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Description

Table 1A

Angle of Twist (rad) 12

93 mm) 8

94 mm) 9

74 mm) 10

39 mm) 12

Torque (Nm)

Angle of Twist (rad)

Table 1B

Angle of Twist (rad) 12

Torque (Nm)

Angle of Twist (rad)

ME2113-2 TORSION OF CIRCULAR SHAFTS (EA-02-21) (INFORMAL REPORT)

NAME: He Quanjie,

Boey MATRICULATION NUMBER: A0094502L CLASS: 2F1 DATE: 16 October 2013

SEMESTER 3 2013/2014

DEPARTMENT OF MECHANICAL ENGINEERING NATIONAL UNIVERSITY OF SINGAPORE

Objectives • •

To study the torsion of solid and hollow circular shafts evaluate the torsional stiffness and strengths of circular shafts(solid against hollow) ◦ having the same outer diameter or ◦ having the same volume

Table 1: Experimental data for solid and hollow shafts

Radians 7

93 mm 8

94 mm 9

74 mm 10

39 mm 12

003491 0

006981 0

010472 0

013963 1

017453 1

024435 2

027925 2

031416 3

034907 3

56 1144

09 2035

75 1391

63 1633

63 1800

03 1908

Table 2: Strength and stiffness of hollow and solid shafts having the same outer diameter Solid Shaft

% change in torsional stiffness K h−K s'×100 % Ks (experimental)

Ds = 12 mm

Vs = 11309

Hollow Shafts 1

Theoretical % change in maximum shear stress

V h −V s'×100 % Vs

Vh= πL(Dh²-dh²) / 4

ΔK =−(

τ h−τ s'×100 % τs

D −d 1 D'3s

Experimental 1

Theoretical

×100 %

*Deviations from theoretical values due to experimental errors(explained below)

Table 2

8 dh/Dh

ΔK (Experimental) Logarithmic (ΔK (Theoretical)) Δτ

Logarithmic (ΔK (Experimental)) ΔV Exponential (Δτ)

ΔK (Theoretical) Logarithmic (ΔV)

Sample Calculation Vs

πLDs2 / 4 π(100)(12)2 / 4 11309

Experimental % change in torsional stiffness,

K K h−K s'×100 % Ks = (284

19 – 280

38) / 280

Theoretical % change in torsional stiffness,

dh ) ×100 % Ds = – (1/2)4 x 100 = –6

ΔK =−(

V V h −V s'×100 % Vs = [(πL(Dh²-dh²) / 4) – 11309

Theoretical % change in maximum shear stress Δτ =

τ h−τ s'×100 % τs

D −d ×100 % 1 D'3s = [{12 / (124 – 64)} – 1/123] / (1/123) x 100 = 6

Table 3: Strength and stiffness of hollow and solid shafts having the same volume Theoretical % change in maximum shear stress τ −τ Δτ = h s'×100 % τs

% change in torsional stiffness

Experimental K −K s'ΔK = h ×100 % Ks

Theoretical 2 D'2×(1−( s') ) Dh Ds 2 ( ) Dh

D −d 1 D'3s

×100 %

93 mm dia

94 mm dia

Ks= 198

74 mm dia

Ks= 158

39 mm dia

Ks= 187

×100 %

Table 3

Ds/Dh 3 2

5 % Change

-1 Ds/Dh

ΔK (Experimental) ΔK (Theoretical) Δτ(Max Shear Stress)

Sample Calculation Experimental % change in torsional stiffness,

K K h−K s'×100 % Ks = (203

53 – 97

87) / 97

Theoretical % change in torsional stiffness,

K 2×( 1−( ΔK =

Ds 2 )) Dh

D 2 ( s) Dh

×100 %

6612) / 0

Exponential (ΔK (Experimental)) Exponential (ΔK (Theoretical)) Logarithmic (Δτ(Max Shear Stress))

Theoretical % change in maximum shear stress Δτ =

τ h−τ s'×100 % τs

D −d ×100 % 1 D'3s = [12 / (124

Discussion Refering to Table 2(comparing shafts with same outer diameter),

This is because,

solid shafts possesses more surface area to distribute the load resulting in less stress to support

However the majority of the load is handled by the exterior of the shaft and hence,

the strength differences between solid shaft and hollow shaft is not very significant

it can be seen that the ΔK (theoretical) curves are negative

However the rest of experimental values agree with the theory that hollow shafts have lower torsional stiffness as compared to solid shafts of the same outer diameter

As Kh is lesser than Ks this results in negative values according to the equation ΔK = (Kh−Ks) / Ks ×100%

As the central hole of the hollow shaft gets larger (larger inner diameter,

this means that the torsional stiffness of the hollow shaft becomes lower as the solid shaft remains unchanged

Therefore for hollow and solid shafts of the same outer diameter,

solid shafts are stiffer and more rigid than hollow shafts

Refering to Table 3(comparing shafts with same volume),

density is uniform and same for all the shafts,

comparing shafts with same volume is equivalent to comparing shafts of the same mass and material

Hollow shafts with the same volume(and hence mass in this case),

are stronger as the outer diameter is larger

As most of the load is handled by the exterior of the shaft,

this implies that the hollow shaft of equal volume would be stronger than that of the solid shaft

In this experiment,

the hollow shafts all had an outer diameter of 12mm and its respective solid shaft of equivalent volume all had outer diameters of less than 12mm

it can be seen that both the ΔK (experimental) and ΔK (theoretical) curves are positive

This means that the experimental values agrees with the theoretical values that hollow shafts have higher torsional stiffness as compared to solid shafts of the same volume

As Kh is larger than Ks,

resulting in positive values according to the equation ΔK = (K h−Ks) / Ks ×100%

As Ds/Dh increases,

meaning the torsional stiffness of the hollow shaft becomes lower and closer to the torsional stiffness of the solid shaft as the diameter of the solid shaft approaches the diameter of the hollow shaft

Hence for hollow and solid shafts of the same volume,

hollow shafts are stiffer and more rigid than solid shafts

the best fit logarithm curve of experimental results is very similar to the theoretical curve

The deviations could be accounted by the experimental errors and errors in the apparatus and the shaft

However,

both graphs still show a general downward trend

The difference could be attributed to experimental errors and/or errors in the apparatus and shaft

Error exists in the shaft itself ◦ Shaft may have undergone deformation after repeated usages ◦ Shear modulus,

may not be constant for all shaft due to production inaccuracies Systematic errors such as ◦ Calibration errors of the measuring instrument Human errors such as ◦ Parallax errors ◦ Difficulties in taking readings as the equipment is sensitive and readings fluctuate due to external reactions(e

vibration of table or hands) Electrical interferences

we look at the general trend of the data and overlook the discrepancies

Conclusion In conclusion,

we determine theoretically and verify experimentally,

that for the same material 1) With same outer diameter : a hollow shaft will be much lighter and slightly weaker than a solid shaft

it would be more economical in such a case to utilise a hollow shaft,

especially in scenarios where weight of the structure is important(e

Vehicles)

hence selection of shaft would be dependent on cost against strength required,

as solid shafts are often less expensive than thick tubed shafts

For a given amount of material,

would you fabricate it to a hollow or solid shaft

a hollow shaft of the same weight is stronger and since,

the amount of material is a given and cost is no longer a factor,

I would fabricate it to a hollow shaft for added strength