WAVES AND VIBRATIONS IN INHOMOGENEOUS STRUCTURES ...

1608 J. S. JENSEN
Cramer’s rule [17] is applied to Equation (5):
u =
2Nd−2 i=0 ãii 2Nd ˜bi i
i=0
where Nd is the total number of degrees of freedom in the discretized model. It should be noted
that Equation (9) is an exact representation of u. But for normal finite element models this
formulation requires the computation of a large number of expansion coefficients. This is not
only inconvenient but also practically impossible due to numerical errors (as will be discussed
in Section 2.3). However, an accurate approximation in the vicinity of the expansion frequency
( = 0) can be obtained with series containing fewer expansion terms in both the numerator and
the denominator:
Ni=0 ãi
u ≈
i
Ni=0 ˜bi i = ã0 + ã1 +···+ãN N ˜b0 + ˜b1 +···+ ˜bN N (10)
where N terms have been used in the expansion. It should be noted that the choice of retaining the
same number of terms in the numerator and denominator is not the only possible one. For further
details the reader is referred to a separate publication on PAs, e.g. [2].
Finite response for = 0 ensures that ˜b0 = 0 (can be ensured by non-zero damping in the system).
Hence, all coefficients can be divided by ˜b0 to obtain a new set of coefficients b1,..., bN and
a1,...,aN
u = a0 + a1 +···+aN N
1 + b1 +···+bN N
in which the approximation sign has now been replaced by an equality sign for simplicity. Thus, u
now represents the expansion in Equation (11) and not the exact solution of the original equation.
The coefficient a0 is easily determined by considering the system at the expansion frequency
( = 0). Equation (11) reduces for = 0to
u( = 0) = a0
so that this coefficient is recognized as the solution at the expansion frequency 0 (for clarity
referred to as u0). Thus, the final expression for the expansion is
2.1. Gradients at = 0
u = u0 + N i=1 ai i
1 + N i=1 bi i
The gradients at = 0 are used to determine the unknown coefficients. To do this u is written in
form of its truncated Taylor series (with Nt terms)
(9)
(11)
(12)
(13)
u = u0 + u ′ 1
0 +
2! u′′ 02 +···+ 1
Nt! u(′ Nt)
0 Nt (14)
Copyright q 2007 John Wiley & Sons, Ltd. Int. J. Numer. Meth. Engng 2007; 72:1605–1630
DOI: 10.1002/nme