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Momentum and energy equations for a car rolling over a step 
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#1
Jan2013, 10:30 AM

P: 2

A model car with a main body mass M and four wheels each with mass m is rolling down a track at velocity v, when it encounters a step of height b, which is less than the wheel radius r. The wheels are rolling, not sliding, and we know the moment of inertia around their respective centers and thus the angular velocity and angular momentum of each. The distance & direction of each pair of axles from the COM is known. (See schematic)
Presuming frictionless axles and a totally elastic collision, I am trying to use conservation of momentum and angular momentum to figure out how much of the translational and rotational energy is converted into a vertical "bounce" when the front pair of wheels hits the step, and later when the rear wheels hit the step. The COM is slightly in front of the rear axles, and we can presume that the tipping of the car will not cause the rear of the main body to drop far enough to hit the step, nor will the first impact have enough energy to flip the front up over the back. I have only seen problems like this for a single wheel, nothing for a car that has a COM far behind the wheel center. I'm confused about how to write the equations when there are multiple masses and multiple (car / wheel) types of rotation. Do I have to calculate all angular momentum about a single point? If so, is the impact point the best choice? I think this can be done with instantaneous impulses and the conservation of momentum, but I'm having a tough time drawing the diagram and the forces, velocities and angular velocities before and after each impact. Can someone help me set up the system of equations to solve? P.S. we can assume a flat section of track, if it makes any difference, with the acceleration of gravity pointing straight downward. This may affect how long it takes for the bounce to dampen out, but b<<r, so I'm considering that the magnitude of the bounce will be small, almost imperceptible  with the car coming back to a normal roll very soon after the rear wheels get over the step. 


#2
Jan2013, 02:46 PM

P: 2

I think the solution will require me to calculate the moment of inertia of the main body around some axis, as some of the angular momentum at the wheels gets converted to angular momentum of the main body as the front end bounces up.
That is further complicated by the fact that the main body is not a constant density. For simplicity, it can be approximated by two sections (with rectangular crosssection) as shown in this revised schematic. The rear section will have a higher density than the front, for reasons that aren't really related to this problem. I am hoping to find (for a given mass and overall length) a relationship between wheelbase (L1+L2), COM placement, and the energy loss response to the step impulse. It appears that a long wheelbase and a low COM will result in the lowest energy loss, as that gives the minimum angle between the impact point and the COM, and thus the lowest vertical component to the impulse. 


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