138 S. Peter et al. 0 0.2 0.4 0.6 0.8 1 0 2 4 6 8 10 12 14 Frequency [Hz] Amplitude [m] harm1 harm2 harm3 harm4 harm1/2 0 0.2 0.4 0.6 0.8 1 −1.4 −1.2 −1 −0.8 −0.6 −0.4 −0.2 0 Frequency [Hz] Amplitude [m] Fig. 12.7 Harmonics (left) and mean position (right) for the SDOF-Oscillator with one-sided impact and positive cubic stiffness 0 500 1000 1500 2000 −15 −10 −5 0 5 10 15 Time [s] Displacement [m] 0 500 1000 1500 2000 −15 −10 −5 0 5 10 15 Time [s] Displacement [m] Fig. 12.8 Time integration results for the SDOF-Oscillator with one-sided impact and positive cubic stiffness (left) and negative cubic stiffness (right) Table 12.3 Parameters for SDOF-Oscillator with one-sided impact and negative cubic stiffness Parameter Value Unit Parameter Value Unit m 1 kg z0 12 m d 0.2 Ns/m k0 10 N/m k 10 N/m d0 0.2 Ns/m Ofexc 10 N ˇ 0.02 N=m3 In this example there are also multiple solutions at a certain frequency range, as the FRF for an oscillator with negative cubic stiffness is bent to the left. This effect is even stronger when the cubic nonlinearity is combined with an impact which can be observed in Fig. 12.9 for the harmonics with D f1;2;3;4g (left). For the mean position of the vibration, shown in Fig. 12.9 (right), the same holds as for the previous example. However, an additional sharp bend occurs at the position where the oscillator detaches from the impact. This bend can be explained by looking at the subharmonic response for D1 and D2 which is displayed in Fig. 12.10. So, for this example the subharmonic response suddenly becomes dominant at the moment when the oscillator detaches from the impact, which causes additional solutions. In contrast, this cannot be observed in the time signals for the displacement with sweep excitation which is displayed in Fig. 12.8. 12.6 Conclusion and Future Work This paper presented a method for the calculation of FRFs under consideration of sub- and higher harmonics. For the solution a Continuation Method based on tangent prediction and Gauss-Newton/Levenberg-Marquardt correction is adopted.
RkJQdWJsaXNoZXIy MTMzNzEzMQ==