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Unformatted text preview: ( x ) dx = – ∫ q ⎛ -- ⎞ dx = ------ ( L – x )
xB = L
L = –∫ Printed from: http://www.me.mtu.edu/~mavable/MoM2nd.htm 3. The internal torque can also be found using a free-body diagram. We can make an imaginary cut at some location x and draw the freeL
body diagram of the right side. The distributed torque represented by ∫ t ( x ) dx is the area of the trapezoid BCDE, and this observation
can be used in drawing a statically equivalent diagram, as shown in Figure 5.41. Equilibrium then gives us the value of the internal
torque as before. We can find the internal torque as shown.
----L T C
q D E 1 ⎛ qx
-- ----- + q⎞ ( L – x ) = ------ ( L – x )
2L B T Figure 5.41 Internal torque by free-body diagram in Example 5.11.
4. The free-body diagram approach in Figure 5.41 is intuitive but more tedious and difficult than the use of Equation (5.14). As the function representing the distributed torque grows in complexity, the attractiveness of the mathematical approach of Equation (5.14)
grows correspondingly. January, 2010 M. Vable Mechanics of Materials: Torsion of Shafts 5 230 MoM in Action: Drill, the Incredible Tool
Drills have been in use for almost as long as humans have used tools. Early humans knew from experience that friction
generated by torquing a wooden shaft could start a fire—a technique still taught in survivalist camps. Archeologists in
Pakistan have found teeth perhaps 9000 years old showing the concentric marks of a flint stone drill. The Chinese used
larger drills in the 3rd B.C.E. to extract water and oil from earth. The basic design—a chuck that delivers torque to the drill
bit—has not changed, but their myriad uses to make holes from the very small to the very large continues to grow.
Early development of the drill was driven by the technology of delivering power to the drill bit. In 1728, French dentist
Pierre Fauchard (Figure 5.42a) described how catgut twisted around a cylinder could power the rotary movement as a bow
moved back and forth. However, hand drills like these operated at only about 15 rpm. George F. Harrington introduced the
first motor-driven drill in 1864, powered by the spring action of a clock. George Green, an American dentist, introduced a
pedal-operated pneumatic drill just four years later—and, in 1875, an electric drill. By 1914 dental drills could operate at
Other improvements took better understanding of the relationship between power, torsion, and shear stress in the drill bit
(problems 5.45—5.47) and the material being drilled:
• The sharper the drill tip, the higher the shear stresses at the point, and the greater the amount of material that can be
sheared. For most household jobs the angle of the drill tip is 118o. For soft materials such as plastic, the angle is sharper,
while for harder material such as steel the angle is shallower. • For harder materials low speeds can prolong the life of drill bit. However, in dentistry higher speeds, of up to 500,000
rpm, reduce a patient’s pain.
(a) (b) Figure 5.42 (a) Pierre Fauchard drill. (b) Tunnel boring machine Matilda (Courtesy Erikt9). Printed from: http://www.me.mtu.edu/~mavable/MoM2nd.htm • Slower speeds are also used to shear a large amount of material. Tunnel boring machines (TBM) shown in Figure 5.42b
may operate at 1 to 10 rpm. The world’s largest TBM, with a diameter of 14.2 m, was used to drill the Elbe Tunnel in
Hamburg, Germany. Eleven TBM’s drilled the three pipes of the English Channel, removing 10.5 million cubic yards of
earth in seven years. • Drill bits can be made of steel, tungsten carbide, polycrystalline diamonds, titanium nitrate, and diamond powder. The
choice is dictated by the material to be drilled as well as the cost. Even household drills have different bits for wood,
metal, or masonry.
Delivery and control of power to the drill bit are engineering challenges. So is removal of sheared material, not only to prevent the hole from plugging, but also because the material carries away heat, improving the strength and life of a drill bit.
Yet the fundamental function of a drill remains: shearing through torsion. January, 2010 M. Vable Mechanics of Materials: Torsion of Shafts 5 231 PROBLEM SET 5.2
5.19 The torsional shear stress at point A on a solid circular homogenous cross-section was found to be τA= 120 MPa. Determine the
maximum torsional shear stress on the cross-section.
300 60 mm Figure P5.19 100 mm The torsional shear strain at point A on a homogenous circular section shown in Figure P5.20 was found to be 900 μ rads. Using a
shear modulus of elasticity of 4000 ksi, determine the torsional shear stress at point B. 5.20 B
300 550 1.5 in. Figure P5.20 2.5 in. 5.21 An aluminum shaft (Gal= 28 GPa) and a steel shaft (GS=82 GPa) are securely fastened to form composite shaft with a cross section
shown in Figure P5.21. If the maximum torsional shear strain in aluminum is 1500 μ rads, determine the maximum torsional shear s...
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This note was uploaded on 04/08/2010 for the course ENGR 232 taught by Professor Smith during the Spring '10 term at Aarhus Universitet.
- Spring '10