An experimental study has been carried out in a controlled environment in a large anechoic chamber at Boeing to clarify certain features associated with the nonlinear distortion of acoustic waves. The test points were chosen to yield both subsonic and supersonic convective Mach numbers. Both spectral and time-domain analyses have been performed to elucidate nonlinear effects. The spectral analysis indicates that there is agglomeration of energy at the higher frequencies as the propagation distance increases. The time-domain analysis shows strong positive peaks in the pressure signals; the skewness values of the acoustic signals jump from ~0.05 to ~0.3, when the convective Mach number is increased from low subsonic values to just above unity. An examination of the Morfey-Howell nonlinear indicator reveals that energy is transferred from the spectral peak to the higher frequencies as a consequence of long-distance propagation. It is established that the convective Mach number is a critical parameter that may be used to identify the onset of nonlinear effects. When the value exceeds unity, there is a dramatic difference in the wave characteristics. In addition, a set of high-quality data has been generated that can be used to guide the development and the validation of prediction methods for nonlinear propagation.
adaptive harmonic balance method for predicting the nonlinear dynamic responses of mechanical systems. Application to bolted structures. Journal of Sound and Vibration, Elsevier, 2010, 329, pp.4048-4067. <10.1016/j.jsv.2010
AbstractAeronautical structures are commonly assembled with bolted joints in which friction phenomena, in combination with slapping in the joint, provide damping on the dynamic behaviour. Some models, mostly non linear, have consequently been developed and the harmonic balance method (HBM) is adapted to compute non linear response functions in the frequency domain. The basic idea is to develop the response as Fourier series and to solve equations linking Fourier coefficients. One specific HBM feature is that response accuracy improves as the number of harmonics increases, at the expense of larger computational time. Thus this paper presents an original adaptive HBM which adjusts the number of retained harmonics for a given precision and for each frequency value. The new proposed algorithm is based on the observation of the relative variation of an approximate strain energy for two consecutive numbers of harmonics. The developed criterion takes the advantage of being calculated from Fourier coefficients avoiding time integration and is also expressed in a condensation case. However, the convergence of the strain energy has to be smooth on tested harmonics and this constitutes a limitation of the method. Condensation and continuation methods are used to accelerate calculation. An application case is selected to illustrate the efficiency of the method and is composed of an asymmetrical two cantilever beam system linked by a bolted joint represented by a nonlinear LuGre model. The practice of adaptive HBM shows that, for a given value of the criterion, the number of harmonics increases on resonances indicating that non linear effects are predominant. For each frequency value, convergence of approximate strain energy is observed. Emergence of third and fifth harmonics is noticed near resonances both on vibratory responses and on approximate strain energy. Parametric studies are carried out by varying the excitation force amplitude and the threshold value of the adaptive algorithm. Maximal amplitudes of vibration and frequency response functions are plotted for three different points of the structure. Non linear effects become more predominant for higher force amplitudes and consequently the number of retained harmonics is increased.
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