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Unformatted text preview: = AS (V U ) + AM (V U )
j U) = AM (V
j Z U) = (V f )j = xj pV U dx (1.3.9c) qV Udx (1.3.9d) V fdx: (1.3.9e) 0 xj;1 Zx j xj;1 Zx j xj;1 (1.3.9b) 0 It is customary to divide the strain energy into two parts with AS arising from internal
energies and AM arising from inertial e ects or sources of energy.
Matrices are simple data structures to manipulate on a computer, so let us write the
restriction of U (x) to xj 1 xj ] according to (1.3.7) as
; U (x) = cj 1 cj ]
; j (x) =
j (x) ; 1 j ; 1 (x) j (x)] cj 1
; x 2 xj 1 xj ]: (1.3.10a)
; We can, likewise, use (1.2.3b) to write the restriction of the test function V (x) to xj 1 xj ]
in the same form
V (x) = dj 1 dj ] j (xx) = j 1(x) j (x)] dd 1
x 2 xj 1 xj ]: (1.3.10b)
; ; ; ; ; ; 12 Introduction Our task is to substitute (1.3.10) into (1.3.9c-e) and evaluate the integrals. Let us begin
by di erentiating (1.3.10a) while using (1.3.4a) to obtain U (x) = cj 1 cj ]
0 ; ;1=hj = ;1=hj 1=hj ] cjc 1
j x 2 xj 1 xj ]: ; 1=hj...
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- Spring '14