Finite Element Analysis. Barna Szabó

Finite Element Analysis

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

where

(1.143) a Subscript i Baseline equals integral Subscript 0 Superscript script l Baseline mu f upper U Subscript i Baseline d x period

The Rayleigh¹⁵ quotient is defined by

(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 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 .

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 .

Example 1.15 Let us consider the eigenvalue problem

(1.148) kappa StartFraction partial-differential squared u Over partial-differential x squared EndFraction equals mu StartFraction partial-differential squared u Over partial-differential t squared EndFraction comma u left-parenthesis 0 right-parenthesis equals u left-parenthesis script l right-parenthesis equals 0 comma t greater-than-or-equal-to 0 period

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 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

satisfies eq. (1.148).

If we approximate the eigenfunctions using uniform mesh, p equals 2 and plot the ratio left-parenthesis omega Subscript upper F upper E Baseline slash omega Subscript upper E upper X Baseline right-parenthesis Subscript n against n slash upper N , where n is the nth eigenvalue, then we get the curves shown in Fig. 1.13. The curves show that somewhat more than 20% of the numerically computed eigenvalues will be accurate. The higher eigenvalues cannot be well approximated in the space upper S Superscript 0 Baseline left-parenthesis upper I right-parenthesis . The existence of the jump seen at Скачать книгу