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Modelling and analysis of suspension systems 213 Equation 2 9f vs 436 6x 3 6z 435.157 (4.160) Equation 3 436f vs 9 6x 3 6y 1979.499 (4.161) This leaves us with four unknowns, 6x , 6y , 6z and f vs , but only three equations. We can use the same approach here as used with the tie rod in the preceding analysis. Since the spin degree of freedom of Body 6 about the axis C–I has no bearing on the overall solution we can again use the vector dot product to enforce perpendicularity of { 6 } 1 to {R CI } 1 as shown in Figure 4.71. This will yield the fourth equation as follows: { 6 } 1 • {R CI } 1 0 (4.162) ⎡ 3 ⎤ [ 6 6 6 ] ⎢ 9 ⎥ x y z 0mm/s ⎢ ⎥ ⎣⎢ 436⎦⎥ (4.163) Equation 4 3 6x 9 6y 436 6z 0 (4.164) The four equations can now be set up in matrix form ready for solution. The solution of a four by four matrix will require a lengthy calculation or access as before to a program that offers the capability to invert the matrix: ⎡ 3 0 436 9 ⎤ ⎡ f vs ⎤ ⎡ 120.555 ⎤ ⎢ 9 436 0 3 ⎥ ⎢ ⎥ ⎢ 6 435.157 ⎥ ⎢ ⎥ ⎢ x ⎥ ⎢ ⎥ (4.165) ⎢436 9 3 0 ⎥ ⎢6y ⎥ ⎢1979.499⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎣ 0 3 9 436⎦ ⎣6z ⎦ ⎣ 0 ⎦ Solving equation (4.165) yields the following answers for the four unknowns: f vs 4.561 s 1 6x 0.904 rad/s 6y 0.245 rad/s 6z 1.159 rad/s This gives us the last two angular velocity vectors upper and lower damper bodies: T { 6} 1 [ 0.904 0.245 1.159 ] rad/s T { 7} 1 [ 0.904 0.245 1.159 ] rad/s From equation (4.151) we now have ⎡ 3 ⎤ ⎡ 13.682 ⎤ VC 6C7 1 fvs RCI 4.561 ⎢ 9 ⎥ ⎢ 41.045 ⎥ mm/s 1 ⎢ ⎥ ⎢ ⎥ ⎣⎢ 436⎦⎥ ⎣⎢ 1988.378⎦⎥ { } { } (4.166) Since the velocity vector {V C6C7 } 1 acts along the axis of the strut C–I the magnitude of this vector will be equal to the sliding velocity V s : V s | V C6C7 | 1988.641 mm/s (4.167)

214 Multibody Systems Approach to Vehicle Dynamics Table 4.10 Comparison of angular velocity vectors computed by theory and MSC.ADAMS Body Angular velocity vectors Theory MSC.ADAMS x (rad/s) y (rad/s) z (rad/s) x (rad/s) y (rad/s) z (rad/s) 2 12.266 0.0 0.0 12.266 0.0 0.0 3 14.039 0.0 0.855 14.040 0.0 0.855 4 8.774 10 3 0.945 1.446 10 3 8.774 10 3 0.945 1.446 10 3 5 14.121 3.881 10 2 1.106 14.121 5.394 10 2 1.106 6 0.904 0.245 1.159 10 3 0.904 0.245 1.163 10 3 7 0.904 0.245 1.159 10 3 0.904 0245 1.163 10 3 Table 4.11 Comparison of translational velocity vectors computed by theory and MSC.ADAMS Point Translational velocity vectors Theory MSC.ADAMS V x (mm/s) V y (mm/s) V z (mm/s) V x (mm/s) V y (mm/s) V z (mm/s) C 120.555 435.157 1979.499 120.495 435.224 1979.570 D 204.345 112.260 3355.321 204.244 112.316 3355.440 G 0.0 110.394 3373.150 0.0 110.393 3373.130 H 252.524 112.968 3219.588 252.210 112.972 3219.588 P 166.468 108.859 3366.0 166.468 108.859 3366.0 C 6 C 7 13.682 41.045 1988.378 13.682 41.046 1988.440 At this stage it can be seen that the sliding velocity V s is realistic in magnitude. Given knowledge of the damper force–velocity relationship it would be possible to determine the damping forces produced and reacted at points I and C in the system. A comparison of the angular velocities found from the preceding calculations and those using an equivalent MSC.ADAMS model is shown in Table 4.10. A comparison of the translational velocities found at points within the suspension system from the preceding calculations and those found using an equivalent MSC.ADAMS model is shown in Table 4.11. 4.10.3 Acceleration analysis As before the approach taken here is to initially ignore the damper assembly between points C and I. Solving for the rest of the suspension system will deliver the acceleration {A C } 1 of point C thus providing a boundary condition allowing a separate analysis of the damper to follow. Before proceeding with the acceleration analysis it is necessary to identify the unknowns that define the problem. The angular accelerations and angular velocities of the rigid bodies representing suspension components can

214 Multibody Systems Approach to Vehicle Dynamics<br />

Table 4.10 Comparison of angular velocity vectors computed by theory and<br />

MSC.ADAMS<br />

Body<br />

Angular velocity vectors<br />

Theory<br />

MSC.ADAMS<br />

x (rad/s) y (rad/s) z (rad/s) x (rad/s) y (rad/s) z (rad/s)<br />

2 12.266 0.0 0.0 12.266 0.0 0.0<br />

3 14.039 0.0 0.855 14.040 0.0 0.855<br />

4 8.774 10 3 0.945 1.446 10 3 8.774 10 3 0.945 1.446 10 3<br />

5 14.121 3.881 10 2 1.106 14.121 5.394 10 2 1.106<br />

6 0.904 0.245 1.159 10 3 0.904 0.245 1.163 10 3<br />

7 0.904 0.245 1.159 10 3 0.904 0245 1.163 10 3<br />

Table 4.11 Comparison of translational velocity vectors computed by theory and<br />

MSC.ADAMS<br />

Point<br />

Translational velocity vectors<br />

Theory<br />

MSC.ADAMS<br />

V x (mm/s) V y (mm/s) V z (mm/s) V x (mm/s) V y (mm/s) V z (mm/s)<br />

C 120.555 435.157 1979.499 120.495 435.224 1979.570<br />

D 204.345 112.260 3355.321 204.244 112.316 3355.440<br />

G 0.0 110.394 3373.150 0.0 110.393 3373.130<br />

H 252.524 112.968 3219.588 252.210 112.972 3219.588<br />

P 166.468 108.859 3366.0 166.468 108.859 3366.0<br />

C 6 C 7 13.682 41.045 1988.378 13.682 41.046 1988.440<br />

At this stage it can be seen that the sliding velocity V s is realistic in magnitude.<br />

Given knowledge of the damper force–velocity relationship it would<br />

be possible to determine the damping forces produced and reacted at points<br />

I and C in the system.<br />

A comparison of the angular velocities found from the preceding calculations<br />

and those using an equivalent MSC.ADAMS model is shown in Table 4.10.<br />

A comparison of the translational velocities found at points within the suspension<br />

system from the preceding calculations and those found using an<br />

equivalent MSC.ADAMS model is shown in Table 4.11.<br />

4.10.3 Acceleration analysis<br />

As before the approach taken here is to initially ignore the damper assembly<br />

between points C and I. Solving for the rest of the suspension system<br />

will deliver the acceleration {A C } 1 of point C thus providing a boundary<br />

condition allowing a separate analysis of the damper to follow.<br />

Before proceeding with the acceleration analysis it is necessary to identify<br />

the unknowns that define the problem. The angular accelerations and angular<br />

velocities of the rigid bodies representing suspension components can

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