Engineering Maths

[scraggle]

Please forgive me if I have posted this in wrong section. I have to complete maths problems but I am not getting anywhere, can someone explain how I figure this out?

Gear A and Gear B are intermeshed, A having a radius of 200mm and B a radius of 150mm. Gear A starts from rest and rotates with a constant angular acceleration of 2 rad/s2. How do I determine the the angular velocity and angular acceleration of Gear B after A has completed 10revs?

I understand how to work out basic ang vel and acc but don't quite understand the after 10revs bit and the intermeshed bit!

[scotstone]

The fact that they are intermeshed means that the peripheral speed of both gearwheels is the same.

The 10revs is the angular displacement whilst accelerating ie 10 x 2Pi = 20Pi rad.

For wheel A :- calculate the final ang vel using the formula that relates final ang vel, initial ang vel, ang accel and ang displacement.

Using this value calculate the time wheel A takes to travel 20Pi rad from the formula that relates final ang vel, initial ang vel, ang accel and time.

As the wheels are intermeshed :-

Peripheral speed V(A) = V(B) AFTER 10 revs

and peripheral speed V(A) = ang vel(A) x Radius (A)

V(B) = ang vel(B) x Radius(B)

equating and solving gives ang vel (B).

For wheel B :-
Using the formula that relates final ang vel(B), initial ang vel(B), ang acc(B) and time with the time calculated for wheel A transpose and calculate ang acc(B).

Let me know your answers and I will tell you if correct.

[scraggle]

Thanks for this but it seems i don't understand much at all... Not sure how i calculate the final velocity for gear A. I am obviously missing something easy but I can't see it. For the final ang vel I get 15.85 rad/s but as i say i am totally unsure i am doing this right. I am not sure of the next formula either. Sorry for not really being much good at this.

[scotstone]

15.85 rad/s is correct. Now calculate the time taken from:-

final ang vel = init ang vel + ang acc x time

Get back when done.

[scraggle]

so if i am getting this right... moving the formula around to get time = init ang vel + ang acc x final ang vel ... t = 0 + 2 x 15.85 = 31.7seconds.

[scotstone]

You have transposed wrongly.

Time = final ang vel/ ang acc.

[scraggle]

OK so 7.93 seconds. but i am sorry i just don't get the rest at all. Find the whole subject of ang acc and vel very difficult.

[Denise S.]

Correct. As the wheels are intermeshed the peripheral speed ie the linear vel where they engage must be the same( or else gear teeth would be flying all over the place) therefore:-
linear vel(A) = linear vel(B) and given that :-

lin vel = ang vel x radius for BOTH wheels, then:-

Lin V(A) = ang vel (A) x radius (A) and
Lin V(B) = ang vel (B) x radius (B) then :- as V(A) = V(B)

ang vel (A) x radius (A) = ang vel (B) x radius (B)

which can be transposed to calculate ang vel (B).

[scraggle]

Thanks Denise...so would i be right with a figure of 21.14 rad/s for the ang vel (B)? Is this figure the final ang vel after the 10 revs?

[scotstone]

Yes . You see it was that easy I got the wife to explain it to you! Now to calculate the ang acc of wheel B:-

For wheel B using :-

fin ang vel(B) = init ang vel(B) + ang acc(B) x time... and that

the time is the same as you calculated for wheel A and init ang velB is zero you should be able to calculate ang acc (B).

[scraggle]

...and so the Ang Acc for B is 2.67 rad/s2?

[scotstone]

Correct!

[scraggle]

I can't thank you enough for that Scot... absolutely excellent. Thanks for your patience!! Now how are you at torque and shear stress?.... because if your not busy....

[scotstone]

OK, I think. What's the question?

[scraggle]

A tubular steel shaft has an external diameter of 50 mm and a wall thickness of 6.5mm. I have to determine the torque that can be transmitted by the shaft if the shear stress in the shaft is not to exceed 15 M.Pa and the power that can be transmitted when the shaft rotates at 50revs/s.

[scotstone]

Calculate the Polar Moment of Inertia(J) of the hollow shaft using the standard formula:-

J = Pi(D4-d4)/32

Let me know your answer.

[scraggle]

Sorry what is the D4-d4 bit? what does the D and d represent?

[scraggle]

is it external diameter and internal diameter?

[scotstone]

Yes and to the power 4

[scraggle]

hmmm... I am getting 262,350... not sure what the unit of measurement is though.