Names - Vittal Surya Lakshya

Suman Mandal

Kalind Baraya

Roll Nos. – 120103087

120103085

120103083

Title – Modification in Whirling of Shafts

Experiment.

Aim : The objective of this report is to select the rods for whirling of shafts in such a way that when it is at its maximum deflection it does not touch the base of apparatus.

Experimental Set-up :

The apparatus consists of a rigid bed which is within a safety canopy. A variable speed motor drives the headstock spindle via three-step pulleys and a belt. The spindle is fitted with a shaft speed sensor. The spindle drives the coupling connector via a kinematic flexible coupling. Test shafts, guards, bearings and shaft weights are set up as required upon the bed to perform the experiments.

Motivation for this report :

When we were operating on the apparatus to study the whirling of shafts, there was a problem of excessive sound after increasing the speed up-to certain level. So we have to discontinue our experiment and take that reading as the final reading even though it was not correct.

That sound was due to the shaft making contact with the base of the apparatus on which the motor was fixed. So to prevent such things from happening in the future, our group decided to do a small report on this experiment and theoretically analyse the model to come up with shafts such there won’t be any such nuisance in the future.

Experimental Procedure :

First decide upon the experiment. A) Bearing type and, B) number of bearings that will support the shaft. Place the bearings on the bench and then between each bearing unit place a guard unit.

Position the bearing units as required. Use a tape measure to set the spacing. The guards should be about mid-way between the bearing units. They can be moved slightly if a disc weight is to be placed there.

Select the shaft to be tested and then select the correct size of adaptors. Place an adaptor and a nut by each bearing unit. Select the weights to be added, as required by the experiment. Place the weights on the bed as well. Insert the shaft through the bearings and guards adding the shaft adaptors, nuts and weights as appropriate. Screw the shaft into the shaft coupling.

Now slide the shaft adaptors into the bearings and clamp with the knurled nuts, tightly, by hand. Position the weights as required. To move a weight slacken the knurled grip, once in position tighten.

A drop of oil may be placed on the shaft where it passes through the shaft adaptors.

Check that the bearings are in line by sight adjusts the bearing housings where necessary.

Finely check that all bearing units, guards and shaft adaptors are securely fixed before commencement of experiment.

Select the test speed range and slip the belt over the pulleys as required.

Power is supplied to the apparatus via a filtered inlet unit at the rear of the base.

For an unloaded plain shaft the critical speed in rpm is given by:-

Ncrit .=

118 .6dL2

i2… (i)

Where d = shaft diameter in mm.

L = distance between the bearing centres in m.

i = number of vibration mode.

For a shaft with lateral loads, e.g. discs on it, the critical speed is given by Dunkerley’s Formula:-

1N crit .2

= 1N02+ 1N 12+ .. .

… (ii)

N0 is the critical speed in rpm of the shaft without any load, so as before

N0 .=

118 .6dL2

N1, N2 etc. are the critical speeds in rpm due to loads m1, m2 etc. in kg at a distance x1, x2 etc. in m from a bearing

N1=

1.67d2

x (L−x ) √Lm1 … (iv)

Where d = shaft diameter in mm.

L = distance between the bearing centres in m.

The critical speeds in rpm at different number of vibration mode (i) is

N crit .=

i2 π2

L2 √EI m … (v)

Where E = Young’s Modulus

I = Second moment of area

i = number of vibration mode.

m = mass per unit length

L = distance between the bearing centres.

The equation (i) & (ii) is derived from the equation (v).

Theoretical Calculations:

In terms of deflection of centre we can write,

Considering deflection of centre, (δ) to be between 2.0mm – 4.0mm without any other mass connected.

Where, g = Acceleration due to gravity (9.8 m/sec2)

= total maximum static deflection.

And without any external load

δstl = =Maximum static deflection of shaft

Where, w = weight of shaft, kg

E = modulus of elasticity, kg/m2

I = moment of inertia, = m4

L = length of shaft, m

Since,

Thus, Taking δst = 2.5 mm,

Nc = 30π √ 9.81(2.5×10−3 ) =598.185 rpm

Taking this value of Nc we can find out the values of E, i.e.

Using, δstl = 5w L3

384 EI

For D = 8mm and L= 1.1 m i.e. same as used in experiment

2 .5×10−3=5×w× (1.13 )384×E×I

I=π D4

64 =π × (8×10−3 )4

64

I=2.011×10−10 m4

Putting the value of I in above equation, we get

Ew=3.447×1010 N/(kg.m2)

Material Value of E (GPa) Value of w (kg)

Aluminium Bronze 120 3.48

Aluminium 69 2.00

Brass 100 2.90

Carbon Fibre Reinforced Plastic

150 4.35

Copper 117 3.39

Grey Cast Iron 130 3.77

Magnesium metal (Mg) 45 1.31

Iron 210 6.09

Steel, stainless AISI 302 180 5.22

Steel, Structural ASTM-A36 200 5.80

Tin 47 1.36

Result :

So, for a Shaft having a diameter of 8 mm and length of 1.1 m, and considering the deflection of only 2.5 mm.

We get,

Nc = 598.185 rpm

And this speed we get is quite reasonable and in a moderate range which our motor can provide.

Now we can use any material having the ratio of E and w to be,

Ew

=3.447×1010

So choosing any material in the given table, if we select the mass around that given in the table we can approximate its critical speed.

In this way we are able to avoid any big deflection of shaft and can perform our experiment by adding weight at the centre of the shaft without damaging the equipment.