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Modelling and analysis of suspension systems 171 Variables such as BZ-AZ are defined using system variables which measure components of displacements between markers, such as DZ(1414,1411). The REQUEST statement REQ/1 demonstrates how to access the information calculated by the VARIABLE statements. The alternative method of writing a FORTRAN subroutine is demonstrated in Table 4.7 by the listing of a user-written REQSUB developed specifically for a double wishbone suspension. The subroutine would be called from the main data set as follows: REQUEST/id,FUNCTIONUSER(1,par1,par2,par3,par4,par5,par6, par7,par8,par9) where the parameters par1, par2, …, par9 are the various items of data outlined in the subroutine. 4.5.8 Calculation of wheel rate The wheel rate for a suspension system can be thought of as the stiffness of an ‘equivalent’ spring acting between the wheel centre and the vehicle body as shown in Figure 4.31. This is the definition most useful for developing basic full vehicle MBS models where the wheel will be modelled as rigid with a separate tyre model. This differs slightly from other definitions sometimes used for wheel (or suspension rate) where the force displacement curve is measured at the centre of the tyre contact patch. In a quarter vehicle MBS model this would simply involve moving the point of jack contact with the wheel from the wheel centre to the tyre contact patch. The wheel rate should also not be confused with the term ride rate. This is associated with the force displacement relationship between the vehicle body, or sprung mass, and the ground. To derive this with a quarter vehicle model it would be necessary to model an additional spring, representing Fw kw VEHICLE BODY w kw Fs ks Equivalent spring acting at the wheel centre w Fw s ls A lw Fig. 4.31 Equivalent spring acting at the wheel centre

172 Multibody Systems Approach to Vehicle Dynamics the stiffness of the tyre, acting between the wheel centre and the jack with contact at the centre of the tyre contact patch. The suspension outputs discussed until this point have been based on the suspension geometry and as such have not required the inclusion of the road spring in the model. By including the road spring and plotting the force against the displacement in the jack translational joint, the wheel rate may be obtained from the slope of the curve at the origin. An estimate of the wheel rate may also be made as follows. Treating the road spring as linear gives the basic force displacement relationship Fs ks s (4.56) For the equivalent spring we also have Fw kw w (4.57) Taking moments about point A gives Fw (Ls/Lw)Fs (4.58) From the suspension geometry we can approximate the displacement in the road spring from s (Ls/Lw)w (4.59) This allows an estimate of the wheel rate, kw, based on the road spring stiffness and suspension geometry from kw Fw/w (Ls/Lw)Fs/(Lw/Ls)s (Ls/Lw) 2 ks (4.60) The introduction of a square function in the ratio can be considered a combination of two effects: (i) The extra mechanical advantage in moving the road spring to the wheel centre. (ii) The extra spring compression at the wheel centre. 4.6 The compliance matrix approach The use of a compliance matrix, in programs such as ADAMS/Car, is a method not commonly described in standard texts on vehicle dynamics but is well suited to an automated computer MBS analysis particularly when the influence of compliance requires consideration. The suspension compliance matrix relates incremental movements of the suspension to incremental forces applied at the wheel centres. The suspension compliance matrix is computed at each solution position as the suspension moves through its range of travel. Characteristics such as suspension ride rate and aligning torque camber compliance are computed based on the compliance matrix. The compliance matrix for a suspension system, [C], is defined as the partial derivatives of displacements with respect to applied forces [C] [/F] (4.61) If a system is assumed to be linear, the compliance matrix can be used to predict the system movement due to force inputs

172 Multibody Systems Approach to Vehicle Dynamics<br />

the stiffness of the tyre, acting between the wheel centre and the jack with<br />

contact at the centre of the tyre contact patch.<br />

The suspension outputs discussed until this point have been based on the<br />

suspension geometry and as such have not required the inclusion of<br />

the road spring in the model. By including the road spring and plotting the<br />

force against the displacement in the jack translational joint, the wheel rate<br />

may be obtained from the slope of the curve at the origin.<br />

An estimate of the wheel rate may also be made as follows. Treating the<br />

road spring as linear gives the basic force displacement relationship<br />

Fs ks s (4.56)<br />

For the equivalent spring we also have<br />

Fw kw w (4.57)<br />

Taking moments about point A gives<br />

Fw (Ls/Lw)Fs (4.58)<br />

From the suspension geometry we can approximate the displacement in the<br />

road spring from<br />

s (Ls/Lw)w (4.59)<br />

This allows an estimate of the wheel rate, kw, based on the road spring stiffness<br />

and suspension geometry from<br />

kw Fw/w (Ls/Lw)Fs/(Lw/Ls)s (Ls/Lw) 2 ks (4.60)<br />

The introduction of a square function in the ratio can be considered a combination<br />

of two effects:<br />

(i) The extra mechanical advantage in moving the road spring to the<br />

wheel centre.<br />

(ii) The extra spring compression at the wheel centre.<br />

4.6 The compliance matrix approach<br />

The use of a compliance matrix, in programs such as ADAMS/Car, is a<br />

method not commonly described in standard texts on vehicle dynamics but<br />

is well suited to an automated computer MBS analysis particularly when<br />

the influence of compliance requires consideration. The suspension compliance<br />

matrix relates incremental movements of the suspension to incremental<br />

forces applied at the wheel centres. The suspension compliance<br />

matrix is computed at each solution position as the suspension moves through<br />

its range of travel. Characteristics such as suspension ride rate and aligning<br />

torque camber compliance are computed based on the compliance matrix.<br />

The compliance matrix for a suspension system, [C], is defined as the partial<br />

derivatives of displacements with respect to applied forces<br />

[C] [/F] (4.61)<br />

If a system is assumed to be linear, the compliance matrix can be used to<br />

predict the system movement due to force inputs

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