2021-01-11T18:11:22Z

FsaeIllumina:

==Forces in 6 Suspension Tubes Per Corner==
In double wishbone suspension systems typically seen on FSAE cars, there are 6 tubes connecting the wheel assembly to the vehicle:
# Upper Wishbone, Fore
# Upper Wishbone, Aft
# Lower Wishbone, Fore
# Lower Wishbone, Aft
# Push/Pull Rod or Spring/Damper (direct suspension)
# Toe Rod or Steering Tie Rod

==Applications==

* Proper design and selection of suspension tubes
* Proper design of upright / knuckle
* Better understanding of forces during different loadcases
* Proper design of bellcrank
* Proper design of mounting brackets of suspension tubes onto chassis
* Reduce failures, while keeping weight low
* Compliance analysis

==Analysis of Only the Push/Pull Rod (Incorrect Method)==
Some teams have assumed that the vertical force of the tire at the contact patch is exactly equal to the vertical component of the push/pull rod force, and used the component forces / similar triangles / Trigonometry method to calculate the force in the push/pull rod, ignoring the additional forces of the other 5 suspension tubes. '''This is incorrect''', and can underestimate forces by a factor of 2 or more. The method is most inaccurate on pull-rod suspension. Here, the upper wishbone applies an additional vertical force to the wheel assembly, which increases the loads on the pull-rod. [[File:Pullrod Forces2.png|right|middle|thumb|Upper Wishbone Increasing Forces on Pull-rod]]

==Static Free Body Diagram 6x6 Matrix Method==
Numerous papers on this method exist. A google search for "Formula SAE Suspension Forces Matrix" shows a couple, or check the [[Suspension Forces#See Also|See Also]] Section. The rest of this section provides only a summary with a few pictures. The video provides a detailed explanation in a very easy to understand format.
{{#ev:youtube|https://youtu.be/XXXXXXXXX|1500|center|Formula SAE Suspension Forces Matrix Video|frame}}

===Assumptions===
Because all of these have spherical bearings on both ends, they are two-force-members, so they will only see tension/compression forces. If the push/pull rod is mounted to a control arm, then it will introduce bending forces in that control arm. In order to calculate the axial forces in all 6 tubes, it will be assumed that they are all two-force members.

Acceleration of the wheel assembly is ignored ignored here for simplicity.

===Theory===
Apply sum of forces equals zero (Fx, Fy, and Fz are 3 equations), and sum of moments (torques) equals zero (Mx, My, and Mz are 3 equations) to the wheel assembly. Break up all 6 suspension tube force vectors into their x, y, and z components multiplied by the unknown magnitude of the force in each arm. The 6 equations and 6 unknowns form a solvable 6x6 linear system. [[File:Free Body Diagram.png|right|middle|thumb|Free Body Diagram Showing 3 of 6 Suspension Arms]]

===Solving===

A summary of the solution is given here. It is a decently long process. It is recommended the reader follow through a paper, or watch a video.

First determine the x, y, z components of each of the 6 tubes. In vector form, this is the same as the unit vectors of each tube, multiplied by the magnitude of the force of each arm. The latter of which will be left as a variable because it is unknown. [[File:Suspension Forces 1.png|right|middle|thumb|Breaking up Force Vectors into x, y, z components]]
[[File:Equations With Highlights.png|right|middle|thumb|Force Equations with Unit Vectors Highlighted in Red and Unknowns Highlighted in Orange]]

Calculating the moments from each tube is best done in vector form as the cross product of the moment arm vector and the force vector. This will result in a moment vector that has a Mx, My, and Mz components, which can be placed into their respective <math display="inline"> \sum M_x = 0, \sum M_y = 0, \sum M_z = 0,</math> equation.

The 6 equations can be rearranged into a matrix equation of the form A*X=b, which can be solved easily with linear algebra.

==See Also==
[http://scholarworks.csun.edu/bitstream/handle/10211.3/123383/Flickinger-Evan-thesis-2014.pdf;sequence=1 DESIGN AND ANALYSIS OF FORMULA SAECAR SUSPENSION MEMBERS]

[http://www.iaeme.com/MasterAdmin/uploadfolder/IJMET FORCE CALCULATION IN UPRIGHT OF A FSAE RACE CAR]

[[Category:Suspension]]

Tires

2021-01-11T16:39:55Z

FsaeIllumina: added analysis section

Good smelling round things
=Theory Stuff=
[[G-g Diagram|G-g Diagram]]

Tires are rubber based, with metal cords to re-enforce the structure both on the contact face, and on the vertical “side walls”. They are most commonly pneumatic (i.e inflated with air), and the result is a substantially stiff rolling structure, with the rubber generating adhesion as well as mechanical “interlocking” with the asphalt, with the result being friction.Why do tires matter? Put simply, they are the only link of the car with the ground. While almost all systems are critical for the car operation, the tires have a special role in the car’s performance - a race-car generates forces and accelerations, both through driver inputs and engine power. In this way, the tires are essentially a “gate-keeper”, where the limit of force (and as such acceleration) has an upper limit. So, more engine power, or more brake power, is only helpful if the tires can handle it. Why is more acceleration better? On a race-track, better acceleration/deceleration in a straight line means higher top speed and later braking before the corner. In a corner, better acceleration means higher corner speeds for a given radius (a=v^2/r). It gets more complex. The tire is not fully rigid. If it was, it would not be able to “store” energy and generate cornering force. It is best to think of it like a spring, and cornering forces generate a torsion around the vertical axis going through the tire centerpoint. The twist angle generated is called “Slip angle”.

[[File:image34.png|center|middle|thumb|Slip angle ]]<a class="image"><img class="thumbimage" src="/images/thumb/d/db/Image34.png/300px-Image34.png" srcset="/images/thumb/d/db/Image34.png/450px-Image34.png 1.5x, /images/thumb/d/db/Image34.png/600px-Image34.png 2x" alt="" width="300" height="420" /></a> Slip angle
As the driver steers, the slip angle rises (and so does the cornering acceleration/force), but it does not do so linearly. This is a very important aspect of tire behavior.

[[File:Image6.png|center|middle|thumb|Tire Comparison (fictional)]]

Tire Comparison (fictional)Another related aspect is “alligning torque”, which is the moment the tire exerts around its centerline to return to neutral position (pre-deformation, i.e zero slip angle). How would it generate a moment? Only if the cornering force acts at a distance from the centerline of the tire.

[[File:Image8.png|center|middle|thumb|Pneumatic trail ]]Now, why is this important? Because, we are not very sensetive to accelerations - i.e, the driver cant just feel in his body that the car is at peak acceleration very accurately. As such, the feedback of the tires through the steering wheel is very useful.

[[File:Image151.png|link=https://www.semanticscholar.org/paper/Modeling-%2C-Analysis-and-Control-Methods-for-Vehicle-Takahashi/3f1dc5d4f82dce2e5a25cf20d470bbe68d4bb2b3|center|middle|thumb]]

This is where it should be noted that this torque is also non linear, relative to slip angle and to cornering force! What the driver would feel is increasing steering force, then gradually decreasing, while the car is still increasing in cornering force. This is due to the slippage of the contact patch changing location.

Another factor is the tire’s relationship to vertical load - i.e, why heavier cars dont corner as well as lighter cars. Tires operate with both “adhesion” and “Hysteresis” that can also be called as “mechanical interlocking” on a macro scale with the asphalt. A sidenote - The adhesion is vastly reduced in the rain, but the Hysteresis is not affected.

[[File:Image14.png|link=https://www.fhwa.dot.gov/publications/research/safety/14065/002.cfm|center|middle|thumb|Mechanisms of grip ]]

Now, the coefficient of friction of the tire is unitless, and it is the single most important measure of tire performance - or at least, the easiest one to see at a glance. We were taught in highschool physics that this friction coefficient is constant, and independent of surface area. That is not true! On some materials it may stay very near constant. But on a rubber tire, the coefficient is actually negatively correlated to vertical load - load goes up, coefficient goes down. This is substantial enough to be the single biggest justification for making a race-car lighter. Below, a primary reason for this behavior is seen. This is the macro level “interlocking” of the rubber with the asphalt, a very conventional friction model. As the vertical (i.e normal) load increases, the rubber expands to fill the road ridges, but it begins to saturate (i.e it cannot go any further into the asphalt).

[[File:Image29.png|link=http://insideracingtechnology.com/tirebkexerpt1.htm|center|middle|thumb|Load sensitivity ]]Consequentially, the coefficient = Fcornering/Fnormal decreases, as 1 unit of Fnormal increase results in a <1 unit of Fcornering increase - contributing to the value (being a ratio) to reduce.

[[File:Image20.png|link=https://en.wikipedia.org/wiki/Tire_load_sensitivity|center|middle|thumb|Load Sensitivity ]]

=Analysis=
==Tire Testing Consortium (TTC)==
Formula SAE tire testing data is available for a one time purchase of $500 from the TTC. http://www.millikenresearch.com/fsaettc.html

==Curve Fitting==
=== The Bill Cobb Approach ===
Bill Cobb has two excellent threads on how to get started fitting. Note: TTC members only
* [http://www.fsaettc.org/viewtopic.php?f=17&t=23 Matlab Tire Processing Code (Report Form)]
* [http://www.fsaettc.org/viewtopic.php?f=13&t=216 Estimating Pacejka model coefficients in Matlab]

=== Non-Dimensional Tire Modeling ===
Based on the Patton 2013 Paper. <ref> Patton, Chris. Development of Vehicle Dynamics Tools for Motorsports. Oregon State University. 2013.</ref>
* Code is available here: https://www.mathworks.com/matlabcentral/fileexchange/67987-analyzing-tire-test-data

== Suggested Reading ==
* Pacejka, Hans. Tyre and Vehicle Dynamics.
* Clark, Samuel K. (Editor), Mechanics of Pneumatic Tires, 2nd Ed. U.S. Dept. of Transportation, National Highway Traffic Safety Administration, 1981. U.S. Govt. Reference #: 050-003-00377-8.
* Hays, D.F. & Browne, A.L., The Physics of Tire Traction. Plenum, 1973. ISBN 0-306-30806-1.
* Kummer, H.W. & Mayer, W.E., Unified Theory of Rubber and Tire Friction. University Park, PA.: Pennsylvania State University, 1966
* Moore, D.F., The Friction of Pneumatic Tyres. Elsevier, 1975. ISBN 0-444-41323-5.

=List of Tires=
[https://www.millikenresearch.com/fsaettc.html Formula SAE Tire Test Consortium]

[https://www.hoosiertire.com/tires/circuit/spec/#panel-spec Hoosier Tires FSAE 2020]

[http://www.avonmotorsport.com/resource-centre/tyre-applications/sae-formula-student Avon Tyres Formula Student]

 

==Dry==

{| class="wikitable"
|-
|'''Make'''
|'''Model'''
|'''Size'''
|'''Weight'''
|'''Comments'''
|'''TTC'''
|-
|Continental
|
|
|
|
|
|-
|Goodyear
|
|
|
|
|
|-
| rowspan="9"|Hoosier <code><ref>https://www.hoosiertire.com/images/content/files/FormulaSAE19(1).pdf</ref></code> 
|
|16.0 x 6.0-10 
|~3.2kg 
|
|
|-
|
|16.0 x 7.5-10
|~3.6kg
|
|
|-
|
|6.0/18.0-10
|~3.6kg
|
|
|-
|
|18.0 x 6.0-10
|~4kg
|
|
|-
|
|18.0 x 7.5-10
|~4.5kg
|
|
|-
|
|19.5 x 6.5-10
|~4.5kg
|
|
|-
|
|19.5 x 7.5-10
|~5kg
|
|
|-
|
|20.5 x 7.0-13
|~5kg
|
|
|-
|
|20.0 x 7.5-13 
|~5.4kg
|
|
|-
|Pirelli 
|
|
|
|
|
|}

==Wet==

{| class="wikitable"
|-
|'''Make'''
|'''Model'''
|'''Size'''
|'''Weight'''
|'''Comments'''
|'''TTC'''
|-
| rowspan="2"|Hoosier
|
|
|
|
|
|-
|
|
|
|
|
|-
|
|
|
|
|
|
|}

[[Category:Suspension]][[Category:Vehicle Dynamics]]

=References=
<code><references /></code>