How is the shaft deflection calculated in the following cases.Shaft dia=2inches.Distance between bearings center to center=60inches
Case1-There is a concentrated load of 4000lbs
at the center.
Case-2-Uniformly distributed load of 4000lbs.
Please post the formula and a sample calculation.Does any one has an Excel sheet to do the calculation automatically?
Thank you,
Simon

Attached is example calculation including formulas for these two cases. (Bearings were assumed infinitely stiff).

Case 1: max deflection at center is 0.76"

Case 2: max ceflection at center is 0.48"

Hopefully you or someone will double-check to make sure I didn't make an error.

beam_static_deflection.pdf (16 Kb, 59 downloads)

By the way, the above was a static deflection calculation. If the force varies with time but far below resonance, it is approximately correct. If frequency is not far below resonance, then another calculation is required to account for dynamic (vs static) stiffness.

My first impression that kind of deflection due to loading might be Ok for rigging to remove a motor or pump for maintenance, but not for rotating machinery.

I agree with that. I did try to clarify that it was a static calculaton in my last post.

Also, it is not clear what physical situation these loads are intended to represent... an applied load in constant direction or in direction rotating with the shaft? Or an attached mass? The dynamic calculations can quickly get much more complex than the static calc presented above.

I am trying to learn the fundamentals of rotordynamics.I wisg to calcualte the 1st critical speed of a single shaft with a center disk.Only stiffness and mass to be considered.
For a machine like Jeffcott rotor.
Simon Peter

load and mass are not necessarily the same thing.
Unfactored Deflection is important or useful only as an indicator of SDOF mass and stiffness.
Resonant frequencies exist in space where there is no deflection due to gravity.
A plucked guitar plays the pretty much the same note whether the guitar neck and string are horizontal (some gravitational induced deflection) or vertical.

Some otherwise very useful vintage German shaft calculation software had a glitch that regarded force and mass as the same thing when calculating resonant frequencies.

I agree with Dan.

In it's simplest form, the Jeffcott rotor is a SDOF system.

The relationship between static deflection (calculated earlier) and resonant frequency for a SDOF system can be derived as follows:

Spring Force = Gravity force (static condition)
k * sd = m * g
k/m = g / sd
sqrt(k/m) = sqrt(g/sd)
w = sqrt(k/m) = sqrt(g/sd)
where w = radian resonant frequency = 2 * pi * f
Attached is calc for Simon Peter's mass-in-center geometry starting with the static deflection and calculating the resonant frequency. Note the assumptions at the beginning. (Neglecting shaft mass, bearing stiffness assumed infinite). The result is 3.6 hz.

This message has been edited. Last edited by: electricpete,

SimpleJeffcot.pdf (17 Kb, 16 downloads)

Here is the spreadsheet I made based on HI calculations.

Centrifugal_Pump_Shaft_Deflection.xls (232 Kb, 18 downloads)

More discussion on the resonant frequency of the Jeffcott rotor:

Attached shows the effects of adding flexibility to the bearings (and bearing support) of this model, including a critical speed map.

Since the bearing stiffness is effectively in series with the "shaft stiffness", the lower of the two stiffnesses will dominate the behavior.

At very low bearing stiffness values, the bearing stiffness dominates, and the system resembles the center-mass directly on the two parallel bearings (no shaft). Therefore the resonant frequency is sqrt(2*Kbearing/m) which increases linearly with Kbearing on a log/log scale (slope = 1/2).

At very high bearing stiffness values, the shaft stiffness dominates, and the curve is flat as bearing stiffness changes. (at the same value previously calculated ~ 3.6 hz).

This message has been edited. Last edited by: electricpete,

7581080683_JeffcottWithBearingsR2.xls (24 Kb, 17 downloads)

Today, there are a number of books on rotordynamics. You may learn more from one of these than you can gather from a forum such as this.

Other subjects like vibration and strength of materials may be useful also.

Some reference papers, like Jeffcott's might be of interest, too.

Methods that will answer your questions can be found in available books. It might be easier and more successful to study some.