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ASE 365 - Lecture 9

# ASE 365 - Lecture 9 - Response to periodic inputs y(t...

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Response to periodic inputs Cam-follower example: k 1 c m x(t ) k 2 ω0 y(t ) FB D fD fS 1 fS 2 m ) ( 2 1 2 1 y x k x c x k f f f x m S D S - - - - = - + = ) ( ) ( ) ( 2 2 2 1 T t y k t y k x k k x c x m + = = + + + : EOM y(t ) t T s. ' and s ' the of ity orthogonal of because found be can series Fourier the in s ' and s ' The where general, In sin cos 2 ), sin cos ( 2 ) ( ) ( ) ( ) ( 0 0 1 0 0 p p p p p b a T t p b t p a a t f T t kf t kf t F kx x c x m π ϖ ϖ ϖ = + + = + = = = + + =

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Orthogonality of basis functions = = = + + + - - + + + = - + + = + + + + + + q p T q p q p t q p t q p q p q p t q p q p t q p dt t q p t q p dt t q t p T t t T t t T t t T t t T t t T t t if if if if , 2 , 0 , ] [ 2 1 ) ( ) sin( 2 1 , ) ( ) sin( 2 1 ) ( ) sin( 2 1 ) cos( ) cos( 2 1 cos cos ˆ ˆ ˆ ˆ 0 0 ˆ ˆ 0 0 ˆ ˆ 0 0 ˆ ˆ 0 0 ˆ ˆ 0 0 ϖ ϖ ϖ ϖ ϖ ϖ ϖ ϖ ϖ ϖ
= = + + - - - = + - - = + + + + q p T q p q p q p t q p q p t q p dt t q p t q p dt t q t p T t t T t t T t t T t t if if (if , 2 , 0 ) ) ( ) sin( 2 1 ) ( ) sin( 2 1 ) cos( ) cos( 2 1 sin sin ˆ ˆ 0 0 ˆ ˆ 0 0 ˆ ˆ 0 0 ˆ ˆ 0 0 ϖ ϖ ϖ ϖ ϖ ϖ ϖ ϖ q p q p q p q p t q p q p t q p dt t q p t q p dt t q t p T t t T t t T t t T t t = = - - - + + + - = - + + = + + + + if or if (if , 0 ) ) ( ) cos( 2 1 ) ( ) cos( 2 1 ) sin( ) sin( 2 1 cos sin ˆ ˆ 0 0 ˆ ˆ 0 0 ˆ ˆ 0 0 ˆ ˆ 0 0 ϖ ϖ ϖ ϖ ϖ ϖ ϖ ϖ

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Multiply f(t) by cos qω0t and integrate over one period: Similarly, multiply f(t) by sin qω0t and integrate: , 2 , 1 , 0 , cos ) ( 2 2 cos ) sin cos ( 2 cos ) ( ˆ ˆ 0 ˆ ˆ 0 0 1 0 0 ˆ ˆ 0 = = = + + = + + = + q dt t q t f T a T a dt t q t p b t p a a dt t q t f T t t q q T t t p p p T t t ϖ ϖ ϖ ϖ ϖ , 3 , 2 , 1 , sin ) ( 2 2 sin ) sin cos ( 2 sin ) ( ˆ ˆ 0 ˆ ˆ 0 0 1 0 0 ˆ ˆ 0 = = = + + = + + = + q dt t q t f T b T b dt t q t p b t p a a dt t q t f T t t q q T t t p p p T t t ϖ ϖ ϖ ϖ ϖ
Even and odd functions axis) to respect with (Symmetry series). (cosine zero are s ' and even is then If ) ( ) ( ), ( ) ( t f b t f t f t f n - = origin) to respect with (Symmetry series). (sine zero are s ' and odd is then If n a t f t f t f ) ( ), ( ) ( - - =

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SDOF response ) sin cos ( 2 ) ( 0 1 0 0 t p b t p a a t f p p p ϖ ϖ
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ASE 365 - Lecture 9 - Response to periodic inputs y(t...

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