12 Identification of Sub- and Higher Harmonic Vibrations in Vibro-Impact Systems 139 0 0.2 0.4 0.6 0.8 1 0 2 4 6 8 10 12 14 16 18 20 Frequency [Hz] Amplitude [m] harm1 harm2 harm3 harm4 0 0.2 0.4 0.6 0.8 1 −7 −6 −5 −4 −3 −2 −1 0 Frequency [Hz] Amplitude [m] Fig. 12.9 Harmonics (left) and mean position (right) for the SDOF-Oscillator with one-sided impact and negative cubic stiffness 0 0.2 0.4 0.6 0.8 1 0 2 4 6 8 10 12 14 Frequency [Hz] Amplitude [m] Fig. 12.10 Subharmonic response for the SDOF-Oscillator with one-sided impact and negative cubic stiffness With these methods it was possible to calculate FRFs with remarkable influence of sub- and higher harmonics as well as sharp bends in solution curves leading to multiple solutions in certain frequency ranges. The comparison of the FRFs with results obtained by time integration shows that the amplitude and mean position are approximated accurately. Even when the vibrations are actually aperiodic, which was shown by the Lyapunov-Exponents, it is possible to find a decent approximation of the resonance frequency, the amplitude and mean position of the vibration. For some systems it is also interesting that the presented method can also find unstable solutions which are difficult to obtain e.g. with time integration. Nevertheless, in some applications it is of interest to determine whether the vibration is periodic or aperiodic, which obviously is not possible by harmonic approximation. Therefore, calculations in the time domain and Lyapunov-Exponents can be beneficial. Consequently, it is often necessary to use a combination of different methods to capture the overall dynamics of a nonlinear system correctly. For future research the concept for calculation of FRFs can be extended to systems with multiple degrees of freedom. In this context also the calculation of backbone curves to capture frequency-energy dependency of nonlinear systems is interesting. Additionally, the influence of the modeling of the impact has to be examined and validated experimentally.
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