The topic of the project is the rear suspension system of the Formula SAE car MG-13.18 of the RaceUP Team of the University of Padua.
Fig. 1: MG 13.18
In particular it is a double wishbone pull rod system with an anti roll bar: the wheel is connected by two bearings to the upright. The last one is linked to the frame by an upper and a lower wishbone. A rod connects the upper wishbone to a rocker, to wich the spring and the damper are linked. Also the anti roll bar is linked to the rocker thanks to a tie beam. Finally, another rod connects the upright to the frame to control the toe angle.
The rules of the competition prescribes at least a wheel travel of 25 mm both in bump and in rebound, while there are no other significant restrictions on the kinematics of the system.
The targets of this project are mainly to study the kinematics of the system, in particular the variation on the camber angle and the toe angle of the rear wheel in maximum bump and maximum rebound situation, the motion ratio between wheel and the group of spring-damper, the rotation of the anti roll bar in maximum pitch and maximum roll and in general the force acting on the rockers.
The modelling problem
The origin of the car’s reference frame is fixed to the projection on the ground of the furthest poin of the frame, while the axes are oriented as: X coincides with the longitudinal direction of the vehicle (pointing to the front) and Y axis points to the right of the car, Z axis points upward. Adams settings: MMKS, gravity along Z downward.
Rear Right Wheel Group
The modelling phase started from the kinematics of the system: the fundamental kinematics point were created in Adams:
|R1||1668.9000||281.2100||172.7100||Lower wishbone front pivot|
|R2||2122.2500||159.3700||118.0000||Lower wishbone rear pivot|
|R3||1995.0000||573.1300||156.0000||Lower wishbone outer ball joint|
|R4||1703.5800||306.3600||316.1000||Upper wishbone front pivot|
|R5||2144.1799||209.1600||253.4400||Upper wishbone rear pivot|
|R6||1995.0000||542.9200||370.5800||Upper wishbone outer ball joint|
|R7||1986.3950||501.1950||341.9960||Push rod wishbone end|
|R8||1930.0160||227.9500||154.6230||Push rod rocker end|
|R9||2095.0000||564.0000||240.0000||Outer track rod ball joint|
|R10||2136.3999||185.7500||176.0000||Inner track rod ball joint|
|R11||1713.8560||284.7930||166.5920||Damper to body point|
|R12||1891.4600||294.6480||194.1540||Damper to rocker point|
|R14||1995.0000||595.0000||256.5000||Wheel center point|
|R15||1898.7520||220.8780||152.2650||Rocker axis 1st point|
|R16||1899.7350||226.3660||143.9640||Rocker axis 2nd point|
Than the main groups of component were imported from the CAD used by the team to design the car. Unfortunately it wasn’t possible to import directly the bodies in standard formats (as parasolid for example), due to some incopatibility between the softwares. The problem was solved importing the bodies as *stl files (stereolitography) and than introducing manually for every body the mass, the position of the center of mass and the inertia tensor calculated by the CAD. The body imported were:
– Wheel: includes the tire, the rim, the hub the brake disc and some other secondary components;
– Upright: includes the upright and the brake caliper;
– Upper wishbone;
– Lower wishbone;
– Pull rod;
– Toe rod;
Fig. 2: rear right wheel group
Remark: the *stl model of every body was exported from the CAD in order to be positioned in the right place and with the right orientation once it has been imported in Adams. Then were created the joints:
– Revolute joint in R4 with the rotation axis pointing through R5 between the ground and the upper wishbone: it defines the rotation axis of the upper wishbone;
– Revolute joint in R2 with the rotation axis pointing through R1 between the ground and the lower wishbone: it defines the rotation axis of the lower wishbone;
– Spherical joints in R3 and R6 between the wishbones and the upright: they define the steering axis;
– Revolute joint in R15 with the rotation axis pointing through R16 between the ground and the rocker: it defines the rotation axis of the rocker;
– Spherical joints in R9 between the toe rod and the upright;
– Hooke joint in R10 between the toe rod and the ground;
– Spherical joint in R7 between the pull rod and the upper wishbone;
– Hooke joint in R8 between the pull rod and the rocker;
– Revolute joint in R14 between the wheel and the upright
A spring element was creted between the R11 point of the ground and the R12 point of the rocker, with a stiffness coefficient of 65,7 N/mm and a pre load of 621 N.
Fig. 3: joints of the model
Finally where created two motions:
– bump_rebound: applied on the ground contact point of the wheel, it moves the wheel in bump and rebound with a sinusoidal trend and a width of 25 mm; the formulation is:
Fig. 4: motion of the rear right wheel
- vertical wheel: it maintains the wheel with the same orientation during the bump_rebound motion
At the end the Gruebler count is:
Fig. 5: Gruebler count of the model
Rear Axis Model with Rigid ARB
This model is an extension of the last one, in fact it was imported also the rear left wheel group of the car. (remark: some changes were brought in the model: the group of the wheel and the group of the upright were united in one body calle “unsprung mass”, mainly to redouce the complexity of the model). Furthermore it has been imported the anti roll bar system, made of three bodies:
– ARB: it is the real anti roll bar (support and elastic elements, in black in the picture);
– Rod ties: there are two rod ties that link the ARB with the two rocker;
Fig. 6: rear axis suspension system
Then were created some additional joints for the anti roll bar system:
– Revolute joint in [2189.46, 0.0, 118.968] between the arb support and the ground, with the joint axis perpendicular to the XZ plane: it defines the rotation axis of the arb support;
– Two spherical joints in [1909.25, ±276.484, 162.591] between the rod and the rocker;
– Two Hooke joints in [2185.75, ±272.0, 162.326] between the rod and the arb support.
Fig. 7: joint of the anti roll bar system
The same sinusoidal motion located on the ground contact point of the rear right wheel was mantained on this model. At the end the Gruebler count was:
Fig. 8: Gruebler count of the model
Rear Axis Model with Flexible ARB
This model is an upgrade of the previous one, in fact the anti roll bar system gained the flexibility thanks to two FE part; in the picture can be seen the rigid support of the arb (in black) and the two flexible elements (in yellow):
Fig. 9: anti roll system with FE parts
The two FE part (called in the model “edge”) are perfectly the same, the only difference between them is the location. To build the right edge first it was necessary to create a spline curve define as:
Fig. 10: spline creation
Then were set the material (aluminum) and the beam type (3D beam), the reference spline:
Fig. 11: FE part creation
And finally the two terminal sections:
Fig. 12: sections for FE part
Section_arb_1: 13 x 5 mm; Section_arb_2: 26 x 11,5 mm
This configuration of the anti roll bar system is the one with the minimum stiffness, because the bending stiffness of the two flexible element is minimum (minimum moment of inertia of the beam section). It was built another version of the model, with the maximum stiffness arb configuration:
Fig. 13: maximum stiffness configuration of the anti roll system
Regardless of the model version, it was created a fix joint between the edge and the arb support, and a Hooke joint between the edge and the rod tie:
Fig. 14: joints of the model
Finally, a motion was created for each wheel in order to have the two tires moving in counter-phase, even with a wheel travel of 25 mm in bump and in rebound.
Simulations and analysis of results
REAR RIGHT WHEEL GROUP
During the bump phase (positive wheel travel), the camber angle takes negative values (the upper part of the tire points towards the center of the car) while in the rebound phase (negative wheel travel) it takes positive values. This behavior is good because when the car rolls, the outer tire is in bump situation and recovers the loss of camber due to the roll.
Fig. 15: plot of the camber angle as a function of the wheel travel
Positive wheel travel: towards negative values; Negative wheel travel: towards positive values
Fig. 16: plot of the caster angle as a function of the wheel traveL
Negative wheel travel: positive values (the front part of the wheel points towards the center of the car); Positive wheel travel: negative values (the front part of the wheel points towards the outside of the car); The variation of the toe angle for negative wheel tavel is much more marked with respect to the variation for positive wheel travel: it helps the car during the low speed and radius turn, because the configuration of the the inner wheel (in rebound,so pointing toward the center of the car)and of the outer wheel (in bump, so pointing outside the car) reduces the radius of curvature of the vehicle.
Fig. 17: plot of the toe angle as a function of the wheel travel
The rear track of the car changes with the wheel travel, in particular in the bump phase the track shortens, while in the rebound phase the track stretches.
Fig. 18: plot of the track as a function of the wheel travel
The range of the rotation of the rocker is about 29°; for positive wheel travel the rotation is positive (so the spring is compressed) and viceversa.
Fig. 19: plot of the rotation angle of the rocker as a function of the wheel travel
Force and deformation of the spring
In static conditions, the force developed by the sping is not null due to the preload. Positive wheel travel causes the compression of the spring (negative deformation) and positive force and vice versa.
Fig. 20: plot of the deformation and the force of the spring as a function of the wheel travel
REAR AXIS MODEL WITH RIGID ARB
Rotation of the anti roll bar support
The range of the rotation when the whole axis is concurrently in bump and rebound phase is about 35°. For positive wheel travel the rotation is positive, so the arb support rotates towards the center of the vehicle, while for negative wheel travel the rotation is negative.
Fig. 21: plot of the rotation of the anti roll bar support as a function of the wheel travel
REAR AXIS MODEL WITH FLEXIBLE ARB (MINIMUM STIFFNESS CONFIGURATION)
Rotation of the anti roll bar support
The support of the anti roll bar is perfectly vertical when the left and the right wheel are at the same level, while when there is the maiximum opposition of the two wheels, the support take a positive angle of 0,9°.
Fig. 22: plot of the rotation of the anti roll bar support as a function of the wheel travel
The edge deformation is about ±13 mm. The maximum deformation is obtained when there is the maiximum opposition of the two wheels.
Fig. 23: plot of the two wheel travel as a function of the deformation of the left edge
Fig. 24: plot of the two wheel travel as a function of the deformation of the right edge
Force developed by the edge
The force developed by the deformation of the edge has got a linear trend.
Fig. 25: plot of the force developed by the edgel as a function of the deformation
Fig. 26: maximum deformation of the anti roll bar system (minimum stiffness configuration)
REAR AXIS MODEL WITH FLEXIBLE ARB (MAXIMUM STIFFNESS CONFIGURATION)
Rotation of the anti roll bar support
The trend is similar to the previous case, but the maximum rotation angle is bigger.
Fig. 27: plot of the rotation of the anti roll bar support as a function of the wheel travel
Also the edge deformations are pretty similar to the previous case, but there are small variation on the values.
Fig. 28: plot of the two wheel travel as a function of the deformation of the left edge
Fig. 29: plot of the two wheel travel as a function of the deformation of the right edge
Force developed by the edge
Also in this case the force developed by the deformation of the edge has got a similar trend, but the values are higher because the configuration of the system is different (maximum stiffness).
Fig. 30: plot of the force developed by the edgel as a function of the deformation
Fig. 31: maximum deformation of the anti roll bar system (maximum stiffness configuration)
The hardest part of the work was the building phase, when for every body it has been necessary to import the geometry and the mass property manually. But at the end the model works.
However the model are a simplification of the reality, in fact no tire stiffness was introduced, the value of the force developed by the spring are referred to a steady state condition and no damping ratio was considered.
Anyway the kinematic analysis has been well performed and the force values obtained give a first approximation of the stress acting on the componets.