Finding equivalent mass moment of inertia for gear system

[immel]

Homework Statement

Two masses, having mass moments of inertia J1 and J2, are placed on rotating rigid shafts that are connected by gears. If the numbers of teeth on gears 1 and 2 are n1 and n2, respectively, find the equivalent mass moment of inertia corresponding to theta_1.

Homework Equations

Kinetic energy of a rotating shaft = 1/2*J*theta_dot
J = 1/2*m*r^2
n1/n2 = theta_1/theta_2 ?

The Attempt at a Solution

This is for my mechanical vibrations class for ME majors. I assume I need to relate theta_dot_1 to theta_dot_2 somehow. I am also not sure what to do with the number of teeth. It seems as though I may be able to do something with the gear ratio, but I don't know where to begin relating things. Please help!

 

[tiny-tim]

hi immel!

(have a theta: θ and try using the X² and X₂ icons just above the Reply box )

don't you have cogs in your brain?

the ratio of the number of turns equals the ratio of the number of teeth!

 

[immel]

Thanks for the help tiny-tim.

So was I right with my assumption that n₁/n₂ = θ₁/θ₂ ?

I am confused about what variable I am solving for. In the problem statement "...find the equivalent mass moment of inertia corresponding to θ₁." Does this mean I'm solving for θ₁ or some product with the unit of mass*length² (mass moment of inertia has these units)? J=1/2*m*r² has units of mass*length².

It's starting to look like I need to relate the kinetic energy equation and n₁/n₂ = θ₁/θ₂.

 

[tiny-tim]

hi immel!

I am confused about what variable I am solving for. In the problem statement "...find the equivalent mass moment of inertia corresponding to θ₁." Does this mean I'm solving for θ₁ or some product with the unit of mass*length² (mass moment of inertia has these units)? J=1/2*m*r² has units of mass*length².

It's starting to look like I need to relate the kinetic energy equation and n₁/n₂ = θ₁/θ₂.

i think by "equivalent" it means that if you pretend that you can only see one cog, and it's rotating at angular speed θ₁', then you want a formula L = J_equivθ₁', where L is the angular momentum for the whole system, and J_equiv can then be called the equivalent (or effective) mass moment of inertia for the whole system

 

[immel]

I've come up with a solution, but using the equivalent kinetic energy method.

With the relation n₁/n₂ = θ₁/θ₂ = θ₁'/θ₂' (is this valid?), I solved for θ₂' = (θ₁'*n₂)/n₁

I have KE₁=1/2*J₁*θ₁'² and KE₂=1/2*J₂*[(θ₁'*n₂)/n₁]²

I then have KE_system = KE₁ + KE₂ where KE_system = 1/2*J_eq*θ₁'²

Canceling out the 1/2 and θ₁'² I was left with J_eq = J₁ + J₂*(n₂/n₁)²

Sooo, how far off am I?

 

[tiny-tim]

I've come up with a solution, but using the equivalent kinetic energy method.

oh, that's probably right then! (i haven't come across equivalent moment of inertia before )

Canceling out the 1/2 and θ₁'² I was left with J_eq = J₁ + J₂*(n₂/n₁)²

Looks good!

 

[immel]

Thanks tiny-tim! Much appreciated

 

[jtfrizzy]

I know this is about 2 years later but in case anyone comes across this looking for help I wanted to correct an error I noticed in the calculations. When making the relationship between number of teeth, [n][/1] and [n][/2] and angular velocity [θ'][/1] and [θ'][/2] they are inversely proportional. Namely, if [n][/1] < [n][/2] then [θ'][/1] > [θ'][/2]. If you think about it less teeth equals smaller diameter and when a constant torque is applied the smaller gear (and shaft coupled to it will spin faster than the gear with more teeth. Right? So the appropriate relations ship should between the two is:
[n][/1]/[n][/2]=[θ'][/2]/[θ'][/1]

NOTE: often both "n" and "N" are used in gear train calculations but refer to different things. "N" would be the RPMs of the gear and "n" equals the number of teeth of the gear. Adding diameter "d" onto the mix the following relationships all exist:

[d][/1]/[d][/2]=[n][/1]/[n][/2]=[N][/2]/[N][/1]=[θ'][/2]/[θ'][/1]

This is if, of course, the diameter and number of teeth are dirrectly dependent upon one another (which they are if your using gears in the real world)

 

[jtfrizzy]

Sorry about [x][/1] crap in the earlier post I guess I did the X_subscript wrong so I hope anyone reading it can get the drift