12 Identification of Sub- and Higher Harmonic Vibrations in Vibro-Impact Systems 137 0 0.2 0.4 0.6 0.8 1 0 0.005 0.01 0.015 0.02 0.025 Frequency [Hz] Amplitude [m] Fig. 12.4 Subharmonic response for the SDOF-Oscillator with one-sided impact 0 500 1000 1500 2000 −50 −40 −30 −20 −10 0 10 Time [s] Displacement [m] 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 −0.02 0 0.02 0.04 0.06 0.08 0.1 Frequency [Hz] Largest Lyapunov Exp. Fig. 12.5 Time integration results (left) and largest Lyapunov-Exponent (right) for the SDOF-Oscillator with one-sided impact −20 −10 0 10 −20 −10 0 10 20 x 1 x2 −7 −6 −5 −10 −8 −6 −4 x1 x2 −40 −20 0 20 −40 −20 0 20 40 x1 x2 −20 −15 −10 −5 15 20 25 30 35 x1 x2 Fig. 12.6 Phase portrait and Poincaré-Map for the SDOF-Oscillator with one-sided impact at 0.15 Hz (left) and 0.3 Hz (right) Table 12.2 Parameters for SDOF-Oscillator with one-sided impact and positive cubic stiffness Parameter Value Unit Parameter Value Unit m 1 kg z0 12 m d 0.2 Ns/m k0 100 N/m k 1 N/m d0 0.2 Ns/m Ofexc 10 N ˇ 0.05 N=m3 negligible. In comparison, the time integration in Fig. 12.8 (left) provides similar amplitudes but instead of the multiple solutions a jump in amplitude occurs. The time integration also shows that for this example the solution is periodic in the regarded frequency range. Hence, the calculation of the Lyapunov-Exponents is dispensable. This also applies to the subsequently considered example for combined nonlinearities, which is a SDOF-Oscillator with negative cubic stiffness and the parameters listed in Table 12.3.
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