Determination of the natural frequency of a Hartnell governor

AI Thread Summary
The discussion focuses on determining the natural frequency of a Hartnell governor, emphasizing the importance of including ball weight and centrifugal force in the moment balance equation. The user presents their approach to solving the problem, detailing the moment balance about the pivot and the resulting equations. They conclude that the natural frequency is influenced by rotation speed, leading to a derived equation for angular frequency. Feedback suggests that the solution appears correct, and it's recommended to consult the teacher regarding specific questions. The conversation highlights the complexity of the calculations involved in analyzing governor dynamics.
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Homework Statement



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Homework Equations

The Attempt at a Solution


I found this solution for the nature frequency
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but here it does not include the Ball weight and centrifugal force in the moment balance equation about the pivot (O), it is wrong answer...is not it?

I tried to solve the problem this way, moment balance about pivot (O)
$$(mb^2)θ'' = -1/2*k*(1/100+a sinθ)a conθ + mgbsinθ + mrw^2bcosθ$$

where :
$$ -1/2*k*(1/100+a sinθ)a conθ $$ : is the moment of the spring force
$$ mgbsinθ $$: is the moment of the weight of the ball.
$$mrw2bcosθ $$: is the moment of the centrifugal force.
$$a=12cm,b=20cm,k=104N/m,mg=25N$$

for small displacement $$sinθ=θ,cosθ=1$$. then
$$(mb^2)θ′′=−1/2k(1/100+aθ)a+mgbθ+mrw^2b$$

r is the distance between ball center and the center of rotation.
$$r=(16/100+bsinθ)=(16/100+bθ)$$

the equation becomes
$$(mb^2)θ′′=−1/2∗k(1/100+aθ)a+mgbθ+m(16/100+bθ)w^2b$$
rearrangement of the equation
$$(m∗b^2)θ′′+(1/2∗ka^2+−mgb−mb^2w^2)θ+1/2∗k∗1/100∗a−m∗16/100∗w^2b=0$$
from this equation we could say that the nature frequency is
$$(w_n)^2=(1/2∗ka^2−mgb−mb^2w^2)/(mb^2)$$
which shows that the nature frequency changes with rotation speed. Is this a right solution?
 
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Did you have to do part a of the problem? If so what did you get.
 
I think the last two equations are correct. In addition if you set θ and θ'' to zero you can solve for ω the angular velocity at equilibrium. I would not worry about the answers dependence on angular velocity. You look like you got the right answer, good work! Ask your teacher about your question.

Edit, remember ω for your problem is fixed.
 
Thank you for your reply and check of the solution. and when you set θ and θ'' to zero you would get the answer to part (a).
 
Kindly see the attached pdf. My attempt to solve it, is in it. I'm wondering if my solution is right. My idea is this: At any point of time, the ball may be assumed to be at an incline which is at an angle of θ(kindly see both the pics in the pdf file). The value of θ will continuously change and so will the value of friction. I'm not able to figure out, why my solution is wrong, if it is wrong .
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