Shafts -Natural Frequencies / Whirling speeds



Purpose of Calculation
Simple calculations for obtaining the natural frequency  of traverse vibration of shafts and beams which are equal to the shaft whirling speeds.
Calculation Reference /Natural_Vibrations.html
Calculation Validation
Background information on the theory is provided on my website
Important Note:Please only enter data in the identified coloured cells.
The calculations below are simple calculations to establish the natural frequency of traverse vibration of shafts .  It can be easily proved that the naturalfrequency of a shaft is equal to the whirling speed. Whirling is a result of resonance when the shaft rotates at the same speed as one of the shafts natural frequencies of transverse vibration. At this speed the shaft will  whirl at an increasing rate . Increasing or reducing the speed will eliminated the whirling effect.
The calculations are all based on the assumption that the beams are round bars of uniform diameter.
The calculations are only for the lower four modes of vibration.
List Of Calcs
1) Cantilever beam / Cantilever shaft on long bearing.
2) Simply supported beam . Shaft supported between bearing providing minimum angular restraint
3) Beam with fixed end . Shaft supported between long bearings with angular restraint.
4) Beam / Shaft with one end fixed and the other simply supported
5) Beam/ shaft with one end fixed and the other end carrying a significant mass
6) Beam /Shaft simply supported at both ends with a significant mass in the middle
7) Beam /Shaft simply supported at both ends with a significant mass at some point along its length
8)Beam  /Shaft carrying a number of descrete masses

Calculation Reference
Mechanical Engineers Data Handbook Carvill Butterworth
Roark's Formulas for Stress and Strain
Vibration Handbook

Calculation Preview

Submitted On:
18 Nov 2015
File Size:
288.66 Kb
File Version:
File Author:
Roy Beardmore

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Comments: 12
roymech1 8 years ago
If everyone used consistent units systems we everyone could exchange spreadsheets all over the world without worrying about a metric or US units. That's why I think it is so useful for ExcelCalcs authors.
Mass is a fundamental unit not a derived unit so it is tonnes for SI(mm) so that we get Force is in Newtons.
I don't like beng too perscriptive - Thats why our Good Calculation Guide is a guide and not a Good Calculation Law :) In any case we are delighted that you grace us with your presence Roy in addition to all the fantastic work on your own Roymech website.
roymech1 8 years ago
Hello again
I am sorry but I have not used the template before.
I have now been able to do the necessary unit formatting
and I can achieve the necessary units both ways.
I have seen the light, that using the consistent units system resolves my
initial problem and I do not require a correction constant in the
equation. For this example it simply requires ensuring the mass units
are consistent. (its a bit daunting converting lbs/in^2 to slugs/in^3 density to lbf.s^2/in^4 .
Please correct me if I am in error but, on the template I downloaded,
the mass units for the SI(mm) are shown as t surely this should be
N.s^2 /mm.
I will definitely update my worksheet within a couple of weeks. There
a bit of work in fitting it in with the template. I do not see this as
urgent because the worksheet I have done seems OK .
roymech1 8 years ago

Roy you could just enter the formula "=Length&"4". The variable Length is "mm" if you selected SI(mm) or "in" if you have selected US(in). The use of the superscripted 4 character is very useful when working with units (you can copy if from cell AU8 in the ExcelCalcs Calculation Template).
Alternatively you can if this term is used often you can add a column to the table. Just insert a column just before the last column in the table (i.e. just before the column headed stress). I also define the Excel name "MomOfArea" to the 'Selected' cell.
roymech1 8 years ago
Hello again,
I have just tried to incorporate my worksheet into the exelcalcs template as
suggested by John Doyle. Unfortunately this did not solve the essential problem
that within the equation there is a need to incorporate a constant for converting
a force unit to a mass.accelaration. For fully consistent units this is unity. Also I had a bit of difficulty adding a column for Moment of Area (I) in the units table.
I will definitely use the template for my next submission though. It includes so
many advantages.
For this worksheet I have chickened out because of time limitations. I have simply created four pages all fairly identical which complete the same calcs using different unit systems. The pages (m.kilogram,s) and (ft, slug, s) are
consistent and do not need corrections. the other two (mm, kilogram, s) and (inch, lb, s ) are not consistent and need a correction factor.
roymech1 8 years ago
I have been caught out with units previously - getting 'g' in the right units when solving dynamics problems is an all too easy mistake to make. I manage to avoid these problems using consistent units as set out in the table below:
Just use the units in any of columns but you can't pick units from different columns. So if you want to work in mm use the SI (mm) column but acceleration due to gravity is in mm/s² not m/s² which is in the SI column. What's more if you have a problem encoded in one consistent system you can change with any other consistent system.
This is encoded in the ExcelCalcs Calculation Template and incudes a dropdown box to change between US and metric consistent units systems. It is covered in our Good Calculation Guide.
roymech1 8 years ago
Hell again,
I have now modified this spreadsheet . I think it is essentially OK now although
I am still reviewing it in terms of the basic problem that it is not convenient
to use other units than m (metres). It may be better if I replace M with W/g.
But this results in slightly more complicated equations and, for a shaft of wieght 1kg, to include W = 9.81 N instead of 1 kg is more confusing.
Roy Beardmore
roymech1 8 years ago
I created this calculation sheet . I am sorry for not replying earlier I have just been away on holiday.
Thanks DRT for your comment. I have quickly checked my worksheet and you are of course correct. I shall modify the spreadsheet within 24 hours ( when I have spend a little more time reviewing the calc.)
My initial intention is to simply convert the length units
to m. (from mm) this should then allow for the fact that a kg mass has
an accelaration of 1m/s when a force of 1 N is applied and not 1mm/s
as implied by the calc.
I am sorry for this error.
roymech1 8 years ago
I think its becasue tehre is a factor of 1000 missing in the SQRT term which accounts for acceleration being 9.81ms-2 and NOT 9.81mms-2.
If using mm terms you need to multiply the top term in the SQRT by 1000 to account for gravity being 9810mms-2??
roymech1 8 years ago
DRT wrote:
roymech1 8 years ago
Also I think that to get the frequencies correct you need to use metres and not mm? The problem is that the square root of mm isnt the same as the sqrt of m.
Certainly when I use metres I get the same results you would expect using
n= pi/2 * sqrt(EI/ML^3)
Do you agree?
roymech1 8 years ago
Thanks Roy -
I think though that for examples 2 3 and 4 the Freq needs to refer to mass and not Area as it does?
DRT 8 years ago
Thanks Roy
I think though that on examples 2 3 and 4 the F needs to refer to mass (not Area as it does)?