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Stuart Tang A0125640 Lab Group 2F2 ME2113-2 Torsion of Circular Shafts

Objective: The objective for this experiment is to study how a hollow shaft would behave compared with a solid shaft, in terms of strength and rigidity.

Sample Calculations: π πΌπ

=

πΊπ πΏ

β

π π

=

πΊπΌπ πΏ

=πΎβ

used to calculate theoretical values of K in Table 1; G = 40GPa for brass

For experimental values of K, we plot the graphs of Torque (Nm) against Angular Displacement (rad) for all the shafts, and calculate the gradient of each graph. In Table 2, to find experimental values of βπΎ, only experimental values are used. These can be taken from Graphs 1 and 2, by choosing the appropriate gradient. Similarly, for theoretical values of βπΎ only theoretical values are used. These values can be found from Table 1. Volume of Solid Shaft =

π

Volume of Hollow Shaft =

π·π 2 4

πΏ

2 π·β2 βπβ

4

πΏ ; L = 0.1m for all shafts.

Results: Angular Displacements Degrees Radians 0 0 0.20 0.003491 0.40 0.006981 0.60 0.010472 0.80 0.013963 1.00 0.017453 1.20 0.020944 1.40 0.024435 1.60 0.027925 1.8 0.031416 2.0 0.03907 Ip (mm4) K (Nm/rad)

Torque (Nm) for Solid Shafts 7.93mm 0 0.46 0.93 1.43 1.92 2.43 2.95 3.47 3.99 4.55 5.08 388.23 155.292

8.94mm 0 0.56 1.10 1.70 2.34 3.02 3.73 4.43 5.20 5.95 6.75 627.12 250.848

9.74mm 0 0.61 1.32 2.05 2.92 3.79 4.71 5.69 6.64 7.67 8.70 883.56 353.424

Table 1: Experimental Data for Solid and Hollow Shafts

10.39mm 0 0.64 1.40 2.15 2.99 3.84 4.80 5.85 6.91 8.03 9.12 1144.09 457.636

Torque (Nm) for Hollow Shafts 12.0mm 0 0.86 1.87 3.04 4.43 5.74 7.27 8.84 10.50 12.25 13.91 2035.75 814.300

12/9mm 0 0.65 1.36 2.14 3.00 3.90 4.86 5.85 6.88 7.95 9.05 1391.63 556.652

12/8mm 0 0.62 1.33 2.14 2.98 3.93 4.88 5.88 6.94 8.02 9.15 1633.63 653.452

12/7mm 0 0.55 1.23 2.01 2.92 3.84 4.95 6.03 7.20 8.37 9.56 1800.03 720.012

12/6mm 0 0.82 1.78 2.79 3.88 5.01 6.21 7.47 8.74 10.03 11.40 1908.52 763.408

Graph 1: Torque (Nm) against Angular Displacement (rad) for Solid Shafts 16 14

y = 382.64x - 0.5768

Torque (Nm)

12 10

y = 248.35x - 0.2712

8

y = 237.99x - 0.2347 y = 183.06x - 0.1026

6

y = 137.54x + 0.0211

4 2 0 -2

0

0.005

0.01

0.015

0.02

0.025

0.03

0.035

0.04

0.045

Angular Displacement (rad) 7.93mm

8.94mm

9.74mm

10.39mm

12.0mm

Graph 2: Torque (Nm) against Angular Displacement (rad) for for Hollow Graph 2: Torque (Nm) against Angular Displacement (rad) Hollow Shafts Shafts 14 y = 310.85x - 0.2585

12

y = 263.65x - 0.4596

Torque (Nm)

10

y = 249.93x - 0.2867

8

y = 246.76x - 0.251

6 4 2 0 -2

0

0.005

0.01

0.015

0.02

0.025

0.03

Angular Displacement (rad) 12/9 mm

12/8 mm

12/7 mm

12/6 mm

0.035

0.04

0.045

Theoretical % Change in Maximum Shear Stress

Solid Shaft Ds = 12mm

% Change in Torsional Stiffness

% Change in Volume

Vs = 45.239Β΅m2 Ks = 814.3 (Theoretical)

βπΎ =

πΎβ β πΎπ β 100% πΎπ

βπ =

βπ =

πβ β ππ β 100% ππ

πβ β ππ β 100% ππ

π·β 1 β π·β4 β πβ4 π·π 3 = β 100% 1 3 π·π

Ks = 382.64 (Experimental) Experimental

Theoretical

Hollow Shafts 1. 12/6mm dh/Dh = 0.5

-18.76%

-6.25%

-25%

6.667%

2. 12/7mm dh/Dh = 0.583

-31.10%

-11.58%

-34.03%

13.095%

-34.68%

-19.75%

-44.44%

24.615%

-35.51%

-31.64%

-56.25%

46.285%

3. 12/8mm dh/Dh = 0.667 4. 12/9mm dh/Dh = 0.75

Table 2: Strength and Stiffness of Hollow and Solid Shafts Having the Same Outer Diameter

Graph 3: Strength and Stiffness of Hollow and Solid Shafts with the Same Outer Diameter 60 40

% Change

20 0 0.45

0.5

0.55

0.6

0.65

0.7

-20 -40 -60 -80

dh/Dh ΞK (Experimental)

ΞK (Theoretical)

ΞV

ΞΟ

0.75

0.8

% Change in Torsional Stiffness βπΎ =

1. 2. 3. 4.

solid, 7.93mm hollow, 12/9mm solid, 8.94mm hollow, 12/8mm solid, 9.74mm hollow, 12/7mm solid, 10.39mm hollow, 12/6mm

πΎβ β πΎπ β 100% πΎπ

Theoretical % Change in Maximum Shear Stress πβ β ππ βπ = β 100% ππ π·β 1 β π·β4 β πβ4 π·π 3 = β 100% 1 π·π 3

Experimental

Theoretical

79.41%

258.46%

-57.78%

36.53%

160.50%

-48.47%

10.78%

103.72%

-39.52%

25.17%

66.82%

-30.76%

Table 3: Strength and Stiffness of Hollow and Solid Shafts having the Same Volume

Graph 4: Strength and Stiffness of Hollow and Solid Shafts with the Same Volume 300 250

% Change

200 150 100 50 0 -50

0.6

0.65

0.7

-100

0.75

0.8

Ds/Dh ΞK (Experimental)

ΞK (Theoretical)

ΞΟ

0.85

0.9

Discussion: From Table 1, we can see that theoretical values of K increases as the diameter of solid shafts increases. This is because Ip increases, hence more torque is needed to produce the same angular displacement. The same logic applies for hollow shafts with the same outer diameter. As the inner diameter decreases (shaft becomes thicker), Ip increases, and theoretical values of K increases as well, implying that more torque is required to produce the same angular displacement among hollow shafts. We observe similar trends in Graphs 1 & 2. For solid shafts, K increases as diameter increases, and for hollow shafts, K increases as thickness increases. However, we observe that our experimental values of K are significantly lower than our theoretical values. This can be attributed to wear and tear of the brass shafts. We noticed that as we unloaded and loaded the same shaft to the same angular displacement, torque readings became slightly lower every time. An accumulation of wear and tear from previous lab groups might have caused the extreme differences that we see between our experimental values and theoretical values of K. From Table 2 and Graph 3, we want to compare the strength and stiffness of solid and hollow shafts when Ds = Dh. From our data, we can observe that as dh increases (thickness decreases), K decreases, implying that the hollow shaft is weaker and less rigid than a solid shaft of the same outer diameter. Logically, this is because there is less mass in the hollow shaft to sustain the same load as a solid shaft. Hence such a result is unsurprising. The data for the max shear stress gives the same conclusions β as dh/Dh increases, the maximum shear stress within the shaft increases, since less mass is required to sustain the same amount of load. Hence, a hollow shaft is weaker and less rigid than a solid shaft if Ds = Dh. In Table 3, we investigate the effects of how stiffness and maximum shear stress of two shafts (one solid and one hollow) would be changed if they have the same volume. These data are then plotted onto Graph 4. From our results, it is apparent that K is much higher in a hollow shaft compared to a solid shaft if volume was the same in both shafts. This implies that more torque is required to produce the same angular displacement, hence a higher rigidity in the hollow shaft. We also observe that as thickness of the hollow shaft increases, K decreases exponentially. It is hence more cost effective to have hollow shafts that have a larger outer diameter, but lower thickness if we were to increase the volume of the shaft. Similar conclusions can be drawn from our data for shear stress. Hollow shafts would experience lower shear stresses compared to solid bars if they had the same volume and underwent the same loading. As Ds/Dh decreases, maximum shear stress experience in the bar decreases as well, implying that the hollow shaft can sustain heavier loads if it has a lower thickness, but volume remains the same (higher strength to weight ratio). Hence, it is indeed more cost effective to produce hollow shafts with a larger outer diameter, but lower thickness.

Conclusion: In conclusion, we can conclude that as the diameter of the solid shaft increases, more torque is required to produce the same angular displacements. Similarly, as the inner diameter of the hollow shaft increases, less torque is required to produce the same angular displacements. However, if we were to keep volume a constant, hollow shafts with larger outer diameters and lower thickness are the most cost effective as they have a much larger strength to weight ratio (βπ is much lower), as well as rigidity to weight ratio (βπΎ is much higher).

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Objective: The objective for this experiment is to study how a hollow shaft would behave compared with a solid shaft, in terms of strength and rigidity.

Sample Calculations: π πΌπ

=

πΊπ πΏ

β

π π

=

πΊπΌπ πΏ

=πΎβ

used to calculate theoretical values of K in Table 1; G = 40GPa for brass

For experimental values of K, we plot the graphs of Torque (Nm) against Angular Displacement (rad) for all the shafts, and calculate the gradient of each graph. In Table 2, to find experimental values of βπΎ, only experimental values are used. These can be taken from Graphs 1 and 2, by choosing the appropriate gradient. Similarly, for theoretical values of βπΎ only theoretical values are used. These values can be found from Table 1. Volume of Solid Shaft =

π

Volume of Hollow Shaft =

π·π 2 4

πΏ

2 π·β2 βπβ

4

πΏ ; L = 0.1m for all shafts.

Results: Angular Displacements Degrees Radians 0 0 0.20 0.003491 0.40 0.006981 0.60 0.010472 0.80 0.013963 1.00 0.017453 1.20 0.020944 1.40 0.024435 1.60 0.027925 1.8 0.031416 2.0 0.03907 Ip (mm4) K (Nm/rad)

Torque (Nm) for Solid Shafts 7.93mm 0 0.46 0.93 1.43 1.92 2.43 2.95 3.47 3.99 4.55 5.08 388.23 155.292

8.94mm 0 0.56 1.10 1.70 2.34 3.02 3.73 4.43 5.20 5.95 6.75 627.12 250.848

9.74mm 0 0.61 1.32 2.05 2.92 3.79 4.71 5.69 6.64 7.67 8.70 883.56 353.424

Table 1: Experimental Data for Solid and Hollow Shafts

10.39mm 0 0.64 1.40 2.15 2.99 3.84 4.80 5.85 6.91 8.03 9.12 1144.09 457.636

Torque (Nm) for Hollow Shafts 12.0mm 0 0.86 1.87 3.04 4.43 5.74 7.27 8.84 10.50 12.25 13.91 2035.75 814.300

12/9mm 0 0.65 1.36 2.14 3.00 3.90 4.86 5.85 6.88 7.95 9.05 1391.63 556.652

12/8mm 0 0.62 1.33 2.14 2.98 3.93 4.88 5.88 6.94 8.02 9.15 1633.63 653.452

12/7mm 0 0.55 1.23 2.01 2.92 3.84 4.95 6.03 7.20 8.37 9.56 1800.03 720.012

12/6mm 0 0.82 1.78 2.79 3.88 5.01 6.21 7.47 8.74 10.03 11.40 1908.52 763.408

Graph 1: Torque (Nm) against Angular Displacement (rad) for Solid Shafts 16 14

y = 382.64x - 0.5768

Torque (Nm)

12 10

y = 248.35x - 0.2712

8

y = 237.99x - 0.2347 y = 183.06x - 0.1026

6

y = 137.54x + 0.0211

4 2 0 -2

0

0.005

0.01

0.015

0.02

0.025

0.03

0.035

0.04

0.045

Angular Displacement (rad) 7.93mm

8.94mm

9.74mm

10.39mm

12.0mm

Graph 2: Torque (Nm) against Angular Displacement (rad) for for Hollow Graph 2: Torque (Nm) against Angular Displacement (rad) Hollow Shafts Shafts 14 y = 310.85x - 0.2585

12

y = 263.65x - 0.4596

Torque (Nm)

10

y = 249.93x - 0.2867

8

y = 246.76x - 0.251

6 4 2 0 -2

0

0.005

0.01

0.015

0.02

0.025

0.03

Angular Displacement (rad) 12/9 mm

12/8 mm

12/7 mm

12/6 mm

0.035

0.04

0.045

Theoretical % Change in Maximum Shear Stress

Solid Shaft Ds = 12mm

% Change in Torsional Stiffness

% Change in Volume

Vs = 45.239Β΅m2 Ks = 814.3 (Theoretical)

βπΎ =

πΎβ β πΎπ β 100% πΎπ

βπ =

βπ =

πβ β ππ β 100% ππ

πβ β ππ β 100% ππ

π·β 1 β π·β4 β πβ4 π·π 3 = β 100% 1 3 π·π

Ks = 382.64 (Experimental) Experimental

Theoretical

Hollow Shafts 1. 12/6mm dh/Dh = 0.5

-18.76%

-6.25%

-25%

6.667%

2. 12/7mm dh/Dh = 0.583

-31.10%

-11.58%

-34.03%

13.095%

-34.68%

-19.75%

-44.44%

24.615%

-35.51%

-31.64%

-56.25%

46.285%

3. 12/8mm dh/Dh = 0.667 4. 12/9mm dh/Dh = 0.75

Table 2: Strength and Stiffness of Hollow and Solid Shafts Having the Same Outer Diameter

Graph 3: Strength and Stiffness of Hollow and Solid Shafts with the Same Outer Diameter 60 40

% Change

20 0 0.45

0.5

0.55

0.6

0.65

0.7

-20 -40 -60 -80

dh/Dh ΞK (Experimental)

ΞK (Theoretical)

ΞV

ΞΟ

0.75

0.8

% Change in Torsional Stiffness βπΎ =

1. 2. 3. 4.

solid, 7.93mm hollow, 12/9mm solid, 8.94mm hollow, 12/8mm solid, 9.74mm hollow, 12/7mm solid, 10.39mm hollow, 12/6mm

πΎβ β πΎπ β 100% πΎπ

Theoretical % Change in Maximum Shear Stress πβ β ππ βπ = β 100% ππ π·β 1 β π·β4 β πβ4 π·π 3 = β 100% 1 π·π 3

Experimental

Theoretical

79.41%

258.46%

-57.78%

36.53%

160.50%

-48.47%

10.78%

103.72%

-39.52%

25.17%

66.82%

-30.76%

Table 3: Strength and Stiffness of Hollow and Solid Shafts having the Same Volume

Graph 4: Strength and Stiffness of Hollow and Solid Shafts with the Same Volume 300 250

% Change

200 150 100 50 0 -50

0.6

0.65

0.7

-100

0.75

0.8

Ds/Dh ΞK (Experimental)

ΞK (Theoretical)

ΞΟ

0.85

0.9

Discussion: From Table 1, we can see that theoretical values of K increases as the diameter of solid shafts increases. This is because Ip increases, hence more torque is needed to produce the same angular displacement. The same logic applies for hollow shafts with the same outer diameter. As the inner diameter decreases (shaft becomes thicker), Ip increases, and theoretical values of K increases as well, implying that more torque is required to produce the same angular displacement among hollow shafts. We observe similar trends in Graphs 1 & 2. For solid shafts, K increases as diameter increases, and for hollow shafts, K increases as thickness increases. However, we observe that our experimental values of K are significantly lower than our theoretical values. This can be attributed to wear and tear of the brass shafts. We noticed that as we unloaded and loaded the same shaft to the same angular displacement, torque readings became slightly lower every time. An accumulation of wear and tear from previous lab groups might have caused the extreme differences that we see between our experimental values and theoretical values of K. From Table 2 and Graph 3, we want to compare the strength and stiffness of solid and hollow shafts when Ds = Dh. From our data, we can observe that as dh increases (thickness decreases), K decreases, implying that the hollow shaft is weaker and less rigid than a solid shaft of the same outer diameter. Logically, this is because there is less mass in the hollow shaft to sustain the same load as a solid shaft. Hence such a result is unsurprising. The data for the max shear stress gives the same conclusions β as dh/Dh increases, the maximum shear stress within the shaft increases, since less mass is required to sustain the same amount of load. Hence, a hollow shaft is weaker and less rigid than a solid shaft if Ds = Dh. In Table 3, we investigate the effects of how stiffness and maximum shear stress of two shafts (one solid and one hollow) would be changed if they have the same volume. These data are then plotted onto Graph 4. From our results, it is apparent that K is much higher in a hollow shaft compared to a solid shaft if volume was the same in both shafts. This implies that more torque is required to produce the same angular displacement, hence a higher rigidity in the hollow shaft. We also observe that as thickness of the hollow shaft increases, K decreases exponentially. It is hence more cost effective to have hollow shafts that have a larger outer diameter, but lower thickness if we were to increase the volume of the shaft. Similar conclusions can be drawn from our data for shear stress. Hollow shafts would experience lower shear stresses compared to solid bars if they had the same volume and underwent the same loading. As Ds/Dh decreases, maximum shear stress experience in the bar decreases as well, implying that the hollow shaft can sustain heavier loads if it has a lower thickness, but volume remains the same (higher strength to weight ratio). Hence, it is indeed more cost effective to produce hollow shafts with a larger outer diameter, but lower thickness.

Conclusion: In conclusion, we can conclude that as the diameter of the solid shaft increases, more torque is required to produce the same angular displacements. Similarly, as the inner diameter of the hollow shaft increases, less torque is required to produce the same angular displacements. However, if we were to keep volume a constant, hollow shafts with larger outer diameters and lower thickness are the most cost effective as they have a much larger strength to weight ratio (βπ is much lower), as well as rigidity to weight ratio (βπΎ is much higher).

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