Mechanical vibration -> proving a larger motor will vibrate more

[rock.freak667]

Homework Statement

I have a task of essentially proving that bigger motor = bigger vibration. I have certain data such as the running speed of the motor (), moment of inertia of the rotor, weight of the motor and shaft and the diameter of the shaft.

Homework Equations

$$|X| = \frac{me}{M} \frac{r^2}{\sqrt{(1-r^2)^2+(2 \zeta r)^2}}$$

$$k = \frac{384EI}{L^3}$$

$$I = \frac{\pi d^4}{64}$$

$$\zeta = \frac{c}{\sqrt{mk}}$$

$$r = \frac{\omega}{\omega _n}$$

The Attempt at a Solution

The motor is vertically mounted so I am not sure if this will affect my theoretical calculations below:
Mass of rotor = m
Mass of motor = M
Polar moment of inertia of rotor = J
Running speed = ω
Natural frequency = ω_n
Stiffness of shaft = k
Modulus of elasticity of shaft = E
Eccentricity of shaft = e (I will put this as some small number as the shaft may not be perfectly round but round enough to run properly)

Since the motor will essentially be running with an eccentric load (albeit quite small) the magnitude of the vibration should be given by

$$|X| = \frac{me}{M} \frac{r^2}{\sqrt{(1-r^2)^2+(2 \zeta r)^2}}$$
Where r = w/wn
So the main issue is to get the values of r and z. So here is where I think I’ve gone wrong.
I know that ζ = c/√(mk) and I’ve assumed c = 2000 kg/s (I have no idea how much this value is to be on average).

To find the stiffness I considered the rotor to be like a simply supported beam (supported by the two bearings) with a uniform load w (or WL)
The displacement would be given by
$$y = \frac{5WL^3}{384EI}$$
$$k = \frac{W}{y} = \frac{384EI}{L^3}$$
I = 2J, with E assumed to be 200 GPa
Also confused here as I know the diameter of the shaft so I can use I = πd⁴/64 which is not the same value as in the data sheet of the motor.
From the formula above, I get k (which is a very large number from my numbers – in the order of 10^8 N/m if I remember correctly) and use
$$ω_n = \sqrt{\frac{k/m}}$$

so I can get 'r' and 'ζ'.

The problem comes in that I think my values for k is too large which causes my value for ζ to be too small and subsequently r is also too small.

The second problem I come into contact with is the fact that changing the values to reflect a smaller motor ( so smaller m and M) shows me that my vibration amplitude decreases! Can anyone give me some sort of guidance as to where I am going wrong or have I made this case too simple for complex vibrations?

 

[LightHero]

I am not sure of how to correctly calculate the stiffness or how to correctly use the formula for the magnitude of vibration. Any help would be much appreciated.