7 damped oscillations r1 2 2m 1 4km 1 2 4km i 1 2 4km

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Unformatted text preview: ,2 ￿ smaller than 1 or complex 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 ￿ negative or complex smaller than 1 or complex 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 ￿ We have the usual three cases... negative or complex smaller than 1 or complex Applications - vibrations (3.7) • Damped oscillations 4km (i) <1 2 γ 4km (ii) =1 2 γ 4km (iii) >1 2 γ r1 ,2 γ = 2m ￿ −1 ± ￿ 4km 1− 2 γ ￿ Applications - vibrations (3.7) • Damped oscillations 4km (i) <1 2 γ 4km (ii) =1 2 γ 4km (iii) >1 2 γ ⇒ r1 ,2 γ = 2m ￿ −1 ± ￿ 4km 1− 2 γ r1, r2 < 0, exponential decay only (over damped - γ large) ￿ Applications - vibrations (3.7) • Damped oscillations r1 ,2 γ = 2m ￿ −1 ± ￿ 4km 1− 2 γ 4km (i) <1 2 γ ⇒ 4km (ii) =1 2 γ r1, r2 < 0, exponential decay only (over damped - γ large) ⇒ r1=r2, exp and t*exp decay (critically damped) 4km (iii) >1 2 γ ￿ Applications - vibrations (3.7) • Damped oscillations r1 ,2 γ = 2m ￿ −1 ± ￿ 4km 1− 2 γ 4km (i) <1 2 γ ⇒ 4km (ii) =1 2 γ r1, r2 < 0, exponential decay only (over damped - γ large) ⇒ r1=r2, exp and t*exp decay (critically damped) 4km (iii) >1 2 γ ⇒ r = α...
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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 The University of British Columbia.

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