Energy transfer in semi-active suspension

C. Collette, A. PreumontU.L.B., Department of Mechanical Engineering and Robotics,50, av. F.D. Roosevelt, B-1050, Brussels, Belgiume-mail: ccollett@ulb.ac.be

AbstractThis paper investigates the effect of unintended high frequency excitations generated by the semi-activesky-hook control algorithm on the isolation properties of a car suspension. Using a quarter car model, theenergy transfer from low to high frequency is established. To decrease the effect of these excitations, anelastic mount of the semi-active actuator (using rubber bushing at both of its ends) is investigated. Theresults presented show that the rubber bushing can play a role of mechanical filter, improving the isolationperformances of the semi-active suspension at high frequency.

1 Introduction

A classical (passive) suspension is constituted of elements having constant stiffness and damping properties.The design of the suspension (i.e. the choice of the most suitable values of the stiffness and damping) isconcerned with (i) the car body acceleration for ride comfort, (ii) the tyre deflection for road holding and(iii) the suspension travel (rattle space) which must remain within fixed limits. An ideal suspension wouldminimize these three quantities for any road and operating condition, which is not achievable for a suspensionhaving constant stiffness and damping. Actually, there is a trade-off between ride comfort and road holding,or between the ride comfort and the suspension deflection [1, 2].

A remedy to relax the trade-off is the use of active suspension [1, 3]. In this case, the suspension can be eitherfully active, i.e. constituted of a wide band actuator capable of controlling the system in the full bandwidth,or passive on which is added such an actuator. Up to now, active suspensions are not widely used in theautomotive industry because of their high external energy demand and complex actuation systems [4].

An alternative solution is the use of semi-active suspensions [5, 6]. In this case, the stiffness and dampingproperties can be changed in real time according to a defined strategy, but cannot bring energy to the system.Requiring a minimum of effort, they are theoretically also able to reduce the car body resonance withoutdegrading the road holding. This attractive solution leads unfortunately to some disturbances like jerk,chatter or rattling noise [7]. These problems have three origins: (i) a poor knowledge of the state variablesof the system leading to some error of the command; (ii) an inability of the controlling device (servo valves)to follow the command, introducing a time delay; (iii) intrinsic to the semi-active strategy.

This paper focuses on the third origin. Actually, this case arise for instance when fast switches of thecommand generate high variations of the damper force, leading to an unpleasant rattle sound each time thevalve is opened or closed [8, 9]. The valve motion does not only cause sound problems but also the entirebodywork of the car starts to vibrate at a rather high frequency due to these abrupt control actions. A lot ofstrategies and technological solutions have been developed during the last two decades to avoid or alleviatethese problems [9, 4, 10], some of which are reviewed in this work.

The paper is organized as follows. Section two compares the performances of a passive, active and semi-active suspension using a single degree of freedom (d.o.f.) model. Section three considers the same set ofsuspensions in a two d.o.f. model, that takes into account the mass of the wheel and the stiffness of the

tyre. The effect of the unintended high frequency force excitations generated by the control algorithm on thesuspension properties is pointed out.

Section four reviews existing solutions to decrease the effect of these excitations, and investigates the effectof a resilient mounting of the semi-active actuator (using rubber bushing at both of its ends) as an alternativesolution. Section six draws the conclusions.

2 One d.o.f. suspension

2.1 Passive and active

Figure 1(a) shows the model of a single d.o.f. passive suspension. The variable is the vertical displacementof the sprung mass ms, representing one quarter of the car body mass; k and c are the stiffness and dampingcoefficient of the suspension, v is the vertical velocity of the excitation approximated by a white noise. In thismodel, the mass of the wheel and the stiffness of the tyre are neglected. Using the state variables x1 = xs−wand x2 = xs, the dynamics of the system reads

msx2 + c(x2 − v) + kx1 = 0

x1 = x2 − v

Figure 1: Three single d.o.f. isolators: (a) Passive; (b) Active; (c) Semi-active.

Figure 2 shows the magnitude of the transmissibility Txsv for two values of the damping coefficient: c =100 Ns/m and c = 1000 Ns/m; ms = 240 Kg and k = 16000 N/m. The reduction of the resonancepeak is obtained by increasing the value c, but at the cost of degradading of the isolation at high frequency.One way to improve it is to use a small value of c, and insert in the suspension an active element (Fig. 1(b)),whose force is chosen according to a defined strategy. A classical control force is

fc = −gxs (1)

where g is the gain. Here, the control law is called sky-hook, because it is identical to a viscous damper ofconstant g, connecting the mass and a fixed point in space (the sky).

Figure 2 shows the magnitude of the transmissibility Txsv for g = 1000 Ns/m and c = 100 Ns/m. Theactive isolator can reduce the resonance peak, while still providing a good isolation at high frequency.

2.2 Semi-active

The semi-active suspension (Fig. 1(c)) is made of a spring, in parallel with a damper whose coefficient canvary in a fixed range. Sensors measure the state of the system in real time, and the control unit can change

Figure 2: Transmissibility Txsv of three isolators: passive (c = 100 Ns/m and c = 1000 Ns/m), active(g = 1000 Ns/m and c = 100 Ns/m) and semi-active (Cmin = 100 Ns/m, Cmax = 1000 Ns/m andg = 1000 Ns/m).

the value of the damping coefficient in a range between two extremal values Cmin and Cmax, according toa defined strategy. As a non-linear device, the response of such a shock absorber depends on the excitationamplitude and on its frequency content, and it has the capability to transfer energy from one frequency toanother. Various classes of semi-active strategies are available in the literature [11, 12, 13, 14]. The mostoften used is the semi-active sky-hook damper that tries to emulate the active sky-hook damper describedin the previous section. Actually, the force in the damper is the product of its damping coefficient and therelative velocity of both ends:

fc = −c(u)(xs − v) (2)

where c(u) is the damping coefficient, xs the velocity of the sprung mass and v the velocity of the excitation.Comparing Eq.(2) and Eq.(1), the command that emulates the sky-hook damper is

c(u) =gxs

xs − v(≥ 0) (3)

When xs and xs − v have opposite sign, Eq.(3) leads to a negative value of c(u). In this case, the damperforce is fixed at its minimum value Cmin. Accounting that the maximum achievable coefficient is Cmax, thefinal control algorithm, often called SA1 in the literature, is

c(u) = max{Cmin, min[gxs

xs − v, Cmax]} (4)

Figure 2 shows the transmissibility Txv of the semi-active suspension using the strategy SA1; Cmin =100 Ns/m, Cmax = 1000 Ns/m and g = 1000 Ns/m. It has been obtained from a time domain simulationof 800 s, a time step of 0.003 s and a white noise road velocity. The figure shows that semi-active damperis able to produce a better isolation than the equivalent passive suspension. The resonant peak is attenuatedwithout damage to the high frequency isolation.

The next section compares the same set of passive, active and semi-active suspensions on a 2 d.o.f. model,that takes into account the mass of the wheel, as well as the tyre stiffness.

3 2 d.o.f. suspension

3.1 Passive

Figure 3(a) shows the two d.o.f. quarter-car model with a passive suspension. The variables are: the verticaldisplacement of the sprung mass ms (representing one quarter of the car body mass), and the vertical dis-placement of the unsprung mass mus (representing the wheel); kt is the stiffness of the tyre, k and c are thestiffness and damping coefficient of the suspension.

Figure 3: Quarter-car model: (a) passive suspension; (b) active suspension; (c) semi-active suspension.

Using the set of state variables described in Fig. 3, the dynamics of the system read

msx2 = −kx1 + c(x4 − x2)

musx4 = −ktx3 + kx1 + c(x2 − x4)

x1 = x2 − x4

x3 = x4 − v

where x2 represents the ride comfort, x3 the road holding, x1 the rattle space and v the input groundvelocity, approximated as a white noise. The following numerical values have been used in this study:ms = 240 Kg;mus = 36 Kg; kt = 160000 N/m; k = 16000 N/m; c = 1000 Ns/m. A convenientquantity to compare the different designs is the RMS value of the relevant quantities; the RMS value of avariable q is defined as

σq = {∫ ∞

0Φq(ω)dω}1/2 (5)

where ω is the frequency and Φq(ω) the power spectral density of q. Figure 4(a) shows σxs as a functionof σx3 when the damping coefficient of the suspension varies from c = 200 Ns/m to c = 5000 Ns/m;Figure 4(b) shows σxs as a function of σx1 when the damping coefficient of the suspension varies fromc = 200 Ns/m to c = 5000 Ns/m.

Low values of σx3 and σx1 are obtained by increasing the damping coefficient, at the cost of an increaseof σxs . In other words, there exist a trade-off between the comfort and the road holding, and between thecomfort and the suspension deflection. A remedy to relax the trade-offs is the use of active suspension [1, 3]discussed in the next section.

Figure 4: (a) Trade-off curves between RMS sprung mass acceleration σxs (representing the comfort) andRMS tyre deflection σx3 (representing the road holding): (b) Trade-off between RMS sprung mass acceler-ation σxs and RMS suspension deflection σx1 . Solid lines correspond to a passive suspension for which thevalue of c increases from 200 Ns/m to 5000 Ns/m; dashed lines correspond to an active suspension forwhich the value of the gain g increases from 0 Ns/m to 2000 Ns/m.

3.2 Active

The active suspension can be either fully active, i.e. constituted of a wide band actuator capable of controllingthe system in the full bandwidth, or a passive suspension supplemented with an ideal force actuator andcontrolled with a partial state feedback (e.g. sky-hook damper), like depicted in the Fig. 3(b). The dynamicsof the system read

msx2 = f − kx1 + c(x4 − x2)

musx4 = −f − ktx3 + kx1 + c(x2 − x4)

x1 = x2 − x4

x3 = x4 − v

f = −gx2

where f is the force of the actuator. Using this last model, Fig. 4(a) shows, for each value of the dampingcoefficient considered, the evolution σxs as a function of σx3 when the gain of the actuator increases fromg = 0 Ns/m to g = 2000 Ns/m. Similarly, Fig. 4(b) shows the evolution of σxs as a function of σx1 whenthe gain of the actuator increases from g = 0 Ns/m to g = 2000 Ns/m. One sees that, whatever the valueof the damping coefficient, an increase of the gain always increase the comfort. However, very high valuesof the gain can sometimes slightly degrades the suspension deflection or the road holding (Fig. 4(a) and (b)).

3.3 Semi-active

The two d.o.f. model of the semi-active suspension is illustrated in Fig. 3(c). The dynamics of the systemis governed by the same set of equations as the passive one, except that the damper coefficient can varyaccording to the sky-hook strategy

c(u) = max{Cmin, min[gxs

xs − xus, Cmax]} (6)

Time domain simulations have been conducted with the same numerical data as the passive suspension. Theshock absorber constant is supposed to vary between Cmin = 100 Ns/m and Cmax = 2000 Ns/m. Theroad velocity v is assumed to be a white noise. Figure 5 shows various time histories of the quarter-carresponse, respectively the tyre force ktx3, the body velocity xs = x2, the relative velocity between thebody and the wheel x1 = xs − xus, the required control force f = −gxs and the actual control forcefc = −c(u)(xs − xus), and the damper constant c(u) .

Figure 5: Time history of a quarter-car model with a semi-active suspension using algorithm SA1: (a) Tyreforce ktx3; (b) Body velocity xs = x2; (c) Relative velocity between the body and the wheel x1 = xs− xus;(d) Required force f = −gxs and control force fc = −c(u)(xs − xus); (e) Damper constant c(u).

Figure 5 shows that each switch of the command creates a sudden change in force and acceleration of thesuspended mass. Any sudden change of the seat acceleration is perceived by the person in the seat as ajerk. Figure 6 (a) compares the transmissibility between the road velocity and the body acceleration, Txsv

of, respectively the passive suspension (c = 200 Ns/m), the sky-hook control (g = 2000 Ns/m) and thesemi-active sky-hook with Cmin = 100 Ns/m and Cmax = 2000 Ns/m. The transmissibility has beenobtained from a time simulation of 200 s, with a time step of 0.0005 s and a white noise road velocity.

The semi-active control is successful in reducing the body resonance, and the transmissibility of the bodyacceleration is comparable to that of the active control up to the wheel resonance ωt = 70 rad/s. Beyondthat, however, the transmissibility rolls off much slower than in the active case, and one observes peaks atvarious harmonics of the wheel resonance, which are likely to excite flexible modes of the vehicle if nothingis done to attenuate them. Figure 6(b) shows the transmissibility between the road velocity and the tyredeflection Tx3v. No spurious high frequency components are observed and the road holding properties of the

Figure 6: Quarter car model with SA1 algorithm. (a) Transmissibility Txsv between the sprung mass accel-eration and the road velocity: Passive suspension (c = 200 Ns/m), active suspension (g = 2000 Ns/m),semi-active sky-hook with Cmin = 100 Ns/m and Cmax = 2000 Ns/m. (b) Transmissibility Tx3v betweenthe road velocity and the tyre deflection.

semi-active suspension seem to match closely to the active suspension.

Due to the abrupt changes of the damping coefficient of the suspension when the sign of the relative velocityof the sprung and unsprung mass changes imposed by the SA1, harmonics of the second resonance of thesuspension are visible on the sprung mass acceleration spectrum (Fig. 6(a)). The frequency of variouspeaks appearing in the transmissibility Txsv is indicated. All components are at harmonics of the unsprungmode (70 rad/s) except for the second harmonic. The two peaks at 132 rad/s and 148 rad/s seem toresult from the beat of the second harmonic of the wheel at 2ωt = 140 rad/s with the body mode atωn =

√k/ms = 8 rad/s. The other peaks are odd harmonics of the second resonance, and the Fourier

coefficients come directly from the decomposition of a square function f(t) with a period T into a sum ofsine functions

f(t) =∞∑

n=0

Cnsin(2π

T(2n + 1)t) (7)

where Cn are constant coefficients. Figure 7 shows the frequency spectrum of the sprung mass accelerationunder a band limited random noise. Above the frequency range of excitation, no significant fall is visible,neither on the spectral density of the sprung mass acceleration, nor on the spectral density of the damperforce (Fig. 7). This means that the high frequency content of the spectrum comes from an excitation of thesystem at low frequencies, which means an energy transfer from low frequency to high frequency throughthe semi-active device.

Figure 7: Power spectral density of (a) the road velocity Φvv, (b) the sprung mass acceleration Φxsxs and (c)the damping force ΦFcFc for the semi-active suspension under a low frequency random excitation.

Further evidence of the non-linear energy transfer between low and high frequency can be obtained from thecoherence function between the road velocity v(t) and the sprung mass acceleration xs(t) defined by

γ2vxs

(ω) =|Φvxs(ω)|2

Φvv(ω)Φxsxs(ω)(8)

where Φvv(ω) and Φxsxs(ω) are the power spectral densities of the road roughness velocity and the sprungmass acceleration, and Φvxs(ω) is the cross power spectral density (Fig. 8(a)). γ2

vxsis equal to 1 for a perfect

linear system; it measures the causality of the signal at every frequency. According to Fig. 8, the coherenceis very good up to the tyre mode, and falls rapidly to zero above 100 rad/s, which indicates that at thosefrequencies, the energy content of the body acceleration is not due to the road profile. The existing solutionsto mitigate the jerk inherent to the algorithm SA1 are to modify the control algorithm, change the damperparameters or filter the control input. These are explained in the next section.

4 Anti-jerk Solutions

4.1 Modification of the SA1 algorithm

Using the algorithm SA1, it has been pointed out in the previous section that the damping force is a discon-tinuous function of xs and xs − xus [5]. All of the discontinuities cause sudden variations that may lead tojerk. In order to reduce them, the control surface fc(xs, xs− xus) can be replaced by a continuous one, usinga shaping function [15, 7].

Another possible modification of the algorithm is to switch the valves only in those cases where the relativevelocity between sprung and unsprung mass is zero [16]. In these instances, the suspension (spring + damper)force remains unchanged independent of the damper settings because no damper force can be generated [8].

Figure 8: Quarter car model with SA1 algorithm. Coherence function γ2vxs

between the road velocity v(t)and the body acceleration xs(t).

4.2 Change the damper parameters

The range of variation of the damper constant has also an impact on the isolation properties of the suspension.High differences between Cmin and Cmax generate steps in the command and, as a consequence, highfrequency components in the damper force. As the value of Cmax increases, the saturation decreases. Asa consequence, it increases slightly the damping of the first resonance, and increases the excitations at highfrequencies. If the value of Cmin increases, the curvature of the damping force during the off period alsoincreases. As a consequence, the switches between on and off show less sharp angles in the time history ofthe damper force. It decreases also the excitation of high frequencies and the semi-active damper becomesmore and more passive.

4.3 Filter the control input

The introduction of a filter has an effect of smoothing the control force [4]. Figure 9 shows the effect of afirst order filter 1/(1 + τs) for τ = 0.01sec. and τ = 0.001sec. As long as the bandwidth of the filter (1/τ )is greater than the frequency of the second resonance (

√kt/mus), the higher the value of τ , the smoother the

command, and the better the isolation of the sprung mass. The transmissibilities (Fig.9) are compared withthe case of the passive suspension (c = 100 Ns/m) and the active suspension (c = 100 Ns/m and g =1000 Ns/m). Whatever the constant of the filter, the level of the transmissibility at high frequency remainshigher for the semi-active suspension than for the active (and passive for c = 100 Ns/m) suspension.

In practice, the time response of hydraulically powered digital servo valves lies in the range of 10 ms [8] andsometimes a little more [17]. As an alternative to an analog filter, the next section investigates if an elasticmount of the damper, using rubber bushing at both ends, could play a role of mechanical filter to smooth thecontrol force.

5 Effect of the rubber bushing compliance

In order to point out the effect of the bushing on the isolation properties of the suspension, it is first studiedon the passive suspension in the next section.

5.1 Passive suspension

The bushing is modelled as a spring (Fig. 10, [10]). In this case, the dynamics of the system reads

msx2 = −kx1 − k1x5

Figure 9: Effect of the Time delay on the suspension dynamics (g = 2000 Ns/m; Cmin =100 Ns/m; Cmax = 2000 Ns/m): (a) Detail of the damper force fc over a short period of time andcorresponding value of the damping coefficient; (b) Transmissibility Txsv between the road velocity andthe sprung mass acceleration. The transmissibilities are compared with the case of the passive suspension(c = 100Ns/m) and the active suspension (c = 100 Ns/m and g = 1000 Ns/m).

Figure 10: State variables of the rubber bushing model.

musx4 = kx1 + k1x5 − ktx3

x1 = x2 − x4

x3 = x4 − v

x5 = x2 − x4 − k1

cx5

and upon defining ω2n = k/ms, µ = mus/ms and ω2

t = kt/mus they are rewritten in matrix form

x1

x2

x3

x4

x5

=

0 1 0 −1 0−ω2

n 0 0 0 −ω2nκ

0 0 0 1 0ω2

n/µ 0 −ω2t 0 κω2

n/µ0 1 0 −1 −k1/c

+

00−v00

(9)

Figure 11(a) and (b) show the effect of the bushing stiffness on the transmissibility Txsxus between the wheelvelocity and the car body velocity, and the transmissibility Txsv between the road velocity and the sprungmass acceleration for c = 2000 Ns/m; the lower the value of k1, the better the high frequency isolation.Figure 12 shows the transmissibility Txsv of the suspension for different values of the damping coefficient.For a given value of the bushing stiffness, it shows that the value of the damper coefficient does not affectthe isolation at high frequencies.

5.2 Semi-active suspension

This section investigates the effect of the bushing stiffness on the semi-active suspension using the algorithmSA1. The set of equations is the same as in the previous section, but the damping coefficient varies asdefined by Eq.(4). Figure 13 shows the effect k1 on the suspension transmissibility Txsxus and the sprungmass acceleration transmissibility Txsv for two values of Cmin. Whatever the value of Cmax, a decrease ofthe bushing stiffness improves the isolation at high frequency. A low value of Cmin generates high damperforce variations each time the command switches and the peaks corresponding to harmonics of the secondresonance are more sharp (Fig. 13, also section 4.2). In this case, the system is highly non-linear and theeffect of the bushing on the isolation is low. When the value of Cmin is high (Fig. 13(b)), the suspensionis more passive and of the rubber bushing improves significantly the isolation, acting as a filter at highfrequency. Conversely, a high value of Cmin generates low variations of the damping force, the suspension

Figure 11: (a) Transmissibility Txsxus between the wheel velocity and the car body velocity; (b) Transmissi-bility Txsv between the road velocity and the sprung mass acceleration for c = 2000 Ns/m. The lower thevalue of k1, the better the isolation at high frequencies.

Figure 12: Transmissibility Txsv between the body acceleration and the road velocity for various values ofthe damper constant; k1 = 160 kN/m. For all value of the damping coefficient, the slope of the roll off isfixed by the value of k1.

Figure 13: Effect k1 on the suspension transmissibility Txsxus and Txsv for (a) Cmin = 100Ns/m; (b)Cmin = 500Ns/m (Cmax = 2000Ns/m). The lower the value of k1, the better the isolation.

is less non-linear, and behaves more like the passive suspension discussed in section 5.1. For a given valueof k1, the bushing provides a better isolation for high values of Cmin. This tends to say that the value ofk1 should be as low as possible. However, when the value k1 becomes too low, the relaxation spring has aneffect of increasing the level of the transmissibility Txsv for frequencies below the second resonance (Figs.12 and 13). In order to choose the most suited value of k1, Fig. 14 compares the effect of the filter andthe bushing on the mean square value of the sprung mass acceleration σ(xs). Figure 14(a) shows the meansquare value of the sprung mass acceleration σ(xs) when the command passes through a first order low passfilter as a function of the delay τ . As the value of τ increases, the mean square value of the sprung massacceleration decreases. For large values of τ , it increases again. In this case, the optimal delay giving thelowest mean square of the sprung mass acceleration is τ = 0.005 s. Actually, such a criteria for the design ofthe filter is unrealistic because the optimal value is lower than the time response of the system (section 4.3).In the case of a damper with elastic mounts, Fig. 14(b) shows that an increase of the bushing compliance(= 1/k1) increases the mean square of the sprung mass acceleration, for all value of the bush compliance.

In summary, both the filter and the bushing improve the quality of the isolation at high frequency, but bothof them also increase the mean square value of the sprung mass acceleration. However, the slope of thisincrease remains very low as long as the corner frequency of the filter remains above the frequency of thesecond resonance of the quarter car model.

Figure 14: (a) Mean square value of the sprung mass acceleration σ(xs) when the command passes througha first order low pass filter as a function of the delay τ (the damper is rigidly mounted); (b) Mean squarevalue of the sprung mass acceleration σ(xs) as a function of the bushing compliance (= 1/k1).

6 Conclusion

Through a simple model of a quarter-car, the isolation properties of a semi-active suspension has been stud-ied using the SA1 sky-hook like algorithm. It has been shown that the isolation provided by the suspensionis degraded by the non-linearity of the semi-active algorithm, which tends to generate high frequency com-ponents at the harmonics of the unsprung mass resonance. These high frequency vibrations may potentiallygenerate noise through the excitation of flexible modes of the vehicle. This energy transfer phenomenon maybe alleviated to some extent by filtering the control signal or adding a resilient mounting to the semi-active

damper. Both the filter and the bushing have been examined; they improve the quality of the high frequencyisolation without deteriorating the behavior below the unsprung mass resonance, as long as the corner fre-quency of the filter is higher than the frequency of the second resonance of the suspension. However, noneof these approaches seem to reduce the RMS sprung mass acceleration.

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