# lecture210 - ME 563 Mechanical Vibrations Lecture #2...

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ME 563 Mechanical Vibrations Lecture #2 Newton-Euler Laws (Derivation of equations of motion)

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Newton-Euler Laws 1
Flavors of Euler’s Law 2 What point “A” should we use? A = CM A is fixed A moves parallel to CM

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Proof of the Previous Slide (first two lines) 3 Parallel axis theorem (I A =I cm +M × r A/cm 2 )
Example Rolling Disc on Incline 4 K is massless and linear No slip (velocity of point in disc at incline is zero)

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Example Rolling Disc on Incline 5 F CM = M A CM ( ) j + K x x u ( ) f + Mg sin α ( ) i = M ˙ ˙ x ( ) i M ˙ ˙ x + Mg sin α− K x x u ( ) = f Euler’s Law about Center of Mass Newton’s Law
Applying Constraints 6 x=-a ⋅φ (rolling constraint) Two equations, two unknowns Ma ˙ ˙ φ + Mg sin α + K a + x u ( ) = 1 a I cm ˙ ˙ I cm + Ma 2 ( ) ˙ ˙ + Mga sin + Ka a + x u ( ) = 0 I cm + Ma 2 ( ) ˙ ˙ + + Ka 2 = Mga sin α− Kax u

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Example Rolling Disc on Incline 7 Euler’s Law about Point A M A = d H A dt
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lecture210 - ME 563 Mechanical Vibrations Lecture #2...

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