CA-AA242B-Ch1

CA-AA242B-Ch1 - AA242B: MECHANICAL VIBRATIONS AA242B:...

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Unformatted text preview: AA242B: MECHANICAL VIBRATIONS AA242B: MECHANICAL VIBRATIONS Analytical Dynamics of Discrete Systems These slides are based on the recommended textbook: M. G´ eradin and D. Rixen, “Mechanical Vibrations: Theory and Applications to Structural Dynamics,” Second Edition, Wiley, John & Sons, Incorporated, ISBN-13:9780471975465 AA242B: MECHANICAL VIBRATIONS AA242B: MECHANICAL VIBRATIONS Outline 1 Principle of Virtual Work for a Particle 2 Principle of Virtual Work for a System of N Particles 3 Hamilton’s Principle for Conservative Systems and Lagrange Equations 4 Lagrange Equations in the General Case AA242B: MECHANICAL VIBRATIONS AA242B: MECHANICAL VIBRATIONS Principle of Virtual Work for a Particle Particle mass m Particle force force vector f = [ f 1 f 2 f 3 ] T force component f i , i = 1 , ··· , 3 Particle displacement displacement vector u = [ u 1 u 2 u 3 ] T displacement component u i , i = 1 , ··· , 3 motion trajectory u ( t ) where t denotes time AA242B: MECHANICAL VIBRATIONS AA242B: MECHANICAL VIBRATIONS Principle of Virtual Work for a Particle Particle virtual displacement arbitrary displacement u ? (can be zero) virtual displacement δ u = u ?- u ⇒ arbitrary by definition family of arbitrary virtual displacements defined in a time-interval [ t 1 , t 2 ] and satisfying the variational constraints δ u ( t 1 ) = δ u ( t 2 ) = Important property d dt ( δ u i ) = d dt ( u ? i- u i ) = du ? i dt- du i dt = ˙ u ? i- ˙ u i = δ ˙ u i = ⇒ d dt ( δ ) = δ ( d dt ) (commutativity) AA242B: MECHANICAL VIBRATIONS AA242B: MECHANICAL VIBRATIONS Principle of Virtual Work for a Particle Equilibrium strong form m ¨ u- f = ⇒ m ¨ u i- f i = 0 , i = 1 , ··· , 3 weak form ∀ δ u , ( δ u T ) ( m ¨ u- f ) = 0 ⇒ 3 X i =1 ( m ¨ u i- f i ) δ u i = 0 = ⇒ ( m ¨ u i- f i ) δ u i = 0 , i = 1 , ··· , 3 δ u T ( m ¨ u- f ) = ( m ¨ u- f ) T δ u is homogeneous to a work = ⇒ virtual work ( δ W ) Virtual work principle The virtual work produced by the effective forces acting on a particle during a virtual displacement is equal to zero AA242B: MECHANICAL VIBRATIONS AA242B: MECHANICAL VIBRATIONS Principle of Virtual Work for a System of N Particles N particles: k = 1 , ··· , N Equilibrium m ¨ u k- f k = , k = 1 , ··· , N Family of virtual displacements δ u k = u ? k- u satisfying the variational constraints δ u k ( t 1 ) = δ u k ( t 2 ) = (1) Virtual work m ¨ u k- f k = ⇒ N X k =1 δ u T k ( m ¨ u k- f k ) = N X k =1 ( m ¨ u k- f k ) T δ u k = 0 AA242B: MECHANICAL VIBRATIONS AA242B: MECHANICAL VIBRATIONS Principle of Virtual Work for a System of N Particles Conversely, ∀ δ u k compatible with the variational constraints (1) N X k =1 δ u T k ( m ¨ u k- f k ) = 0 ⇒ N X k =1 3 X i =1 ( m k ¨ u i k- f i k ) δ u i k = 0 (2) If (2) is true ∀ δ u k compatible with (1) ⇒ (2) is true for δ u k = [1 0] T , δ u k = [0 1 0] T , and δ u k = [0 1] T = ⇒ N X k =1 m k ¨ u i k- f i k = 0 , i = 1 , ··· , 3 If the virtual work equation is satisfied for any displacement...
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This note was uploaded on 06/17/2010 for the course AA 242B taught by Professor Charbelfarhat during the Spring '10 term at Stanford.

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CA-AA242B-Ch1 - AA242B: MECHANICAL VIBRATIONS AA242B:...

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