This preview shows pages 1–2. Sign up to view the full content.
This preview has intentionally blurred sections. Sign up to view the full version.
View Full Document
Unformatted text preview: 18.03 Lecture #12 Oct. 5, 2009: notes Today will be about the nice little spring system from Friday, complicated with the addition of a dashpot. This is a mechanical damping device (typically a piston moving inside a cylinder filled with viscous oil). The word damping means slowing down; so a damping device should act always to decelerate. According to Newton, that means it should apply a force pointed away from the velocity. In the case of a dashpot, well assume (as on page 101 of EP) that the damping force is proportional to the velocity: dashpot damping force = cy (Dashpot) When we combine this damping with the Hookes law force exerted by the spring spring force = ky (Hooke) from Fridays lecture, and plug it all into Newtons law F = ma , we get my = cy  ky. (Damped spring) y ( t ) = y (initial position) , y ( t ) = y (initial velocity) . (Spring initial conditions) The physical setting suggests that the constants m , c , and k should all be positive, but mathe matically that doesnt matter; the equation makes sense in any case. Sometimes its convenient to write the equation as my + cy + ky = 0 . (Damped spring normalized) In order to solve this, Ill rewrite it once more. Define D = d dt = operation of taking a derivative . The idea is that D takes a function (of one independent variable t ) and spits out its derivative: Df = df dt , D cos = sin , D ( t 11 ) = 11 t 10 ....
View
Full
Document
 Fall '09
 unknown

Click to edit the document details