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ECS221L26

# ECS221L26 - PRODUCTS OF INERTIA Product of inertia Ixy = xy...

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PRODUCTS OF INERTIA Product of inertia: dA xy I xy = x y A x y I xy may not be positive!

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The Rectangle Compute I xy . dA = dxdy dI xy = xydxdy y x dy h b ∫ ∫ = h 0 b 0 xy ydy xdx I 2 2 xy h b 4 1 I =
Geometrical Meaning Product of inertia I xy = 0 x y I xy = 0 x y x y I xy = h 2 b 2 /2 dA xy I xy =

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Parallel Axis Theorem: Virtually the same as for I x , I y . y x A I I xy xy + = C y’ y dA x y y y’ x’ x’ x x
Principal Axes; Principal Moments of Inertia Consider: dA xy I dA x I dA y I xy 2 y 2 x = = = [ ] = y xy xy x I I I I I x y referred to coordinate system x,y

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Motivation The Rectangle I x = bh 3 /12 x x’ y y’ b h I y = hb 3 /12 I x’ = hb 3 /12 I y’ = bh 3 /12 I x , I y , I xy values change when axes are rotated!
Problem 1 How does [I] change when we refer it to a rotated set of axes x’,y’, i.e., how are I x’ , I y’ , I x’y’ related to I x , I y , I xy ?

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ECS221L26 - PRODUCTS OF INERTIA Product of inertia Ixy = xy...

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