[green],Dahl:[blue])

damper has increased to about two times of the Dahl model.This is mainly because the linear damping in the used Dahlmodel, does not have a linear damping coefficient proportionalto the current such as our Bouc-Wen.

VIII. RESULTS

These experiments show us that Bouc-Wen is more adapt-able to our system. Even though the Dahl hysteresis is quitesimilar to the Bouc-Wen, it could not fit the system as good.With no current on the damper the differences were quitesmall, and could probably been even smaller if someone hadused a specific scientific method to estimate the parameters.When the current was set to one, the Dahl model could notmanage to increase the linear slope of the hysteresis. Thisproblem could easily be solved by adding a term for the lineardamping dependent on current. Like, ki · i

A source of error in addition to inaccurate parameters couldbe that the neglect of the tire damping makes our simplifiedmodel too inaccurate.

IX. CONCLUSION

The determined parameters using different models hasbe utilized to determine the dynamic hysteresis loop, theprocess was done using different models which simulate thehysteresis implemented in a SAS system.

• The methodology of identifying the hysteresis parametershas been implemented using two different models : Dahland Bouc-Wen

• The Semi Active Suspension system has proven to beefficient for studying the rotational MR damper behavior,thus including the vibration and hysteresis analysis.

• The identified parameters which has been harvested fromthe passive and active vibration data has been used todetermine the hysteresis dynamics in different models

• The Dahl, Lugre and Bouc-Wen have all an adaptablehysteresis, but the Bouc-Wen is found to be the bestmodel for illustrating the MR damper. The Dahl is alsopretty good in the passive part but it needs to be modifiedto fit the active properties. This may be done by addinga voltage dependent damping coefficient similar to theBouc-Wen and Lugre models.

ACKNOWLEDGMENT

The authors would like to thank the engineering staff atINTECO Sp., Krakow, Poland for supplying us with theadequate Semi Active Suspension system which enabled us toarchive the goal of studying and identifying the static, dynamicvibration and hysteresis. Moreover, special thanks to AndrzejTurnau for supplying us with valuable knowledge from thearea of semi active damping, his dedication to this work ishighly appreciated.

REFERENCES

[1] J. Sethna, “What’s hystereis,” 1994, cornell University, Loboratory ofAtomic and Solid State Physics.

[2] M. F. Z. D. la Hoz, “Semiactive control strategies for vibration mitigationin adaptronic systems equipped with magnetorheological dampers,” Ph.D.dissertation, 2009.

[3] “Static and dynamic phenomena,” http://www.20sim.com/webhelp/modeling tutorial/friction/staticdynamicphenomena.htm,.

[4] C. C.-d.-W. K.J. Aastrom, “Revisiting the lugre model; stick-slip motionand rate dependence,” IEEE Control Systems Magazine 28, 2008.

[5] H. R. Karimi, “Lecture 15, ma417-friction,” 2009, university of Agder,Dept of Engineering.

[6] C. C. d. W. M. G. H. Olsson, K.J. strom and P. Lischinsky, “Frictionmodels and friction compensation,” pp. 10–11, 1997.

[7] J. R. D. W. S. N. N. Aguirre, F. Ikhouane, “Viscous + dahl model for mrdampers characterization: A real time hybrid test (rtht) validation,” 14theuropean conference on earthquake engineering, 2010.

[8] V. K. A.E Charalampakis, “A bouc-wen model compatible with plasticitypostulates,” ScienceDirect - Journal of Sound and Vibration, 2008.

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