ME 4 Engineering Mechanics Lecture 8: Moment of nertia Ahmad Shahedi Shakil Lecturer, Dept. of Mechanical Engg, BUET E-mail: sshakil@me.buet.ac.bd, shakil679@gmail.com Website: teacher.buet.ac.bd/sshakil Courtes: Vector Mechanics for Engineers, Beer and Johnston
Moment of nertia of an Area Second moments or moments of inertia of an area with respect to the and aes, da da Evaluation of the integrals is simplified b choosing da to be a thin strip parallel to one of the coordinate aes. For a rectangular area, da h 0 bd bh The formula for rectangular areas ma also be applied to strips parallel to the aes, d d d da d
Polar Moment of nertia The polar moment of inertia is an important parameter in problems involving torsion of clindrical shafts and rotations of slabs. J 0 r da The polar moment of inertia is related to the rectangular moments of inertia, J r da da da 0 da
Radius of Gration of an Area Consider area A with moment of inertia. magine that the area is concentrated in a thin strip parallel to the ais with equivalent. A k A k k = radius of gration with respect to the ais Similarl, A J k A k J A k A k O O O O O k k k
Sample Problem 9. SOLUTON: A differential strip parallel to the ais is chosen for da. d da da l d Determine the moment of inertia of a triangle with respect to its base. For similar triangles, l b h h h l b h ntegrating d from = 0 to = h, da b h h h 0 4 4 h b d h h 0 h da b d h b h h 0 h d bh
Prob # 9.9 and 9. Determine b direct integration the moment of inertia of the shaded area with respect to the and aes.
Parallel Ais Theorem Consider moment of inertia of an area A with respect to the ais AA da The ais BB passes through the area centroid and is called a centroidal ais. da da d d da da d da Ad parallel ais theorem
Parallel Ais Theorem Moment of inertia T of a circular area with respect to a tangent to the circle, 4 Ad r r r T 5 4 r 4 4 Moment of inertia of a triangle with respect to a centroidal ais, AA BB BB AA 6 bh Ad Ad bh bh h
Moments of nertia of Composite Areas The moment of inertia of a composite area A about a given ais is obtained b adding the moments of inertia of the component areas A, A, A,..., with respect to the same ais.
Sample Problem 9.5 Determine the moment of inertia of the shaded area with respect to the ais. SOLUTON: Compute the moments of inertia of the bounding rectangle and half-circle with respect to the ais. The moment of inertia of the shaded area is obtained b subtracting the moment of inertia of the half-circle from the moment of inertia of the rectangle.
Sample Problem 9.5 SOLUTON: Compute the moments of inertia of the bounding rectangle and half-circle with respect to the ais. Rectangle: 6 4 40 0 8. 0 bh mm 4 4r 90 a 8. mm b 0 - a 8.8 mm A r.7 0 90 mm Half-circle: moment of inertia with respect to AA, 4 4 6 r 90 5.76 0 mm AA 8 8 4 moment of inertia with respect to, AA Aa 7.0 0 6 mm 6 5.76 0.7 0 moment of inertia with respect to, Ab 9.0 6 mm 4 4 7.0 0 6.7 0 8.8
Sample Problem 9.5 The moment of inertia of the shaded area is obtained b subtracting the moment of inertia of the half-circle from the moment of inertia of the rectangle. 4 8. 0 6 mm 4 9.0 6 mm 45.9 0 6 mm 4
Prob # 9.6 Determine the moments of inertia of the shaded area shown with respect to the and aes when a =0 mm.
Prob # 9.46 Determine the polar moment of inertia of the area shown with respect to (a) point O, (b) the centroid of the area.
Moment of nertia of a Mass Angular acceleration about the ais AA of the small mass Dm due to the application of a couple is proportional to r Dm. r Dm = moment of inertia of the mass Dm with respect to the ais AA For a bod of mass m the resistance to rotation about the ais AA is Dm r Dm r Dm r r dm mass moment of inertia The radius of gration for a concentrated mass with equivalent mass moment of inertia is k m k m
Moment of nertia of a Mass Moment of inertia with respect to the coordinate ais is r dm dm z Similarl, for the moment of inertia with respect to the and z aes, z dm z dm n S units, r dm kg m n U.S. customar units, lb s slug ft ft ft lb ft s
Parallel Ais Theorem For the rectangular aes with origin at O and parallel centroidal aes, dm z dm z z dm dm z dm z z dm z z m m m z z z Generalizing for an ais AA and a parallel centroidal ais, md
Moments of nertia of Thin Plates For a thin plate of uniform thickness t and homogeneous material of densit r, the mass moment of inertia with respect to ais AA contained in the plate is AA r rt dm rt AA, area r da Similarl, for perpendicular ais BB which is also contained in the plate, BB rt BB, area For the ais CC which is perpendicular to the plate, CC AA BB rt JC, area rt AA, area BB, area
Moments of nertia of Thin Plates For the principal centroidal aes on a rectangular plate, AA BB CC rt rt a b, rt ma AA area BB ab, area rt mb AA, mass BB, mass m a b For centroidal aes on a circular plate, AA BB rt 4 r, rt mr AA area 4 4 CC AA BB mr
Moments of nertia of Common Geometric Shapes
Sample Problem 9. SOLUTON: With the forging divided into a prism and two clinders, compute the mass and moments of inertia of each component with respect to the z aes using the parallel ais theorem. Add the moments of inertia from the components to determine the total moments of inertia for the forging. Determine the moments of inertia of the steel forging with respect to the z coordinate aes, knowing that the specific weight of steel is 490 lb/ft.
Sample Problem 9. SOLUTON: Compute the moments of inertia of each component with respect to the z aes. clindersa in., L in.,.5in., in. : ma m 0.089 0.089.59 0 lb ft s each clinder : V m g m 0.089 lb s 490 lb/ft in 78 in ft.ft s ft m a 0.089 0.089.5 4.7 0 m L m lb ft s a L m 0.089 0.089.5 6.48 0 lb ft s
Sample Problem 9. prism : m V g m 0. lb s 490 lb/ft 6 78 in ft.ft s ft in prism (a = in., b = 6 in., c = in.): z 4.88 0 c 0. 6 m b lb ft s a 0. m c 0.977 0 lb ft s Add the moments of inertia from the components to determine the total moments of inertia. 4.88 0.59 0 z 0.977 0 4.88 0 4.7 0 6.48 0 z 0.06 0 lb ft s 9. 0 lb ft s 7.84 0 lb ft s
Prob # 9.4 Determine the mass moments of inertia and the radii of gration of the steel machine element shown with respect to ais. (The densit of steel is 7850 kg/m )