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Technical Report - 009
Lotus Elan Rear SuspensionThe Effect of Halfshaft Rubber Couplings
by
T. L. Duell
Prepared forThe Elan Factory
May 2004
Terry DuellTerry Duellconsultingconsulting
19 Rylandes Drive, Gladstone ParkVictoria 3043, Australiaph. +61 3 9338 [email protected]
http://www.planet.net.au/~tduell/cons.html
© 2004 Terry DuellPublished on a RISC OS system using !TechWriter Pro+
Figures and diagrams prepared in !Tau and !Draw
1 IntroductionThe general arrangement of the Lotus Elan rear suspension is as shown in Fig. 1
Fig. 1 Lotus Elan rear suspension
The driveshaft rubber couplings, which were original fitment, contribute to drivelineharshness and can cause some surge or shudder when starting from rest. This form ofcoupling is rarely used in more modern drive shaft arrangements.Alternate replacement driveshaft arrangements including hookes joints with sliding shaft,and constant velocity joints, have been available from various 3rd party suppliers.The rubber couplings resist any difference in angle between the two coupled shafts, andhence contribute to the suspension forces and roll resistance. This report details aninvestigation into the effect of the halfshaft rubber couplings on the rear suspension.
2 MethodThe study method has been as follows;
• measure the characteristics of sample couplings• analyse the rear suspension arrangement to determine coupling angularity at
values of wheel travel• analyse the effect of the coupling moments on suspension spring forces• analyse front and rear suspensions to determine roll centres• build a 3D vehicle model and analyse the effect the couplings on vehicle handling
for a discreet set of manoeuvres
It should be noted that a number of derived data, particularly relating to suspensiongeometry and wheel travel, are approximate only, as a result of working from small scaledrawings of variable accuracy. These drawings show the spring lengths for the normal rideposition as well as at bump and droop conditions, but there is no information to confirm thewheel loads or the spring rates assumed for these drawings. Springs were measured and rates calculated. The distance between spring seats on the rearstrut was also measured, along with front anti roll bar geometry. Calculations of wheelloads, from the spring deflections derived from the drawings are at odds with the reportedkerb weight of the vehicle. As a result the estimated wheel loads and front to rear weightdistribution may not be accurate, which will influence the accuracy of any predictions of the
vehicle dynamic response.
3 Coupling characteristicsFour rubber couplings, designated A, B, C, and D, were tested to establish indicativebending moment-angle characteristics. The details of the couplings are as follows;
• A-Imp style, new (non interleaved)• B-Lotus (Dunlop), new• C-Lotus (Dunlop), used• D-Metalastic, used
The couplings are shown in Fig. 2
Fig. 2 Test couplings
To measure the bending moment-angle characteristics, each coupling was mounted on adummy differential output shaft which was secured, vertically, and a halfshaft was thenmounted to the coupling. A normal force was applied to the top of the halfshaft, andmeasured by a spring balance. The angle of the halfshaft was derived by measuring thedeflection of the top of the halfshaft. The test setup is shown in Fig.3.
Fig.3 Test setup
The force was applied normal to the axis of the shaft, on a radial line through one of themounting holes and repeated at 180 degrees. It would have been desirable to also apply theforce on a radial line between mounting holes, but that was not possible with the test setup.The derived bending moment-angle data is graphed data in Fig. 4.
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Shaft angle (deg.)
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C1-Lotus (Dunlop), used
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Shaft angle (deg.)
Ben
ding
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D2-Metalastik, used
Fig. 4. Derived bending moment-angle data
All couplings exhibit a linear bending moment-angle relationship. The increase in slope athigh angles shown in the results of tests A1, B2, and to a lesser degree for C1 and D2, isprobably caused by the reaching the angular limit for that particular setup (E.G. the end of amounting bolt on halfshaft fouling the dummy axle shaft).If the first and second tests are averaged, and a best fit straight line applied, the results are asshown in Fig. 5.
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Shaft angle (deg.)
Ben
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Nm
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✳ A■ B❏ C❉ D
Fig. 5. Best Fit line applied to averaged test data
It is noted that couplings A, C, and D could be considered to exhibit effectively the samebending moment resistance per degree of shaft angularity. Coupling B (Lotus, new) showsnoticeably less resistance to bending. As coupling C is the same as A, only used, it ispossible that the new Lotus couplings become more resistant to bending with use.For the purpose of this study, it will be assumed that, on average, a coupling will have amaximum 'bending stiffness' of 4.7 Nm/deg, and a minimum of 2.9 Nm/deg.
4 Analysis of suspension geometryAnalysis of the rear suspension geometry (layout shown in Fig. 6) indicates that thecoupling angles vary with wheel travel, as shown in Fig. 7.The suspension strut angle varies with wheel travel as shown in Fig. 8.The angular deflection of the couplings induces a resisting moment on the unsprungassembly (wheel, strut, inner coupling etc.). This moment can be resolved as a vertical forceat the wheel vertical centreline, applied over the horizontal distance from the wheel verticalcentreline to the vertical centreline through the inner coupling. The variation of the momentarm with vertical wheel travel, is shown in Fig. 9.The combined effects of the coupling angles and moment arm with wheel travel, and theeffect of the assumed upper and lower bounds of coupling stiffness on vertical wheel force,are shown in Fig. 10.
Fig. 6 Layout of rear suspension
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Cou
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g an
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.)
❍ Inner coupling■ Outer coupling
Fig. 7 Coupling angle and wheel travel
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Vertical wheel travel from droop (mm)
Str
ut a
ngle
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eg.)
Fig. 8 Suspension strut angle and wheel travel
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Vertical wheel travel from droop (mm)
mom
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rm (
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Fig. 9 Moment arm and wheel travel
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Vertical wheel travel from droop (mm)
Cou
plin
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rce
at w
heel
(N
)
■ Max. coupling stiffness❍ Min. coupling stiffness
Fig. 10 Vertical wheel force resulting from coupling deflection
Table 1 shows derived suspension geometry data.
Front Rear
Camber change on bump (deg/mm) -0.062 -0.09
Spring/damper angle at bump (deg) 17.7 21.1
Spring/damper angle at ride (deg) 16.0 18.8
Spring/damper angle at droop (deg) 13.5 15.7
Roll centre height, above groundat ride (mm)
75.4 59.0
Table 1. Derived suspension geometry data (approximate)
Table 2 shows relevant measured and calculated suspension data, and table 3 showsmeasured and calculated spring data.
Front Rear
Hub/strut unsprung mass (kg) 13.6 13.6
Wheel/tyre mass (kg) 11.8 11.8
Total unsprung mass per wheelstation (kg)
25.4 25.4
Spring mean diameter (mm) 57.8 99.0
Spring wire diameter (mm) 8.23 10.9
Active coils 17.6 11.25
Spring free length (mm) 408.4 388.6
Spring rate (N/mm) 13.37 12.89
Wheel rate at ride height (N/mm) 7.8 12.89
Auxiliary roll stiffness (Nm/deg) 107.0 -
Table 2. Measured and calculated suspension data (approximate)
Front Rear
Wire diameter (in) [mm] 0.324 [8.3] 0.430 [10.9]
Mean coil diameter (in) [mm] 2.276 [57.8] 3.9 [99.06]
Number of coils 19.6 11.25
Ends Squared& ground
Plain
Number of active coils 17.6 11.25
Free length (in) [mm] 16.08 [408.4] 15.3 [388.6]
Nominal rate (lbf/in) [N/mm] 75 [13.14] 75 [13.14]
Calculated rate (lbf/in) [N/mm] 76.3 [13.4] 73.6 [12.9]
Table 3. Measured and calculated spring data (approximate)The front suspension geometry is shown in Fig. 11, and the front and rear roll centres, atnormal ride, in Figs. 12, 13 respectively.The effect of the couplings in resisting body roll can be considered by rotating the sprungmass about the rear roll centre and deriving the effective wheel travel and angle of body roll.That analysis indicates that a body roll of 2.1 degrees will result in approximately 25mm ofwheel travel (bump travel at the outer wheel and rebound travel at the inner). Referring to Fig. 7 it can be shown that the effective wheel travel results in a change inangle at each coupling of approximately 3 degrees. Hence, the effective auxiliary rollstiffness provided by the couplings is approximately 27 Nm/deg, and 17 Nm/deg, for themaximum and minimum assumed coupling stiffnesses, respectively. The couplings, at theirstiffest, represent 25% of the auxiliary roll stiffness provided by the anti-roll bar fitted fittedat the front of the Elan.Analysis of the suspension spring deflections and the drawings (Figs. 6 and 11) to arrive at
wheel and axle loads (ground reactions), is inconclusive. To arrive at a model, in order toexamine the possible influence of the couplings on the vehicle dynamics, some assumptionshave had to made. Available Lotus data indicates that the kerb mass of the Elan is 641 kg.
Fig. 11. Front suspension geometry
Fig. 12. Front roll centre at normal ride
Fig. 13. Rear roll centre at normal ride
That value is noted as unladen, and probably means no driver and very little, if any fuel. Ifwe allow 80 kg for the driver mass, and approx. 23 kg for fuel ( about 27 litres), the mass ofthe vehicle is likely to be about 745 kg. If we now assume a mass distribution, front/rear, of 50%/50%, it is possible to test theseassumptions by calculating the axial spring forces and vertical wheel forces.The vertical wheel force at the rear is 1824N (25% of total), and the vertical component ofthe sprung weight is 1575N. The axial spring force as a result, will be about 1491N (thestrut is at angle of about 18.8 deg.). Hence the spring deflection, from no load, will be about115.6mm. Therefore we would expect the distance between the spring seats to be the springfree length less the deflection (388.6-115.6), which is 273mm. The drawing shows about224mm between spring seats at normal ride. Hence we must conclude from this that eitherthe weight on the rear wheels is higher than assumed; the spring rate is lower, or possiblyboth.If we apply the same argument to the front suspension we arrive at an axial spring deflectionof about 157.5mm, and an axial spring length of about 250mm. The drawing shows the axialspring length to be about 206mm at normal ride. Hence again we must conclude that theweight on the front wheels is higher; the spring rate lower, or both. The vehicle is not likely to be significantly heavier, and changing the mass distributioncannot help. The assumed spring rates are the calculated rates which are very close to thereported nominal rates, hence it is concluded that the drawings may not be applicable (iethey may be for a prototype version using different springs. It is noted that rear suspensionlayout shows a variable rate spring which was not fitted in production), and that theassumptions are reasonable enough to use for the analysis. Data for the model 26R (theracing version of the Elan) states that the mass distribution is 48%/52% (F/R) with two up,hence an assumption of 50/50 with one person is probably not much in error.To further assist understanding the effect of the driveline couplings on the suspension, Fig.14 shows the rear wheel force-deflection curve for suspension spring, bump stop, drivelinecouplings, and the combination of the two.Full suspension droop is arbitrarily set at zero deflection. The effect of the couplings can beseen as an increase in the effective spring rate, albeit small. Fig. 14 is approximate only,being estimated from a combination of conflicting data from the suspension generalarrangement diagrams and measurements. The bump stop characteristic have been assumed.
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Zero coupling deflection
Normal ridedeflection
Full droopdeflection
Combined spring &coupling force
Bump stopcontact
Full bump
Fig. 14 Rear wheel force-deflection curve
5 Vehicle dynamics analysisModels of the Elan, using the derived data, were built in CarSimEd. CarSimEd is areasonable comprehensive 3d handling analysis simulation, limited only by a restriction onspring and damper characteristics being linear, hence it is not possible to model varying ratesprings or varying rate damper bump and rebound characteristics. These limitations vary insignificance depending upon the condition being simulated. Rebound damping was set to be the equivalent of 0.6 of critical damping (vertical, at thewheel) for both vehicle models. A typical rebound damping rate was used, as rebounddamping is more likely to influence handling manoeuvres than bump damping.One Elan model reflected the original specification and included the rear drive shaft rubbercouplings as an auxiliary rear roll stiffness, using the maximum bending stiffness that hadbeen established from the tests. The other model did not include any auxiliary rear rollstiffness, reflecting the situation where the rubber couplings had been replaced with CVjoints or similar arrangement.The handling of the vehicle models was compared for constant speed step steer, at 50 km/h.The step steer involves driving the vehicle straight and then rapidly making a step steerinput attempting to follow a 40m diameter circle. This allows the transient behaviour to beexamined when the step steer is applied, and also the steady state handling under constantlateral acceleration whilst negotiating the circular path. The vehicle speed was chosen to avoid severe rear bump stop deflection, and hence staywithin the regime approximating a linear spring force-deflection characteristic.The data used to describe the vehicle models is as shown in table 4.
Elan with driveline couplings
Elan withoutdriveline couplings
Wheelbase (mm) 2133.6 2133.6
Track front (mm) 1196.0 1196.0
Track rear (mm) 1230.0 1230.0
Gross mass (kg) 745.4 745.4
Unsprung mass, front (kg) 50.9 50.9
Unsprung mass, rear (kg) 50.9 50.9
Front spring rate (N/mm) 13.4 13.4
Rear spring rate (N/mm) 12.0 12.9
Front damper rate (Nsec/mm) 4.0 4.0
Rear damper rate (Nsec/mm) 1.9 1.9
Front spring/damperinstallation ratio
0.777 0.777
Rear spring/damper installation ratio
1.0 1.0
Front auxiliary rollstiffness (Nm/deg)
107.0 107.0
Rear auxiliary rollstiffness (Nm/deg)
27.0 0.0
Camber change onbump (deg/mm)
-0.062 -0.09
Centre of gravityheight (mm)
400 400
Front roll centreheight (mm)
75.4 59.0
Table 4. Estimated data for CarSimEd modelsResults from CarSimEd, for the Elan with driveline couplings, are shown in Figs. 15 to 18,and for the Elan without driveline couplings in Figs. 19 to 22.The figures show the understeer plot, tyre vertical forces, tyre slip angles, and roll angle.
-10
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-0.1 0 0.1 0.2 0.3 0.4 0.5 0.6
Vehicle lateral acceleration - g's
Steering-wheel angle - deg
Fig.15 Understeer plot, Elan with couplings
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Tire vertical load - N
Fz_LFFz_LRFz_RFFz_RR
Fig. 16 Tyre vertical forces (N), Elan with couplings
-2.5
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-1.5
-1
-0.5
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Slip angle - deg
Alpha_LFAlpha_LRAlpha_RFAlpha_RR
Fig. 17 Tyre slip angles (deg), Elan with couplings
-0.5
0
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3.5
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Body roll - deg
Fig. 18 Roll angle (deg), Elan with couplings
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-0.1 0 0.1 0.2 0.3 0.4 0.5 0.6Vehicle lateral acceleration - g's
Steering-wheel angle - deg
Fig. 19 Understeer plot, Elan without couplings
1000
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2400
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2800
0 2 4 6 8 10 12 14 16 18 20 22 24 26Time - sec
Tire vertical load - N
Fz_LFFz_LRFz_RFFz_RR
Fig. 20 Tyre vertical forces (N), Elan without couplings
-2.5
-2
-1.5
-1
-0.5
0
0.5
0 2 4 6 8 10 12 14 16 18 20 22 24 26Time - sec
Slip angle - deg
Alpha_LFAlpha_LRAlpha_RFAlpha_RR
Fig. 21 Tyre slip angles (deg), Elan without couplings
-0.5
0
0.5
1
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3.5
4
0 2 4 6 8 10 12 14 16 18 20 22 24 26Time - sec
Body roll - deg
Fig. 22 Roll angle (deg), Elan without couplings
6 DiscussionThe CarSimEd simulations do leave a bit to be desired regarding absolute accuracy. The realvehicle would exhibit non linear suspension force-deflection and non linear bump andrebound damping. It is also possible that the real vehicle mass and mass distribution will notbe in accordance with the data used for the models. These differences between the modelsand real vehicles will affect the dynamic response. The simulations that have been conducted have provided an indication of the relativedynamic response, with all parameters constant other than the rear auxiliary roll stiffness. Inthese circumstances, it is reasonable to draw some conclusions about the relative effect ofremoving the driveline couplings.There are no striking differences in the predicted dynamic response of the two CarSimEdmodels, and given the minor differences in the data describing the models, that is not
surprising.The understeer plots (Figs 15 and 19) show that the maximum lateral acceleration is 0.5g,which is quite high for normal driving. In both cases the understeer plots show that thevehicles are exhibiting neutral steer.Body roll, as shown in Figs. 18 and 22, is a maximum of 3.7 degrees with the couplings, and3.9 degrees without the couplings, which indicates that the auxiliary stiffness of thedriveline couplings does contribute to the overall roll control of the vehicle. The differences are not expected to be sufficient to make any noticeable difference duringnormal driving, but may effect control at or near the limits of performance.
7 ConclusionsTesting of driveshaft rubber couplings shows bending stiffness varies from 2.9 Nm/deg to4.7 Nm/deg. Used couplings were stiffer than a new coupling.The driveshaft rubber couplings contribute some auxiliary roll stiffness to the rearsuspension of the vehicle. The auxiliary roll stiffness of the stiffest couplings could addapproximately 25% of the roll stiffness of the anti-roll bar fitted to the front suspension.If the driveshafts are replaced with shafts which do not include rubber couplings (e.g. shaftswith constant velocity joints), the change in auxiliary roll stiffness is not expected to makeany noticeable difference during normal driving, but may effect control at or near the limitsof performance.
