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Variational Methods For The Solution of Problems of Equilibrium and Vibration

The document discusses variational methods for solving equilibrium and vibration problems using finite element analysis. It introduces variational problems involving functionals for kinetic and potential energies. It describes using the Rayleigh-Ritz method to find minimizing sequences of functions to solve variational problems by substituting linear combinations of coordinate functions. It also briefly mentions other numerical methods like finite differences, random statistical methods, and gradients.

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handikajati kusuma
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90 views1 page

Variational Methods For The Solution of Problems of Equilibrium and Vibration

The document discusses variational methods for solving equilibrium and vibration problems using finite element analysis. It introduces variational problems involving functionals for kinetic and potential energies. It describes using the Rayleigh-Ritz method to find minimizing sequences of functions to solve variational problems by substituting linear combinations of coordinate functions. It also briefly mentions other numerical methods like finite differences, random statistical methods, and gradients.

Uploaded by

handikajati kusuma
Copyright
© © All Rights Reserved
We take content rights seriously. If you suspect this is your content, claim it here.
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FINITE ELEMENT INTRODUCTION

HOMEWORK 1:
VARIATIONAL METHODS FOR THE SOLUTION OF PROBLEMS OF EQUILIBRIUM AND VIBRATION
MARJADI HANDIKAJATI KUSUMA (2173528)

Problems of equilibrium and vibrations lead to linear self-adjoint differential


equations for an unknown function u ( x , y):
L (u )=f ,∨¿L (u )+ λu=0
in a two-dimensional domain or equivalent variational for the kinetic and
potential energies of the system.
1. THE VARIATIONAL PROBLEM
Assuming domain B bounded by a piecewise smooth curve C with arc length
of s and differentiation in the direction of the inward normal by ∂ /∂ n or by
subscript n. The variational problems refer to quadratic functionals
Q ( v )=Q (v , v ) is defined by symmetric bilinear expression such as

D ( v , w ) =∬ ( v x w x + v y w y ) dxdy ,
B

M ( v , w )=∬ [ ∆ v ∆ w+ α ( v xx w yy + v yy w xx )−2 v xy w xy ¿ ] dxdy ,


B

H ( v , w ) =∬ vwdxdy , K ( v , w )=∫ vwds ,


B C

R ( v , w )=∫ v xx w xx dx ,
L
where L is a line y=const. in B. Thus, consider functional as
Q ( v , w ) =aD ( v , w ) +bM ( v , w ) + cK ( v . w ) +dR ( v . w ) ,
where a, b, c, d are constants.
The stable equilibrium of a plate or membrane under an external pressure f is
characterized by,
Q ( v ) +2 H ( v , f )=minimum ,
for the deflection v, whereas vibrations of plates and membranes correspond
to the problem of finding stationary values, v 2=λ, of Q(v)/H(v). Then v
defined are the natural frequencies of the system.
The Euler differential equations of the variational problems must be
supplemented by appropriate boundary conditions. Physically, rigid conditions
correspond to rigid constraints of the system at the boundary C while natural
conditions express equilibrium of the system of C if along C partial or full
freedom of motion is allowed.
2. RAYLEIGH-RITZ METHOD
When finding the minimum d of an integral expression or any other
variational expressionI (ϕ) then start with minimizing sequence of functions
ϕ 1 , ϕ2 , … . ,ϕ n ,… . ,
admissible in variational problem, for which
lim I (ϕn )=d ,
n→∞
d being the lower bound of the functional I (ϕ). However, the problem in
application is just the practical construction of such a minimizing sequence.
Starting with coordinate functionω 1 , ω 2 , .. ,ω n ,.. , which should satisfy
conditions that any linear combination ϕ n=c 1 ω1 +c 2 ω 2+ …+c n ωn
is admissible in the variational problem and should form a complete system of
functions. Considering any function ϕ n and substitute it in our variational
problem, thus
I ( ϕn )=¿ F ( c1 , c 2 , … , c n )=minimum
3. METHOD OF FINITE DIFFERENCES. GENERAL RANDOM
STATISTICAL METHODS
4. METHOD OF GRADIENTS
5. NUMERICAL TREATMENT OF THE PLANE TORSION
PROBLEM FOR MULTIPLY-CONNECTED DOMAINS

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