Finite Element Analysis. Barna Szabó

Finite Element Analysis - Barna Szabó


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E Superscript 0 Baseline left-parenthesis upper I right-parenthesis"/> can be written as a linear combination of the eigenfunctions:

      (1.142)double-vertical-bar f minus sigma-summation Underscript i equals 1 Overscript infinity Endscripts a Subscript i Baseline upper U Subscript i Baseline left-parenthesis x right-parenthesis double-vertical-bar Subscript upper L squared left-parenthesis upper I right-parenthesis Baseline equals 0

      (1.144)upper R left-parenthesis u right-parenthesis equals StartFraction upper B left-parenthesis u comma u right-parenthesis Over upper D left-parenthesis u comma u right-parenthesis EndFraction dot

      Eigenvalues are usually numbered in ascending order. Following that convention,

      (1.145)omega 1 squared identical-to omega Subscript min Superscript 2 Baseline equals min Underscript u element-of upper E Superscript 0 Baseline left-parenthesis upper I right-parenthesis Endscripts upper R left-parenthesis u right-parenthesis equals upper R left-parenthesis upper U 1 right-parenthesis

      that is, the smallest eigenvalue is the minimum of the Rayleigh quotient and the corresponding eigenfunction is the minimizer of upper R left-parenthesis u right-parenthesis on upper E Superscript 0 Baseline left-parenthesis upper I right-parenthesis. This follows directly from eq. (1.140). The kth eigenvalue minimizes upper R left-parenthesis u right-parenthesis on the space upper E Subscript k Superscript 0 Baseline left-parenthesis upper I right-parenthesis

      (1.146)omega Subscript k Superscript 2 Baseline equals min Underscript u element-of upper E Subscript k Superscript 0 Baseline left-parenthesis upper I right-parenthesis Endscripts upper R left-parenthesis u right-parenthesis equals upper R left-parenthesis upper U Subscript k Baseline right-parenthesis

      where

      (1.147)upper E Subscript k Superscript 0 Baseline left-parenthesis upper I right-parenthesis equals left-brace u vertical-bar u element-of upper E Superscript 0 Baseline left-parenthesis upper I right-parenthesis comma upper B left-parenthesis u comma upper U Subscript i Baseline right-parenthesis equals 0 comma i equals 1 comma 2 comma ellipsis comma k minus 1 right-brace period

      When the eigenvalues are computed numerically then the minimum of the Rayleigh quotient is sought on the finite‐dimensional space upper S Superscript 0 Baseline left-parenthesis upper I right-parenthesis. We see from the definition upper R left-parenthesis u right-parenthesis that the error of approximation in the natural frequencies will depend on how well the eigenfunctions are approximated in energy norm, in the space upper S Superscript 0 Baseline left-parenthesis upper I right-parenthesis.

      The following example illustrates that in a sequence of numerically computed eigenvalues only the lower eigenvalues will be approximated well. It is possible, however, at least in principle, to obtain good approximation for any eigenvalue by suitably enlarging the space upper S Superscript 0 Baseline left-parenthesis upper I right-parenthesis.

      This equation models (among other things) the free vibration (natural frequencies and mode shapes) of a string of length script l stretched horizontally by the force kappa greater-than 0 (N) under the assumptions that the displacements are infinitesimal and confined to one plane, the plane of vibration, and the ends of the string are fixed. The mass per unit length is mu greater-than 0 (kg/m). We assume that κ and mu are constants. It is left to the reader to verify that the function u defined by

      (1.149)u equals sigma-summation Underscript i equals 1 Overscript infinity Endscripts left-parenthesis a Subscript i Baseline cosine left-parenthesis omega Subscript i Baseline t right-parenthesis plus b Subscript i Baseline sine left-parenthesis omega Subscript i Baseline t right-parenthesis right-parenthesis sine left-parenthesis lamda Subscript i Baseline x right-parenthesis

      where ai, bi are coefficients determined from the initial conditions and

      (1.150)lamda Subscript i Baseline equals i StartFraction pi Over script l EndFraction comma omega Subscript i Baseline equals lamda Subscript i Baseline StartRoot StartFraction kappa Over mu EndFraction EndRoot