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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 = ω + ω ω  ω &¡¢¡£ 1 C = n p p 1 ω = ω > x = ¤ n p q , ω = ω ¡ 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 ω = ω > ¥ n p q ω = ω 1 F (t) 1 2 F (t) F (t) + 2 1 n 1 n 1 1 x 2 x x F (t) + ζω + ω = ¤¤ ¤ 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 + + ζω + + ω + = + && && & & ¡¢£¢¤ ¡¢£¢¤ ¡¢£¢¤ && & 2 with x (0) x , = 2 2 n 2 n 2 2 also x 2 x x F (t) + ζω + ω = && & 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) + ζω + ω + = && & 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, + ζω + ω + = && & ¡ 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 = + + + + ζω + + ω + + + = + && && & & 2 2 2 1 2 n 1 2 n 1 2 1 2 (x x ) 2 (x x ) (x x ) x x & + + ζω + + ω + + + && && & & 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 , , ), = λ λ & ¡ ¡ 2 2 1 1 x f (x , , ....), = λ & ¢ N N 1 1 x f (x , ,....) = λ & ¢ 1 1 1 N N p x f x , f , and x f & ¡ λ & ¡ & ¡ ¢ £ ¢ £ ¢ £ = = λ = ¢ £ ¢ £ ¢ £ ¢ £ ¢ £ ¢ £ λ ¤ ¥ ¤ ¥ ¤ ¥ £ £ £ ¤ ¤ ¤ x f(x, ,t) = λ & ¤ ¤ ¤ ¤ £ ¡ 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...
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This note was uploaded on 12/28/2011 for the course ME 580 taught by Professor Na during the Fall '10 term at Purdue.
 Fall '10
 NA

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