Area Moments of Inertia by Integration

Transcription

1 Area Moments of nertia ntegration Second moments or moments of inertia of an area with respect to the and aes, da da Evaluation of the integrals is simplified choosing da to e a thin strip parallel to one of the coordinate aes ME0 - Division

2 Area Moments of nertia Products of nertia: for prolems involving unsmmetrical cross-sections and in calculation of M aout rotated aes. t ma e +ve, -ve, or zero Product of nertia of area A w.r.t. - aes: da and are the coordinates of the element of area da= When the ais, the ais, or oth are an ais of smmetr, the product of inertia is zero. Parallel ais theorem for products of inertia: A - + Quadrants + - ME0 - Division

3 Area Moments of nertia Rotation of Aes Product of inertia is useful in calculating inclined aes. Determination of aes aout which the M is a maimum and a minimum da Moments and product of inertia w.r.t. new aes and? Note: cos sin da cos sin da ' ' ' ' ' ' da da ' ' da cos sin cos sin cos / sin ' cos sin da cos sin da cos sin cos sin da cos cos sin cos sin cos cos cos sin sin ME0 - Division 3

4 Area Moments of nertia Rotation of Aes ' sin cos cos cos sin sin Adding first two eqns: + = + = z The Polar O Angle which makes and either ma or min can e found setting the derivative of either or w.r.t. θ equal to zero: 0 d' sin cos d Denoting this critical angle α tan two values of α which differ π since tanα = tan(α+π) two solutions for α will differ π/ one value of α will define the ais of maimum M and the other defines the ais of minimum M These two rectangular aes are called the principal aes of inertia ME0 - Division

5 Area Moments of nertia Rotation of Aes cos sin sin cos sin cos ' cos sin tan Sustituting in the third eqn for critical value of θ: = 0 Product of nertia is zero for the Principal Aes of inertia Sustituting sinα and cosα in first two eqns for Principal Moments of nertia: min ma 5 ME0 - Division

6 Area Moments of nertia Mohr s Circle of nertia :: Graphical representation of the M equations - For given values of,, &, corresponding values of,, & ma e determined from the diagram for an desired angle θ. cos sin sin cos sin cos ' tan min ma ave ave R R At the points A and B, = 0 and takes the maimum and minimum values R ave min ma, 6 ME0 - Division

7 Area Moments of nertia Mohr s Circle of nertia: Construction ME0 - Division tan cos cos ' sin cos ma 0 sin sin Choose horz ais M Choose vert ais P Point A known {, } Point B known {, - } Circle with dia AB Angle α for Area Angle α to horz (same sense) ma, min Angle to = θ Angle OA to OC = θ Same sense Point C, Point D 7

8 Area Moments of nertia Eample: Product of nertia SOLUTON: Determine the product of inertia using direct integration with the parallel ais theorem on vertical differential area strips Appl the parallel ais theorem to evaluate the product of inertia with respect to the centroidal aes. Determine the product of inertia of the right triangle (a) with respect to the and aes and () with respect to centroidal aes parallel to the and aes. ME0 - Division 8

9 Area Moments of nertia Eamples SOLUTON: Determine the product of inertia using direct integration with the parallel ais theorem on vertical differential area strips h d h d da h el el ntegrating d from = 0 to =, el el h d h d h da d h 9 ME0 - Division

10 Area Moments of nertia Eamples SOLUTON Appl the parallel ais theorem to evaluate the product of inertia with respect to the centroidal aes. 3 3 With the results from part a, h h A h h 3 3 h 7 ME0 - Division 0

11 Area Moments of nertia Eample: Mohr s Circle of nertia SOLUTON: Plot the points (, ) and (,- ). Construct Mohr s circle ased on the circle diameter etween the points. The moments and product of inertia with respect to the and aes are = 7.06 mm, =.606 mm, and = mm. Using Mohr s circle, determine (a) the principal aes aout O, () the values of the principal moments aout O, and (c) the values of the moments and product of inertia aout the and aes Based on the circle, determine the orientation of the principal aes and the principal moments of inertia. Based on the circle, evaluate the moments and product of inertia with respect to the aes. ME0 - Division

12 Area Moments of nertia Eample: Mohr s Circle of nertia SOLUTON: Plot the points (, ) and (,- ). Construct Mohr s circle ased on the circle diameter etween the points. OC CD R ave CD DX mm 6 mm 6 mm mm mm 6 mm Based on the circle, determine the orientation of the principal aes and the principal moments of inertia. tan m DX.097 m 7. 6 CD m 3. 8 ME0 - Division

13 Area Moments of nertia Eample: Mohr s Circle of nertia OC ave R mm 6 mm Based on the circle, evaluate the moments and product of inertia with respect to the aes. The points X and Y corresponding to the and aes are otained rotating CX and CY counterclockwise through an angle θ (60 o ) = 0 o. The angle that CX forms with the horz is f = 0 o o = 7. o. OF OC CX cos Rcos7. ' OG OC CYcos Rcos7. ' ave ave 6 mm o o mm FX CYsin Rsin 7. ' o mm ME0 - Division 3

14 Mass Moment of nertia Application in rigid od dnamics - Measure of distriution of mass of a rigid od w.r.t. the ais (constant propert for that ais) = r dm r = perpendicular distance of the mass element dm from the ais O-O r Δm :: measure of the inertia of the sstem ME0 - Division

15 Mass Moment of nertia Aout individual coordinate aes ME0 - Division 5

16 Mass Moment of nertia Parallel Ais Theorem ME0 - Division 6

17 Mass Moment of nertia 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 ME0 - Division 7

18 Mass Moment of nertia Moments of nertia of Thin Plates For the principal centroidal aes on a rectangular plate, AA BB CC rt rt AA BB 3 a, area rt ma 3 a, area rt m AA, mass BB, mass m a For centroidal aes on a circular plate, AA BB rt AA r, area rt mr ME0 - Division 8

19 Mass Moment of nertia Moments of nertia of a 3D Bod ntegration Moment of inertia of a homogeneous od is otained from doule or triple integrations of the form r r dv For odies with two planes of smmetr, the moment of inertia ma e otained from a single integration choosing thin slas perpendicular to the planes of smmetr for dm. The moment of inertia with respect to a particular ais for a composite od ma e otained adding the moments of inertia with respect to the same ais of the components. ME0 - Division 9

20 Mass Moment of nertia M of some common geometric shapes ME0 - Division 0