Proceedings of the 42nd IEEE Conference on Decision and Control Maui, Hawii USA, December 2003 WeM08-3

Performance Benefits in Passive Vehicle Suspensions Employing Inerters

Malcolm C. Smith and Fn-Cheng Wang

Abstract-A new ideal mechanical nne-port network ele- ment named the inerter was recently introduced, and shown to be realisable, with the property that the applied force is proportional to the relative acceleration across the element. This paper makes a comparative study of several simple passive sus- pension strue, each containing at most one damper and inerter as a preliminary investigation into the potential performance advantages of the element. Improved performance for several different measures in a quarter-car model is demonstrated here in comparison with a conventional passive suspension S t r u t A study of a full-car model is also undertaken where performance improvements are also shown in comparison to conventional passive suspension struts. A protntgpe inerter has been built and tested. Experimental results are presented which demonstrate a characteristic phase ad\,ance property which cannot he achiwed with conventional passive struts consisting of springs and dampers only.

I . , INTRODUCTION

In [7] an alternative to the traditional electrical- mechanical analogies was proposed in the'context of syn- thesis of passive mechanical networks. Specifically, a new two-terminal element called the inerter was introduced, as a substitute for the mass element, with the property that the force across the element is proportional io the relative acceleration between the terminals. It was argued in [7] that such an element is necessary for the synthesis of the full class of physically realisable passive mechanical impedances. In- deed, the traditional suspension strut employing springs and dampers only and avoiding the mass element has dynamic characteristics which are greatly limited in comparison. The consequence is that, potentially, there is scope to improve the vehicle dynamics of a passively suspended vehicle by using suspension struts employing inerters as well as springs and dampers. It is the purpose of the present paper to give more detailed consideration to these possible performance benefits using some standard performance measures for quarter-car and full-car vehicle models. In addition, some experimental test results on a prototype inerter will be reporied.

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11. BACKGROUND ON THE INERTER The force-current analogy. between mechanical. and elec-

force ft cuiient velocity ct voltage

mechanical ground ct 'electrical ground

trical 'networks has the following correspondences:

spring U inductor damper ct resistor.

This work was rupponed in pan by the EPSRC. Tel. t 4 4 1223 332745, Email: mcs@eng.cam.ae.uk. Depanment of En-

Tel. +886 2 23630231 ex1 4079. Emaii: fcw@ntu.edu.tw. Depanment of ginccnnp, Univerrily of Cambndp, Cambridge CB2 1PZ U.K.

Mechanical Enpineering. National Taiwan University. Taipei. Taiwan

0-7803-7924-1/03/$17.00 02003 IEEE 2258

rack pinions

I I gear ayw'heel terminal 1

i temunal 2

Fig. I. Schematic of a mechanical model of an inener.

Additionally the mass element has always been taken as the analogue of the capacitor, even though it has been appreciated (6, p. 1111, [2, p. 10-51 that the mass is strictly analogous only to a capacitor with one terminal connected to ground. This is due to the fact that Newton's Second Law refers the acceleration of the mass to a fixed point in an initial frame, i.e. mechanical ground. The restrictive nature of the mass element in networks has the disadvantage that electrical circuits with ungrounded capacitors do not have a direct spring-mass-damper analogue. This imposes a restriction on the class of passive mechanical impedances which can be physically realised. A further problem is that the suspension strut needs to have small mass compared to that of the vehicle body and wheel hub, which itself imposes further restrictions on the class of mechanical impedances which may be practically realised using the classical spring-mass- damper analogue.

To remedy the situation a network element called the inerter was introduced in [7] with the following definition. The (ideal) inener is a two-terminal mechanical device with the propeny that the equal and opposite force F applied at the terminals is proportional to the relative acceleration between the nodes, i.e. F = b(C2 - G I ) where q, u2 are the velocities of the two terminals and b is a constant of proportionality called the inerfance which has units of kilograms. The stored energy in the inerter is equal to $ b ( v 2 - v 1 ) ~ .

A simple approach to provide a physical realisation of an inener is to take a plunger sliding in a cylinder which drives a flywheel through a rack, pinion and gears (see Fig. I). Such a realisation may be viewed as approximating its mathematical ideal in the same way that real springs, dampers, capacitors etc. approximate their mathematical ideals.

A table of the circuit symbols of the six basic electrical and mechanical elements, with the inerter replacing the mass, is shown in Fig. 2. The symbol chosen for the inerter rep- resents a flywheel. The impedance of a mechanical element, in the force-current analogy, is defined by Z ( s ) = C ( s ) / F ( s )

I Mechanical I Electrical I

incner i capacitor

F = C ( V ~ - - U I ] resistor

Fig. 2. admicunee Y(r1.

Circuit symbols and correspondences with defining equations and

(where-denotes Laplace transform, U is the relative velocity across the element and F is the force) and the admittance is given by Y ( s ) = l /Z(s).

111. SUSPENSION STRUTS

We now introduce a few simple networks as candidates for a suspension strut each of which contains at most one damper and one inener. While this does not exploit the full class of positive-real impedances/admittances, it nevertheless provides a number of new possibilities to investigate which are relatively simple to realise in practice.

Fig. 3(a) shows the conventional parallel spring-damper arrangement. In Fig. 3(b) there is a relaxation spring kb in series with the damper. Figs. 3(c), 3(d) show a parallel spring- damper augmented by an inerter in parallel or in series with the damper. When the spring stiffness k is fixed it often proves relatively straightfonvard to optinuse over the two remaining parameters b and c in these configurations.

The series arrangement of Fig. 3(d) has a potential disadvantage in that the node between the damper and inerter has an absolute location which is indeterminate in the steady- state. This could give rise to drift of the damper and/or inerter to the limit of travel in the course of operation. To remedy this the arrangement of Fig. 3(e) is proposed with a pair of springs of stiffness kl, which we call cenuing springs, which may be preloaded against each other. Fig. 3(f) is sinular but allows for unequal springs kl and k z . Figs. 3(h), 3(g) differ from Figs. 3(9, 3(e) by having an additional relaxation spring kb.

Iv. THE QUARTER-CAR MODE1 An elementary model to consider the theory of suspen-

sion systems is the quarter-car of Figure 4 consisting of the sprung mass ms, the unsprung mass mu and a tyre with spring stiffness k,. The suspension sfruf provides an equal and opposite force on the sprung and unsprung masses and is assumed here to be a passive mechanical admittance Q ( s ) which has negligible mass. The equations of motion in the

(a) layout SI

(C) layout s3

(e) layout S5

k

(gi layout S l

Fig. 3. The eight suqmsion layouts

k (f (b) layout S2

(d) layout S4

k

(f) layout S6

ri

k

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Fig. 4 . Qumersar vehicle model

Laplace transformed domain are:

mrs2Zs = kx-~Q(~)(?s-?,,), (1) m,s?, = sQ(7)(tS - & ) + k , ( % (2)

In this paper we will fix the parameters of the quarter-car model as follows: i n s = 250 kg, mu = 35 kg, k, = 150 N/mm.

A. Petjormance measures There are a number of practical design requirements for a

suspension system such as passenger comfort, handling, tyre norntal loads, h u t s on suspension travel etc. which require careful optimisation. In the simplified quaner-car model these can be translated approximately into specifications on the disturbance responses from Fs and e, to is and zu . We now introduce several basic measures.

We first consider road disturbances. Following [SI a time- varying displacement i ( r ) is derived from traversing a rigid road profile at velocity V . Further, let z(1) have the form :(x) where x is the distance in the direction of motion. Thus ;(f) = z ( V r ) . Moreover the corresponding spectral densities are related by

1 ., V

Sf) = -s- (n) where f is frequency in cycleslsecond, n is the wavenumher in cycleslnietre and f = I I V . Now consider an output variable ?(r) u8hich is related to ;(t) by the transfer function H ( s ) . Then the expectation of y2(t) is given by:

E[yZ(t)] = Lw m IH(jZnf)I S(f)df

Here we will use the following spectrum [5]

S; (n) = K J ~ J - (d /cyc le)

where K is a road roughness parameter. We take V = 25 ms- and K = 5 x mcycle-I. The r.ni.s. body vertical accel- eration paranteter J I (ride comfort) is defined by

where T;,? denotes the transfer function from .? to and I l f ( jo)112 = ( & j - z I f ( j o ) I Z d o ) l / is the standard Hz- norm. Similarly the r.ni.s. dynamic tyre load parameter 53 is defined by

Another factor to be considered is the ability of the sus- pension to withstand loads on the sprung mass, e.g. those induced by braking, accelerating and cornering. Following [SI we make use of the following measure for this purpose:

where 11.lIm represents the 3f,-norm, which is the supre- mum of the modulus over all frequency. Note that this n o m equals the maximal power transfer for square integrable signals. so it is a measure of dynamic load carrying.

B. Optimisation .S individual performance measures This section considers only the optimisation of individ-

ual performance measures. Some results on multi-objective optimisation are given in [IO].

Our approach is to fix the static stiffness of the suspension strut and then optimise over the remaining parameters. This will be done for a range of static stiffness settings from k = 10 N/mm to k = 120 N / m . This covers a range from softly sprung passenger cars through sports cars and heavy goods vehicles up to racing cars. It should be noted that the static stiffness in SI to S4 is equal to k hut not for the other four struts. For example, for layout S8 the static stiffness is equal to: k + (k; + k; + k ; ) - I ,

I ) Optimisarion of J1 (ride quality): The results of optinusation are shown in Fig. 5. It was found that the relaxation spring kb did not prove helpful to reduce J I . This left five of the eight smuts in Fig. 3 to he considered. Optimisation for layouts SI , S3, and S4 appears to be convex in the free parameters. Both the parallel (S3) and series (S4) arrangements gave improvements over the conventional strut (SI) for the full range of static stiffness with S4 giving the biggest improvement for stiff suspensions. It should he noted that the parallel arrangement gives lower values of inertance than the series arrangement. For example, at the midrange value of k = 60 Nlmm we have b = 31.27 kg and h = 333.3 kg respectively.

For layouts S5 and S6 the optimisation problem appears no longer to he convex in the parameters. The Nelder- Mead simplex method was used for various starting points. Solutions were found which gave a clear improvement on the series arrangement S4 particularly for softer suspensions. For

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. ,b static stiffness (b) Percentage improvement in I!

(c) optimal damper selling

s , . . . , ,

.,b stauc stirmess

(d) optimal inener setting

the arrangement S6 the improvement was at least 10% across the whole stiffness range. For much of the range kl and k2 were about 113 and 1112 of the static stiffness respectively.

2) Opriniisarion .f Jz (tyre loads): The results of op- timisation are given in [lo]. Here it was found that the relaxation spring kb helped to reduce J? for lower values of static stiffness. The results show an improvement in Jz with parallel (S3) and series (S4) arrangements if the static stiffness is large enough, with the series arrangement again giving the biggest improvement.

3) Oprimisarion of Js (dynamic load carrying): The results of optimisation of J5 are given in [IO]. There is a theoretical minimum for Js equal to the d.c. gain of the transfer function Tp,+:,, which is equal to (k;' + k; ' ) - ' where ko is equal to the static stiffness of the suspension. This can be achieved using layout SI fo rk less than about 68 NI" and using layout S3 for k up to about 100 Nlmm. Using layout S4 the theoretical minimum for Js is not achievable beyond k = 68 Nlnun. In contrast to JI and J3 it appears to be the parallel arrangement (S3) which is more effective than the series one (S4) to reduce Jr.

V. THE FULL-CAR MODEL

We now consider a standard full-car model (see [9] for diagram and equations). The following parameters taken from 191 will be used ins = 1600 kg, Io = 1000 kg m', la = 450kgm2, r f = 0 . 7 5 m , r,=0.75m, I f = 1 . 1 5 m , I ,= 1.35 m, m i =SO kg, m, = 50 kg. ktf = 250 N/mm, k,, = 250 N/mm. We use a full-car stochastic road disturbance performance measure based on a method of Heath [I].

Our approach is to optimise JI and Jz (defined below) over choices of the front and rear dampers c l and c, for the conventional suspension (layout SI), or over choices of the front and rear inerters and dampers b f , b,, c f and c, for layout S3 at each comer, and including centring springs kif and k l , for layout SS. We will take a fixed static stiffness for each suspension ~ t ~ t equal to 100 N/mm. We will assume the vehicle has a forward velocity V = 25 d s e c (90 k").

A. The oprimisation of J I (ride quality)

We will compute the r.m.s. body acceleration param- eter JI = yms for the transntission path Ti+$ with U = [Z,,.Z,~.Z~~,Z,~~' and y = [$,ie.T0]'. The results are il- lustrated in Table I. It is noted that for layout SI the optimisation of J , over c f and c, appears to be convex. But the optimisations for layouts S3, S5~do not necessarily find a global optimum. Similar to the quarter-car case we observe an improvement in both parallel and series arrangements.

Fio. 5 . layout S1 (dot-dashed). layout S5 (dotted) and layout S6 (solid).

The optimisation of It on: layout SI (bold). layout S3 (dashed),

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I I layout I optimal J I I Parameter settings

-. 9.26% improvement

SI 1 J1 = 2.7358 S3 1 JI =2.5122 I h i = 3 1 . 0 7 , 6 , = 4 4 . 2 3

I c f = 2.98, c, = 3.70

, , ~, cf = 3.24. cr = 3.94 kj, =7.85, k i , = 14.'10

I 8.17% improvement I c ( =2.32, c, =3.16 s5 I .I , =2.4823 I h i = 332.82. h. = 374.03

layout

SI S3

optimal Ji Parameter settings

53 = 1.6288 J x = 1.6288

cf = 3.82, c, = 3.85 h i = 0 , h, = O

B. The optimisation of 51 (tyre loads) We now conipute the r.m.s. dynanuc tyre load pa-

rameter 53 = yms for the transmission path %,,, with U [:rL,~r2.~r3,~r41' and Y = Iktt(Zul - ~ r , ) . k r , ( ~ u ~ - z,J k,,(zu, - zri), k,,(z,,, - z r4 ) ] ' . The results a e illustrated in Table 11. Again it is noted that for layout SI the opti- misation of 53 over c, and c, appears to be convex. But the optinusations for layouts S 3 , S5 do not necessarily find a global optimum. Similar to the quarter-car case at some values of static stiffness an improvement is obtained with the series arrangement but not with the parallel.

6 , = 710.74, b, = 418.42

TABLE I1

PliRFOKMASCE INDEX 13 fxIO-3) WITH VAKIOUS ILAYOUTS AT EACII Wll l i l l l . STATION, PliKCI(NTAGE IMPKOVl iMBNI ASD PAKAMBTEK

SITTINGS ( k ' s A R E I N N I M M , c's AKE IN N S I M M . 6's A R E IN KG).

VI. EXPERIMENTAL RESULTS A prototype inerter of rack and pinion design has been

built and tested at Cambridge University Engineering De- partment. A picture of the inerter mechanism is shown in Fig. 6. There are two gearing stages with combined ratio of 19.54:l. The flywheel has a mass of 0.225 kg and the total inerfarice of the device is approxiniately 726 kg. A clutch safety mechanism is integrated into the flywheel to prevent loads in excess of 1.5 kN being delivered to the piston. The device has a stroke of about 80 mm.

The inerter was tested in a series arrangement with centring springs as shown in Fig. 7 using the Cambridge University mechanics laboratory Schenck hydraulic ram. A series of single sinewave excitations was applied at a set of discrete frequencies from 0.05 to 20 Hz. Three signals were measured the total force in the SINI, the total displacement,

and the relative displacement across the inerter. Gains and phase shifts for the different signal paths were calculated frequency by frequency [31.

The ideal linear model of the strut is that of layout S5 but without the main spring k . The admittance Q of the strut is given by the following expression:

(4)

It is noted that there is a zero at the frequency w = m. Let D, and Db represent the displacements of the damper and inerter respectively, and let the total strut displacement be D = Db + D,. Then the following transfer functions can be derived

(bs'+ k)(cs + k l ) s ( b s 2 + c s + k + k l ) '

Ideal frequency responses were calculated for'each of the transfer functions in (4). ( 5 ) and ( 6 ) with the following parameters, which were estimated by measurements on the individual physical components: k = 5.632 NI", kl = 9.132 NI", c = 4.8 Ndmm, b = 726 kg. In addition, stiction nonlineaities were incorporated into the model in parallel with the inerter and damper by adding a fofce 20 sign(Db) to the inerter force, and a force 30 sign(&) to the damper force, corresponding to physically measured stiction forces. Sinewave tests on a nonlinear simulation model were carried out at the same set of frequencies as the practical experinients. The resulting time response data was analysed 'in a similar way to produce a corresponding set of frequency responses for comparison. The Bode plot corresponding to the transfer function in (4) for (i) ideal linear, (ii) nonlinear sinidation and (iii) experimental results, is shown in Fig. (8). It was felt that the agreement between simulation and experiment was relatively good-in particular the phase advance was clearly in evidence in the admittance Q-and optimisation of parameters to get a closer fit between simulation and experiment was not attempted.

VIl. CONCLUSIONS This paper represents a preliminary optimisation study

of the possible benefits of the inerter in vehicle suspension systems. For some relatively simple struts it was shown that improvements could be obtained in a quarter-car vehicle model across a wide range of static suspensions stiffnesses. Improvements of about 10 % or greater were shown for measures of ride, tyre normal load and handling. For certain combinations of these measures, good simultaneous improve- ment was obtained. Improvements were also shown for a full- car model. A prototype inerter was built and tested in a series arrangement with centring springs and shown to exhibit the expected phase advance property.

VIII. ACKNOWLEDGEMENTS We are most grateful to Samuel Lesley, Peter Long,

Neil Houghton, John Beavis, Barry Puddifoot and Alistair

Fig. 6. Pmlolype inenel

Bode plat of admittance Q

I

10. 100 Frequency (Hz)

1 0

Fig. 8. nonlinear simulation with friction (darhedl. experimental dam (solid).

Bode plot of ihe addmitiance Q(r): linear model (dash-doned).

Ross for their work in the design and manufacture of the inener prototype. We would also like to thank David Cebon for making the Vehicle Dynamics Groups hydraulic ram available to us, and to Richard Roebuck for his assistance in the experiments.

IX. REFERENCES [ I ] A.N. Heath Application of the isotropic road rough-

ness assummion. Journal of Sound and Vibration, i i5(i) , 1 3 i : i ~ , i9s7.

[21 E.L. Hixson, Mechanical Impedance, Shock arid Vibration Handbook, 2nd edition, C.M. Harris, C.E. Crede (Fds.), Cha ter 10, McGraw-Hill, 1976.

[3] L. Ljnng System fdentification, Theory for the User, Prentice-Hall, 1987.

[4] A. Papoulis Probability. Random Variables, and Stochastic Process, McCraw-Hill,. 1991.

[5] J.D. Robson Road surface descnption and vehicle response, International Joumal of Vehicle Design, Vol. 1, no. 1, 25-35. 1979.

[6] J.L. Shearer, A.T. Murphy and H.H. Richardson, In- tmduction to s stem dwaniics, Addison-Wesley, 1967.

[71 M.C. Smith hyntheiis of mechanical networks: the inerter, IEEE Transactions on Automatic Contml, 47. 164-1662, 2002.

[SI M.C. Smith and G.W. Walker, A mechanical network approach to performance capabilities of passive sus- pensions, Proceedings of the Workshop on Modelling and Control of Mechanical Systems, Imperial College, London, 17-20 June 1997, pp. 103-117, Imperial .. College Press.

191 M.C. Smith and F-C. Wann. 2002. Controller Pa- -. . _ rameterization for Disturbance Response Decoupling: Application to Vehicle Active Suspension Control, IEEE Trans. on Con1r.S 1st. Tech., 10, 393407.

[lo] M.C. Smith and F-C. d--, an- 2003, Performance Ben- efits in Passive Vehicle Suspensions Employing Inert- ers, submitted for publication.

Fig. 7. lnener in sener wiih damper with eentnng springs

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