Moment of Inertia: Non Uniform Rod

In summary, the conversation discusses the determination of the moment of inertia for a cantilever beam with two sections of variable thickness. The known variables are L, D, P, and t, while the dependent variables are T2 and L2 and the design variables are T1 and L1. The standard moment of inertia equation is modified to include the two sections, and it is mentioned that moments of inertia are additive. The overall solution for the moment of inertia is obtained by breaking the beam into convenient shapes and adding up their individual moments of inertia.
  • #1
Andrew VanFossen
1
0

Homework Statement


Hello,

I am looking to determine the moment of inertia for the cantilever beam pictured below. I want I to be a function of L1 and T1.

Known variables: L, D, P, t
Dependent variables: T2, L2
Design variables: T1, L1

rod.PNG


Homework Equations


L2 = L - L1
T2 = T1 - t

Standard Moment of Inertia: I = π/4 * (Ro^4 - Ri^4)
I am looking to modify the above equation to be include 2 sections of variable thickness

The Attempt at a Solution


I1 = π/64 * (D^4 - (D - T1)^4)
I = I1 * (1 - L1/L)^4

The above solution is valid for a linearly tapered beam, not a beam with 2 discrete sections. I have not been able to find any derived moments of inertia for the above shape. Any help would be greatly appreciated!
 
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  • #2
Andrew VanFossen said:
I have not been able to find any derived moments of inertia for the above shape.
Moments of Inertia are additive. Just break it up into pieces of convenient shape, find the MoI of each (about the same axis) and add them up.
 

1. What is the moment of inertia of a non-uniform rod?

The moment of inertia of a non-uniform rod, also known as rotational inertia, is a measure of an object's resistance to changes in its rotational motion. It is a property that depends on the object's mass distribution and the axis of rotation.

2. How do you calculate the moment of inertia of a non-uniform rod?

The moment of inertia of a non-uniform rod can be calculated by dividing the rod into small infinitesimal segments, finding the moment of inertia of each segment, and then using the parallel axis theorem to sum up the individual moments of inertia.

3. What factors affect the moment of inertia of a non-uniform rod?

The moment of inertia of a non-uniform rod is affected by the mass distribution of the rod, the distance of the mass from the axis of rotation, and the shape of the rod. Generally, the further the mass is from the axis of rotation, the larger the moment of inertia will be.

4. How does the moment of inertia of a non-uniform rod differ from that of a uniform rod?

The moment of inertia of a non-uniform rod is different from that of a uniform rod because the mass distribution of a uniform rod is evenly distributed, while a non-uniform rod has varying mass distributions along its length. This results in a more complex calculation for the moment of inertia of a non-uniform rod.

5. What are the practical applications of understanding moment of inertia of a non-uniform rod?

Understanding the moment of inertia of a non-uniform rod is important in various engineering and physics applications such as designing structures, calculating the stability of rotating objects, and understanding the dynamics of rotational motion. It is also crucial in fields such as robotics, aerospace engineering, and sports biomechanics.

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