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Lecture17-Prof.Ju

# Lecture17-Prof.Ju - CEE M237A MAE M269A Lect Lecture 17 17...

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CEE M237A / MAE M269A Lecture 17: Global Approximation Methods Professor J. Woody Ju © Prof. J.W. Ju, 2010

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Global Approximate Methods y Once a variational principle has been established for a particular problem, it constitutes a basis for seeking what are called approximate solutions y There are a host of problems where exact solutions are extraordinarily difficult, if not impossible, to obtain y Approximate analysis procedures based on variational principles belong to the mathematical class of solutions called the direct method of the calculus of variation © Prof. J.W. Ju, 2010 2
Global Approximate Methods (Cont’d) y Rayleigh’s quotient and Ritz method belong to this solution class y The main approach in approximate techniques is the representation of the structural behavior by a suitably chosen set of functions y These functions must meet the essential boundary conditions of the problem y For our problems, the essential boundary conditions are the prescribed kinematic (or geometric) boundary conditions © Prof. J.W. Ju, 2010 3

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lobal Approximate Methods (Cont’d) Global Approximate Methods (Cont’d) y The analysis then reverts to a minimization task as opposed to the task of enforcing equilibrium y The central idea of employing a select set of functions for such representations apply equally well other approximate methods that are not to other approximate methods that are not predicated on variational principles, such as Galerkin’s (FEM) and collocation methods y From the perspective of using suitably chosen functions for representing the behavior, the Galerkin and collocation methods are somewhat akin to Rayleigh’s quotient and the Ritz method as there is lso some sort of inimization requirement © Prof. J.W. Ju, 2010 also some sort of minimization requirement 4
Rayleigh’s Quotient y Provides a means for estimating the fundamental (lowest) natural frequency of vibration, ω 1 , of a structural system y Based on the fact that the total energy E in a conservative system in free vibration is constant over time: totol E TV =+ y Although the total energy E is a constant at all times, its distributions in kinetic and potential energies changes continuously with time © Prof. J.W. Ju, 2010 5

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ayleigh’s Quotient (Cont’d) Rayleigh s Quotient (Cont d) y In the free vibration of a spring-mass SDOF system, the displacement and velocity differ in phase by π /2 or one-quarter of a period y Thus, when the stored potential energy in the spring is maximum, the velocity is zero and hence there is no kinetic energy y When the kinetic energy is maximum, the configuration of the system at this instance in time is its undeformed position so that the potential energy is zero © Prof. J.W. Ju, 2010 6
Rayleigh’s Quotient (Cont’d) y The basis of Rayleigh’s quotient ax max 00 c o n s t a n t tal E TV =+ = += max max max total ⇒∴ = y General form of the displacement for a structural ystem in free vibration system in free vibration ( ) ( ) , it Ux t Xxe = ω © Prof. J.W. Ju, 2010 7

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ayleigh’s Quotient (Cont’d) Rayleigh s Quotient (Cont d) y
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Lecture17-Prof.Ju - CEE M237A MAE M269A Lect Lecture 17 17...

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