Frequency of gears (given frequency of shaft)

[Reefy]

Homework Statement

Homework Equations

tangential velocity v = ωr

angular velocity ω = 2πf

The Attempt at a Solution

Not really looking for help to solve the problem. I'm just confused about the speed/angular velocity and frequency

If the frequency of the motor is given as 24 Hz, then that means the shaft AB has that frequency, correct?

Then does that mean the gear A has that same frequency? And then gear B has the same frequency? Or does A share the same frequency as AB and share the same tangential velocity as gear B (meaning different frequencies and obviously different radii)?

More specifically, I'm confused about shaft CD. If I know the frequency of shaft CD, does that mean I know the tangential velocity of gears C and D? Are they the same? How do I relate the frequency of a shaft with its respective gear?

Thank you

 

[2nafish117]

any object that is rigidly connected to each other will have the same angular velocity(but not all of the gears and shafts have the same frequency)
the shaft AB will have the same frequency as that of gear B(because they are rigidly connected).
similarly shaft CD has same angular velocity as that of gears C and D(but not necessarily same angular velocity as gear B)
angular freq of output F is the same as angular freq of gear E.

now to find the angular freq of gears connected to another gear the trick is that the two gears at the point where they touch have no relative velocity(tangential)

 

[Reefy]

any object that is rigidly connected to each other will have the same angular velocity(but not all of the gears and shafts have the same frequency)
the shaft AB will have the same frequency as that of gear B(because they are rigidly connected).
similarly shaft CD has same angular velocity as that of gears C and D(but not necessarily same angular velocity as gear B)
angular freq of output F is the same as angular freq of gear E.

now to find the angular freq of gears connected to another gear the trick is that the two gears at the point where they touch have no relative velocity(tangential)

In order to find frequency, I took the tangential velocity of shaft AB and set it equal to shaft CD so that (2πf_AB)r_B = (2πf_CD)r_C.

Then the 2π cancels out and I'm left with a relationship between just the gear radius and frequency (f_AB)r_B = (f_CD)r_C

. Which would mean that shaft AB frequency of 24 Hz results in a shaft CD frequency of 9.8 Hz. Is that correct? No relative tangential velocity means that the tangential velocity of both are equal? If so, why does this method give two different results for the frequency of shaft EF. You said gears C and D have the same angular velocity (meaning same frequency as angular velocity is ω = 2πf) but using this method will give different values of frequency for shaft EF as the radius of C and D are different

(f_CD)r_C = (f_EF)r_E

(f_CD)r_D = (f_EF)r_E I know this is correct as the gears of D and E make contact but then this means that the frequency of C and D are different, right?

Thank you once again in advance

 

[2nafish117]

In order to find frequency, I took the tangential velocity of shaft AB and set it equal to shaft CD so that (2πf_AB)r_B = (2πf_CD)r_C.

Then the 2π cancels out and I'm left with a relationship between just the gear radius and frequency (f_AB)r_B = (f_CD)r_C

. Which would mean that shaft AB frequency of 24 Hz results in a shaft CD frequency of 9.8 Hz. Is that correct? No relative tangential velocity means that the tangential velocity of both are equal? If so, why does this method give two different results for the frequency of shaft EF. You said gears C and D have the same angular velocity (meaning same frequency as angular velocity is ω = 2πf) but using this method will give different values of frequency for shaft EF as the radius of C and D are different

(f_CD)r_C = (f_EF)r_E

(f_CD)r_D = (f_EF)r_E I know this is correct as the gears of D and E make contact but then this means that the frequency of C and D are different, right?

Thank you once again in advance

(f_CD)r_C = (f_EF)r_E
is not correct .
this equation tells that the tangential velocity of the gear C(at its edge) is the same as the tangential velocity of gear E(which is not true)

this can be seen by taking an example

consider a disk rotating about its central point(along an axis perpendicular to its plane)
now all points on the disk have the same angular velocity(also same freq)
but they DO NOT have the same velocity at every point.

as you go to the edge of the disk their linear velocity increases(in a linear fashion from v=rω).
main point is the linear velocity of a point depends on how far the point is from the axis of rotation.

so the tangential velocity of the point of gear D which is in contact with gear E, is equal to the tangential velocity of gear E which is in contact with gear D.
this sentence is a bit wordy so ill write it as

(f_CD)r_D = (f_EF)r_E (same as what you wrote)

now you can see why

(f_CD)r_C = (f_EF)r_E is not correct.

 

[2nafish117]

one more point to clarify the tangential velocity of gear C is not equal to tangential velocity of gear D

ie (f_CD)r_C ≠ (f_CD)r_D

 

[Nidum]

You are both completely lost - look up 'Gear train calculations'

 

[2nafish117]

You are both completely lost - look up 'Gear train calculations'

aren't our angular freq calculations correct?? it should be shouldn't it??

 

[Reefy]

(f_CD)r_C = (f_EF)r_E
is not correct .
this equation tells that the tangential velocity of the gear C(at its edge) is the same as the tangential velocity of gear E(which is not true)

this can be seen by taking an example

consider a disk rotating about its central point(along an axis perpendicular to its plane)
now all points on the disk have the same angular velocity(also same freq)
but they DO NOT have the same velocity at every point.

as you go to the edge of the disk their linear velocity increases(in a linear fashion from v=rω).
main point is the linear velocity of a point depends on how far the point is from the axis of rotation.

so the tangential velocity of the point of gear D which is in contact with gear E, is equal to the tangential velocity of gear E which is in contact with gear D.
this sentence is a bit wordy so ill write it as

(f_CD)r_D = (f_EF)r_E (same as what you wrote)

now you can see why

(f_CD)r_C = (f_EF)r_E is not correct.

Considering just a single gear, you're saying the tangential velocity is different at every point? How is that possible when ω and r are both constant?

You are both completely lost - look up 'Gear train calculations'

These only show me calculations when I know the teeth of the gears. In this problem, I don't know the teeth.

 

[Nidum]

Say that the 60 mm gear has N teeth and work out the other gear teeth numbers in terms of N .

The N will cancel in the end anyway - it is only the ratios of the teeth numbers that matter in this problem .

Really it is only the ratios of the PCD's that matter .

 

[2nafish117]

Considering just a single gear, you're saying the tangential velocity is different at every point? How is that possible when ω and r are both constant?

tangential velocity is different at parts which are at different distances from the centre.