Technical Report - 9 Lotus Elan Rear Suspension The Effect of Halfshaft Rubber Couplings by T. L. Duell Prepared for The Elan Factory May 24 Terry Duell consulting 19 Rylandes Drive, Gladstone Park Victoria 343, Australia ph. +61 3 9338 464 tduell@planet.net.au http://www.planet.net.au/~tduell/cons.html
24 Terry Duell Published on a RISC OS system using!techwriter Pro+ Figures and diagrams prepared in!tau and!draw
1 Introduction The 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 driveline harshness and can cause some surge or shudder when starting from rest. This form of coupling 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, and hence contribute to the suspension forces and roll resistance. This report details an investigation into the effect of the halfshaft rubber couplings on the rear suspension. 2 Method The 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 suspension geometry and wheel travel, are approximate only, as a result of working from small scale drawings of variable accuracy. These drawings show the spring lengths for the normal ride position as well as at bump and droop conditions, but there is no information to confirm the wheel loads or the spring rates assumed for these drawings. Springs were measured and rates calculated. The distance between spring seats on the rear strut was also measured, along with front anti roll bar geometry. Calculations of wheel loads, from the spring deflections derived from the drawings are at odds with the reported kerb weight of the vehicle. As a result the estimated wheel loads and front to rear weight distribution may not be accurate, which will influence the accuracy of any predictions of the
vehicle dynamic response. 3 Coupling characteristics Four rubber couplings, designated A, B, C, and D, were tested to establish indicative bending 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 a dummy differential output shaft which was secured, vertically, and a halfshaft was then mounted to the coupling. A normal force was applied to the top of the halfshaft, and measured by a spring balance. The angle of the halfshaft was derived by measuring the deflection 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 the mounting holes and repeated at 18 degrees. It would have been desirable to also apply the force 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.
8 A1, Imp style, new (non interleaved) 15 A2-Imp style, new (non interleaved) Bending moment (Nm) 6 4 2 Bending moment (Nm) 1 5 5 1 15 2 Shaft angle (deg.) 1 B1-Lotus (Dunlop), new 1 2 3 Shaft angle (deg.) 15 B2-Lotus (Dunlop), new 8 Bending moment (Nm) 6 4 Bending moment (Nm) 1 5 2 1 2 3 Shaft angle (deg.) 1 C1-Lotus (Dunlop), used 1 2 3 Shaft angle (deg.) 15 C2-Lotus (Dunlop), used 8 Bending moment (Nm) 6 4 Bending moment (Nm) 1 5 2 1 2 3 Shaft angle (deg.) 1 D1-Metalastik, used 1 2 3 Shaft angle (deg.) 15 D2-Metalastik, used 8 Bending moment (Nm) 6 4 Bending moment (Nm) 1 5 2 1 2 3 1 2 3 Shaft angle (deg.) Shaft angle (deg.) Fig. 4. Derived bending moment-angle data All couplings exhibit a linear bending moment-angle relationship. The increase in slope at high angles shown in the results of tests A1, B2, and to a lesser degree for C1 and D2, is probably caused by the reaching the angular limit for that particular setup (E.G. the end of a mounting 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 as shown in Fig. 5.
1 8 A B C D Bending moment (Nm) 6 4 2 1 2 Shaft angle (deg.) 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 same bending moment resistance per degree of shaft angularity. Coupling B (Lotus, new) shows noticeably less resistance to bending. As coupling C is the same as A, only used, it is possible 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 a maximum 'bending stiffness' of 4.7 Nm/deg, and a minimum of 2.9 Nm/deg. 4 Analysis of suspension geometry Analysis of the rear suspension geometry (layout shown in Fig. 6) indicates that the coupling 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 unsprung assembly (wheel, strut, inner coupling etc.). This moment can be resolved as a vertical force at the wheel vertical centreline, applied over the horizontal distance from the wheel vertical centreline to the vertical centreline through the inner coupling. The variation of the moment arm with vertical wheel travel, is shown in Fig. 9. The combined effects of the coupling angles and moment arm with wheel travel, and the effect of the assumed upper and lower bounds of coupling stiffness on vertical wheel force, are shown in Fig. 1.
2 15 Fig. 6 Layout of rear suspension Inner coupling Outer coupling 1 Coupling angle (deg.) 5-5 -1-15 -2-25 5 1 15 2 Vertical wheel travel from droop (mm) Fig. 7 Coupling angle and wheel travel
22 2 Strut angle (Deg.) 18 16 14 5 1 15 2 Vertical wheel travel from droop (mm) Fig. 8 Suspension strut angle and wheel travel 445 44 moment arm (mm) 435 43 425 5 1 15 2 Vertical wheel travel from droop (mm) Fig. 9 Moment arm and wheel travel
1 Max. coupling stiffness Min. coupling stiffness Coupling force at wheel (N) 5-5 -1 5 1 15 2 Vertical wheel travel from droop (mm) Fig. 1 Vertical wheel force resulting from coupling deflection Table 1 shows derived suspension geometry data. Front Rear Camber change on bump (deg/mm) -.62 -.9 Spring/damper angle at bump (deg) 17.7 21.1 Spring/damper angle at ride (deg) 16. 18.8 Spring/damper angle at droop (deg) 13.5 15.7 Roll centre height, above ground at ride (mm) 75.4 59. Table 1. Derived suspension geometry data (approximate) Table 2 shows relevant measured and calculated suspension data, and table 3 shows measured 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 wheel station (kg) 25.4 25.4 Spring mean diameter (mm) 57.8 99. Spring wire diameter (mm) 8.23 1.9 Active coils 17.6 11.25 Spring free length (mm) 48.4 388.6 Spring rate (N/mm) 13.37 12.89 Wheel rate at ride height (N/mm) 7.8 12.89 Auxiliary roll stiffness (Nm/deg) 17. - Table 2. Measured and calculated suspension data (approximate) Front Rear Wire diameter (in) [mm].324 [8.3].43 [1.9] Mean coil diameter (in) [mm] 2.276 [57.8] 3.9 [99.6] Number of coils 19.6 11.25 Ends Squared & ground Plain Number of active coils 17.6 11.25 Free length (in) [mm] 16.8 [48.4] 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, at normal ride, in Figs. 12, 13 respectively. The effect of the couplings in resisting body roll can be considered by rotating the sprung mass 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 of wheel 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 in angle at each coupling of approximately 3 degrees. Hence, the effective auxiliary roll stiffness provided by the couplings is approximately 27 Nm/deg, and 17 Nm/deg, for the maximum and minimum assumed coupling stiffnesses, respectively. The couplings, at their stiffest, represent 25% of the auxiliary roll stiffness provided by the anti-roll bar fitted fitted at 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 to examine the possible influence of the couplings on the vehicle dynamics, some assumptions have 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. If we allow 8 kg for the driver mass, and approx. 23 kg for fuel ( about 27 litres), the mass of the vehicle is likely to be about 745 kg. If we now assume a mass distribution, front/rear, of 5%/5%, it is possible to test these assumptions 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 of the sprung weight is 1575N. The axial spring force as a result, will be about 1491N (the strut is at angle of about 18.8 deg.). Hence the spring deflection, from no load, will be about 115.6mm. Therefore we would expect the distance between the spring seats to be the spring free length less the deflection (388.6-115.6), which is 273mm. The drawing shows about 224mm between spring seats at normal ride. Hence we must conclude from this that either the weight on the rear wheels is higher than assumed; the spring rate is lower, or possibly both. If we apply the same argument to the front suspension we arrive at an axial spring deflection of about 157.5mm, and an axial spring length of about 25mm. The drawing shows the axial spring length to be about 26mm at normal ride. Hence again we must conclude that the weight 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 distribution cannot help. The assumed spring rates are the calculated rates which are very close to the reported nominal rates, hence it is concluded that the drawings may not be applicable (ie they may be for a prototype version using different springs. It is noted that rear suspension layout shows a variable rate spring which was not fitted in production), and that the assumptions are reasonable enough to use for the analysis. Data for the model 26R (the racing version of the Elan) states that the mass distribution is 48%/52% (F/R) with two up, hence an assumption of 5/5 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, driveline couplings, and the combination of the two. Full suspension droop is arbitrarily set at zero deflection. The effect of the couplings can be seen 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 general arrangement diagrams and measurements. The bump stop characteristic have been assumed.
6 Zero coupling deflection Full droop deflection Full bump Vertical wheel force (N) 4 2 Spring force Bump stop contact Driveline coupling force Normal ride deflection Combined spring & coupling force -2 2 4 6 8 1 12 Vertical wheel deflection (mm) Fig. 14 Rear wheel force-deflection curve 5 Vehicle dynamics analysis Models of the Elan, using the derived data, were built in CarSimEd. CarSimEd is a reasonable comprehensive 3d handling analysis simulation, limited only by a restriction on spring and damper characteristics being linear, hence it is not possible to model varying rate springs or varying rate damper bump and rebound characteristics. These limitations vary in significance depending upon the condition being simulated. Rebound damping was set to be the equivalent of.6 of critical damping (vertical, at the wheel) for both vehicle models. A typical rebound damping rate was used, as rebound damping is more likely to influence handling manoeuvres than bump damping. One Elan model reflected the original specification and included the rear drive shaft rubber couplings as an auxiliary rear roll stiffness, using the maximum bending stiffness that had been established from the tests. The other model did not include any auxiliary rear roll stiffness, reflecting the situation where the rubber couplings had been replaced with CV joints or similar arrangement. The handling of the vehicle models was compared for constant speed step steer, at 5 km/h. The step steer involves driving the vehicle straight and then rapidly making a step steer input attempting to follow a 4m diameter circle. This allows the transient behaviour to be examined when the step steer is applied, and also the steady state handling under constant lateral acceleration whilst negotiating the circular path. The vehicle speed was chosen to avoid severe rear bump stop deflection, and hence stay within 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 without driveline couplings Wheelbase (mm) 2133.6 2133.6 Track front (mm) 1196. 1196. Track rear (mm) 123. 123. Gross mass (kg) 745.4 745.4 Unsprung mass, front (kg) 5.9 5.9 Unsprung mass, rear (kg) 5.9 5.9 Front spring rate (N/mm) 13.4 13.4 Rear spring rate (N/mm) 12. 12.9 Front damper rate (Nsec/mm) 4. 4. Rear damper rate (Nsec/mm) 1.9 1.9 Front spring/damper installation ratio Rear spring/damper installation ratio Front auxiliary roll stiffness (Nm/deg) Rear auxiliary roll stiffness (Nm/deg) Camber change on bump (deg/mm) Centre of gravity height (mm) Front roll centre height (mm).777.777 1. 1. 17. 17. 27.. -.62 -.9 4 4 75.4 59. Table 4. Estimated data for CarSimEd models Results 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.
Steering-wheel angle - deg 6 5 4 3 2 1-1 -.1.1.2.3.4.5.6 Vehicle lateral acceleration - g's Tire vertical load - N 26 Fig.15 Understeer plot, Elan with couplings 24 22 2 18 16 14 Fz_LF Fz_LR Fz_RF Fz_RR 12 1 2 4 6 8 1 12 14 16 18 2 22 24 26 Time - sec Fig. 16 Tyre vertical forces (N), Elan with couplings
Slip angle - deg.5 -.5-1 -1.5 Alpha_LF Alpha_LR Alpha_RF Alpha_RR -2-2.5 2 4 6 8 1 12 14 16 18 2 22 24 26 Time - sec 3.5 3 2.5 2 1.5 1.5 -.5 Body roll - deg 4 Fig. 17 Tyre slip angles (deg), Elan with couplings 2 4 6 8 1 12 14 16 18 2 22 24 26 Time - sec Fig. 18 Roll angle (deg), Elan with couplings
Steering-wheel angle - deg 6 5 4 3 2 1-1 -.1.1.2.3.4.5.6 Vehicle lateral acceleration - g's Tire vertical load - N 28 26 24 22 2 Fig. 19 Understeer plot, Elan without couplings 18 16 14 Fz_LF Fz_LR Fz_RF Fz_RR 12 1 2 4 6 8 1 12 14 16 18 2 22 24 26 Time - sec Fig. 2 Tyre vertical forces (N), Elan without couplings
Slip angle - deg.5 -.5-1 -1.5 Alpha_LF Alpha_LR Alpha_RF Alpha_RR -2-2.5 2 4 6 8 1 12 14 16 18 2 22 24 26 Time - sec 3.5 3 2.5 2 1.5 1.5 -.5 Fig. 21 Tyre slip angles (deg), Elan without couplings Body roll - deg 4 2 4 6 8 1 12 14 16 18 2 22 24 26 Time - sec Fig. 22 Roll angle (deg), Elan without couplings 6 Discussion The CarSimEd simulations do leave a bit to be desired regarding absolute accuracy. The real vehicle would exhibit non linear suspension force-deflection and non linear bump and rebound damping. It is also possible that the real vehicle mass and mass distribution will not be in accordance with the data used for the models. These differences between the models and real vehicles will affect the dynamic response. The simulations that have been conducted have provided an indication of the relative dynamic response, with all parameters constant other than the rear auxiliary roll stiffness. In these circumstances, it is reasonable to draw some conclusions about the relative effect of removing the driveline couplings. There are no striking differences in the predicted dynamic response of the two CarSimEd models, 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.5g, which is quite high for normal driving. In both cases the understeer plots show that the vehicles are exhibiting neutral steer. Body roll, as shown in Figs. 18 and 22, is a maximum of 3.7 degrees with the couplings, and 3.9 degrees without the couplings, which indicates that the auxiliary stiffness of the driveline couplings does contribute to the overall roll control of the vehicle. The differences are not expected to be sufficient to make any noticeable difference during normal driving, but may effect control at or near the limits of performance. 7 Conclusions Testing of driveshaft rubber couplings shows bending stiffness varies from 2.9 Nm/deg to 4.7 Nm/deg. Used couplings were stiffer than a new coupling. The driveshaft rubber couplings contribute some auxiliary roll stiffness to the rear suspension of the vehicle. The auxiliary roll stiffness of the stiffest couplings could add approximately 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. shafts with constant velocity joints), the change in auxiliary roll stiffness is not expected to make any noticeable difference during normal driving, but may effect control at or near the limits of performance.