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Me2113 Lab

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Words 916
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Objective
The objective of this experiment is to study how a hollow shaft would behave compared with a solid shaft, in terms of strength and rigidity.

Sample Calculation
For Table 2: Comparing strength and torsional stiffness of solid and hollow shafts with the same outer diameter (12mm)
For hollow shaft 12/6 mm,
Experimental % change in torsional stiffness,

= 305.05-343.64343.64 *100% = -11.23%
Theoretical % change in torsional stiffness,

=-(612)4*100% = -6.25%
%change in volume,

=πDh2-πdh2h4-πDs2h4πDs2h4*100% =Dh2-dh2-Ds2Ds2*100%
=122-62-122122*100%
= -25%
Theoretical % change in maximum shear stress,

=12124-64-11231123*100%
=6.67%
For Table 3:Comparing strength and torsional stiffness of solid and hollow shafts for same volume.
For solid shaft of 7.93 mm in diameter and hollow shaft 12/9 mm,
Experimental %change in torsional stiffness,

= 222.67-105.92105.92 *100% = 110.2%
Theoretical % change in torsional stiffness,
2*1-(DsDh)2(DsDh)2*100%

=2*1-(7.9312)2(7.9312)2 *100% =258.0%

Theoretical % change in maximum shear stress,

=12124-94-17.93317.933*100%
=-57.78%

Results (Tables and Graphs)

Table 1 refer to appendix 1.

Table 2: Strength and stiffness of hollow and solid shafts having the same outer diameter Solid ShaftDs = 12 mmVs=0.00001131m³Ks=343.64Nm/rad | % change in torsional stiffness | % change in volume | Theoretical % change in maximum shear stress | | Experimental | Theoretical | | | Hollow Shafts12/6 mmdh/Dh=0.500 | -11.23 | -6.250 | -25.00 | 6.670 | 12/7 mmdh/Dh=0.583 | -17.38 | -11.58 | -34.03 | 13.10 | 12/8 mmdh/Dh=0.667 | -20.71 | -19.75 | -44.44 | 24.62 | 12/9 mmdh/Dh=0.750 | -35.20 | -31.64 | -56.25 | 46.29 |

Table 3: Strength and stiffness of hollow and solid shafts having the same volume | % change in torsional stiffness | Theoretical % change in maximum shear stress -57.78 -48.47 -39.52 -30.76 | | Experimental | Theoretical | | 1.solid, 7.93 mm dia. hollow 12/9 mm2.solid, 8.94 mm dia. hollow 12/8 mm3.solid, 9.74 mm dia. hollow 12/7 mm4. solid, 10.39 mm dia. hollow 12/6 mm | 110.2 46.86 22.11 19.03 | 258.0 160.3 103.6 66.78 | |

From Table 1,
Graph 1:

Graph2:

From Table 2,
Graph 3:
Graph 4:

Discussion
2) Comparing the results for the shafts having the same outer diameter (Table2) and for shafts having the same volume (Table3)
By comparing the results for shafts having the same outer diameter, we can see that the % changes in torsional stiffness are all negative, which shows that the stiffness of a hollow shaft is smaller than that of a solid shaft with the same diameter. Therefore shows that a hollow shaft is weaker than a solid shaft with the same diameter. Furthermore, the values of % change of torsional stiffness ∆K increase as the inner diameter of the shaft increases. This shows that the bigger the inner diameter of the hollow shaft, the weaker it becomes.
By comparing the results for shafts having the same volume. (Table 3), we can see that the values of ∆K are all positive, showing that a hollow shaft is stronger than a solid shaft with the same volume.

3) In Table 2 and graph 3, compare your experimental results with the theoretical results. Would a hollow shaft be stronger and more rigid than a solid shaft if Ds=Dh?
The graph of experiment results is very similar to the theoretical curve. Any differences could be due to experimental errors and errors in the apparatus and the shaft.
A hollow shaft would not be stronger and more rigid than a solid shaft. Since when Ds=Dh, as the inner diameter of the hollow shaft increases, the percentage change in the torsional stiffness becomes more negative while the percentage changes in maximum shear stress becomes more positive. The more negative ∆K shows a hollow shaft with larger inner diameter is less stiff while the more positive ∆τ shows a larger maximum shear stress is experienced by the hollow shaft, which is more likely to deform. Therefore, solid shafts are more rigid and stiffer than hollow shafts with the same diameter.

4) In Table 3 and graph 4, compare your experimental results with the theoretical curves. For a given amount of material, would you fabricate it to a hollow or solid shaft?
The graph of experimental results differs from the theoretical curve. However, both graphs still show a general downward trend. The big difference could be due to experimental errors and errors in the apparatus and the shaft.
For a given amount of material, I would fabricate it into a hollow shaft. From table 3, when solid shaft and hollow shaft having the same volume, the positive ∆K values show that a hollow shaft is stiffer than a solid shaft with the same volume while the negative ∆τ values show that a less maximum shear stress would be experienced by the hollow shaft than by a solid shaft with the same volume. Therefore, if other factors like cost of drilling a hole of suitable size or risk of making the wall too thin to be practical are not considered, a hollow shaft is preferable.
Conclusion
In conclusion, given a same diameter, solid shafts are much more rigid and stiff as compared to hollow shafts. However, with the same amount of material (same volume), hollow shafts are much stiffer than solid shafts.

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