Dunkerly vs Rayleigh Clarification

[cmmcnamara]

Hey guys, I was hoping someone could set me straight on the two main methods of calculating shaft critical speeds. I have a final coming up but I can't seem to get these two to agree.

What I know:
-Dunkerly underestimates, Rayleigh overestimates
-Deflections used in Dunkerly are those caused by each single mass (eg remove all other masses during calculations)
-Deflections used in Rayleigh are those calculated with all masses "attached"

What I can do:
-Dunkerly's method very well. Wrote a VBA program for it. All calculations agree with all example problems I've run across as well as HW vs solution manual verifications

What I can't do:
-Rayleigh's method. The answers I get from this do overestimate, but are far, far off from that of Dunkerly, when they should be fairly close to one another. I can follow examples very easily and get the correct answers but I'm missing something in practice myself. For example a HW problem I am working pegs Dunkerly speed at 450 rpm, while Rayleigh gets 1050 rpm which is more than x2 Dunkerly


Can anyone point out what I am doing wrong here, at least by my thinking method? I'd be happy to post some of my worked out examples to see if I am just messing math up, but I am positive I am not.


I appreciate any help, thanks all!

 

[SteamKing]

Please post your work. It would help us to see the details of your problem.

 

[cmmcnamara]

Hey SteamKing thanks for the quick reply.

I attached a pdf of my work from MathCAD which also has a crudely drawn picture of the beam in question. The deflections that are given come from an Excel program that my professor gave us that will solve the beam deflections.

For Dunkerly I found the deflections by only analyzing the beam with the first force and then only with the second force. For Rayleigh I found the deflections by analyzing the beam with both forces applied just as the picture shows. The rest of the equations should explain the rest of what I am doing.

I've used this same exact method on a bunch of practice examples following them to a tee but whenever I work these problems myself I see a huge disparity between the speeds/frequencies coming out of either method when all examples show usually a +/- of about 100 rpm maximum. I know it can't be a unit problem because MathCAD handles all the unit reduction for you and I've also worked it out by hand just in case and I get the same result as MathCAD. I've also confirmed with my professor that my answer with Dunkerly is correct.

I've been struggling all quarter to get Rayleigh's down properly but I can't seem to get it and have already bugged my professor countless times regarding this and its a subject from a pre-requisite class anyways. I appreciate your help on this greatly!

 

[SteamKing]

What is I for your shaft?

 

[cmmcnamara]

The shaft has a 2" diameter so it should be pi/3

 

[SteamKing]

Could you also provide the deflection calculations from the Excel spreadsheet?

 

[cmmcnamara]

Yea sorry about that.

For local deflections used by Dunkerly I got:

Left mass deflection=7.317E-4 in
Right mass deflection=1.311E-3 in


For total deflections used by Rayleigh I got:

Left mass deflection=2.927E-3 in
Right mass deflection=4.332E-3 in

 

[SteamKing]

Yes, I saw those numbers on your calculations. What I am looking for is a print of the actual spreadsheet calculations.

 

[SteamKing]

I have checked your calculations and I found a discrepancy for the deflection due to the overhung weight of 50 lbs.

According to my calculations, the overhung deflection should be estimated by the formula:
d = WL^3/(3EI), which gives d = 0.0003748 in.

Correcting for this value in the Dunkerly Eq. gives wnd = 590.5 rad/s. The max. shaft op. speed becomes:

590.5 * 0.8 * 60 / (2*pi) = 4500 RPM

One other minor point: omega has units of radians/sec. not Hertz. Strictly speaking, Hertz is used to measure frequency and is derived from cycles/sec., thus

omega = 2*pi*f

I checked the Rayleigh deflections and obtained dr1 = 0.0002911 in and dr2 = 0.0004314 in., which agree with your spreadsheet calculations.

I think the Dunkerly calculation varies from the Rayleigh calculation so much because the estimates of shaft deflection in the former are so much higher than the deflections a proper beam analysis gives. For this shaft, the presence of the overhung weight influences the deflection of the 100 lb weight to a large extent, and this is reflected in the critical shaft speed.