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Unformatted text preview: 1 & Reviewing: We are considering the use of Multiple Scales method on the Duffings oscillator We assumed a solution of the form: Substituting in the equation, and collecting terms Here A has to be determined using secularity conditions in u(0) = & 2 1 2 1 2 u u (T ,T ,T ) u u = + + 2 3 u u u 1 u(0) a , u(0) + + = << = = && & } 2 2 O( ) : D u u u (0,0,0) a , D u + = = = 0 0 i T 1 2 The solution is : u A(T ,T ) e C.C . = + 1 O( ) EOM! 0 0 0 0 i T 1 2 2 2 1 1 1 3i T 3 O( ) : D u u [2i D A 3A A] e 3A e C.C. + =  + + & Then, a firstorder approximation means where is A determined by equation. We then use initial conditions to determine a and 2 1 2 2 1 2 1 2 3 a(T ,T ) a(T ) ; (T ,T ) a T (T ) 8 = = + u u O( ) = + 2 a(T ) const! & = 2 1 Sec.producing terms 2i D A 3A A 0. = & + = 1 O( ) 2 (T ) const! = i 1 Letting A a e , we get 2 = 2 & For the firstorder nonlinear approximation, we have but What do we need to satisfy initial conditions? Imposing these on the solution u u O( ) = + 2 1 i(3a T /8 ) i T . 1 a e e C.C 2 + = + 0 0 i T u A e C.C. = + 1 u (T 0,T 0) a , D u (0,0) = = = = i 1 u (0,0) a a e C.C. a 2 & = & + = a cos a & = The velocity condition gives: & = D u (0,0) = & = & = a a asin = + 2 1 0 0 i(3a T /8 ) i T 1 u a e e C.C 2 i 1 a (i ) e C.C. 2 & + = 3 & Going back to the solution in terms of original time: & We capture frequencyamplitude relationship with the firstorder approximation. = + 2 1 0 0 i(3a T /8 ) i T 1 u a e e C.C 2 + = + 2 i( 3a /8 )t 1 a e C.C 2 = + + 2 3 or u(t) a cos( a )t O( ) 8 Ex 2: Multiple scales method for the van der pol equation (Note : Read section 3.1 in text on damping mechanisms) Step 1: Let = + + + + & 2 2 2 2 1 1 2 2 d D 2 D D (D 2D D ) dt = + + + 2 1 2 1 1 2 2 1 2 3 u u (T ,T ,T ) u (T ,T ,T ) u (T ,T ,T ) O( ) +  = 2 2 2 d u du The van der Pol equation is : u (1 u ) dt dt = + + & 2 1 2 d set D D D dt 4 & Step 2: Step 3: = + iT 1 2 The solution is : u A(T ,T )e C.C = D u (0,0,0) + = = 2 O( ): D u u u (0,0,0) a 1 2 [Please confirm the O( ), O( ) and O( ) equations!] + =  + 1 2 2 1 1 1 O( ): D u u 2D D u (1 u )D u + = + + iT 2 2 1 1 1 Then D u u i( 2 D A A A A) e C.C + 3iT 3 i A e C.C Now, Eliminate terms that will result in secular terms in the solution: Separate real and imaginary parts = = 2 1 1 2 2D A A A A, where A A(T ,T )! & + i i 1 1 1 a a 2 e i e 2 T 2 T = i 1 set A a e 2  = i 2 2i i 1 1 a e a e a e 2 8 5 & Try to work out qualitative behavior using velocity functions Interlude: if we are only interested in firstorder approximation, then & = = + = = 1 3 2 1 T 2 2 1 a 1 1 4 a a a T 2 8 1 c(T )e (T ) T = = const...
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 Fall '10
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