This preview shows pages 1–4. 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: 1 & Forced, linear oscillations of an undamped oscillator. If we choose an initial condition , the response is & It is a periodic response only if or when & where p & q are integers & Consider & The resulting response contains frequency components that have lower frequency than the excitation term; called subharmonic response o 1 n 2 2 n F x(t) C cos t cos t m( ) B = +  & 1 C = n p p 1 = > x = n p q , = If we have super or ultraharmonic generation If we have ultra subharmonics response Note that for this to happen in a linear oscillator, the damping = 0. However, for nonlinear oscillators, sub & super harmonics can be generated even for damped systems. Superposition fails in nonlinear systems If are the solutions of a linear oscillator with identical initial conditions, but different forcing is a solution to the system with forcing . i.e. with 1 2 x (t)& x (t) n /q q 1 = > n p q = 1 F (t) 1 2 F (t) F (t) + 2 1 n 1 n 1 1 x 2 x x F (t) + + = 1 x (0) x = 1 x (0) x = 2 1 2 &F (t), it is easily seen that x (t) x (t) + 2 & and Adding these & & Thus, the principle of superposition holds. However, in the presence of some nonlinearity, say a quadratic term , 2 1 2 n 1 2 n 1 2 1 2 (x x ) 2 (x x ) (x x ) F (t) F (t) x x x + + + + + = + && && & & && & 2 with x (0) x , = 2 2 n 2 n 2 2 also x 2 x x F (t) + + = && & 2 x (0) x = & & 1 2 t 1 2 t and (x x ) x (x x ) x = = + = + = & & & 2 2 1 n 1 n 1 1 1 consider x 2 x x x F (t) + + + = && & 2 2 2 n 2 n 2 2 2 1 2 and x 2 x x x F (t), and x (t) and x (t) are the two solutions, + + + = && & Let us again add the two equations & Since i.e., the idea of superposition fails in nonlinear systems . 1 2 1 2 2 2 1 2 n 1 2 n 1 2 1 2 1 2 F F , where as, (x x ) will satisfy the equation (x x ) 2 (x x ) (x x ) (x x ) F F = + + + + + + + + + = + && && & & 2 2 2 1 2 n 1 2 n 1 2 1 2 (x x ) 2 (x x ) (x x ) x x & + + + + + + + && && & & 2 2 2 1 2 1 2 x x (x x ) , the two equations don't have the same solution, + + 3 & Existence & Uniqueness of Solutions Consider a system of ordinary differential equations: Let us write Then, the system can be written as: 1 1 1 N 1 p x f (x , x , , ), = & 2 2 1 1 x f (x , , ....), = & N N 1 1 x f (x , ,....) = & 1 1 1 N N p x f x , f , and x f & & & = = = x f(x, ,t) = & A local existence/unique theorem: If are continuous, and their derivatives with respect to and t are also continuous, then there exists a small window of time...
View Full
Document
 Fall '10
 NA

Click to edit the document details