# Physics of gears

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I'm having trouble with torques, angular velocities and gears. Let A and B be gears of respective radiuses rA and rB. Let wA be the the angular velocity of gear A, and wB be B's. Then wA and wB are related by wA*rA = -wB*rB. If we derive with respect to time, we get that aA*rA = -aB*rB, where aA and aB are the angular accelerations. We also know that: Sum of external torques on a solid = angular acceleration * inertia of the solid. If we apply that to our gears A and B we get: Sum (Torques on A) = aA*Ia and Sum (Torques on B) = aB*Ib, where Ia and Ib are the inertias of A and B. If we sum these two sums, we get: Sum (Torques on A and B) = aB*(Ib - (rB/rA)*Ia) If Ib and Ia are equal, and if rA and rB are also equal, then the right hand side is zero, which can't be right. What I am doing wrong? If you wonder where this comes from, I am trying to model the transmission of a go-kart, including the engine, the gearbox, the clutch, the brakes, the axle with the wheels and the ground.

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Your equation is right, total angular momentum L is always null(Everything is symmetric and we have wA=-Wb).
dL/dt=Sum (Torques on A and B) => Sum (Torques on A and B)=0

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The torques shouldn't be 0 unless the system stopped.

Also, 0 is not null, it's 0.

You're problem lies in your conceptualization of the two gears. If gear A and gear B are linked together and still, the resultant torque is 0. (nothing is moving)

If you being to rotate gear A only, it will also rotate gear B. gear B's rotation is purely based on gear A. so the magnitude of the torque of gear A = the magnitude of the torque of gear B.

If you want to turn both gears at one time, then tA + tB = tT, where tA is the torque on A, tB is the torque on B, and tT is the total torque on the system. Bare in mind, this mandates that there is a positive direction of motion and a negative direction. In other words, if the two gears are rotating in opposite directions, you need them to subtract (or well, they will have oppositely signed angular velocities)

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I'm confused at what the problem is exactly.

So here's a thread where I struggled with the same stuff. Maybe it will help you.

http://www.gamedev.net/community/forums/topic.asp?topic_id=364810&whichpage=1�

Or maybe not.

Good luck.

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Kambiz is right, my equation was correct, but I misinterpreted it.
Sum(Torques on A) + Sum(Torques on B) is not the same as Sum(Torques on A and B).
The fact that Sum(Torques on A and B) is zero does not imply that the respective angular velocities of A and B are constant.

I did my calculations again, the way my teachers 10 years ago always insisted I should do:

System A is composed of gear A with radius rA and inertia Ia
System B is composed of gear B with radius rB and inertia Ib

External forces on A: force F due to B (I leave out the ground and brakes for clarity)
External forces on B: force -F due to A (action/reaction principle) and torque due to the engine.

For A we get: rA*F = Ia * d(wA)/dt
For B we get: rB*F + T(engine) = Ib * d(wB)/dt

Since we know that rA*d(wA)/dt = rB*d(wB)/dt, we can solve the problem.
The result I get is:

d(wB)/dt = T(engine)/(Ib + R^2*Ia) where R=rB/rA

I'm a bit suprised that R is still in the picture, but that does seem too shocking.

CombatWombat: Thanks for the link. Seems interesting, but I haven't digested it yet. Makes me wonder, would be nice if the forum allowed to write formulas using e.g. latex syntax. A "drawing board" using e.g. a Java applet would be nice too.

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