Finding equivalent mass moment of inertia for gear system

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Discussion Overview

The discussion revolves around finding the equivalent mass moment of inertia for a gear system involving two masses with known moments of inertia, J1 and J2, connected by gears with a specified number of teeth, n1 and n2. Participants explore the relationships between angular displacements and velocities, as well as the implications for kinetic energy in the context of mechanical vibrations.

Discussion Character

  • Homework-related
  • Mathematical reasoning
  • Technical explanation
  • Debate/contested

Main Points Raised

  • One participant suggests relating angular velocities through the gear ratio, proposing that n1/n2 = θ1/θ2.
  • Another participant expresses confusion about the variable to solve for, questioning whether it is θ1 or the equivalent mass moment of inertia.
  • A participant proposes that "equivalent" refers to a formula for angular momentum, suggesting Jequiv can be defined for the entire system.
  • One participant derives an expression for the equivalent mass moment of inertia using kinetic energy, arriving at Jeq = J1 + J2*(n2/n1)².
  • A later reply challenges the relationship between the number of teeth and angular velocities, stating they are inversely proportional, and provides a corrected relationship involving diameters and RPMs.
  • Another participant acknowledges the correction and clarifies the notation used in gear calculations.

Areas of Agreement / Disagreement

Participants express varying levels of understanding and agreement regarding the relationships between the variables involved. There is no consensus on the correctness of the initial assumptions or the derived expressions, particularly concerning the relationships between angular velocities and gear teeth.

Contextual Notes

Participants note the potential confusion arising from the notation used for teeth and RPMs, as well as the dependence of relationships on physical properties like diameter. There are unresolved aspects regarding the assumptions made in the derivations and the implications of the corrections suggested.

immel
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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.

1.jpg


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!
 
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hi immel! :smile:

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

don't you have cogs in your brain? :rolleyes:

the ratio of the number of turns equals the ratio of the number of teeth! :wink:
 
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.
 
hi immel! :smile:
immel said:
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 = Jequivθ1', where L is the angular momentum for the whole system, and Jequiv can then be called the equivalent (or effective) mass moment of inertia for the whole system :wink:
 
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 KEsystem = KE1 + KE2 where KEsystem = 1/2*Jeq*θ1'2

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

Sooo, how far off am I?
 
immel said:
I've come up with a solution, but using the equivalent kinetic energy method.

oh, that's probably right then! :biggrin: (i haven't come across equivalent moment of inertia before :redface:)
Canceling out the 1/2 and θ1'2 I was left with Jeq = J1 + J2*(n2/n1)2

Looks good! :smile:
 
Thanks tiny-tim! Much appreciated
 
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)
 
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
 

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