WAVES AND VIBRATIONS IN INHOMOGENEOUS STRUCTURES ...

1614 J. S. JENSEN
h
2h
Α
fcos Ωt
Figure 4. An unsupported 2-D elastic body (plane stress) subjected to a harmonic load.
error is measured as
1 N
error =
N
i=1
2 bi − (bi)ref
(bi)ref
in which the reference values have been computed with 100 significant digits in the computations.
Figure 3 reveals a close to direct relation between the choice of numerical precision and log(error)
of the bi coefficients. For this specific system and choice of parameters the high-order expansion
(N = 7) leads to a significant error also for computations with standard double-precision variables
(16 significant digits). If quad precision variables (32 significant digits) are used, the error is seen
to be reduced to a sufficiently small level.
2.4. Numerical example: 2-D forced vibration
The second example demonstrates the numerical scheme for a structure with many degrees of
freedom. An unsupported 2-D elastic body (Figure 4) is subjected to a harmonic load in the
middle point of the lower boundary and the response is computed in the middle point A of the
upper boundary. The following material parameters are used for the elastic body:
E = 1, = 1, = 0.3
with a plane stress model and unit thickness. A Rayleigh damping model is assumed so that the
damping matrix C is a linear combination of the mass- and the stiffness-matrices:
(34)
C = M + K (35)
with damping coefficients = 0.5 and = 0.005.
The body is discretized with 40 × 20 bi-linear quadratic elements and the size of the body
is normalized so that the total mass is unity. All computations are carried out with standard
double-precision numerics.
The response ln(u Aū A) is plotted in Figure 5 versus the excitation frequency . The expansion
frequency is 0 = 1. As reference, the direct solution is computed for a high number of frequencies
and shown with a thick solid line. As appears from the figure an accurate approximation of the
response is obtained in the frequency range from ≈ 0.7 to1.3 if a sufficient number of expansion
terms are included (N = 9). However, if N is chosen larger than 9, the accuracy can be shown
Copyright q 2007 John Wiley & Sons, Ltd. Int. J. Numer. Meth. Engng 2007; 72:1605–1630
DOI: 10.1002/nme