Moments and Products of Inertia
Moments and Products of Inertia
Moments and Products of Inertia
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Three-DimensionalKinetics <strong>of</strong> a Rigid Body
<strong>Moments</strong> <strong>and</strong> <strong>Products</strong> <strong>of</strong> <strong>Inertia</strong>• Moment <strong>of</strong> <strong>Inertia</strong>dmzydmrdIxxx222mmzzzmmyyymmxxxdmyxdmrIdmzxdmrIdmzydmrI222222222
• Product <strong>of</strong> <strong>Inertia</strong>dI xy xydmIxyIyxmxydmIyzIzymyzdmIxzIzxmxzdm<strong>Moments</strong> <strong>and</strong> <strong>Products</strong> <strong>of</strong> <strong>Inertia</strong>
• Parallel-Axis TheoremIIIxxyyzz 2 2Ix'x' m yG zG 2 2Iy'y' m xG zG 2 2I m x y z'z'GG<strong>Moments</strong> <strong>and</strong> <strong>Products</strong> <strong>of</strong> <strong>Inertia</strong>
• Parallel-Planes TheoremIIIxyyzzxIIx'y'y'z'GGIz'x' mzGxGGmxmyGGyzGG<strong>Moments</strong> <strong>and</strong> <strong>Products</strong> <strong>of</strong> <strong>Inertia</strong>
• <strong>Inertia</strong> Tensor◦ The inertialproperties <strong>of</strong> abody are thereforecompletelycharacterized bynine terms, II IxxyxzxIx00 IIIyyI0y0xyzy00IzIIIzzxzyz<strong>Moments</strong> <strong>and</strong> <strong>Products</strong> <strong>of</strong> <strong>Inertia</strong>
H A vH A Amiviivi vA H mv AiAiAAm ivA A Ami dm dmAm A mA AAAAngular Momentum
• Fixed Point OH0 m00dmAngular Momentum
• Center <strong>of</strong> Mass GH G m G GdmAngular Momentum
• Arbitrary Point AHA G /AmvGHGAngular Momentum
• Rectangular Components <strong>of</strong> HHxiH0 m0 dmH0 mH G dm H j H k yzm Gm Gdmxi yj zki j kxi yj zkxyzdmAngular Momentum
• Rectangular Components <strong>of</strong> HHHHxyz Ixx I I Ixyxzxxxxy I I Iyyzyyyyxz I IyzzzzzzIf x, y, z become principalaxes <strong>of</strong> inertia,Angular MomentumHHHxyzIIIxyzxzy
• Principle <strong>of</strong> Impulse <strong>and</strong> MomentumMay be used to solve problems which involve force, velocity<strong>and</strong> 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 <strong>and</strong> the otherthree relate the body’s angular impulse <strong>and</strong> 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 <strong>of</strong> MassT12mv2GT121 mv H G2Ix2x1212Iy02y12Iz2zKinetic Energy
• Principle <strong>of</strong> Work <strong>and</strong> EnergyHaving the formulated the kinetic energy for a body, theprinciple may be used to solve problems which involve force,velocity <strong>and</strong> displacement T U T1 1 2 2Kinetic Energy
• Equations <strong>of</strong> Translational MotionF ma GF ma x G xFy m aGy F z m aGzEquations <strong>of</strong> Motion
• Equations <strong>of</strong> Rotational MotionM H rr i / Gmiai / G miriGaGHG/FEquations <strong>of</strong> Motion i i / GiviGmG /0Hi ri / Gm ivi/ G ri/ Gmivi / G0GHH i ri/ GiaiGGGm /rOH Oai / Gmi i / GM H G G
• Equations <strong>of</strong> Rotational MotionMMOGHO HOxyzHG HGxyzEquations <strong>of</strong> Motion
• x, y, z Axes Having Motion Ω=0MMOH OGH G xyz xyzEquations <strong>of</strong> 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, <strong>and</strong> therefore themoments <strong>and</strong> products <strong>of</strong> inertia <strong>of</strong> the body would have to be expressedas function <strong>of</strong> time
• x, y, z Axes Having Motion Ω=ωMMOGHOHOxyzHGHGxyzThe axes may be chosen suchthey are fixed in <strong>and</strong> move withthe bodyThe moments <strong>and</strong> products <strong>of</strong> inertia <strong>of</strong> the body relative to these axeswill be constant during the motionEquations <strong>of</strong> 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 <strong>of</strong> inertia, the product <strong>of</strong>inertia are zerox Iyy Izz yz I xyyzx2 2yzIzx zxy Mz ....Equations <strong>of</strong> 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 <strong>of</strong> ω, is convenientto choose the x, y, z axes having an angular velocity Ω which isdifferent from the angular velocity <strong>of</strong> the bodyMMMxyzIII xyzx zy IIIyxzzxyyzxIIIzxyyzxzxyEquations <strong>of</strong> Motion