#
Front Suspension
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*Mechanical Systems* Model of a Transverse-Shock, Anti-roll Automotive
Front Suspension
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*by Robert Beretta*
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*Suspension Graphic*
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Reference
**Context**
Mech`Mech3D`
**Package Version**
1.3
*Mathematica* Version
2.2
**Copyright**
Copyright 1996, Dynamic Modeling
###
Discussion
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Suspension Function
This notebook describes the motion of a pull-rod, inboard damper automotive
front suspension system. The suspension system is analyzed to show
how changes in the linkage geometry affect the velocity ratio between wheel
and damper motion. A static force analysis is done to determine the loads
on each of the elements of the suspension linkage.
The pull-rod front suspension mechanism is a rising-rate, inboard damper
automotive suspension system that has inherent anti-roll characteristics.
Function is as follows:
The wheel carrier is primarily attached to the chassis by a classic
double A-arm. A primary tie rod attaches to the top of the wheel carrier
and extends downward toward the center and bottom of the car, such that
it is in tension when supporting the weight of the car. This tie rod is
attached to the rocker arm, which is forced to pivot about its axis on
the chassis as the suspension travels.
As the rocker arm pivots it pulls on the secondary tie rod, which passes
from the rocker arm across the bottom of the car to the toggle. Thus, the
right-hand wheel directly actuates the left toggle, and vice-versa.
The two damper-spring assemblies (shocks absorbers) lie horizontally
across the bottom of the vehicle, one in front of the other. Thus, the
suspension is asymmetrical in the vehicle, although its operation is symmetrical
with respect to the compression of the shocks relative to wheel travel.
Each of the two shocks attaches to both of the toggles, being controlled
simultaneously from both ends. If one wheel were to go over a bump, while
the other stayed still, the rotation of the toggle attached to that wheel
would cause one of the two shocks to be compressed, and the other shock
to be released.
The unique anti-roll characteristics of this suspension system can be
described qualitatively as follows:
Imagine that the end of each toggle that compresses the shock is 3 times
longer than the end that releases the shock. The vehicle then goes over
a bump that moves both wheels upward simultaneously 1 unit of travel. Call
it a 1 unit bump. The compression end of each toggle compresses its shock
3 units, while the release end of each toggle releases its shock 1 unit.
This results in a net compression of each shock of two units.
Now imagine that the vehicle is rolling in a corner such that the right
wheel travels upward 1 unit, and the left wheel travels downward 1 unit.
The toggle associated with the right wheel compresses its shock 3 units,
while the toggle associated with the left wheel compresses the same shock
1 more unit, for a total compression of 4 units. Similarly, the other shock
is released 4 units by the chassis roll.
To summarize, a 3:1 ratio between the length of the compression and
release ends of the toggles results in effective suspension stiffness that
is two times greater in pure roll than in pure bump. The magnification
of the anti-roll characteristic is adjusted simply by changing the ratio
of the lengths of the two ends of the toggles.
####
Suspension Model
#####
Bodies
The `Mech3D` model of the inboard damper suspension consists
of eight bodies, five of them moving:
1. The ground; this body serves as the
stationary reference frame for the entire model.
2. The chassis
3. The right carrier; since the wheel
does not need to spin in this model, this single body serves as both the
right wheel and right wheel carrier (the king pin).
4. The left carrier; this body serves as the
left wheel and left wheel carrier.
5. The right rocker; the rocker arm
that links the two tie rods for the right suspension.
6. The left rocker; the rocker arm that links
the two tie rods for the left suspension.
7. The left toggle; the final rocker
arm that ties the motion of the right wheel to its shock.
8. The right toggle; the final rocker
arm that links the motion of the left wheel to its shock.
#####
Coordinates
The 3D coordinate system of the model is as follows.
+X is out the back of the vehicle
+Y is out the right side of the vehicle
+Z is straight up
The origin of the global coordinate system is located on the surface
of the ground (Z), at the center of the car (Y), and directly between the
centers of the two front wheels (X). All units are assumed to be in inches.
The overall physical scale of the model is that of a 3/4 scale Formula
One car.
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Kinematic Model
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Preparation
This loads the required `Mech3D` package into *Mathematica*.
Names are defined for each of the body numbers for clarity.
####
Body Definitions
#####
The Ground
Two points are defined on the ground body:
P1. A point directly above the origin (+Z direction)
to define a vertical translation axis for the chassis.
P2. A point directly to the right side of the origin
(+Y direction) to use as a rotational orientation reference for the chassis.
#####
The Chassis
The local origin of the chassis is located at the bottom surface of the
chassis, at the center of the car, directly below the wheel axis. Thus,
the origin of the chassis is coincident with the origin of the ground when
the car's suspension is bottomed out.
Sixteen points are defined on the chassis.
P1. The center of rotation of the right upper A-arm on the chassis.
The upper ball joint on the right wheel carrier travels on a circular path
centered at this point.
P2 and P3. The two attachment points of the right upper A-arm
on the chassis. These are immediately in front of and behind point P1.
P4. The center of rotation of the left upper A-arm on the chassis.
This point is mirrored from point 1 across the plane normal to the local
Y axis.
P5 and P6. The two attachment points of the left upper A-arm
on the chassis. These are immediately in front of and behind P4.
P7. The center of rotation of the right lower A-arm on the chassis.
The lower ball joint on the right wheel carrier travels on a circular path
centered at this point.
P8 and P9. The two attachment points of the right lower A-arm
on the chassis. These are immediately in front of and behind P7.
P10. The center of rotation of the left lower A-arm on the chassis.
This point is mirrored from P7.
P11 and P12. The two attachment points of the left lower A-arm
on the chassis. These are immediately in front of and behind P10.
P13. A point on the pivot axis of the right rocker on the chassis.
P14. A point on the pivot axis of the left rocker on the chassis.
P15. A point on the pivot axis of the right toggle on the chassis.
The Z coordinate of this point sets the height of the center-line of the
right toggle.
P16. A point on the pivot axis of the left toggle on the chassis.
The InitialGuess for the chassis
corresponds to the chassis being 1.5" off of the ground (normal ride height).
Only the mass of the chassis is included in this model, because the masses
of the other components are negligible.
#####
The Right and Left Carriers
The local origins of the two wheel carriers are at the attachment points
of the lower A-arms.
Four points are defined on each wheel carrier:
P1. The attachment point of the upper A-arm
on the carrier.
P2 and P3. Two points that define the ends
of the axle on the wheel carrier.
P4. A point at the bottom of the tire, where
the tire touches the ground. This point is used to constrain the bottom
of the tire to be in contact with the ground.
The points on the left wheel carrier are a mirror of the points on the
right wheel carrier, mirrored across the local Y plane relative to the
right wheel carrier.
#####
Right and Left Rockers
The local origins of the rockers are located at the points where they attach
to the chassis.
Two points are defined on the each of the two rockers:
P1. The attachment point of the upper tie
rod on the rocker.
P2. The attachment point of the lower tie
rod on the rocker.
The left rocker is a mirror of the right rocker.
SetParameters is used to set the
value of a constant in the model; in this case the value of RockerLength.
#####
The Right and Left Toggles
The local origins of the toggles are located at the points where they attach
to the chassis.
Three points are defined on each toggle:
P1. The attachment point of the shock absorber
that is extended during bump.
P2. The attachment point of the lower tie
rod on the toggle.
P3. The attachment point of the shock absorber
that is compressed during bump.
The left toggle is a mirror of the right toggle.
#####
Incorporate the Body Objects into the Model
####
Constraint Definitions
#####
Constraints for the Chassis
1. A translational joint that allows the chassis to translate on
a plane that is normal to the global X axis.
2. A relative Y constraint controls the side-to-side position
of the car.
3. A relative Z constraint to set the height of the chassis by
specifying the Z component of the origin of the chassis. Positive `T`
causes the car to move towards the ground, `T` = 0 is equal
to 1.5 inch ride height.
4. A relative angle constraint controls the roll of the car.
Positive `T` causes the car to roll to the left.
Driving constraints 3 and 4 can be turned "on and off" by setting the
values of `Roll` and `Bounce` equal to 0 or
1.
#####
Constraints for the Right and Left Carriers
5. A constraint to model the right upper A-arm.
Two degrees of freedom constrained.
6. A constraint to model the right lower A-arm.
Two degrees of freedom constrained.
7. A constraint to set the steering angle
of the right wheel by controlling the included angle between the axle and
a fore-aft line on the ground. One degrees of freedom constrained.
8. A constraint to place the bottom of the
right tire on the surface of the road. One degrees of freedom constrained.
Constraints 9, 10, 11, and 12 control the left wheel carrier in a similar
fashion.
#####
Constraints for the Right and Left Rockers
13. A revolute constraint to model the pivot axis
of the right rocker. Five degrees of freedom constrained.
14. A constraint to model the right upper
right tie rod. One degree of freedom constrained.
15. A revolute constraint to model the pivot
axis of the left rocker. Five degrees of freedom constrained.
16. A constraint to model the left upper left
tie rod. One degree of freedom constrained.
#####
Constraints for the Left and Right Toggles
17. A constraint to model the lower right tie
rod. One degree of freedom constrained.
18. A revolute constraint to model the pivot
axis of the left toggle. Five degrees of freedom constrained.
19. A constraint to model the lower left tie
rod. One degree of freedom constrained.
20. A revolute constraint to model the pivot
axis of the left toggle. Five degrees of freedom constrained.
#####
Incorporate the Constraint Objects into the Model with SetConstraints
Any subset of the entire set of defined constraints may be used, so long
as the set adequately constrains some subset of the bodies in the mechanism.
Set 1: Constrain the chassis only.
Set 2: Constrain the chassis and right carrier.
Set 3: Constrain the chassis and left and right carriers.
Set 4: Constrain the chassis and right carrier and rocker.
Set 5: Constrain everything but the right toggle.
Set 6: All bodies.
###
Running the Model
#####
CheckSystem
`CheckSystem` tests for certain mathematical errors. A return
value of `True` means that no errors were found.
#####
SolveMech
`SolveMech[`*t*`]` seeks a solution that
satisfies all of the constraints with the value of time `T`
set to *t*. The solution is returned as a list of rules specifying
the position of the origin of each body, and the Euler parameters *Eo*,
*Ei*, *Ej*, and *Ek* that uniquely specify the angular orientation
of each body.
###
Output Functions
The length of the shock absorber is equal to the distance between the ends
of the two toggles. The following function returns the left and right shock
lengths as functions of the model's dependent variables. The right shock
is defined as the shock that gets compressed when the right wheel bumps.
In point of fact, the right shock is the rear shock.
Both shocks are the same length in bounce.
The degree of CCW body roll that would provide left and right wheel bounce
of 0.25 and -0.25 inches, respectively, is calculated as follows.
Shock lengths differ with 1 degree of body roll to the left.
It can be seen that body roll induces greater deflection of the shocks
than does an equivalent degree of bounce.
###
Velocity Solution
The following calculates the length of the shock at ride height, and the
rate of change of the length of the shock at ride height, with respect
to changes in `T`. The negative rate of change implies that
the shock is getting shorter as the car bounces. The
option for `SolveMech` causes the first time derivatives
of all of the dependent variables to be calculated.
###
Tabular Output
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Variations in Time
Tables of solution points can be created using `SolveMech`.
These tables can then be used to generate plots of various functions vs.
time.
`SetBodies` is first called with no arguments to reset
all of the initial guesses to the values defined previously in their respective
`Body` objects. `SolveMech` is then used
to move the initial guesses up to top-out in 2 steps.
`SolveMech` uses a special form of iterator: the following
input causes `SolveMech` to generate 11 solution points,
with values of time varying from -1.0 to 1.5; top-out to full bounce.
#####
Plot of Wheel to Shock Leverage Ratio as a Function of Bump Height:
This plot shows that this suspension is a "rising rate" suspension.
#####
Plot of Shock Length as a Function of Bump Height:
#####
Plot of Camber Angle as a Function of Bump Height:
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Plot of Tire Scrub as a Function of Bump Height:
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Variations in Rocker Length
To create a table of solution points while varying some parameter other
than time, the *Mathematica* `Table` function is used.
#####
Plot of Wheel to Shock Leverage as a Function of Rocker Length, with Bump
Held Constant at Zero:
###
Graphic Images
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Mech3D Graphic Objects
Here is a graphic object of chassis and ground plane.
Here is a graphic objects of wheel carriers.
Here is a graphic objects of right and left rockers.
Here is a graphic object of left and right toggles.
####
Graphics Displays
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Static Loads
Loads are applied to the model to simulate the shock absorbers. Load 1
simulates the right shock spring, with a specified spring free length and
spring rate. Load 2 models the left shock spring.
`SetLoads` applies the load objects to the model.
The following two inputs return the reaction forces that the ground applies
to the bottom of each tire at ride height; the ground is pushing up on
each tire with 99 pounds of force.
returns the reaction forces that constraint number 8 applies to the right
wheel carrier.
Since there are not, as yet, any gravitational loads applied to the chassis,
the vertical driving constraint (constraint 3) must apply a 197 pound downward
load to the chassis, to counter the 98 pound upward force on each tire.
A load representing gravity is added so that the equilibrium position of
the chassis can be found. The mass of the chassis was already defined in
the chassis `Body` object. Total weight on front of car is
193.2 pounds.
The gravitational loads now almost balance the suspension loads. Constraint
3 is the constraint that controls the vertical motion of the chassis.
Constraint 4 is the constraint that controls the roll of the chassis. There
is no load on this constraint because the chassis loading is symmetrical.
Two loads representing the side loads induced by a right hand corner are
added to the bottom of each tire. The magnitude of the side load on each
tire is proportional to the normal force on the tire, representing kinetic
friction. This condition would probably only exist during a slide, but
the normal case is indeterminate, so this has to do.
###
Equilibrium
A `FreeSystem` is built that has had constraints 3 and 4
dropped, the constraints that control the vertical position and the rotation
of the chassis. This allows the chassis to move to its equilibrium position.
The car is in equilibrium at a ride height of 1.5339 inches.
There is no chassis roll in this load state.
The car is now put into a 1 G right-hand turn.
The ride height of the car has gone up a bit.
which results in, 0.22 degrees of body roll to the left.
###
Component Loads
__Note:__ All of the following loads are based on the results
of the 1 G right-hand corner equilibrium calculation.
The load vectors, in global coordinates, applied to the right and left
wheel carrier upper A-arm attachment points by the upper A-arms:
The load vectors applied to the right and left wheel carrier upper A-arm
attachment points by the upper tie rods:
The magnitudes of the tensile forces in the right and left upper tie rods.
The load vectors, in global coordinates, applied to the right and left
wheel carrier lower A-arm attachment points by the lower A-arms:
The magnitudes of the tensile forces in the right and left lower tie rods.
The load vectors, in global coordinates, applied to the right and left
toggles by the chassis, at their pivot axes on the chassis.
The loads in the right and left shocks.
`SpringRate * (FreeLength - rightshocklength)/.sol
SpringRate * (FreeLength - leftshocklength)/.sol`
###
Other Forces
Converted by *Mathematica*
July 21, 1999 |