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Lecture 20

# Lecture 20 - Section 4.9 Section 5.6 Free Mechanical...

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Section 4.9; Section 5.6 Free Mechanical Vibrations/Couple Mass-Spring System April 14, 2009 Free Mechanical Vibrations/Couple Mass-Spring System

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Today’s Session A Summary of This Session: (1) Free Mechanical Vibration (no forcing term). (2) Coupled Mass-Spring systems (3) Our first exposure to systems of differential equations (4) Eigenvalues and Eigenvectors Free Mechanical Vibrations/Couple Mass-Spring System
Free Mechanical Vibrations Free mechanical vibration=no forcing function, so f ( t ) = 0. We are dealing with my ′′ + by + ky = 0 where m = mass attached to a spring of stiffness k , subject to friction (or damping) proportional to speed with damping constant b . Four cases: (1) undamped free case; (2) underdamped case; (3) overdamped case; (4) critical damped case. Free Mechanical Vibrations/Couple Mass-Spring System

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Case 1: Undamped mass-spring system: b = 0. The equation is given by: my ′′ + ky = 0 or y ′′ + k m y = 0 Let ω 2 = k m . The quantity ω = radicalBig k m is called the angular frequency (measured in radians per second) Period: T = 2 π ω (measured in seconds) Frequency: f = 1 T = ω 2 π (measured in Hertz=1/seconds=# cycles per second) The solution is given by: y = C 1 cos ω t + C 2 sin ω t = A sin( ω t + φ ).
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Lecture 20 - Section 4.9 Section 5.6 Free Mechanical...

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