Explanation of the mechanics of a rack and pinion system?

[shannonxtreme]

Hello everybody, I'm a final year student in mechanical engineering. For my final year project, I'm designing a spring system to store energy when stopping a hand wagon and use this to boost the wagon when starting. This involves a pinion attached to the axle of rotation for the wheels, and a rack connected to the spring such that the rack drives the pinion which in turn moves the wheels.
In my calculations, I need to know what the relationship between the downwards movement of the rack and the torque of the pinion is. For example, assuming I need the wagon to accelerate at "a" metres per second, with a load of M, I will need a thrust of F, obviously. To provide this thrust, I know the wheels have to turn at a certain angular velocity, but I do not know how to calculate this, nor do I know how to relate the necessary torque to this velocity or to the downwards force required by the rack.
Any help would be immensely appreciated.

 

[Alex Mercer]

Hello there!

I am a high school student. Obviously, I don't know much about physics, other than F=MA, L=Iω, so unfortunately, I can't help you with the equations.

However, fortunately I remembered and I thought I'd share some good program with you; my favorite. It is called Algodoo, from Algoryx Incs. It is totally free physics simulator for anyone with a personal computer. Here's the link>http://www.algodoo.com/download/

Although it is in 2D, this physics simulator is really fun to play around and do your research at the same time. Easy to learn. Plotting of data is available. Exact measures can vary with simulation frequency and so on.

Thus, I made a special scene for you. I don't know if it fits your description, but I tried. It took me about a good 2 hours though. You can use that created mechanism to use it in your wagon with your own design.
Here's a link to my creation for you> http://goo.gl/F6Mzyn
Remember, to open the file, you will need Algodoo.Good luck mate! Hope this helped!

P.S: P.M me if you need help.

 

[felmon38]

Hello, going in other direction, I'll try to find the equations.
K spring constant
δ length of compression
R radius wheel of train
r primitive radius of the gear
x train displacement .Origin end of the spring without tension.
y rack displacement "
M train mass
Ms spring mass
Mr rack mass
I inertial moment of the wheel with respect to the instantaneous center.
v final train speed
H height of the train mass center

In order to know what can do the spring, you can apply the energy equation:

K.δ²/2=Mv²/2

neglecting the wheel mass and the spring and rack mass
If you see this will be O.K. you can solve the four following equations :

w.r=[dot]y[/dot]-[dot]x[/dot]
w.R=[dot]x[/dot]
-F=(Ms/2+Mr).[ddot]y[/ddot]
F.(r+R)=I.[ddot]x[/ddot]/R+M.[ddot]x[/ddot]

Sorry; I think that something is wrong with LATEX, when I mean with [dot]x[/dot] is the first derivative of x.

 

[Dr.D]

When felmon38 speaks of the "primitive radius of the gear" I think he is referring to what is more commonly known as the Pitch Radius of the gear.

Before saying much more, I'd really like to seek a sketch of the system you want to analyze.