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ASE 365 - Lecture 25

ASE 365 - Lecture 25 - Lagrange’s equations ns eq s...

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Unformatted text preview: Lagrange’s equations , : ns eq' s Lagrange' n j Q q V q T q T dt d nc j j j j , , 1 , = = ∂ ∂ + ∂ ∂- ∂ ∂ : system MDOF a For s coordinate d generalize of set a choose • s coordinate d generalize of terms in and , , write nc W V T δ • j n j nc j nc q Q W δ δ ∑ = = • 1 , note forces. constraint or FBDs need t don' We system. MDOF the for EOMs the are These n Lagrange’s equations F x x x = + + K C M : as y immediatel EOMs the write can we then , and , , where , and , , write can we If : conclusion this Remember T T T T nc T T K K C C M M C W K V M T = = =- = = = ) ( 2 1 2 1 x F x x x x x δ δ Review: MDOF undamped free vibration ). (or : problem Eigenvalue u u u =- = ) ( 2 2 M K M K ϖ ϖ x x = + K M : EOMs ∑ ∑ = = + → = j j ij j j ij u m u k C Cf f t f t , n? sol' s Synchronou ) ( ) ( u x ). if (linear harmonic is , so , : n cons' Energy 2 = = ≥ ϖ ϖ f C C . eigenpairs in degree of polynomial becomes DOF, With n r n n M K n r r , , 1 }, , { ) det( 2 2 2 = → =- u ϖ ϖ ϖ ). ( ) ( ) ( t U t t r η x = as solution form , for problems response modal uncoupled Solve η ) ( ) sin cos ( ) ( 1 t U t b t a t r r r r n r r η u x = + = ∑ = ϖ ϖ : solution vibration Free diagonal K KU U diagonal M MU U T T = ′ = = ′ = , : ity Orthogonal 2 ), ( ) ( , , , 1 ), ( : ) ( ) ( ) ( r r r T r r r r r r r T m k t t N n r t N k m U t U t t ϖ η η = ′ ′ = = = ′ + ′ = ≠ F u η x F by EOMs multiply , : for analysis Modal ) ( ] [ ), ( 1 1 1 x b x a --- = = U U ϖ : ICs from Constants Properties of K and M others. all in nt displaceme zero a and in nt displaceme unit a in results that vector force the is of column The : tion interpreta Physical i th x K i F x = K : problem static Consider . K i K K i th i i th i of column the , Then zero. equal others all while one equals in entry the i.e., , Suppose k e x x e x = = = else. everywhere force zero a and in force unit a applying from results that vector nt displaceme the is of column The : tion interpreta Physical : problem Static ....
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ASE 365 - Lecture 25 - Lagrange’s equations ns eq s...

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