Slide1: Bruce Mayer, PE Licensed Electrical & Mechanical Engineer BMayer@ChabotCollege.edu Engineering 35 Mass Moment of Inertia


Mass Moments of Inertia: Mass Moments of Inertia The Previously Studied “Area Moment of Inertia” does Not Actually have True Inertial Properties The Area Version is More precisely Stated as the SECOND Moment of Area Objects with Real mass Do have inertia i.e., an inertial Body will Resist Rotation by An Applied Torque Thru an F=ma Analog


Mass Moment of Inertia: Mass Moment of Inertia The Angular acceleration, , about the axis AA’ of the small mass m due to the application of a couple is proportional to r2m. r2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


Mass Radius of Gyration: Mass Radius of Gyration Imagine the entire Body Mass Concentrated into a single Point Now place this mass a distance k from the rotation axis so as to create the same resistance to rotation as the original body This Condition Defines, Physically, the Mass Radius of Gyration Mathematically


Ix, Iy, Iz: Ix, Iy, Iz Mass Moment of inertia with respect to the y coordinate axis Similarly, for the moment of inertia with respect to the x and z axes Units Summary SI US Customary Units


Parallel Axis Theorem: Parallel Axis Theorem Consider Centriodal Axes (x’,y’,z’) Which are Translated Relative to the Original CoOrd Systems (x,y,z) The Translation Relationships In a Manner Similar to the Area Calculation Two Middle Integrals are 1st-Moments Relative to the CG → 0 The Last Integral is the Total Mass Then Write Ix


Parallel Axis Theorem cont.: Parallel Axis Theorem cont. So Ix Similarly for the Other two Axes or In General for any axis AA’ that is parallel to a centroidal axis BB’ Also the Radius of Gyration


Thin Plate Moment of Inertia: Thin Plate Moment of Inertia 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 Similarly, for perpendicular axis BB’ which is also contained in the plate For the axis CC’ which is PERPENDICULAR to the plate note that This is a POLAR Geometry


Thin Plate Examples: Thin Plate Examples For the principal centroidal axes on a rectangular plate For centroidal axes on a circular plate Area = ab Area = πr2


3D Mass Moments by Integration: 3D Mass Moments by Integration The Moment of inertia of a homogeneous body is obtained from double or triple integrations of the form 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.


Common Geometric Shapes: Common Geometric Shapes


Example 1: Example 1 Determine the moments of inertia of the steel forging with respect to the xyz coordinate axes, knowing that the specific weight of steel is 490 lb/ft3 (0.284 lb/in3) SOLUTION PLAN 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.


Example 1 cont.: Example 1 cont. For The Symmetrically Located Cylinders Referring to the Geometric-Shape Table for the Cylinders a = 1” L = 3” Xcentriod = 2.5” ycentriod = 2” Then the Longitudinal (x) Moment of Inertia


Example 1 cont.2: Example 1 cont.2 Now the Transverse (y & z) Moments of Inertia


Example 1 cont.3: Example 1 cont.3 For The Block (Prism) Referring to the Geometric-Shape Table for the Block a = 2” b = 6” c = 2” Then the Transverse (x & z ) Moments of Inertia


Example 1 cont.4: Example 1 cont.4 And the Longitudinal (y) Moment of Inertia Add the moments of inertia from the components to determine the total moment of inertia.


Arbitrary-Axis Moment of Inertia: Arbitrary-Axis Moment of Inertia Wish to Define the Moment of Inertia w.r.t. an Arbitrary Axis OL dm  element of mass p  perpendicular distance from OL to dm Now Let r  Position Vector for dm with Coords (x,y,z)   OL Unit-Vector with direction cosines (x,y,z) Next p = |r|sin; which is the magnitude of xr or Then


Arbitrary Axis M of I cont.: Arbitrary Axis M of I cont. Now expand this expression and group terms The 1st Three Integrals are the Moments of Inertia of the Body w.r.t. the CoOrd Axes Now Expand IOL in terms of Rectangular components


Arbitrary Axis M of I cont.2: Arbitrary Axis M of I cont.2 The Last Three Integrals are the PRODUCT of INERTIA w.r.t Axes PAIRS; Let Now Rewrite IOL in much more compact form And The definition of the mass products of inertia of a is an extension of the definition of Area product of inertia, Thus


Arbitrary Axis M of I cont.3: Arbitrary Axis M of I cont.3 Rewrite the Product of Inertia Terms Therefore Or Thus Can Calc IOL if the CoOrd-Axes I’s, and the Centroidal values are known


Board Example: Board Example Let’s Work Problem 9.130