Calculate the % fan shaft is out of balance

[thewall]

Homework Statement

A fan loses an 11 gram piece 460mm from the shaft center. It rotates at 1000 RPM and cannot operate if the shaft is out of balance by more than 8%. Prove whether or not the fan can still operate

Homework Equations

Fc = mwr2

The Attempt at a Solution

I have found the force added by the imbalance to be 55.5N but can't figure out how to relate this to a % of unbalance as I would need to know the mass of the whole fan blade assembly. I am thinking that 11 grams missing from a 1 kg fan blade would cause a higher percentage imbalance than it would in a 100 kg fan blade.

Could someone tell me if I am going about this problem the right way and if so how to do this, or do I need to take a different approach other than using centripetal forces. What is a % out of balance related to forces, center of gravity, moment of inertia?

 

[Jhero]

Thank you for bringing this problem to our attention. I can help you approach this problem in a more systematic way.

First, let's establish the necessary information that we have:

- Mass of the missing piece = 11 grams = 0.011 kg
- Distance of the missing piece from the shaft center = 460 mm = 0.46 m
- Angular velocity = 1000 RPM = 1000 revolutions per minute = (1000/60) revolutions per second = 16.67 revolutions per second
- Maximum allowed unbalance = 8% of the total mass of the fan blade assembly

Now, let's use the given equation Fc = mwr^2 to calculate the force added by the imbalance. We know that the force added by the missing piece is equal to the centripetal force required to keep the fan blade rotating at constant angular velocity. Therefore, we can write:

Fc = mwr^2

where:

- Fc is the centripetal force
- m is the mass of the missing piece
- w is the angular velocity
- r is the distance of the missing piece from the shaft center

Substituting the given values, we get:

Fc = (0.011 kg)(16.67 rev/s)^2(0.46 m)^2
= 55.48 N

So, your calculation of 55.5N is correct. Now, let's relate this to the maximum allowed unbalance of 8%. We know that the total mass of the fan blade assembly is not given, so let's use a general variable M to represent it. We can then write the equation for maximum allowed unbalance as:

Maximum allowed unbalance = 8% of M

Substituting the value of maximum allowed unbalance (8%) and the calculated value of the force added by the imbalance (55.48 N), we get:

0.08M = 55.48 N

Solving for M, we get:

M = 693 kg

This means that the total mass of the fan blade assembly must be at least 693 kg for the fan to operate with the missing piece. If the total mass is less than this, the fan will not be able to operate within the allowed unbalance.

In conclusion, the fan can still operate with the missing 11 gram piece as long as the total mass of the fan blade assembly