Performance Benefits in Passive Vehicle
Suspensions Employing Inerters
1MALCOLM C. SMITH2AND FU-CHENG WANG3
SUMMARY
A new ideal mechanical one-port network element 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 suspension struts, 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 strut. A study of a full-car model is also undertaken where performance improvements are also shown in comparison to conventional passive suspension struts. A prototype inerter has been built and tested. Experimental results are presented which demonstrate a characteristic phase advance property which cannot be achieved with conventional passive struts consisting of springs and dampers only.
1. INTRODUCTION
In [1] an alternative to the traditional electrical-mechanical analogies was proposed in the context of synthesis 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 to the relative acceleration between the terminals. It was argued in [1] that such an element is neces-sary for the synthesis of the full class of physically realisable passive mechanical impedances. Indeed, 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
1
This work was supported in part by the EPSRC.
2
Address correspondence to: Malcolm C. Smith, Department of Engineering, University of Cambridge, Cambridge CB2 1PZ, UK. E-mail: [email protected]
3Department of Mechanical Engineering, National Taiwan University, Taipei, Taiwan.
Vehicle System Dynamics
ISSN 0042-3114 print; ISSN 1744-5159 online # 2004 Taylor & Francis Ltd. http://www.tandf.co.uk/journals
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 reported.
2. BACKGROUND ON THE INERTER
The force-current analogy between mechanical and electrical networks has the following correspondences:
force $ current velocity $ voltage
mechanical ground $ electrical ground spring $ inductor
damper $ resistor:
Additionally, the mass element has always been taken as the analogue of the capacitor, even though it has been appreciated [2, p. 111], [3, p. 10-5] 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 impedences 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 [1] with the following definition. The (ideal) inerter is a two-terminal mechanical device with the property that the equal and opposite force F applied at the terminals is proportional to the relative acceleration between the nodes, i.e. F¼ bð_vv2 _vv1Þ where
v1, v2 are the velocities of the two terminals and b is a constant of proportionality
called the inertance which has units of kilograms. The stored energy in the inerter is equal to12bðv2 v1Þ2.
A variety of different physical realisations of an inerter are possible (see [4]). A simple approach is to take a plunger sliding in a cylinder which drives a flywheel through a rack, pinion and gears (see Figure 1). 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 Figure 2. The symbol chosen for the
inerter represents a flywheel. The impedance of a mechanical element, in the force-current analogy, is defined by ZðsÞ ¼ ^vvðsÞ=^FFðsÞ (where ^ denotes the Laplace trans-form, v is the relative velocity across the element and F is the force) and the admittance is given by YðsÞ ¼ 1=ZðsÞ.
The inerter mechanical element, and the use of the force-current analogy, allows a classical theorem on synthesis of electrical one-ports in terms of resistors, capacitors and inductors to be translated directly into the mechanical context. Although we will not exploit this result directly in the present paper, it is nevertheless useful to cite it. A network is defined to be passive if it cannot supply energy to the environment. If a one-port mechanical network has an impedance ZðsÞ which is real-rational, then it is passive if and only if ZðsÞ is analytic and ZðsÞ þ ZðsÞ 0 in Re(s) > 0 where denotes complex conjugation. A classical theorem of electrical circuit synthesis, due to Brune, Bott and Duffin, now translates directly over to the following result. See [1] for further details and references as well as a discussion of why this class of impedances is significantly wider than can be obtained using springs and dampers only.
Fig. 1. Schematic of a mechanical model of an inerter.
Theorem 1. Consider any real-rational function ZðsÞ which is positive real. There exists a one-port mechanical network whose impedance equals ZðsÞ which consists of a finite interconnection of springs, dampers and inerters.
3. 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 inerter. While this does not exploit the full class of impedances/admittances of Theorem 1, it nevertheless provides a number of new possibilities to investigate which are relatively simple to realise in practice.
Figure 3a shows the conventional parallel spring-damper arrangement. In Figure 3b there is a relaxation spring kbin series with the damper. Figures 3c and 3d
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 straightfor-ward to optimise over the two remaining parameters b and c in these configurations. The series arrangement of Figure 3d 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 Figure 3e is proposed with a pair of springs of stiffness k1, which we call centring springs, which
may be preloaded against each other. Figure 3f is similar but allows for unequal springs k1 and k2. Figures 3h and 3g differ from Figures 3f and 3e by having an
additional relaxation spring kb.
The mechanical admittance YðsÞ for two of these layouts (layout S3 and S7) is now given for illustration:
Y3ðsÞ ¼ k sþ c þ bs and Y7ðsÞ ¼ k sþ 1 s kbþ s csþk1þ s bs2þk1 :
4. THE QUARTER-CAR MODEL
An elementary model to consider the theory of suspension systems is the quarter-car of Figure 4 consisting of the sprung mass ms, the unsprung mass mu and a tyre with
spring stiffness kt. The suspension strut provides an equal and opposite force on the
sprung and unsprung masses and is assumed here to be a passive mechanical
admittance YðsÞ which has negligible mass. The equations of motion in the Laplace transformed domain are:
mss2^zzs¼ ^FFs sYðsÞð^zzs ^zzuÞ; ð1Þ
mus2^zzu ¼ sYðsÞð^zzs ^zzuÞ þ ktð^zzr ^zzuÞ: ð2Þ
In this paper we will fix the parameters of the quarter-car model as follows: ms¼ 250 kg, mu¼ 35 kg, kt¼ 150 kN/m.
4.1. Performance Measures
There are a number of practical design requirements for a suspension system such as passenger comfort, handling, tyre normal loads, limits on suspension travel etc. which require careful optimisation. In the simplified quarter-car model, these can be translated approximately into specifications on the disturbance responses from Fsand
zrto zs and zu. We now introduce several basic measures.
We first consider road disturbances zr. Following [5] a time-varying displacement
zðtÞ is derived from traversing a rigid road profile at velocity V. Further, let zðtÞ have the form z0ðxÞ where x is the distance in the direction of motion. Thus zðtÞ ¼ z0ðVtÞ.
Moreover, the corresponding spectral densities are related by Szðf Þ ¼1
VS
z0
ðnÞ
where f is frequency in cycles/second, n is the wavenumber in cycles/metre and f ¼ nV. Now consider an output variable yðtÞ which is related to zðtÞ by the transfer function HðsÞ. Then the expectation of y2ðtÞ is given by:
E½y2ðtÞ ¼ Z 1 1 jHð j2f Þj2Szðf Þ df ¼ 1 2 Z 1 1 jHð j!Þj21 VS z0 ðnð!ÞÞ d!: Here we will use the following spectrum [5]
Sz0ðnÞ ¼ jnj2 ðm3=cycleÞ
where is a road roughness parameter. We take V¼ 25 m s1 and ¼
5 107m3
cycle1. The r.m.s. body vertical acceleration parameter J1 (ride
comfort) is defined by J1¼ 1 2V Z 1 1 T^zz r!^€€zzzzsð j!Þ 2 nð!Þ2 d! 1=2 ¼ 2ðVÞ1=2sT^zzr!^zzs 2 ð3Þ
where T^xx!^yy denotes the transfer function from ^xx to ^yy and kf ðj!Þk2¼ ð1 2 R1 1jf ðj!Þj 2
d!Þ1=2 is the standard H2-norm. Similarly the r.m.s. dynamic tyre
load parameter J3is defined by
J3¼ 2ðVÞ 1=2 1 sT^zzr!ktð^zzu^zzrÞ 2 :
Another factor to be considered is the ability of the suspension to withstand loads on the sprung mass, e.g. those induced by braking, accelerating and cornering. Following [6] we make use of the following measure for this purpose:
J5¼ kTFF^s!^zzsk1
wherek k1 represents theH1-norm, which is the supremum of the modulus over
all frequency. Note that this norm equals the maximal power transfer for square integrable signals, so it is a measure of dynamic load carrying.
4.2. Optimisation of Individual Performance Measures
Although suspension design will usually involve a trade-off between various performance measures, it is useful to consider first how much improvement can be obtained in individual performance measures for various different struts.
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 k¼ 10 kN/m to k ¼ 120 kN/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 S1 to S4 of Figure 3 is equal to k but not for the other four struts. For example, for layout S8 the static stiffness is equal to: kþ ðk1
b þ k11 þ k21Þ 1
.
4.2.1. Optimisation of J1(Ride Quality)
The results of optimisation are shown in Figures 5 and 6. It was found that the relaxation spring kbdid not prove helpful to reduce J1. This left five of the eight struts
in Figure 3 to be considered. Optimisation for layouts S1, 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 (S1) for the full range of static stiffness with S4 giving the biggest improvement for stiff suspensions. It should be noted that the parallel arrangement gives lower values of inertance than the series arrangement. For example, at the midrange value of k¼ 60 kN/m we have b¼ 31:27 kg and b ¼ 333:3 kg, respectively.
Fig. 5. The optimisation of J1on: layout S1 (bold), layout S3 (dashed), layout S4 (dot-dashed), layout S5
(dotted) and layout S6 (solid).
Fig. 6. The optimisation of J1: k1 in layout S5 (solid), k1 in layout S6 (dashed) and k2 in layout S6
For layouts S5 and S6, the optimisation problem appears no longer to be convex in the parameters. The Nelder–Mead simplex method was used for various starting points. Solutions were found which gave a clear improvement
Fig. 7. The optimisation of J3on: layout S1 and S2 (bold), layout S3 (dashed), layout S4 (dot-dashed),
on the series arrangement S4 particularly for softer suspensions. For the arrange-ment S6 the improvearrange-ment was at least 10% across the whole stiffness range. For much of the range, k1 and k2 were about 1/3 and 1/12 of the static stiffness,
respectively.
4.2.2. Optimisation of J3(Tyre Loads)
The results of optimisation are shown in Figure 7. Here it was found that the relaxation spring kbhelped to reduce J3for lower values of static stiffness. Indeed, the
conventional strut S2 is a noticeable improvement on S1 for softer suspensions. Again optimisation for layouts S1, S2, S3, and S4 appears to be convex in the free parameters. The results show an improvement in J3with parallel (S3) and series (S4)
arrangements if the static stiffness is large enough, with the series arrangement again giving the biggest improvement.
For layouts S5 and S6, the optimisation problem appears no longer to be convex in the parameters. The Nelder–Mead simplex method was again used for various starting points. As before, the use of centring springs in layouts S5 and S6 gave further improvements over the ordinary series arrangements S4. The use of a relaxation spring kb in S7 was needed to extend the benefits to softer
suspensions.
4.2.3. Optimisation of J5(Dynamic Load Carrying)
In Figure 8 the optimisation of J5 is illustrated for S1, S3 and S4 only. There is a
theoretical minimum for J5 equal to the d.c. gain of the transfer function T^FFs!^zzs,
which is equal to ðk10 þ k1t Þ1 where k0 is equal to the static stiffness of the
suspension. This can be achieved using layout S1 for k less than about 68 kN/m. The upper and lower bounds for c to achieve this are shown in Figure 8c. Using layout S3, the theoretical minimum for J5 can be achieved for k up to about
100 kN/m. The upper and lower bounds for c and b to achieve this are shown in Figures 8c and 8d. Using layout S4, the theoretical minimum for J5 is not
achievable beyond k 68 kN/m. In contrast to J1 and J3 it appears to be the
parallel arrangement (S3) which is more effective than the series one (S4) to reduce J5.
4.3. Multi-Objective Optimisation
In suspension design it is usually necessary to consider several performance objectives. It is interesting to ask if the inerter can give improvements to more than one objective simultaneously. In this section, we will consider J1 and J5
Our approach is to work with a combined performance index as follows: J:¼ J1=J1;0þ ð1 ÞJ5=J5;0;
for 0 1, with J1;0¼ 1:76 and J5;0¼ 2:3333 105 which are the optimal
values for suspension layout S1. The optimisation results for a static stiffness of k¼ 60 kN/m are illustrated in Figure 9. Firstly, it is noted that for each layout the optimisation appears to be Pareto optimal, i.e. it is not possible to improve them together in a given layout, as shown in Figure 9. Secondly, the use of inerters (layouts S3, S4) gives the possibility of improving J1and J5 together in comparison
to layout S1.
5. THE FULL-CAR MODEL
We now consider the full-car model as shown in Figure 10. The following param-eters taken from [7] will be used: ms¼ 1600 kg, I ¼ 1000 kg m2, I¼ 450 kg m2,
Fig. 9. The optimisation of J1 and J5 together: layout S1 (solid), layout S3 (dashed) and layout S4
(dot-dashed).
tf ¼ 0:75 m, tr ¼ 0:75 m, lf ¼ 1:15 m, lr ¼ 1:35 m, mf ¼ 50 kg, mr¼ 50 kg,
ktf ¼ 250 kN/m, ktr ¼ 250 kN/m.
5.1. Road Disturbances
Our general approach to obtain a full-car stochastic performance measure is based on a method of Heath [8]. This section briefly describes our approach to obtain a simple approximation to this measure which can be evaluated in Matlab.
Consider a full-car model moving at a speed of V with road inputs zri ¼ xi, for
i¼ 1; . . . ; 4 and with c ¼ 2tf ¼ 2trand L¼ lfþ lr. Suppose we select a set of outputs
determined by ^yy¼ PðsÞ^xx, where ^xx ¼ ð^xx1; ^xx2; ^xx3; ^xx4Þ t
. If the vehicle is running in a straight line, that is, the road inputs to the rear wheels are regarded as time delays of the inputs to the front wheels, then
^ x x3 ^ x x4 ¼ esT ^xx1 ^xx2 ; where T ¼ L=V. Hence the system outputs satisfy
^yyðsÞ ¼ PðsÞ I esTI ^ u u¼: HðsÞ^uu; ð4Þ where ^uu¼ ð^xx1; ^xx2Þt. If x is WSS (Wide Sense Stationary), then the power spectral
density functions Suuðj!Þ and Syyðj!Þ are related by [9, Sec. 10-3]
Syyðj!Þ ¼ Hðj!ÞSuuðj!ÞHðj!Þ: ð5Þ
By definition the autocorrelation of the (front) road surface is given by: Ruuð Þ ¼ Eðuðt þ ÞuðtÞ t Þ ¼ E x1ðt þ Þx1ðtÞ x1ðt þ Þx2ðtÞ x2ðt þ Þx1ðtÞ x2ðt þ Þx2ðtÞ : ð6Þ
If we suppose the road surface is isotropic, then Eðx1ðt þ Þx1ðtÞÞ ¼ Eðx2ðt þ Þx2ðtÞÞ
and Eðx1ðt þ Þx2ðtÞÞ ¼ Eðx2ðt þ Þx1ðtÞÞ. Let RDð Þ ¼ Eðx1ðt þ Þx1ðtÞÞ and
RXð Þ ¼ Eðx1ðt þ Þx2ðtÞÞ. An auto-spectrum SDand a cross-spectrum SX are defined
as the Fourier transforms of the correlations RDð Þ and RXð Þ respectively. Then the
power spectral density of the road inputs becomes Suuðj!Þ ¼ Z Ruuð Þej! d ¼ SD SX SX SD : ð7Þ
The relation between SDand SX is given by a normalised, real cross-spectrum gðj!Þ
in [5, 8] as
gðj!Þ ¼SXðj!Þ SDðj!Þ
:
Next suppose we can find a spectral factorisation of the following matrix: 1 g
g 1
¼ MM: ð8Þ
Then the power spectrum of the road inputs becomes
Suu ¼ WMMW; ð9Þ where W¼ W ¼ ffiffiffiffiffiffi SD p 0 0 ffiffiffiffiffiffiSD p : From Equations (5) and (4) we then obtain
Syy¼ HSuuH¼ HWMMWH:
Then we define the performance measure of interest as
yrms¼ 1 2 Z traceðSyyÞd! 1=2 ¼ 1 2 Z traceð½HWM ½HWM Þd! 1=2 ¼ P I esTI WM 2 : ð10Þ
As in Section 4 we consider a time-varying displacement xðtÞ derived from traversing a rigid road profile at velocity V. Further, let xðtÞ have the form x0ðzÞ where z is the distance in the direction of motion. Thus xðtÞ ¼ x0ðVtÞ. Moreover, the
corresponding spectral densities are related by [5] Sxðf Þ ¼1
VS
x0
ðnÞ;
where f ¼ nV. A similar relationship holds for the cross-spectrum. Here we will use the following spectrum
Sx0 ¼ n2; ðm
where is a road roughness parameter. We take ¼ 5 107m3cycle1. We
there-fore obtain
SDðj!Þ ¼
Vð2Þ2
!2 : ð11Þ
In [8] the following expression is obtained for the normalised cross-spectrum (in terms of displacement)
2jcnj
ð1Þ K1ðj2cnjÞ;
where K1is the modified Bessel function of the second kind of order 1, and is the
gamma function. We therefore see that gðj!Þ ¼ jc!j
ð1ÞVK1ðjc!=VjÞ
¼ !K1ð !Þ ¼: g ðj!Þ; ð12Þ
where ¼ c=V. We note that gðj!Þ is a real function of ! and is irrational.
To calculate the performance measure numerically the time-delay esT was approximated as follows: esT ’ n 1ð2n=T sÞ n 1ð2n=T þ sÞ ;
with n¼ 6. In addition the following approximation was used in the matrix W in (10): 1
s’ sþ 1;
where ¼ 102. This makes little difference in the calculation value of y rms
because, in cases of interest, PðsÞ has zeros at the origin. To deal with gðj!Þ a par-ticular value ¼ 0:05 was selected and ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffig0:05ðj!Þ
p
was evaluated frequency by frequency. The command fitmag in Matlab was used to give an approximation: g0:05ðj!Þ ’ hh ¼: ggðj!Þ which allows the approximation g ðj!Þ ’ ggðj0:05 !Þ to be
used. A spectral factorisation was then calculated M0:05¼ M M11 MM12 M M12 MM11 ¼ A B C D ; ð13Þ where M M11 ¼ ðs þ 75:30 116:35jÞðs þ 53:96 42:74jÞðs þ 23:05Þðs þ 7:05Þ ðs þ 75:29 116:36jÞðs þ 54:03 43:26jÞðs þ 28:43Þðs þ 8:00Þ; M M12 ¼ 14:3836ðs þ 75:88 117:70Þðs þ 63:80 39:97jÞðs þ 9:26Þ ðs þ 75:29 116:36jÞðs þ 54:03 43:26jÞðs þ 28:43Þðs þ 8:00Þ:
Then, the approximation for any speed V and wheel track c was taken to be: M ’ A ffiffiffiffi p B ffiffiffiffi p C I ; ð14Þ where ¼ V 20c.
5.2. Optimisation of Full-Car Performance Measures
In this section, we shall compare the performance improvement of the full-car model with inerters in the suspension struts. Our approach is to optimise J1 and J3(defined
below) over choices of the front and rear dampers cf and cr for the conventional
sus-pension (layout S1), or over choices of the front and rear inerters and dampers bf,
br, cf and cr for layout S3 at each corner, and including centring springs k1f and k1r
for layout S5. We will take a fixed static stiffness for each suspension strut equal to 100 kN/m. We will assume the vehicle has a forward velocity V¼ 25 m/sec (90 km/h). 5.2.1. The Optimisation of J1(Ride Quality)
We will compute the r.m.s. body acceleration parameter J1¼ yrmswhere P¼ T^uu!^yy
with u¼ ½zr1; zr2; zr3; zr4 0
and y¼ ½€zzs; €zz ; €zz 0. From Equation (10) we can calculate
and compare the performance of the full-car model with various layouts. The results are illustrated in Table 1. It is noted that for layout S1 the optimisation of J1 over cf
and crappears to be convex, as shown in Figure 11. 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.
5.2.2. The Optimisation of J3(Tyre Loads)
We now compute the r.m.s. dynamic tyre load parameter J3¼ yrms from (10)
where P¼ T^uu!^yy with u¼ ½zr1; zr2; zr3; zr4
0 and y¼ ½k
tfðzu1 zr1Þ; ktfðzu2 zr2Þ,
ktrðzu3 zr3Þ; ktrðzu4 zr4Þ 0
. The results are illustrated in Table 2. Again it is noted
Table 1. Performance index J1with various layouts at each wheel station, percentage improvement and
parameter settings (k’s are in kN/m, c’s are in kNs/m, b’s are in kg).
Layout Optimal J1 Parameter settings
Conventional (layout S1) 2.7358 cf ¼ 2:98, cr¼ 3:70
Parallel inerter 2.5122 bf ¼ 31:07, br¼ 44:23
(layout S3) (8.17% improvement) cf ¼ 2:32, cr¼ 3:16
Series inerter with centring springs 2.4823 bf ¼ 332:82, br¼ 374:03
(layout S5) (9.26% improvement) cf ¼ 3:24, cr¼ 3:94
that for layout S1 the optimisation of J3over cf and crappears to be convex. But the
optimisations for layouts S3, 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. EXPERIMENTAL RESULTS
A variety of different embodiments of an inerter are possible (see [4]). A prototype of rack and pinion design has been built and tested at Cambridge University Engineering
Fig. 11. The optimisation of J1over cf and crfor layout S1.
Table 2. Performance index J3 (103) with various layouts at each wheel station, percentage
improvement and parameter settings (k’s are in kN/m, c’s are in kNs/m, b’s are in kg).
Layout Optimal J3 Parameter settings
Conventional (layout S1) 1.6288 cf ¼ 3:82, cr¼ 3:85
Parallel inerter 1.6288 bf ¼ 0, br¼ 0
(layout S3) (0% improvement) cf ¼ 3:82, cr¼ 3:85
Series inerter with centring springs 1.5224 bf ¼ 710:74, br¼ 418:42
(layout S5) (6.53% improvement) cf ¼ 3:16, cr¼ 3:71
Department (see Figure 12). There are two gearing stages with combined ratio of 19.54:1. The flywheel has a mass of 0.225 kg and the total inertance of the device is approximately 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 Figure 13 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 strut, the total displacement, and the relative displacement across the inerter. Gains and phase shifts for the different signal paths were calculated frequency by frequency [10].
The ideal linear model of the strut is shown in Figure 14. The admittance Y of the strut is given by the following expression:
Y¼ ðbs 2þ kÞðcs þ k 1Þ sðbs2þ cs þ k þ k 1Þ : ð15Þ
It is noted that there is a zero at the frequency !¼pffiffiffiffiffiffiffiffik=b. As in Figure 14, let Dc
and Db represent the displacements of the damper and inerter respectively, and let
Fig. 13. Inerter in series with damper with centring springs.
the total strut displacement be D¼ Dbþ Dc. Then the following transfer functions can be derived: ^ D DcðsÞ ¼ bs2þ k bs2þ cs þ k 1þ k ^ D DðsÞ; ð16Þ ^ D DbðsÞ ¼ csþ k1 bs2þ cs þ k 1þ k ^ D DðsÞ: ð17Þ
Ideal frequency responses were calculated for each of the transfer functions in Equations (15), (16) and (17) with the following parameters, which were esti-mated by measurements on the individual physical components: k¼ 5:632 kN/m, k1¼ 9:132 kN/m, c ¼ 4:8 kNs/m, b ¼ 726 kg. In addition, stiction nonlinearities
were incorporated into the model in parallel with the inerter and damper by adding a force 20 signð _DDbÞ to the inerter force, and a force 30 signð _DDcÞ 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 experiments. The resulting time response data were analysed in a similar way to produce a corresponding set of frequency responses for comparison. The Bode plots corresponding to each of the transfer functions in
Fig. 15. Bode plot of the admittance YðsÞ: linear model (dash-dotted), nonlinear simulation with friction (dashed), experimental data (solid).
Fig. 16. Bode plot of transfer-function ^DDcðsÞ= ^DDðsÞ: linear model (dash-dotted), nonlinear simulation with
friction (dashed), experimental data (solid).
Fig. 17. Bode plot of transfer-function ^DDbðsÞ= ^DDðsÞ: linear model (dash-dotted), nonlinear simulation with
Equations (15), (16) and (17), for (i) ideal linear, (ii) nonlinear simulation and (iii) experimental results, are shown in Figures 15, 16 and 17. 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 Y – and optimisation of parameters to get a closer fit between simulation and experiment was not attempted.
7. 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 improvement 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.
ACKNOWLEDGEMENTS
We are most grateful to Samuel Lesley, Peter Long, Neil Houghton, John Beavis, Barry Puddifoot and Alistair Ross for their work in the design and manufacture of the inerter prototype. We would also like to thank David Cebon for making the Vehicle Dynamics Group’s hydraulic ram available to us, and to Richard Roebuck for his assistance in the experiments.
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