Lecture 8 Notes

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Unformatted text preview: t) + ω0 B cos(ω0 t)) x￿￿ (t) = −ω0 A sin(ω0 t) + ω0 B cos(ω0 t) p +(−ω0 A sin(ω0 t) + ω0 B cos(ω0 t)) 2 2 +t(−ω0 A cos(ω0 t) − ω0 B sin(ω0 t)) Forced vibrations (3.8) • Case 2: ω = ω0 F0 ω0 = x+ = cos(ω0 t) m xp (t) = t(A cos(ω0 t) + B sin(ω0 t)) ￿￿ ￿ xp (t) 2 ω0 x ￿ k m = A cos(ω0 t) + B sin(ω0 t) +t(−ω0 A sin(ω0 t) + ω0 B cos(ω0 t)) x￿￿ (t) = −ω0 A sin(ω0 t) + ω0 B cos(ω0 t) p +(−ω0 A sin(ω0 t) + ω0 B cos(ω0 t)) 2 2 +t(−ω0 A cos(ω0 t) − ω0 B sin(ω0 t)) Forced vibrations (3.8) ω = ω0 • Case 2: F0 ω0 = x+ = cos(ω0 t) m xp (t) = t(A cos(ω0 t) + B sin(ω0 t)) ￿￿ ￿ xp (t) 2 ω0 x ￿ k m = A cos(ω0 t) + B sin(ω0 t) +t(−ω0 A sin(ω0 t) + ω0 B cos(ω0 t)) x￿￿ (t) = −ω0 A sin(ω0 t) + ω0 B cos(ω0 t) p +(−ω0 A sin(ω0 t) + ω0 B cos(ω0 t)) A=0 2 2 +t(−ω0 A cos(ω0 t) − ω0 B sin(ω0 t)) Forced vibrations (3.8) ω = ω0 • Case 2: F0 ω0 = x+ = cos(ω0 t) m xp (t) = t(A cos(ω0 t) + B sin(ω0 t)) ￿￿ ￿ xp (t) 2 ω0 x ￿ k m = A cos(ω0 t) + B sin(ω0 t) +t(−ω0 A sin(ω0 t) + ω0 B cos(ω0 t)) x￿￿ (t) = −ω0 A sin(ω0 t) + ω0 B cos(ω0 t) p +(−ω0 A sin(ω0 t) + ω0 B cos(ω0 t)) A=0 F0 F0 B= =√ 2ω 0 m 2 km 2 2 +t(−ω0 A cos(ω0 t) − ω0 B sin(ω0 t)) Forced vibrations (3.8) ω = ω0 • Case 2: F0 ω0 = x+ = cos(ω0 t) m xp (t) = t(A cos(ω0 t) + B sin(ω0 t)) ￿￿ ￿ xp (t) 2 ω0 x ￿ k m = A cos(ω0 t) + B sin(ω0 t) +t(−ω0 A sin(ω0 t) + ω0 B cos(ω0 t)) x￿￿ (t) = −ω0 A sin(ω0 t) + ω0 B cos(ω0 t) p +(−ω0 A sin(ω0 t) + ω0 B cos(ω0 t)) A=0 F0 F0 B= =√ 2ω 0 m 2 km 2 2 +t(−ω0 A cos(ω0 t) − ω0 B sin(ω0 t)) F0 xp (t) = √ t sin(ω0 t) 2 km Forced vibrations (3.8) • Plot of the amplitude of the particular solution as a function of ω. • Calculated: F0 A= 2 m(ω0 − ω 2 ) • Plotted: • Recall that for ω = ω0 , the amplitude grows without bound. Forced vibrations (3.8) • With damping (on the blackboard) Review questions Review questions Review questions • A dye diffuses between two chambers at a rate proportional to the difference in concentrations (c1 and c2) between the chambers (with proportionality constant k>0). Write down a differential equation for the concentration in the first chamber. • Solve: y ￿ − 2ty = t Review questions • A dye diffuses between two chambers at a rate proportional to the difference in concentrations (c1 and c2) between the chambers (with proportionality constant k>0). Write down a differential equation for the concentration in the first chamber. dc1 = k (c2 − c1 ) dt • Solve: y ￿ − 2ty = t Review questions • A dye diffuses between two chambers at a rate proportional to the difference in concentrations (c1 and c2) between the chambers (with proportionality constant k>0). Write down a differential equation for the concentration in the first chamber. dc1 = k (c2 − c1 ) dt • Solve: y ￿ − 2ty =...
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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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