Force, linear and angular acceleration, car & wheels help

[Chris42163]

I'm building an excel dragstrip model. I already have a working model that incorporates force exerted at the rear tires, force of drag, and force of rolling resistance. Now, I want to take into account the linear force that acts on the front tires that is required to accelerate them. There is both a linear component and an angular component. The linear component is already accounted for, because the mass of the wheel, tire, brakes, and spindle are included in the overall mass of the car. The angular inertia of these components is not accounted for.

I am not a regular here, and am not good at typing stuff out. I did some beer math and a free body diagram, and am surprised at the result. I have never seen it calculated this way, and have searched google all day to no avail. Please confirm or shoot down my work. Thanks!

All of the other forces are independent of angular acceleration, but the force at the front wheels, "f_f," must be calculated as part of the acceleration equation.

equation 1:
f_net=m_c*a, where m_c is the mass of the entire car, and a is the total acceleration of the car

equation 2:
f_net=f_r-f_d-f_rr-f_f, where the forces are defined above, and I prefer to use the absolute value of the forces, but I couldn't find the abs val symbol. So I will treat all forces with positive values that should be subtracted if they act against thrust, or f_r.

equation 3:
torque = Iα

equation 4:
Also, torque = f_f*r_f, where r_f is the radius of the front tire and f_f is the linear force applied at the front tire.

equation 5:
I=moment of inertia of the wheel, which =m_f*r_i², where m_f is the mass of the front wheel, tire, brake rotor, spindle, and r_i is the effective inertial radius

r_i is going to be some radius between the axis of rotation and r_f where, for the purposes of calculating the moment of inertia, all the mass of the rotating assembly could be concentrated to a single radius. Therefore, equation 6: r_i=r_f*C_i, where C_i is some coefficient between 0 and 1 that represents the percentage of the distance. substituting that back into the equation for I, above:

equation 7:
I=m_f*r_f²*C_i²

equation 8:
α=angular acceleration, which = a/r_f, where a is the linear acceleration of the car. α is given in rad/s

I've worked out things to substitute for torque, I, and α. So,

equation 9
(f_f*r_f)=(m_f*r_i²)*(a/r_f)

substituting equation 6 for r_i into equation 9, we get
equation 10
(f_f*r_f)=(m_f*r_f²*C_i²)*(a/r_f)
Simplifying:
f_f=(m_f*r_f²*C_i²*a)/r_f²
cancel like terms:
f_f=m_f*C_i²*a

Now for f_f, plug equation 10 into equation 2
equation 11
f_net=f_r-f_d-f_rr-(m_f*C_i²*a)

For f_net, substitute equation 1 to get:
equation 12
m_c*a=f_r-f_d-f_rr-(m_f*C_i²*a)
now solve for a:
(m_c*a)+(m_f*C_i²*a)=f_r-f_d-f_rr
a*(m_c+m_f*C_i²)=f_r-f_d-f_rr
and finally:
a=(f_r-f_d-f_rr)/(m_c+m_f*C_i²)

If I've done this correctly, I have all of the values on the right hand side of the equation in my model and can solve for a. It didn't make as big a difference in the model as I expected, and I was surprised by the cancellation of r_f in equation 10, because it means that the radius of the front tire doesn't actually matter, only the ratio between the radius and the effective inertial radius, r_i.

Thanks for any insight or corrections.

 

[Chris42163]

BTW, here are couple of shots from the model inputs and outputs. I've scaled my car, and put it on the dyno. Then I took it to the track two days ago and ran a:
60': 2.2422
1/4 ET: 13.35s
1/4 Trap: 105.49mph

 

[Chris42163]

All I could find on google or elsewhere on the internet regarding angular velocity, acceleration, and the like were kinetic energy formulas like 1/2*mv^2, and others that substituted I for the mass, and angular velocity for v. I also found several integration formulas, that I didn't try to use, because my model treats acceleration in each gear as constant. This helps when you don't have a formula to model dyno results. There might be some way to convert those other formulas to work, and I wasn't sure if there was anything wrong with the way I did things.

I was hoping someone on this forum might have some feedback or input as to whether or not the direction I went looks right. There seems to be no basis for my method outside of my own project. If I'm right, this formula should work to describe the effects of adding a uniform linear force to a wheel that does have mass. It should allow the simultaneous calculation of both the acceleration of the object and of the angular acceleration of the wheel. Hasn't anyone else done that calculation in a physics class before, or something?