fa_09lecture_set2_[Compatibility_Mode]

fa_09lecture_set2_[Compatibility_Mode] - 1 &...

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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 = + - & 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...
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fa_09lecture_set2_[Compatibility_Mode] - 1 &...

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