ADS Homework 6 - Homework 6: mv + cv + kv = p 0 sin t The...

Info iconThis preview shows pages 1–2. Sign up to view the full content.

View Full Document Right Arrow Icon

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon
This is the end of the preview. Sign up to access the rest of the document.

Unformatted text preview: Homework 6: mv + cv + kv = p 0 sin t The solution of the diferantial equation for the given system has two parts. These are homogeneous solution and particular solution. v c (t ) = e -wt ( A. sin D t + B. cos D t ) v p (t ) = E. sin t + F . cost : : Homogeneous Solution Particular Solution Determine the constants (E and F) of the particular solution. Solution: v p (t ) = E. sin t + F . cost v p (t ) = E. cost - F . sin t v p (t ) = - 2 E. sin t - 2 F . cost mv + cv + kv = p 0 sin t v + p p c k v + v = 0 sin t v + 2v + 2 v = 0 sin t m m m m (- 2 E. sin t - 2 F . cost ) + 2(E. cost - F . sin t ) + 2 ( E. sin t + F . cost ) = p0 sin t m p0 sin t m (1) (- 2 E - 2 + 2 E ) sin t + (- 2 F + 2 + 2 F ) cost = F E From equation 1 we can get; - 2 E - 2 + 2 E = F p0 m (2) (3) - 2 F + 2 + 2 F = 0 E By using equation 2 we can write F in terms of E. F= 2 E 2 -2 (4) 1 If we put F in equation 1 we get; - 2 E - 2 p 2 E + 2E = 0 2 2 m - (5) We know that m = k / 2 . If we put this expression in equation 5, we can get; 2 (2 ) 2 p 2 E 2 - 2 - 2 + 1 = 0 k -2 We know that = / . If we put this expression in equation 6, we can get; (6) (2 ) 2 p 0 E - 2 - + 1 = 1- 2 k and also we can obtain the expression of E. E= p0 1- 2 k (1 - 2 ) 2 + (2 ) 2 (7) We know that; F= 2 E 2 -2 2 2 2 2 F= 2 E F = 2 2 E F = - E 2 - 1- 2 - 2 2 If we put equation 7 into the expression of F that is written above; F =- p 2 p 0 1- 2 2 F =- 0 2 2 2 2 2 2 k (1 - ) + (2 ) 2 1 - k (1 - ) + (2 ) we can obtain the expression of F. So vp(t) can be written as; p0 p 1- 2 2 v p (t ) = . sin t - 0 . cost 2 2 2 2 2 k (1 - ) + (2 ) k (1 - ) + ( 2 ) 2 v p (t ) = p k 1 - 2 sin t - ( 2 ) cost 2 (1 - ) + (2 ) 0 2 2 [( ) 2 ...
View Full Document

This note was uploaded on 11/23/2010 for the course DEP 503E 10852 taught by Professor Prof.zekaicelep during the Spring '10 term at Istanbul Technical University.

Page1 / 2

ADS Homework 6 - Homework 6: mv + cv + kv = p 0 sin t The...

This preview shows document pages 1 - 2. Sign up to view the full document.

View Full Document Right Arrow Icon
Ask a homework question - tutors are online