STATICS. Distributed Forces: Moments of Inertia VECTOR MECHANICS FOR ENGINEERS: Eighth Edition CHAPTER. Ferdinand P. Beer E. Russell Johnston, Jr.

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1 007 The McGraw-Hill Companies, nc. All rights reserved. Eighth E CHAPTER 9 VECTOR MECHANCS FOR ENGNEERS: STATCS Ferdinand P. Beer E. Russell Johnston, Jr. Lecture Notes: J. Walt Oler Texas Tech University Distributed Forces: Moments of nertia

2 Moment of nertia of a Mass Angular acceleration about the axis AA of the small mass m due to the application of a couple is proportional to r m. r m = moment of inertia of the mass m with respect to the axis AA For a body of mass m the resistance to rotation about the axis AA is r1 m r m r3 m r dm mass moment of inertia The radius of gyration for a concentrated mass with equivalent mass moment of inertia is k m k m 007 The McGraw-Hill Companies, nc. All rights reserved. 9 -

3 Moment of nertia of a Mass Moment of inertia with respect to the y coordinate axis is y r dm z x dm Similarly, for the moment of inertia with respect to the x and z axes, x z y x z y dm dm n S units, r dm kg m 007 The McGraw-Hill Companies, nc. All rights reserved. 9-3

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6 is the nertia matrix which has a size of 3 x3 and is symmetric 007 The McGraw-Hill Companies, nc. All rights reserved.

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13 007 The McGraw-Hill Companies, nc. All rights reserved. Since r is independent of inclination of the coordinate axes and depends only on the position of the origin. Therefore sum of moment of inertia at a point in space for a given body is an invariant with respect to rotation of axes.

14 007 The McGraw-Hill Companies, nc. All rights reserved. ighth Mirror Symmetry & mixed moments of nertia xz of each half gives the contribution of same magnitude but opposite sign. This conclusion is also true for xy.

15 Similarly xy = 0 f two axes form a plane of symmetry for the mass distribution of a body, the products of inertia having as an index the coordinate that is normal to the plane of symmetry will be zero. 007 The McGraw-Hill Companies, nc. All rights reserved.

16 Body of Revolution 007 The McGraw-Hill Companies, nc. All rights reserved. 9-16

17 007 The McGraw-Hill Companies, nc. All rights reserved. ighth Body of Revolution Let z axis coincide with the axis of symmetry. This is true all possible xy axis formed by rotating the z axis at O.

18 007 The McGraw-Hill Companies, nc. All rights reserved. ighth Radius of gyration k x, k y and k z are radius of gyration.

19 Parallel Axis Theorem For the rectangular axes with origin at O and parallel centroidal axes x y z, 007 The McGraw-Hill Companies, nc. All rights reserved. 9-19

20 The coordinates of any point are (x,y,z) xbar, ybar, zbar (the quantities with bar on them give the coordinates of CM from O x, y, z are distances of any point fromcm 007 The McGraw-Hill Companies, nc. All rights reserved. 9-0

21 x y z dm y y z z dm x y z dm y y dm z z dm y z dm x x m y z 007 The McGraw-Hill Companies, nc. All rights reserved. 9-1

22 x y z dm y y dm z z dm y z dm 007 The McGraw-Hill Companies, nc. All rights reserved. 9 -

23 Definition of CM y dm y cm y dm 0 z dm 0 Ycm and Zcm are the coordinates of CM from origin. Here CM is at the origin itself. 007 The McGraw-Hill Companies, nc. All rights reserved. 9-3

24 Parallel Axis Theorem x x m y z y y m z x z z m x y Generalizing for any axis AA and a parallel centroidal axis, md 007 The McGraw-Hill Companies, nc. All rights reserved. 9-4

25 Parallel axis for product moments mx y xy x y ' 007 The McGraw-Hill Companies, nc. All rights reserved. 9-5

26 Moments of nertia of Thin Plates For a thin plate of uniform thickness t and homogeneous material of density, the mass moment of inertia with respect to axis AA contained in the plate is AA r t dm t AA, area r da Similarly, for perpendicular axis BB which is also contained in the plate, BB t BB, area For the axis CC which is perpendicular to the plate, CC t JC, area t AA, area BB, area AA BB 007 The McGraw-Hill Companies, nc. All rights reserved. 9-6

27 t is mass per unit area 007 The McGraw-Hill Companies, nc. All rights reserved. 9-7

28 Moments of nertia of Thin Plates For the principal centroidal axes on a rectangular plate, AA BB t t AA BB , area t a b ma , area t ab mb CC 1 AA, mass BB, mass m a b 1 For centroidal axes on a circular plate, AA BB t AA , area t r mr CC AA BB 1 mr 007 The McGraw-Hill Companies, nc. All rights reserved. 9-8

29 Moments of nertia of a 3D Body by ntegration Moment of inertia of a homogeneous body is obtained from double or triple integrations of the form r dv For bodies with two planes of symmetry, the moment of inertia may be obtained from a single integration by choosing thin slabs perpendicular to the planes of symmetry for dm. The moment of inertia with respect to a particular axis for a composite body may be obtained by adding the moments of inertia with respect to the same axis of the components. 007 The McGraw-Hill Companies, nc. All rights reserved. 9-9

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33 007 The McGraw-Hill Companies, nc. All rights reserved. ighth Part b

34 007 The McGraw-Hill Companies, nc. All rights reserved. ighth Part c

35 Moments of nertia of Common Geometric Shapes 007 The McGraw-Hill Companies, nc. All rights reserved. 9-35

36 Moments of nertia of Common Geometric Shapes 007 The McGraw-Hill Companies, nc. All rights reserved. 9-36

37 Sample Problem 9.1 Determine the moments of inertia of the steel forging with respect to the xyz coordinate axes, knowing that the density of steel is 7850 kg/m 3. SOLUTON: With the forging divided into a prism and two cylinders, compute the mass and moments of inertia of each component with respect to the xyz axes using the parallel axis theorem. Add the moments of inertia from the components to determine the total moments of inertia for the forging. 007 The McGraw-Hill Companies, nc. All rights reserved. 9-37

38 Sample Problem 9.1 Determine the moments of inertia of the steel forging with respect to the xyz coordinate axes, knowing that the density of steel is 7850 kg/m The McGraw-Hill Companies, nc. All rights reserved. 9-38

39 Sample Problem The McGraw-Hill Companies, nc. All rights reserved. 9-39

40 Sample Problem 9.1 With the forging divided into a prism and two cylinders, compute the mass and moments of inertia of each component with respect to the xyz axes using the parallel axis theorem. Add the moments of inertia from the components to determine the total moments of inertia for the forging. 007 The McGraw-Hill Companies, nc. All rights reserved. 9-40

41 Sample Problem 9.1 each cylinder : V (0. 05 m) m ( kg -4 m 10 ( m) 3-4 m 3 )(7850 kg/m 3 ) 007 The McGraw-Hill Companies, nc. All rights reserved. 9-41

42 Sample Problem 9.1 cylinders a 5mm, L 75mm, x 6.5mm, y 50mm : x 1 ma my m kg m 007 The McGraw-Hill Companies, nc. All rights reserved. 9-4

43 Sample Problem 9.1 cylinders a 5mm, L 75mm, x 6.5mm, y 50mm : y 1 1 m 3a L mx kg m 007 The McGraw-Hill Companies, nc. All rights reserved. 9-43

44 Sample Problem 9.1 z 1 1 m 3a L m x y kg.m 007 The McGraw-Hill Companies, nc. All rights reserved. 9-44

45 Sample Problem 9.1 prism (a = 50 mm., b = 150 mm, c = 50 mm): x z 1 1 m b c kg kg m y 1 1 m c a kg kg m 007 The McGraw-Hill Companies, nc. All rights reserved. 9-45

46 Sample Problem 9.1 Add the moments of inertia from the components to determine the total moments of inertia. x x kg m 3 y y kg m z z kg m 007 The McGraw-Hill Companies, nc. All rights reserved. 9-46

47 007 The McGraw-Hill Companies, nc. All rights reserved. ighth Moment of nertia With Respect to an Arbitrary Axis kk is the arbitrary axis r is the vector from origin to point (x,y,z)

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52 007 The McGraw-Hill Companies, nc. All rights reserved. ighth Transformation of product M

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59 007 The McGraw-Hill Companies, nc. All rights reserved. Find z z and x z

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