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Modelling and analysis of suspension systems 175 matrix approach used here, however, this is given by Az RW . In this system a positive steer angle results when the wheel turns to the left, which in Figure 4.33 is consistent with a positive rotation Az RW about the z-axis for the right wheel. In this case for the right wheel the steer angle would be given by Az RW C 12,6 Tz LW C 12,12 Tz RW . Similarly Figure 4.34 illustrates the determination of camber angle, Ax RW , resulting at the right wheel due to unit aligning torques acting through both the left and right wheel centres. In this system a positive camber angle results when the top of the wheel tilts away from the body, which in Figure 4.34 would actually be a negative rotation Ax RW about the x-axis for the right wheel. In this case for the right wheel the camber angle would be given by Ax RW C 10,12 Tz RW C 10,6 Tz LW . Needless to say the sign convention used to define positive steer and camber angles always requires careful consideration particularly when considering the definitions given here using a compliance matrix approach to measure movement of the road wheels relative to the vehicle body. 4.7 Case study 1 – Suspension kinematics The following case study is provided to illustrate the application of the methodology described in the previous sections to calculate the suspension characteristics as the suspension moves between the bump and rebound positions. Examples of the plotted outputs described here are shown in Figures 4.38 to 4.43. These plots were from a study based on the front suspension of a passenger car, considering the suspension connections to be joints, linear or non-linear bushes. The assembly of parts used to make up the front suspension system is shown schematically in Figure 4.35. Example data sets for this model are provided in Appendix A together with more detailed system schematics. Upper arm Tie rod Upper damper Road wheel Wheel knuckle Lower damper Lower arm Tie bar Jack part Fig. 4.35 Assembly of parts in the front suspension system example

176 Multibody Systems Approach to Vehicle Dynamics UNI SPH REV BUSH SPH CYL REV BUSH BUSH MOTION SPH FIX BUSH INPLANE MOTION TRANS Fig. 4.36 Modelling the front suspension example using bushes The modelling of the suspension system using bushes is shown in Figure 4.36. The upper link is attached to the body using a connection that is rigid enough to be modelled as a revolute joint. Bushes are used to model the connection of the lower arm and the tie bar to the vehicle body. Bushes are also used to model the connections at the top and bottom of the damper unit. Where the tie bar is bolted to the lower arm a fix joint has been used to rigidly connect the two parts together. This joint removes all six relative degrees of freedom between the two parts creating in effect a single lower control arm. The modelling issue raised here is that rotation will take place about an axis through these two bushes where the bushes are not aligned with this axis. As rotation takes place the bushes must distort in order to accommodate this. The modelling of these connections as non-linear, linear or as a rigid joint was therefore investigated to establish the effects on suspension geometry changes during vertical movement. For the suspension modelled in this manner it is possible to calculate the degrees of freedom for the system as follows: Parts 9 6 54 Fix 1 6 6 Trans 1 5 5 Rev 2 5 10 Uni 1 4 4 Cyl 1 4 4 Sphs 3 3 9 Inplane 1 1 1 Motion 2 1 2 ΣDOF 13

176 Multibody Systems Approach to Vehicle Dynamics<br />

UNI<br />

SPH<br />

REV<br />

BUSH<br />

SPH<br />

CYL<br />

REV<br />

BUSH<br />

BUSH<br />

MOTION<br />

SPH<br />

FIX<br />

BUSH<br />

INPLANE<br />

MOTION<br />

TRANS<br />

Fig. 4.36<br />

Modelling the front suspension example using bushes<br />

The modelling of the suspension system using bushes is shown in Figure<br />

4.36. The upper link is attached to the body using a connection that is rigid<br />

enough to be modelled as a revolute joint. Bushes are used to model the<br />

connection of the lower arm and the tie bar to the vehicle body.<br />

Bushes are also used to model the connections at the top and bottom of the<br />

damper unit. Where the tie bar is bolted to the lower arm a fix joint has<br />

been used to rigidly connect the two parts together. This joint removes all<br />

six relative degrees of freedom between the two parts creating in effect a<br />

single lower control arm.<br />

The modelling issue raised here is that rotation will take place about an axis<br />

through these two bushes where the bushes are not aligned with this axis.<br />

As rotation takes place the bushes must distort in order to accommodate<br />

this. The modelling of these connections as non-linear, linear or as a rigid<br />

joint was therefore investigated to establish the effects on suspension<br />

geometry changes during vertical movement. For the suspension modelled<br />

in this manner it is possible to calculate the degrees of freedom for the system<br />

as follows:<br />

Parts 9 6 54<br />

Fix 1 6 6<br />

Trans 1 5 5<br />

Rev 2 5 10<br />

Uni 1 4 4<br />

Cyl 1 4 4<br />

Sphs 3 3 9<br />

Inplane 1 1 1<br />

Motion 2 1 2<br />

ΣDOF 13

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