10 Nonlinear Modal Interaction Analysis for a Three Degree-of-Freedom System with Cubic Nonlinearities 129 0.985 0.99 0.995 1 1.005 1.01 1.015 0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5 S1 S3 S2 S4± S5± BP 1 BP2,3 U1 Ω 0.96 0.97 0.98 0.99 1 1.01 0 0.1 0.2 0.3 0.4 0.5 S1 S2 S3 S4+ S4− S5± BP1 BP2 BP3 X1 Ω 0.975 0.98 0.985 0.99 0.995 1 1.005 1.01 1.015 0 0.1 0.2 0.3 0.4 0.5 S1 S2 S3 S4± S5± BP1 BP3 BP2 U2 Ω 0.98 0.985 0.99 0.995 1 1.005 1.01 1.015 0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5 S1 S2 S3 S4± S5± BP1 BP2 BP3 X2 Ω 0.985 0.99 0.995 1 1.005 1.01 1.015 0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5 S1 S2 S3 S4± S5± BP1 BP3 BP2 U3 Ω 0.96 0.97 0.98 0.99 1 1.01 0 0.1 0.2 0.3 0.4 0.5 S1 S2 S3 S4− S4+ S5± BP 1 BP2 BP3 X3 Ω Fig. 10.3 Backbone curves for the oscillator with the physical parameters k1 Dk2 Dk3 D1, k3 Dk4 D0:01 and 1 D 2 D 0:05, so the modal natural frequencies are !n1 D1, !n2 D1:005 and !n3 D1:015. The panels in the first and second column show the modal and physical results respectively. Stable solutions are shown with solid lines, whereas unstable solutions are represented by dashed line. Bifurcation points are noted by BP. Note that as S5˙would overlap S1, S5˙backbone curves are indicated byshort cross lines for distinction To further demonstrate the ability of the backbone curve in determining the response of the system to an external forcing, we show a brief example of the relationship between the forced response and the backbone curves. The same fundamental system as above is used and it is forced in the second mode, ŒPm1;Pm2;Pm3 DŒ0;15;0 10 4 (corresponding physical mass force is ŒP1;P2;P3 DŒ15;0; 15 10 4) with a damping ratio of D0:001. The forced response has been computed from an initial steady state solution, found with numerical integration in MATLAB, which is then continued in forcing frequency using the software AUTO-07p [4].
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