3d modeling of a suspension arm

[Mech75]

Hello,
I am trying to model a basic automobile suspension arm in 3 dimensions. The model consists of basically a triangle made up of 3 vectors a, b, and c. Vector a is "ground" and magnitude and direction is known. Magnitudes of vectors b and c are also known. So, I can write the following equations.

a + b + c = 0

and the writing the scalar forms gives me 3 equations:

a*cos(alpha_a) + b*cos(alpha_b) + c*cos(alpha_c) = 0
a*cos(beta_a) + b*cos(beta_b) + c*cos(beta_c) = 0
a*cos(gamma_a) + b*cos(gamma_b) + c*cos(gamma_c) = 0

I have 6 unknown angles and they are:
alpha_b
beta_b
gamma_b
alpha_c
beta_c
gamma_c

I had the model working when the grounded vector a was directed along the x-axis say
[1 0 0]. Because in this case, i could use the law of cosines to find alpha_b and alpha_c.
Also, in this case the following two additional constraints can also be written.
beta_b + beta_c = pi
gamma_b + gamma_c = pi

But when the grounded vector is rotated say 30 degrees about y and 30 degrees about z to
[0.750 -0.433 0.5000] then all my additional constraints break down and I am left with only 3 equations and 6 unknowns. I suspect that the additional equations i need are somehow related to the rotation of the grounded vector, but i am not certain.

Any help is appreciated.

Thanks,
Mech75

 

[eljose79]

Dear Mech75,

Thank you for sharing your question with us. It sounds like you are working on an interesting and challenging project. I have experience with mathematical modeling and I would be happy to offer some guidance.

Firstly, it is important to note that the additional constraints you mentioned (beta_b + beta_c = pi and gamma_b + gamma_c = pi) are only valid when the grounded vector a is aligned with the x-axis. When the vector is rotated, these constraints may no longer hold true. This is because the angles between the vectors will change with the rotation.

To solve this problem, you will need to incorporate the rotation of the grounded vector into your equations. One approach could be to represent the rotation using a rotation matrix or quaternion, and then apply this transformation to your equations. This will allow you to calculate the new angles between the vectors after the rotation.

Alternatively, you could use a different set of equations to define the angles between the vectors. For example, you could use the dot product between the vectors to find the angle between them. This approach may be more flexible and easier to incorporate the rotation.

I hope this helps you in your modeling project. As always, it is important to carefully consider your assumptions and to validate your model against experimental data. Good luck with your project!