This preview shows pages 1–5. Sign up to view the full content.
This preview has intentionally blurred sections. Sign up to view the full version.
View Full DocumentThis preview has intentionally blurred sections. Sign up to view the full version.
View Full Document
Unformatted text preview: Linear secondorder differential equations with constant coefficients and nonzero righthand side We return to the damped, driven simple harmonic oscillator d 2 y dt 2 + 2 b dy dt + 2 y = F sin t We note that this differential equation is linear We call y c the general solution which solves the homogeneous equation d 2 y c dt 2 + 2 b dy c dt + 2 y c = 0 We call y p the particular solution which solves the inhomogeneous equation d 2 y p dt 2 + 2 b dy p dt + 2 y p = F sin t Linear equations with nonzero righthand side continued Because the differential equation is linear, in general y ( t ) = y c ( t ) + y p ( t ) For the damped, driven simpleharmonic oscillator, with F ( t ) = F sin t driving force, y ( t ) = y c ( t ) + F q ( 2 2 ) 2 + 4 b 2 2 sin ( t ) The y c depends on whether the system is underdamped, overdamped, or critically damped To find a solution to a given situation, we need to know the position and velocity of the oscillator at some time (eg t = 0), or alternately the position at two different times Example: Damped, driven simple harmonic oscillator Example: At t = 0 we have y ( t = 0) = 0 and dy dt  t =0 = 0 The system is underdamped, so that b < The equation of motion is given by d 2 y p dt 2 + 2 b dy p dt + 2 y p = F sin t The solution is y ( t ) = y c ( t ) + y p ( t ) y c ( t ) = ce bt sin ( t + ) Here = q 2 b 2 y p ( t ) = F q ( 2 2 ) 2 + 4 b 2 2 sin ( t ) tan = 2 b 2 2 Example continued So we have expressions for y ( t ) and dy dt , y ( t ) = ce bt sin ( t + ) + F q ( 2 2 ) 2 + 4 b 2 2 sin ( t ) dy dt = ce bt [ b sin ( t + ) cos ( t + )]+ F q ( 2 2...
View
Full
Document
 Spring '03
 Staff

Click to edit the document details