WAVES AND VIBRATIONS IN INHOMOGENEOUS STRUCTURES ...

1612 J. S. JENSEN
Table I. Coefficients ai and bi computed for m = k = f = 1,
c = 0.01 and for the centrefrequency 0 = 1.
i bi ai
0 — −0.997905−0.0598893i
1 5.98175+0.628924i 6.00628−0.209591i
2 −45.0048+0.449545i 38.9141+2.42528i
3 −119.742−9.18382i −84.0400+2.49613i
4 280.119−9.43850i −250.510−13.2525i
5 712.978+32.9055i 230.125−11.5939i
6 −385.688+46.9032i 616.065+20.4211i
7 −1670.33−30.2742i −56.4125+23.7608i
8 −493.363−79.0623i −636.239−5.67282i
9 1428.97−21.4553i −297.395−16.6529i
10 1285.16+38.5100i 154.880−6.12446i
11 −0.339896+32.7067i 195.702+1.25137i
12 −545.314+6.37730i 76.8638+1.39493i
13 −349.447−3.13286i 13.9728+0.348890i
14 −104.813−1.94300i 0.997905+0.0299102i
15 −15.9689−0.398731i —
16 −0.997905−0.0299102i —
and by inserting the found expression for H88 this leads to the exact form of u8
u8 = a0 + a1 + a2 2 +···+a14 14
1 + b1 + b2 2 +···+b16 16
in which the solution a0 at the expansion frequency and the 30 additional expansion coefficients
a1 − a14 and b1 − b16 are complicated functions of the system parameters and 0. The expressions
are formidable even for this simple system so only numerical approximate values are given in
Table I for a specific set of parameters (m = k = f = 0 = 1andc = 0.01).
The vibration amplitude of the rightmost mass is illustrated in Figure 2. The response ln(u8ū8),
in which the bar denotes the complex conjugate, is plotted versus the excitation frequency . The
response shows the eight peaks that correspond to the natural frequencies of the system as well
as seven anti-resonances. For >2 the response drops off rapidly.
In Figure 2 the exact response is also compared to results obtained with the proposed numerical
scheme. Even with a small number of expansion terms (N = 3) the accuracy of the approximation is
remarkably good in a large frequency interval and visually indistinguishable from the exact solution
in the range from = 0.75–1.25. With two additional terms in the numerator and denominator
the high-accuracy frequency range is increased and three anti-resonances and 3–4 resonances are
well captured. However, with N = 7 the accuracy significantly deteriorates. The reason is that the
P matrix in Equation (28) becomes severely ill-conditioned.
The problem with ill-conditioning can be overcome if higher precision numerical procedures
are available. In Figure 2 the response with N = 7 is shown when computed with quad precision
numerics (32 significant digits). The accuracy of the approximation is significantly improved,
whereas for N = 3 and 5 the responses are practically unchanged (not shown in the figure). The
Copyright q 2007 John Wiley & Sons, Ltd. Int. J. Numer. Meth. Engng 2007; 72:1605–1630
DOI: 10.1002/nme
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