M+0.5*m for adding shaft mass to Jeffcott rotor 4

[electricpete]

For a massless beam with simply-supported end conditions and a mass in the center, the Euler Beam model tells us we can easily find the stiffness (force divided by centerdisplacement) is
K = 48*E*I/L^3
and the first resonant frequency therefore
w1=sqrt(48*E*I/[M*L^3]))

Many references (Harris’ Shock and Vib Handbook, Mark’s Mechanical Engineering Handbook, Rao’s Mechanical Vibration, and Ehrlich’s Rotordynamics Handbook) suggest that you can enlarge the above formulation to provide for mass in the shaft as follows:
w1=sqrt(48*E*I/[(M + 0.5*m)*L^3]))
where M is center lumped mass and m is distributed mass of the beam (excluding center lumped mass).

Questions:
1 – Does anyone seen any proof or justification for M+0.5*m?

2 – What assumptions / approximations are made to arrive at this formulation (beyond Euler/Bernoulli beam assumptions)?

Note some more related discussion in thread384-156111


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[electricpete]

Attached is the exact solution using method described above (similar to S&V handbook).

http://home.houston.rr.com/electricpete/JeffcotWithDistribMassExact.pdf

The boundary condition that I wrote above as
y’’’’(L/2) = E*I*w^2*(M/2)*Y(L/2)
should have been
y’’’(L/2) = -E*I*w^2*(M/2)*Y(L/2)

In the correction immediately above, there is a change to minus sign on right as expected. Also the force from mass acceleration is related to the 3rd derivative (shear force), not the 4th derivative (force per length) as I had written before. This can be verified by examining the units.

Results for X vary from 0.4857 for very small MassRatio (m<<M) to 0.4927 for very high MassRatio (m>>M) . These match the results which can be derived for the case of distributed mass alone or concentrated mass alone. A table of X vs MassRatio is at the end of the worksheet.

To two decimal places it is 0.49 no matter what, so the values really aren't needed for any practical calculations. It was more of an excercize for me and I learned a lot by working through it and asking questions here.

I still have my assignment from Bill to solve using the transfer matrix which would be a useful excercize as well. Sometime in the near future...

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[electricpete]

On the off-change that someone may actually want these results at some time in the future (long after mywebsite is gone), I'll post the tabular results here;

m/M X
0.001 0.4857168063
0.01 0.4857480390
0.1 0.4860378548
0.2 0.4863333819
0.5 0.4870836626
1 0.4880107197
2 0.4891832972
5 0.4907094280
10 0.4915643189
100 0.4926253580
1000 0.4927527114

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[electricpete]

I tried to use a similar analytical approach to find the resonant frequency of a shaft on bearings... modeled by a beam with uniformly distributed mass and with springs on each end and moment=0 at each end.

The boundary conditions:
> # Equation: y''(0)=0
> # Equation: y''(L)=0
> # Equation 3: y'''(0)=k*y
> # Equation 4: y'''(L/2)=k*y

Maple couldn't find a solution. It looks like I have 4 equations in the 4 uknowns so I don't know why there would be a problem. Maybe I'm overlooking something basic or did something stupid? Any ideas?

http://home.houston.rr.com/electricpete/DistribMassWithBearing.pdf

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[electricpete]

Please disregard my last post. I think I can work through it by checking for errors and trying a few more things. I don't want to waste your time with proofreading my code and using Maple to solve for unknowns. I'll try to stick with vibration questions.

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