Young's Mod and Spring Constant of a disk

[TC9333]

[SOLVED] Young's Mod and Spring Constant of a disk

Hello, I am trying to find the proper thickness of a disk. In the center of the disk I will attach a steel rod which holds a certain fixed mass. The disk must be a fixed diameter and it can be any reasonable material or thickness. I would like to use something like aluminum.
Knowing the Diameter of the disk, mass of the attached wieght, and assuming I have access to any material and it's young's modulus...
I need to find a material and a material thickness that will result in a total system resonance of 110Hz.

So I have a spring-mass system with a set mass, needed resonance, and variables of Material(young's) and Disk thickness with which to find.

This is not a homework problem. I am an engineer (obviously not an experienced one) who is having a brain block. Any usable help (an equation relating all my variables) will be rewarded (if you send me a private email w/name and address).

Thanks a lot.
TC9333

 

[eljose79]

Hello TC9333,

Thank you for sharing your problem with us. This seems like an interesting and challenging engineering problem. I would approach this problem by first setting up the equations for a spring-mass system and then incorporating the variables of material (Young's modulus) and disk thickness.

The equation for the natural frequency of a spring-mass system is given by:

f = 1/2π√(k/m)

Where f is the natural frequency, k is the spring constant, and m is the mass of the attached weight.

To find the spring constant, we can use Hooke's law which states that the force exerted by a spring is directly proportional to its displacement:

F = -kx

Where F is the force, k is the spring constant, and x is the displacement of the spring.

Now, let's incorporate the variables of material (Young's modulus) and disk thickness. The equation for Young's modulus is:

E = (F/A)/(ΔL/L)

Where E is the Young's modulus, F is the force applied, A is the cross-sectional area of the material, ΔL is the change in length, and L is the original length.

Using this equation, we can find the spring constant, k, by rearranging Hooke's law:

k = F/x

Substituting the equation for Young's modulus into this, we get:

k = (E*A/x)/L

Since we know the desired natural frequency, we can substitute this into the equation for natural frequency and solve for the spring constant:

110 = 1/2π√((E*A/x)/L)/m

Solving for x, we get:

x = 2π√(mL/110E*A)

Now, we can use this value of x to find the required thickness of the disk. The equation for the thickness of a disk is:

t = (m*g)/(π*r^2*ρ)

Where t is the thickness, m is the mass of the attached weight, g is the acceleration due to gravity, r is the radius of the disk, and ρ is the density of the material.

Substituting the value of x into this equation, we get:

t = (m*g)/(π*r^2*ρ*(2π√(mL/110E*A)))

Using this equation, you can find the required thickness of the disk for a given