Vehicle Dynamics Estimation

8/13/2019 Vehicle Dynamics Estimation

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Vehicle Centre of Gravity

It all starts with a definition

Locating the Center of Gravity (CG) in three dimensions (length, width and height) is an important first

step to analysing many aspects of a vehicles performance, especially the suspension characteristics. Forinstance, the length and height dimensions are important when analyzing anti-dive and anti-suat

suspension geometry.

!n this article !"m going to try to e#plain the $%ath& 'y deriving the main euations. For those of us who

are challenged to remem'er 'ac as far as high school or first year college math and hysics, don"t worry,

although ! use some trigonometry functions to derive the formulas, the end result uses $real& measura'lenum'ers.

!n order to calculate the length and width dimensions of the CG, the vehicle will need to 'e weighed on a

flat level surface. *sually this entails measuring the weight at each wheel 'y placing a scale under each. +e

careful when lowering the wheels onto the scales, so as not to push the scale sideways. !f this does happen

you ris causing the mechanism to 'ind which may result in a false reading. Commercial scales for

measuring the corner weights of race cars are availa'le that allow the vehicle to 'e driven onto the scalesthus avoiding this concern.

ote you might consider looing around at the local performance car clu's. hese clu's will

often have invested in a set of scales for mem'ers to use. /hops that setup race cars or

performance street suspensions may also have a set of scales.

o find the height dimension of the CG, the vehicle will need to 'e safely raised (as high as possi'le) at one

end and the weight under each wheel again measured while in the raised position. o avoid the previously

identified scale 'inding concern, the rear wheels are often easiest to lift, especially if the vehicle is

euipped with a solid rear a#le. %y formula"s assume that the rear has 'een lifted.

0ith reference to the various figures, we need to first measure the values of the following varia'les

Variable Description

1w' 0heel +ase - measures the distance 'etween front and rear wheel centres. his can 'e

accomplished 'y suspending a plum' 'o' from the centre of a front and rear a#le on the same side

of the vehicle and measuring 'etween the suspended strings.

1tf Front rac - measures the distance 'etween the centres of the front tyre ground contact patches.

1tr 2ear rac - measures the distance 'etween the centres of the rear tyre ground contact patches.



0fl 0eight under the front left wheel ( with vehicle on level ground )

0fr 0eight under the front right wheel ( with vehicle on level ground )

0rl 0eight under the rear left wheel ( with vehicle on level ground )

0rr 0eight under the rear right wheel ( with vehicle on level ground )

1up he height that 'oth rear wheels are lifted off the ground. For convenience and accuracy, this

should 'e measured 'etween the rear wheel centre and the ground. *sually a lift height of 34-35

inches is the ma#imum necessary to achieve usa'le results.

1w'up 0heel +ase lifted - measures the horizontal distance 'etween front and rear wheel centres once the

vehicle is lifted 'y height of 1up. 1w'up should in all cases 'e less than 1w'. he measurement

can 'e accomplished in the same way as 'efore, 'y suspending a plum' 'o' from the centre of a

front and rear a#le on the same side of the vehicle and measuring 'etween the suspended strings.

0flup 0eight under the front left wheel ( after lifting the rear wheels 'y height 1up )

0frup 0eight under the front right wheel ( after lifting the rear wheels 'y height 1up )

0rlup 0eight under the rear left wheel ( after lifting the rear wheels 'y height 1up )

0rrup 0eight under the rear right wheel ( after lifting the rear wheels 'y height 1up )



1wr 1istance of the centre of the a#le to the ground, when all four wheels are sitting level on the

ground.

Table

Length Dimension

0ith reference to the side view of Figure 3, we can see that the distance 1cgl defines the

length co-ordinate of that imaginary point where the total weight of the vehicle would 'alance front to rear.



Figure 1

!f we tae a point 6 some distance 7 in front of the vehicle we find the moment euationfor that point is descri'ed 'y

(7 8 1cgl) 9 0total

Looing at the diagram we can see that the following conditions

hold

0front : 0fl 8 0fr

0rear : 0rl 8 0rr

0total : 0cg : 0front 8 0rear

/u'stituting varia'les and generating a 'alanced euation of moments we

get

( 7 8 1cgl) 9 0total : (7 9 0front) 8 ((7 8 1w') 9 0rear)

!f we now define 7 : ;, the front wheel a#le centre 'ecomes the pivot point. /u'stituting for 0total and

solving for 0cgl we derive our first euation



!uation 1" 1cgl : 1w' 9 0rear

0front 8 0rear

#idth Dimension

Figure $

Looing at the rear view of figure 4 we can see that the following conditions hold

0left : 0fl 8 0rl

0right : 0fr 8 0rr

0total : 0cg : 0left 8 0right

+y the same reasoning used to derive euation 3, we see that we can generate an euation of moments

a'out the point + as follows

(< 8 1cgw) 9 0total : (< 9 0left) 8 ((< 8 1tf) 9 0right)



6s 'efore if we tae <:;, the left wheels 'ecome the pivot point. /u'stituting for 0total and solving for

1cgw we derive our second euation

!uation $" 1cgw

:

1tf 9 0right

0left 8 0right

*sing these two euations together, we see in the top view of figure =, how the co-ordinates relate to each

other.

Figure %

From a practical measurement perspective it may 'e easier to find the centreline of the vehicle and measure

an offset in the transverse direction. his can 'e done 'y calculating 1cgwCentre>ffset as follows

!uation %" 1cgwCentre>ffset : 1cgw ? ( 1tf @ 4 )



otes

3. !f the result is negative we measure left of the longitudinal centre line. 6 positive result ismeasured to the right of the Longitudinal centre line as drawn in figure =.

4. !n this case !"ve specified the front and rear tracs to 'e eual. his is rarely the case and normally

the trac at one end of the car is a little wider than the other.

Figure &

Figure A shows how a wider rear trac effects the calculations. <ou can see that the additional rear width

can 'e descri'ed using two similar triangles of angle theta, where the opposite sides represent the additional

width with adBacent side lengths 1cgl ( ie. 3 ) and 1w' ( ie. 4 )as follows

an ( theta ) : ( 3 @ 1cgl ) : ( 4 @ 1w' )

0e also now that 4 : ( 1tr ? 1tf ) @ 4

/u'stituting 4 and solving for 3 we produce

!uation &" 3 : 1cgl9

( 1tr ? 1tf )

4 9 1w'



1cgwCentre>ffset now 'ecomes

!uation '" 1cgwCentre>ffset : 1cgw ? 3 ? ( 1tf @ 4 )

ote that if the front and rear tracs are the same width, 3 goes to zero and euation reverts to its

previous form as shown in euation =.

(eight Dimension

his one is a little more complicated, 'ut the way it is calculated has 'een around for years and once you

see how it is done, ! thin you"ll agree that the reasoning is rather clever. Dssentially, we assume that the

vehicle is a solid o'Bect and that the CG doesn"t move, within that o'Bect. his is true regardless of the

o'Bects orientation. !f we refer 'ac to Figure 3 E 4, we see that the location of the CG is not only defined

'y varia'les in the length (i.e. 1cgl) and width (i.e. 1cgw), 'ut also in height (i.e. 1cgh).



Figure '

o find the height of the CG, we need to lift one end of the car (in this case the rear) and weigh all four

wheels again using the same scales as 'efore. For 'est results it is necessary to immo'ilize the suspension

so that the springs do not compress. /pring compression changes the relationship of the wheel to the 'ody

and therefore also the CG. +y su'stituting the new weight measurements, we can find the $new& horizontal

dimension of the CG (ie. 1cgup) and produce the following euation

!uation )" 1cgup : 1w'up 9 0rearup

0frontup 8 0rearup

*sing the same reasoning that was applied to the derivation of euation 3, we now that the following

definitions hold

0frontup : 0flup 8 0frup

0rearup : 0rlup 8 0rrup

*sing similar triangles we now that the triangle with hypotenuse : 1w' and 6dBacent side : 1w'up is

similar to the triangle with hypotenuse : 1cgh and >pposite side : 1h# and the triangle with ypotenuse :

1cgl and 6dBacent side : 1cglup. For each triangle, the angle shown as +eta is also eual.

Looing at the diagram, we can see that the following conditions hold

1cgh : 1hy 8 1wr

1h# : 1cglup - 1cgup

/in( +eta ) : 1h# @ 1hy : 1hy : 1h# @ /in( +eta )

Cos( +eta ) : 1cglup @ 1cgl : 1cglup : 1cgl 9 Cos( +eta )

/u'stituting and simplifying

1cgh : (( 1cglup ? 1cgup ) @ /in( +eta ) 8 1wr

1cgh : (( 1cgl 9 Cos( +eta )) @/in( +eta ) ) ? ( 1cgup @ /in( +eta )) 8 1wr

1cgh : ( 1cgl @ an( +eta ) ) ? ( 1cgup @ /in( +eta ) ) 8 1wr

Furthermore we now that

/in( +eta ) : 1up @ 1w'



an( +eta ) : 1up @ 1w'up

/u'stituting further

1cgh : (( 1cgl 9 1w'up ) @ 1up ) ? (( 1cgup 9 1w' ) @ 1up ) 8 1wr

Finally, su'stituting into euation 5 and simplifying a little we derive the final euation

!uation *" 1cgh:

( 1cgl 9 1w'up ) 8 (( 1w'up 9 0rearup ) 9 1w' ) 8 1wr

1up ( 1up 9 ( 0frontup 8 0rearup ) )

Conclusion

here you have it. +y plugging a num'er of relatively simple measurements into the three main euations

( 3, 4 E H ), we can determine ( in theory ) the e#act location of the CG of any four wheeled vehicle.

6uthor %ar Dl'ers

1isclaimer !"m not an engineer, Bust a guy with access to relevant technical documentation and a sense of

curiosity. his article presents my thoughts, 'ased on the research and general reading that !"ve performed

on this topic. !f you find errors in my wor ! would appreciate hearing from you. ! can 'e contacted throughthe 000D1GD e-mail list (trIJmercury.lcs.mit.edu).

Vehicle Dynamics Estimation

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