C2 - Lab Report

Table 1A - Graph of Torsional Stiffness for Solid Shafts Torque (Nm) v.s. Angle of Twist (rad) 12

10

7.93 mm Linear (7.93 mm) 8.94 mm Linear (8.94 mm) 9.74 mm Linear (9.74 mm) 10.39 mm Linear (10.39 mm) 12.0 mm Linear (12.0 mm)

f(x) = 280.38x - 0.38

Torque (Nm)

8 f(x) = 198.89x - 0.11 6

f(x) = 187.04x - 0.21 f(x) = 158.08x - 0.13

4 f(x) = 97.87x - 0.04 2

0 0

0.01

0.01

0.02

0.02

0.03

0.03

0.04

0.04

Angle of Twist (rad)

Table 1B - Graph of Torsional Stiffness for Hollow Shafts Torque (Nm) v.s. Angle of Twist (rad) 12

10

f(x) = 284.19x - 0.28 12/9 mm Linear (12/9 mm) 12/8 mm Linear (12/8 mm) 12/7 mm Linear (12/7 mm) 12/6 mm Linear (12/6 mm)

f(x) = 283.01x - 0.40

Torque (Nm)

8 f(x) = 274.65x - 0.61 6 f(x) = 203.53x - 0.25 4

2

0 0

0.01

0.01

0.02

0.02

Angle of Twist (rad)

0.03

0.03

0.04

0.04

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

Deg. 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 Ip

K

Radians 7.93 mm 8.94 mm 9.74 mm 10.39 mm 12.0 mm 12/9 mm 12/8 mm 12/7 mm 12/6 mm 0 0 0 0 0 0 0 0 0 0 0.003491 0.31 0.64 0.5 0.52 0.81 0.56 0.75 0.6 0.84 0.006981 0.63 1.26 1 1.08 1.59 1.15 1.54 1.23 1.69 0.010472 0.98 1.92 1.5 1.65 2.4 1.78 2.38 2.01 2.56 0.013963 1.3 2.6 2 2.3 3.27 2.46 3.3 2.87 3.52 0.017453 1.66 3.3 2.5 2.94 4.2 3.13 4.26 3.75 4.49 0.020944 2 4 3 3.6 5.19 3.84 5.36 4.85 5.48 0.024435 2.35 4.7 3.66 4.3 6.29 4.61 6.39 5.87 6.53 0.027925 2.69 5.46 4.3 5.01 7.45 5.42 7.5 7 7.62 0.031416 3.05 6.18 4.92 5.74 8.63 6.26 8.64 8.25 8.76 0.034907 3.4 6.93 5.56 6.51 9.82 7.1 9.83 9.57 9.95 388.23 627.12 883.56 1144.09 2035.75 1391.63 1633.63 1800.03 1908.52 (mm4) (Nm/rad) 97.87 198.89 158.08 187.04 280.38 203.53 283.01 274.65 284.19

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

ΔK =

Vs = 11309.73 mm³ Ks = 280.38 Nm/rad L = 100 mm

Hollow Shafts 1. 12/6 mm dh/Dh = 1/2 2. 12/7 mm dh/Dh = 7/12 3. 12/8 mm dh/Dh = 2/3 4. 12/9 mm dh/Dh = 3/4

Theoretical % change in maximum shear stress

% change in volume ΔV =

V h −V s ×100 % Vs

Δτ =

Dh

Vh= πL(Dh²-dh²) / 4

4

dh ) ×100 % Ds (theoretical)

ΔK =−(

τ h−τ s ×100 % τs

=

4 h

4 h

−

D −d 1 D 3s

1 D 3s

Experimental 1.36%*

Theoretical -6.25%

-25.00%

6.67%

-2.04%

-11.58%

-34.03%

13.10%

0.94%*

-19.75%

-44.44%

24.62%

-27.41%

-31.64%

-56.25%

46.29%

×100 %

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

Table 2 - Graph of Strength and Stiffness of Hollow and Solid Shafts with the Same Outer Diameter % Change v.s. dh/Dh 0.6 0.4

% Change

0.2 0 0.45 -0.2

0.5

0.55

0.6

0.65

0.7

0.75

-0.4 -0.6 -0.8 dh/Dh

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

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

ΔK (Theoretical) Logarithmic (ΔV)

0.8

Sample Calculation Vs

= = =

πLDs2 / 4 π(100)(12)2 / 4 11309.73 mm

Experimental % change in torsional stiffness, K K h−K s ×100 % Ks = (284.19 – 280.38) / 280.38 x 100 = 1.36%

ΔK =

Theoretical % change in torsional stiffness, K 4

dh ) ×100 % Ds = – (1/2)4 x 100 = –6.250%

ΔK =−(

% change in volume, V V h −V s ×100 % Vs = [(πL(Dh²-dh²) / 4) – 11309.73] / 11309.73 x 100 = [8482.3 – 11309.73] / 11309.73 x 100 = –25.00%

ΔV =

Theoretical % change in maximum shear stress Δτ =

τ h−τ s ×100 % τs

Dh 4 h

4 h

−

1 D 3s

D −d ×100 % 1 D 3s = [{12 / (124 – 64)} – 1/123] / (1/123) x 100 = 6.67% =

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

Dh =

Experimental K −K s ΔK = h ×100 % Ks

ΔK =

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

4 h

4 h

−

D −d 1 D 3s

1 D 3s

×100 %

1. solid, 7.93 mm dia. hollow, 12/9 mm Ds/Dh = 0.661

Ks= 97.87Nm/rad Kh= 203.53Nm/rad ΔK = 107.96%

257.98%

-57.78%

2. solid, 8.94 mm dia. hollow, 12/8 mm Ds/Dh = 0.745

Ks= 198.89Nm/rad Kh= 283.01Nm/rad ΔK = 42.29%

160.34%

-48.47%

3. solid, 9.74 mm dia. hollow, 12/7 mm Ds/Dh = 0.812

Ks= 158.08Nm/rad Kh= 274.65Nm/rad ΔK = 73.74%

103.58%

-39.52%

4. solid, 10.39 mm dia. hollow, 12/6 mm Ds/Dh = 0.866

Ks= 187.04Nm/rad Kh= 284.19Nm/rad ΔK = 51.94%

66.78%

-30.76%

×100 %

Table 3 - Graph of Strength and Stiffness of Hollow and Solid Shafts of the Same Volume % Change v.s. Ds/Dh 3 2.5 % Change

2 1.5 1 0.5 0 -0.50.65

0.7

0.75

0.8

0.85

-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) / 97.87 x 100 = 107.96%

ΔK =

Theoretical % change in torsional stiffness, K 2×( 1−( ΔK =

Ds 2 )) Dh

D 2 ( s) Dh

×100 %

= 2 x (1- 0.6612) / 0.6612 x 100 = 257.98%

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

0.9

Theoretical % change in maximum shear stress Δτ =

τ h−τ s ×100 % τs

Dh 4 h

4 h

−

1 D 3s

D −d ×100 % 1 D 3s = [12 / (124 - 94) – 1/7.933] / (1/7.933) x 100 = -57.78% =

Discussion Refering to Table 2(comparing shafts with same outer diameter), 1) Strength Comparison of solid and hollow shafts having the same outer diameter Solid shafts with the same outer diameter are stronger. 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. 2) Torsional stiffness comparison of solid and hollow shafts having the same outer diameters From graph 3, it can be seen that the ΔK (theoretical) curves are negative. 2 of the experimental values deviate from the theoretical values due to experimental errors. 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, larger d h/Dh value), the ΔK decreases, 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), 3) Strength Comparison of solid and hollow shafts having the same volume Assuming, 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. 4) Torsional Stiffness Comparison of solid and hollow shafts having the same volume From graph 4, 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, the ΔK value decreases, 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. 5) Comparison of theoretical and experimental results for solid and hollow shafts having the same outer diameter (Table 2) Despite deviating values, 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. 6) Comparison of the theoretical and experimental results for solid and hollow shaft of the same volume (Table 3) The curve of experimental results differs from the theoretical curve. 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.

7) Possible sources of errors •

• •

•

Error exists in the shaft itself ◦ Shaft may have undergone deformation after repeated usages ◦ Shear modulus, G, 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.g. vibration of table or hands) Electrical interferences

Thus, for the discussion of data, 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. Hence, 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.g. Vehicles). 2) And the same final weight(same amount of material): a hollow shaft is stronger than a solid shaft as it has a larger outer diameter, 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? As mentioned in 2), 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.