TORSION_IOWA

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<p>57:019:AAA Intro Mech. Def. Bodies</p><p>Period #9: Torsion</p><p>A. Context</p><p>A torque is a moment applied to mechanical system that causes it to twist or</p><p>rotate.</p><p>In the present circumstances, we’ll be concerned with torsional moments</p><p>(torques) applied to long slender shafts as shown in the figure below.</p><p>We’ll especially want to determine what is the internal stress distribution in a</p><p>shaft subjected to torques, and what is the resulting deformation (or twisting) of</p><p>the shaft itself.</p><p>Initial assumptions:</p><p>• shaft has a circular cross-section;</p><p>• material behaves as a linear, elastic solid.</p><p>9.1 The University of Iowa</p><p>The University of Iowa 57:019:AAA Intro Mech. Def. Bodies 9.2</p><p>B. Kinematics of Torsion</p><p>Kinematics of torsion: how the shaft deforms when subjected to a</p><p>torque.</p><p>Observe that when the shaft is loaded, the rectangular patch</p><p>undergoes essentially pure shear deformation.</p><p>Fig. 9.1 Torsional deformation in a rubber shaft. (a) unloaded, and (b) deformed under torque</p><p>loading.</p><p>(a) (b)</p><p>D’</p><p>D B’</p><p> </p><p>max</p><p>max</p><p>'</p><p>tan</p><p>'</p><p>:ionconfigurat undeformed</p><p>in the AB and AC fibers heConsider t</p><p></p><p></p><p></p><p></p><p></p><p></p><p></p><p></p><p>c</p><p>x</p><p>c</p><p>xx</p><p>BB</p><p>BB</p><p></p><p></p><p></p><p></p><p></p><p></p><p></p><p></p><p></p><p></p><p>57:019:AAA Intro Mech. Def. Bodies 9.3 The University of Iowa</p><p>C. Shear Stresses</p><p>Kinematics of torsion established that =(/c)max.</p><p>Since by Hooke’s Law, t=G it follows that on the cross-section of the shaft: t=(/c)tmax</p><p>max</p><p>max</p><p>c</p><p>c</p><p>G</p><p>t</p><p></p><p></p><p></p><p>t</p><p></p><p></p><p></p><p></p><p></p><p></p><p></p><p></p><p></p><p></p><p></p><p></p><p></p><p></p><p> G</p><p>57:019:AAA Intro Mech. Def. Bodies 9.4 The University of Iowa</p><p>2</p><p>4</p><p>2</p><p>2</p><p>2</p><p>4</p><p>max</p><p>4</p><p>max</p><p>0</p><p>3max</p><p>0</p><p>2</p><p>c</p><p>c</p><p>c</p><p>c</p><p>d</p><p>c</p><p>d</p><p>dAT</p><p>c</p><p>c</p><p>A</p><p>t</p><p></p><p>t</p><p></p><p>t</p><p>t</p><p>t</p><p></p><p></p><p></p><p></p><p></p><p></p><p></p><p></p><p>   </p><p>J</p><p>c</p><p>d</p><p>c</p><p>T</p><p>d</p><p>c</p><p>d</p><p>c</p><p>dAdT</p><p>ddA</p><p>c</p><p>max</p><p>0</p><p>3max</p><p>3max</p><p>max</p><p>2</p><p>2</p><p>2</p><p>2</p><p>t</p><p></p><p>t</p><p></p><p>t</p><p>t</p><p></p><p>t</p><p></p><p></p><p></p><p></p><p></p><p></p><p></p><p></p><p></p><p></p><p></p><p></p><p></p><p></p><p>d</p><p>The University of Iowa 57:019:AAA Intro Mech. Def. Bodies 9.5</p><p>circular cross-section</p><p>(solid shaft)</p><p>annular cross-section</p><p>(tubular shaft)</p><p>2</p><p>4c</p><p>J</p><p></p><p></p><p> </p><p>2</p><p>44</p><p>io cc</p><p>J</p><p></p><p></p><p></p><p>The polar moment of inertia:  dAJ 2</p><p>D. The Polar Moment of Inertia</p><p>E. The Torsion Formula</p><p>F. Torque/Twist Relation:</p><p>co ci</p><p>J</p><p>T</p><p>J</p><p>Tc </p><p>tt  ; max</p><p></p><p></p><p></p><p>dx</p><p>JG</p><p>T</p><p>dx</p><p>d</p><p>JG</p><p>T</p><p>G</p><p>GJ</p><p>T</p><p></p><p></p><p></p><p>t</p><p></p><p>The University of Iowa 57:019:AAA Intro Mech. Def. Bodies 9.6</p><p>G. Power Transmission</p><p>• It is often the case that a source of power such as an engine exists at one location,</p><p>but work must be done at a different location.</p><p>• Power can be transmitted from the engine to the location where work needs to be</p><p>done using drive shafts. The rate of power transmission by shaft is:</p><p>(cycles) sshaft turn heat which t rate theis f</p><p>(radians) sshaft turn heat which t rate theis</p><p>shaft in the torque theis T</p><p>:where</p><p>f2*TTω</p><p>dt</p><p>dθ</p><p>TP</p><p></p><p></p><p>The University of Iowa 57:019:AAA Intro Mech. Def. Bodies 9.7</p><p>H. Example Problems</p><p>Example 9.1: The assembly consists of two sections of</p><p>galvanized steel pipe connected together using a reducing</p><p>coupling at B. The smaller pipe has an outer diameter of 0.75 in.</p><p>and an inner diameter of 0.68 in., whereas the larger pipe has an</p><p>outer diameter of 1 in. and an inner wall diameter of 0.86 in. If</p><p>the pipe is tightly secured to the wall at C, determine the</p><p>maximum shear stress developed in each section of the pipe</p><p>when the couple shown is applied to the handles of the wrench.</p><p>The University of Iowa 57:019:AAA Intro Mech. Def. Bodies 9.8</p><p>Example 9.2: If the applied torque on the shaft CD is</p><p>T’=75N·m, determine the absolute maximum shear stress</p><p>in each shaft. The bearings B, C, and D allow free</p><p>rotations of the shafts, and the motor holds the shafts</p><p>fixed from rotating</p>