Moments and Products of Inertia

Three-DimensionalKinetics of a Rigid Body

Moments and Products of Inertia• Moment of InertiadmzydmrdIxxx222mmzzzmmyyymmxxxdmyxdmrIdmzxdmrIdmzydmrI222222222

• Product of InertiadI xy xydmIxyIyxmxydmIyzIzymyzdmIxzIzxmxzdmMoments and Products of Inertia

• Parallel-Axis TheoremIIIxxyyzz 2 2Ix'x' m yG zG 2 2Iy'y' m xG zG 2 2I m x y z'z'GGMoments and Products of Inertia

• Parallel-Planes TheoremIIIxyyzzxIIx'y'y'z'GGIz'x' mzGxGGmxmyGGyzGGMoments and Products of Inertia

• Inertia Tensor◦ The inertialproperties of abody are thereforecompletelycharacterized bynine terms, II IxxyxzxIx00 IIIyyI0y0xyzy00IzIIIzzxzyzMoments and Products of Inertia

H A vH A Amiviivi vA H mv AiAiAAm ivA A Ami dm dmAm A mA AAAAngular Momentum

• Fixed Point OH0 m00dmAngular Momentum

• Center of Mass GH G m G GdmAngular Momentum

• Arbitrary Point AHA G /AmvGHGAngular Momentum

• Rectangular Components of HHxiH0 m0 dmH0 mH G dm H j H k yzm Gm Gdmxi yj zki j kxi yj zkxyzdmAngular Momentum

• Rectangular Components of HHHHxyz Ixx I I Ixyxzxxxxy I I Iyyzyyyyxz I IyzzzzzzIf x, y, z become principalaxes of inertia,Angular MomentumHHHxyzIIIxyzxzy

• Principle of Impulse and MomentumMay be used to solve problems which involve force, velocityand timemt 1v GFdt m v1G 2t2 tH OM1Odt HO2t12Six scalar equations can be written: three relate the linearimpulse momentum in the x, y, z directions and the otherthree relate the body’s angular impulse and momentumabout the x, y, z axesAngular Momentum

TTii12m 2 1 1m 2 iv i2m iv ivii1212vi vA vA AvA A vA vAmi vAAmi AAmiAT v v dm1 m v2 A 12A A m A Admm AKinetic Energy

• Fixed Point OT 1 H20T12Ix2x12Iy2y12Iz2zKinetic Energy

• Center of MassT12mv2GT121 mv H G2Ix2x1212Iy02y12Iz2zKinetic Energy

• Principle of Work and EnergyHaving the formulated the kinetic energy for a body, theprinciple may be used to solve problems which involve force,velocity and displacement T U T1 1 2 2Kinetic Energy

• Equations of Translational MotionF ma GF ma x G xFy m aGy F z m aGzEquations of Motion

• Equations of Rotational MotionM H rr i / Gmiai / G miriGaGHG/FEquations of Motion i i / GiviGmG /0Hi ri / Gm ivi/ G ri/ Gmivi / G0GHH i ri/ GiaiGGGm /rOH Oai / Gmi i / GM H G G

• Equations of Rotational MotionMMOGHO HOxyzHG HGxyzEquations of Motion

• x, y, z Axes Having Motion Ω=0MMOH OGH G xyz xyzEquations of MotionThe axes may be chosen withorigin at G , such the axes onlytranslate relative to X, Y, ZThe body may have a rotation ω about these axes, and therefore themoments and products of inertia of the body would have to be expressedas function of time

• x, y, z Axes Having Motion Ω=ωMMOGHOHOxyzHGHGxyzThe axes may be chosen suchthey are fixed in and move withthe bodyThe moments and products of inertia of the body relative to these axeswill be constant during the motionEquations of Motion

• x, y, z Axes Having Motion Ω=ωExpressing the previous vector equations as three scalarequations; M I Mxy I ....The axes are chosen as principalaxes of inertia, the product ofinertia are zerox Iyy Izz yz I xyyzx2 2yzIzx zxy Mz ....Equations of MotionxxyxMMMxyzIII xyzx zy IIyzIIzIx Iy x yx y z zx

• x, y, z Axes Having Motion Ω≠ωTo simplify the calculations for the time derivative of ω, is convenientto choose the x, y, z axes having an angular velocity Ω which isdifferent from the angular velocity of the bodyMMMxyzIII xyzx zy IIIyxzzxyyzxIIIzxyyzxzxyEquations of Motion