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Unformatted text preview: Ae/APh 104a Homework Solution Set #3 1/3 (a) Looking at a free body diagram of the piston, we can see that movement in the x direction would cause an opposing force from the spring and the damper. A summation of the forces gives: m ¨ x =- k 1 x- c ( ˙ x- ˙ y )- P A + ( P + P ) A (1) On the damper, where the origin of the y movement is defined, we get the following balance of forces:- k 2 y- c ( ˙ y- ˙ x ) = 0 (2) Note that for the damper, there is no m d ¨ y term because the damper has negligible mass. Since we need a differential equation relating y and P , the variable x must be eliminated. Eqn (2)can be used to substitute into the first equation for the x terms. There is, however, still one x term in eqn(1) that has yet to be differentiated. We can either take the time derivative of eqn(1) or we can integrate eqn(2) to get an expression for x . In general it is easier to differentiate than to integrate, so we get: m ¨ x + c ¨ x + k 1 ˙ x- c ¨ y = ˙ PA and ˙ x = ˙ y + k 2 c y ¨ x = ¨ y + k 2 c ˙ y ... x = ... y + k 2 c ¨ y. 1 Substituting for ... x, ¨ x and ˙ x in eqn(1), we get: m ... y + m k 2 c ¨ y + ( k 1 + k 2 ) ˙ y + k 1 k 2 c y = ˙ PA. (b) If the input P is sinusoidal with frequency ω , find the steady-state am- plitude ratio and the phase angle of y relative to P....
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This note was uploaded on 01/05/2012 for the course AE 104a taught by Professor List during the Fall '09 term at Caltech.
- Fall '09