Force, linear and angular acceleration, car & wheels help

In summary, the author of the summary is building an excel dragstrip model. They already have a working model that incorporates force exerted at the rear tires, force of drag, and force of rolling resistance. Now, they 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.The author has worked out things to substitute for torque, I, and α. So,equation 9(ff*
  • #1
Chris42163
6
0
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!

1.jpg
2.jpg

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

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

equation 2:
fnet=fr-fd-frr-ff, 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 fr.

equation 3:
torque = Iα

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

equation 5:
I=moment of inertia of the wheel, which =mf*ri2, where mf is the mass of the front wheel, tire, brake rotor, spindle, and ri is the effective inertial radius

ri is going to be some radius between the axis of rotation and rf 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: ri=rf*Ci, where Ci 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=mf*rf2*Ci2

equation 8:
α=angular acceleration, which = a/rf, 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
(ff*rf)=(mf*ri2)*(a/rf)

substituting equation 6 for ri into equation 9, we get
equation 10
(ff*rf)=(mf*rf2*Ci2)*(a/rf)
Simplifying:
ff=(mf*rf2*Ci2*a)/rf2
cancel like terms:
ff=mf*Ci2*a

Now for ff, plug equation 10 into equation 2
equation 11
fnet=fr-fd-frr-(mf*Ci2*a)

For fnet, substitute equation 1 to get:
equation 12
mc*a=fr-fd-frr-(mf*Ci2*a)
now solve for a:
(mc*a)+(mf*Ci2*a)=fr-fd-frr
a*(mc+mf*Ci2)=fr-fd-frr
and finally:
a=(fr-fd-frr)/(mc+mf*Ci2)

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 rf 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, ri.

Thanks for any insight or corrections.
 

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  • #2
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

gears.JPG
Results.JPG
 
  • #3
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?
 

1. What is the relationship between force and acceleration?

The relationship between force and acceleration is described by Newton's Second Law of Motion, which states that the acceleration of an object is directly proportional to the net force acting on it and inversely proportional to its mass. This means that the greater the force applied to an object, the greater its acceleration will be, and the more massive the object, the less it will accelerate for a given force.

2. How do you calculate linear acceleration?

Linear acceleration can be calculated by dividing the change in velocity over a given time period by that time period. The formula for linear acceleration is a = (vf - vi)/t, where a is the linear acceleration, vf is the final velocity, vi is the initial velocity, and t is the time period.

3. What is angular acceleration?

Angular acceleration is the rate at which an object's angular velocity changes over time. It is measured in radians per second squared and is calculated by dividing the change in angular velocity by the change in time. The formula for angular acceleration is α = (ωf - ωi)/t, where α is the angular acceleration, ωf is the final angular velocity, ωi is the initial angular velocity, and t is the time period.

4. How does a car's engine create linear acceleration?

A car's engine creates linear acceleration by converting the chemical energy from fuel into mechanical energy, which is used to turn the car's wheels. As the wheels rotate, they push against the ground, creating a reaction force that propels the car forward. This force, combined with the car's weight and friction, causes the car to accelerate.

5. How do wheels help with acceleration?

Wheels help with acceleration by providing a point of contact with the ground, which allows for the transfer of force from the car's engine to the ground. The shape and material of the wheels also affect the amount of friction between the car and the ground, which can impact the car's acceleration. Additionally, wheels with a larger radius can cover more distance with each rotation, resulting in a greater linear velocity and ultimately, a higher acceleration.

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