ENGR-36_Lec-26_Mass_Moment_of_Inertia.pptx 1 Bruce Mayer, PE Engineering-36: Engineering Mechanics - Statics Bruce Mayer, PE Licensed Electrical & Mechanical Engineer Engineering 36 Chp10: Moment of Interia

ENGR-36_Lec-26_Mass_Moment_of_Inertia.pptx 2 Bruce Mayer, PE Engineering-36: Engineering Mechanics - Statics 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

ENGR-36_Lec-26_Mass_Moment_of_Inertia.pptx 3 Bruce Mayer, PE Engineering-36: Engineering Mechanics - Statics Mass Moment of Inertia The Moment of Inertia is the Resistance to Spinning

ENGR-36_Lec-26_Mass_Moment_of_Inertia.pptx 4 Bruce Mayer, PE Engineering-36: Engineering Mechanics - Statics Linear-Rotational Parallels

ENGR-36_Lec-26_Mass_Moment_of_Inertia.pptx 5 Bruce Mayer, PE Engineering-36: Engineering Mechanics - Statics 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 r 2  m. r 2  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

ENGR-36_Lec-26_Mass_Moment_of_Inertia.pptx 6 Bruce Mayer, PE Engineering-36: Engineering Mechanics - Statics 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

ENGR-36_Lec-26_Mass_Moment_of_Inertia.pptx 7 Bruce Mayer, PE Engineering-36: Engineering Mechanics - Statics I x, I y, I z  Mass Moment of inertia with respect to the y coordinate axis  r is the ┴ distance to y-axis  Similarly, for the moment of inertia with respect to the x and z axes  Units Summary SI US Customary Units

ENGR-36_Lec-26_Mass_Moment_of_Inertia.pptx 8 Bruce Mayer, PE Engineering-36: Engineering Mechanics - Statics 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 I x 00 m

ENGR-36_Lec-26_Mass_Moment_of_Inertia.pptx 9 Bruce Mayer, PE Engineering-36: Engineering Mechanics - Statics Parallel Axis Theorem cont.  So I x  Similarly for the Other two Axes  so  In General for any axis AA’ that is parallel to a centroidal axis BB’  Also the Radius of Gyration

ENGR-36_Lec-26_Mass_Moment_of_Inertia.pptx 10 Bruce Mayer, PE Engineering-36: Engineering Mechanics - Statics 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

ENGR-36_Lec-26_Mass_Moment_of_Inertia.pptx 11 Bruce Mayer, PE Engineering-36: Engineering Mechanics - Statics 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 J O to I x & I y Using The Pythagorean Theorem

ENGR-36_Lec-26_Mass_Moment_of_Inertia.pptx 12 Bruce Mayer, PE Engineering-36: Engineering Mechanics - Statics Thin Plate Examples  For the principal centroidal axes on a rectangular plate  For centroidal axes on a circular plate Area = ab Area = πr 2

ENGR-36_Lec-26_Mass_Moment_of_Inertia.pptx 13 Bruce Mayer, PE Engineering-36: Engineering Mechanics - Statics 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.

ENGR-36_Lec-26_Mass_Moment_of_Inertia.pptx 14 Bruce Mayer, PE Engineering-36: Engineering Mechanics - Statics Common Geometric Shapes

ENGR-36_Lec-26_Mass_Moment_of_Inertia.pptx 15 Bruce Mayer, PE Engineering-36: Engineering Mechanics - Statics 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/ft 3 (0.284 lb/in 3 )  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.

ENGR-36_Lec-26_Mass_Moment_of_Inertia.pptx 16 Bruce Mayer, PE Engineering-36: Engineering Mechanics - Statics

ENGR-36_Lec-26_Mass_Moment_of_Inertia.pptx 17 Bruce Mayer, PE Engineering-36: Engineering Mechanics - Statics Example 1 cont.  For The Symmetrically Located Cylinders  Referring to the Geometric-Shape Table for the Cylinders a = 1” (the radius) L = 3” x centriod = 2.5” y centriod = 2”  Then the Axial (x) Moment of Inertia

ENGR-36_Lec-26_Mass_Moment_of_Inertia.pptx 18 Bruce Mayer, PE Engineering-36: Engineering Mechanics - Statics Example 1 cont.2  Now the Transverse (y & z) Moments of Inertia dzdz

ENGR-36_Lec-26_Mass_Moment_of_Inertia.pptx 19 Bruce Mayer, PE Engineering-36: Engineering Mechanics - Statics Example 1 cont.3  For The Sq-Bar  Referring to the Geometric-Shape Table for the Block a = 2” b = 6” c = 2”  Then the Transverse (x & z ) Moments of Inertia

ENGR-36_Lec-26_Mass_Moment_of_Inertia.pptx 20 Bruce Mayer, PE Engineering-36: Engineering Mechanics - Statics Example 1 cont.4  And the Axial (y) Moment of Inertia  Add the moments of inertia from the components to determine the total moment of inertia.

ENGR-36_Lec-26_Mass_Moment_of_Inertia.pptx 21 Bruce Mayer, PE Engineering-36: Engineering Mechanics - Statics 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

ENGR-36_Lec-26_Mass_Moment_of_Inertia.pptx 22 Bruce Mayer, PE Engineering-36: Engineering Mechanics - Statics WhiteBoard Work Some Other Mass Moments  For the Thick Ring

ENGR-36_Lec-26_Mass_Moment_of_Inertia.pptx 23 Bruce Mayer, PE Engineering-36: Engineering Mechanics - Statics WhiteBoard Work Find MASS Moment of Inertia for Prism  About the y-axis in this case

ENGR-36_Lec-26_Mass_Moment_of_Inertia.pptx 24 Bruce Mayer, PE Engineering-36: Engineering Mechanics - Statics Bruce Mayer, PE Registered Electrical & Mechanical Engineer Engineering 36 Appendix

ENGR-36_Lec-26_Mass_Moment_of_Inertia.pptx 25 Bruce Mayer, PE Engineering-36: Engineering Mechanics - Statics WhiteBoard Work Find MASS Moment of Inertia for Roller  About axis AA’ in this case

ENGR-36_Lec-26_Mass_Moment_of_Inertia.pptx 26 Bruce Mayer, PE Engineering-36: Engineering Mechanics - Statics 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 “2 nd Moment of Area”  Massive Objects DO physically have Inertial Properties Finding the true “Moment of Inertia” is very analogous to determination of the 2 nd Moment of Area