Equivalent Moment of Inertia

In summary: The mass on the string is unknown, the acceleration is unknown, and the dimensions of the flywheel are unknown.
  • #1
lukea125
5
0
Hi Guys, I'm having trouble finding the equivalent moment of inertia for a system. Basically it's a mass attached to a string, which is attached to a shaft. As the mass drops, the string unravels imparting some rotation on the shaft, this shaft rotates a small flywheel, and also rotates a second shaft via a geared system. This second shaft is attached to a second larger flywheel.

I've run some experiments and determined the frictional torque of the system. However, I can't work out how to find the equivalent moment of inertia. So far I have the energy balance equation:

(0.5I1w1^2 + 0.5I2w2^2 + 0.5mv^2) - T = 0.5 Ie w1^2

where w = angular velocity
T = frictional torque
I1, I2, and Ie are the two moments of inertia of the flywheel and the equivalent moment
v is the velocity of the weight as it falls
m is the mass of the weight

I have the radii of the flywheels, but not the masses, and I can replace w2 with an expression for w1 using gear reduction. Any tips on how I can perhaps eliminate some variables? I can't seem to find a way of solving this, without having the masses of the flywheels.
 
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  • #2
Where do you want to measure it at? The equivalent rotational inertia will be different on each shaft.

It looks a bit unsolvable without knowing the masses. Unless you know the acceleration.

To include the rotational inertia of the mass on the string, treat it as a point mass located at the radius of the drum. I = m * r2
 
  • #3
Ah sorry, I want to find it for the shaft that the string is wound around. I can eliminate omega since it's a common term through gear ratios etc. It's just those moments of Inertia for each flywheel that's killing me.
 
  • #4
lukea125 said:
Ah sorry, I want to find it for the shaft that the string is wound around. I can eliminate omega since it's a common term through gear ratios etc. It's just those moments of Inertia for each flywheel that's killing me.

What are the known variables? I still don't see why it isn't unsolvable.
 
  • #5
I've worked it out, Thanks for your help. Turns out we needed to estimate the mass from density and it's dimensions. Made it a lot easier haha.
 

What is the equivalent moment of inertia?

The equivalent moment of inertia, also known as the second moment of area, is a measure of an object's resistance to changes in its rotational motion. It is calculated by integrating the squared distance of each element of an object from the axis of rotation.

How is equivalent moment of inertia different from mass moment of inertia?

The equivalent moment of inertia is a property of an object's cross-sectional area, while the mass moment of inertia is a property of an object's mass distribution. They are related by the equation: I = m * r^2, where I is the mass moment of inertia, m is the mass, and r is the distance from the axis of rotation.

Why is equivalent moment of inertia important in engineering?

The equivalent moment of inertia is important in engineering because it is used to determine the stiffness and strength of structural components, such as beams and columns. It also plays a role in the design and analysis of rotating machinery, such as motors and turbines.

How do you calculate the equivalent moment of inertia for irregular shapes?

The equivalent moment of inertia for irregular shapes can be calculated using calculus and the parallel axis theorem. The object is divided into small elements, and the squared distance of each element from the axis of rotation is integrated. The parallel axis theorem is then used to calculate the equivalent moment of inertia about a different axis.

How does the equivalent moment of inertia affect an object's rotational motion?

The equivalent moment of inertia determines an object's resistance to changes in its rotational motion. A higher value of equivalent moment of inertia means it will be more difficult to change the object's rotational motion, while a lower value means it will be easier to change. This is important in designing objects with specific rotational properties, such as stability and maneuverability.

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