Bruce Mayer, PE Licensed Electrical & Mechanical Engineer BMayer@ChabotCollege

1. Engineering 36 Chp10:Moment of Interia Bruce Mayer, PE Licensed Electrical & Mechanical EngineerBMayer@ChabotCollege.edu

2. 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

3. Mass Moment of Inertia The Momentof Inertia isthe Resistanceto Spinning

4. Linear-Rotational Parallels

5. 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

6. 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, k • Mathematically

7. Mass Moment of inertia with respect to the y coordinate axis  ris the ┴distance to y-axis Ix, Iy, Iz • Similarly, for the moment of inertia with respect to the x and z axes • Units Summary • SI • US Customary Units

8. Consider CENTRIODAL Axes (x’,y’,z’) Which are Translated Relative to the Original CoOrd Systems (x,y,z) 0 0 Parallel Axis Theorem • The Translation Relationships • Then Write Ix m • 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

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

10. 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

11. Polar Moment of Inertia • The polar moment of inertia is an important parameter in problems involving torsion of cylindrical shafts, Torsion in Welded Joints, and the rotation of slabs • In Torsion Problems, Define a Moment of Inertia Relative to the Pivot-Point, or “Pole”, at O • Relate JO to Ix & Iy Using The Pythagorean Theorem

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

13. 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.

14. Common Geometric Shapes

15. 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 Square-Bar 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

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

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

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

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

21. T = Iα • When you take ME104 (Dynamics) at UCBerkeley you will learn that the Rotational Behavior of the CrankShaft depends on its Mass Moment of inertia

22. WhiteBoard Work Some OtherMassMoments • For theThick Ring

23. WhiteBoard Work • About the y-axis in this case Find MASSMoment ofInertiafor Prism

24. Engineering 36 Appendix Bruce Mayer, PE Registered Electrical & Mechanical EngineerBMayer@ChabotCollege.edu

25. WhiteBoard Work • About axis AA’ in this case Find MASSMoment ofInertiafor Roller

26. Mass Moment of Inertia • Last time we discussed the “Area Moment of Intertia” • Since Areas do NOT have Inertial properties, the Areal Moment is more properly called the “2nd Moment of Area” • Massive Objects DO physically have Inertial Properties • Finding the true “Moment of Inertia” is very analogous to determination of the 2nd Moment of Area