Lecture19-Prof.Ju

Lecture19-Prof.Ju - CEE M237A / MAE M269A Lecture 19: The...

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CEE M237A / MAE M269A Lecture 19: The Ritz Method II Professor J. Woody Ju
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Example 3–Use of More Coordinate Functions in Example 1 y Consider a series of sinusoidal coordinate functions: 2 () ( ) 1 ,s i n The case for 2 is Example 1. Consider now a series of problems beginning with 3 and incrementing each time by one until 7. By this process, it is possible to observe the step- N n n nx wxt a t L N NN N = = = = = π by-step improvement in the accuracy of the Rayleigh-Ritz analysis. For each value of , an is generated and solved to give all the modal data. algebraic eigenvalue problem N
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Example 3–Use of More Coordinate Functions in Example 1 (Cont’d) y y The above eqn. is obtained first by means of Hamilton’s principle (or Lagrange’s eqn.) and then by invoking simple harmonic motion in time as in Example 1 3 As an example, the algebraic eigenvalue problem for 7 is N = 3 where 2 γ σ =
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Example 3–Use of More Coordinate Functions in Example 1 (Cont’d) y y 4 For this particular example, observe that for 7, the algebraic eigenvalue problem can be obtained by truncation N < Moreover, it is possible by mathematical induction to state the form of the algebraic eigenvalue problem for 7 if further studies are needed or if the higher modes are of interest. N >
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Example 3–Use of More Coordinate Functions in Example 1 (Cont’d) 5 2 N = 3 N =
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Example 3–Use of More Coordinate Functions in Example 1 (Cont’d) 6 4 N = 5 N =
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Example 3–Use of More Coordinate Functions in Example 1 (Cont’d) 7 6 N = 7 N =
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Lecture19-Prof.Ju - CEE M237A / MAE M269A Lecture 19: The...

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