Lecture 7 Notes

7 damped oscillations 4km i 1 2 4km ii 1 2 4km iii 1

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Unformatted text preview: Applications - vibrations (3.7) • Converting from sum-of-sin-cos to a single cos expression: y (t) = C1 cos(ω0 t) + C2 sin(ω0 t) ￿ 2 2 C1 + C2 • Step 1 - Factor out A = • Step 2 - Find the angle and φ for which sin(φ) = ￿ • Step 3 - Rewrite the solution as . C1 cos(φ) = ￿ 2 2 C1 + C2 C2 2 C1 + 2 C2 . y (t) = A cos(ω0 t − φ) . Applications - vibrations (3.7) • Damped mass-spring mx + γ x + kx = 0 ￿￿ ￿ m, γ , k > 0 Applications - vibrations (3.7) • Damped mass-spring m, γ , k > 0 mx + γ x + kx = 0 2 ⇒ mr + γ r + k = 0 ￿￿ ￿ Applications - vibrations (3.7) • Damped mass-spring m, γ , k > 0 mx + γ x + kx = 0 2 ⇒ mr + γ r + k = 0 ￿￿ r1 ,2 γ =− ± 2m ￿ ￿ γ 2 − 4km 2m Applications - vibrations (3.7) • Damped mass-spring m, γ , k > 0 mx + γ x + kx = 0 2 ⇒ mr + γ r + k = 0 ￿ ￿ ￿ ￿ γ 4km γ γ 2 − 4km = −1 ± 1 − 2 =− ± 2m γ 2m 2m ￿￿ r1 ,2 ￿ Applications - vibrations (3.7) • Damped mass-spring m, γ , k > 0 mx + γ x + kx = 0 2 ⇒ mr + γ r + k = 0 ￿ ￿ ￿ ￿ γ 4km γ γ 2 − 4km = −1 ± 1 − 2 =− ± 2m γ 2m 2m ￿￿ r1...
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This note was uploaded on 02/12/2014 for the course MATH 256 taught by Professor Ericcytrynbaum during the Spring '13 term at UBC.

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