Lecture18-Prof.Ju

Lecture18-Prof.Ju - CEE M237A MAE M269A Lecture 18 The Ritz...

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CEE M237A / MAE M269A Lecture 18: The Ritz Method I Professor J. Woody Ju
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The Ritz Method y Whereas Rayleigh’s quotient utilized only one trial function in the estimate of the lowest frequency, the Ritz method allows for the determination of a subset of the lowest frequencies and corresponding modes of vibration y The displacement field is taken in the form of n linearly independent trial function. y Each one in the set of trial functions must satisfy the given kinematic or geometric boundary conditions of the problem 2
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The Ritz Method (Cont’d) y Let denote the total approximate solution: 3 ( ) , Ux t () ( ) ( ) ( ) ( ) ( ) ( ) [] 11 2 2 12 , nn n t a tX x a x a x XX X t t =+ + + = = " " a Xa () () () () where are the undetermined time-dependent coefficients the ' are undetermined time-depend coefficents and ' are assumed trial functions, respectivl gen y, called the eralized T n i i t ta t a t a t at s Xx s ⎡⎤ = ⎣⎦ " a a coordinates coordinate f and t unc he tions
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The Ritz Method (Cont’d) y To implement a Ritz analysis, substitute assumed displacement field into the integral expressions for the kinetic and strain energies of the Lagrangian functional L : 4 () () () ( ) ( ) ( ) ( ) , For linear problems, the expressions for and are quadratic functions of the generlized coordnates and velocities and can be written as 11 22 ii i i TT at T V TV tt t t =− ±± L L L aM a aK a
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The Ritz Method (Cont’d) y K and M are stiffness and mass matrices containing the generalized stiffness and mass components 5 ' a n d ' ij ij ks ms
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The Ritz Method (Cont’d) y Each one of these coefficients involves the integration of the strain and kinetic energy expressions over the entire volume of a given structure using a pair of coordinate functions 6 and . The coefficients of and are symmetric with respect to their indices, i.e., and mass matrix is positive definite, and the . Moreover, t stiffness matrix is non negative- he d ij ij ji ij ji XX kk mm == KM efinite.
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The Ritz Method (Cont’d) y Application of Hamilton’s principle or Lagrange’s equations involves differentiation of L with respect to the generalized coordinates and velocities. This variational process leads to a discrete system of equations of motion: 7 () () () 0 For free vibration, invoke so that the array is of the form: a where column vector is an array of simple t amplitud ime harmo e coeff., nic motion generaliz i ed .e co ., ord the it tt t te += = ±± ω Ka Ma a a a (no longer functions of inates time)
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The Ritz Method (Cont’d) y We therefore obtain the Algebraic eigenproblem y One viewpoint of the Ritz method is that it contracts infinite degrees of freedom in the continuous system to a
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This note was uploaded on 05/18/2011 for the course MAE 269A taught by Professor Ju during the Spring '11 term at UCLA.

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Lecture18-Prof.Ju - CEE M237A MAE M269A Lecture 18 The Ritz...

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