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ASE 365 - Lecture 14

ASE 365 - Lecture 14 - 2DOF systems “Prototype” 2DOF...

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Unformatted text preview: 2DOF systems “Prototype” 2DOF system: k 1 c1 m 1 c2 m 2 k 2 F2 F1 x2 x 1 FBD1 m 1 F1 fS1 fD 1 fD 2 fS2 ) ( ) ( 1 2 2 1 2 2 1 1 1 1 1 2 2 1 1 1 1 1 x x c x x k x c x k F f f f f F x m D S D S - +- +-- = + +-- = FBD2 m 2 F2 fS2 fD 2 ) ( ) ( 1 2 2 1 2 2 2 2 2 2 2 2 x x c x x k F f f F x m D S ---- =-- = Equations of motion: 2 2 2 1 2 2 2 1 2 2 2 1 2 2 1 2 1 2 2 1 2 1 1 1 ) ( ) ( F x k x k x c x c x m F x k x k k x c x c c x m = +- +- =- + +- + + Matrix form: = -- + + -- + + 2 1 2 1 2 2 2 2 1 2 1 2 2 2 2 1 2 1 2 1 F F x x k k k k k x x c c c c c x x m m F x x x = + + K C M : form General Note that mass, damping and stiffness matrices are symmetric, M is diagonal, and C and K are full. EOMs are coupled through C and K, but not through M. Suppose we use different “coordinates”—instead of x1 and x2, let’s use x1 and y2 ≡ x2-x1 : FBD1 m 1 F1 fS1 fD 1 fD 2 fS2 2 2 2 2 1 1 1 1 1 1 2 2 1 2 2 1 1 1 1 1 2 2 1 1 1 1 1 ) ( ) ( y c y k x c x k F x x c x x k x c x k F f f f f F x m D S D S + +-- =- +- +-- = + +-- = FBD2 m 2 F2 fS2 fD 2 ) ( ) ( ) ( 2 1 2 2 2 2 2 2 1 2 2 1 2 2 2 2 2 2 2 2 y x m y c y k F x x c x x k F f f F x m D S + =-- =---- =-- = Equations of motion: 2 2 2 2 2 2 2 1 2 1 2 2 1 1 2 2 1 1 1 1 F y k y c y m x m F y k x k y c x c x m = + + + =- +- + Matrix form: = - + - + 2 1 2 1 2 2 1 2 1 2 2 1 2 1 2 2 1 F F y x k k k y x c c c y x m m m Looks like we’ve lost symmetry. Is there a way to get it back, with x1 and y2 as coordinates? (Without ruining the equations!) What changes are “legal”? We can: • multiply an equation by something, • combine (add?) the two equations, as long as we don’t change the information they express. Here they are again: + = + + + 2 2 1 2 1 2 1 2 1 2 1 2 1 2 2 2 2 1 F F F y x k k y x c c y x m m m m m What if we replace the first equation by the sum of the two?...
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ASE 365 - Lecture 14 - 2DOF systems “Prototype” 2DOF...

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