12 Identification of Sub- and Higher Harmonic Vibrations in Vibro-Impact Systems 133 fundamental response is considered such that the method represents a rough approximation. For a consideration of a higher and subharmonic response like it is done in this contribution the underlying ansatz must be extended to x.t/ Da0 C 1 X D1 1 X D1 a = cos !t Cb = sin !t : (12.4) This ansatz now takes higher harmonics into account for > 1 and subharmonics are represented by > 1. The constant part a0 considers the mean position of the vibration which is necessary for the investigated systems here. Note, that with this ansatz only harmonics in a rational condition are respected. Linear combinations of the harmonics to represent combination resonances [5] are not considered here since the excitation for the presented system is excited with only one frequency. In practice, the choice of the dominant harmonics is an important issue for the efficient calculation of FRFs finding a good trade-off between accuracy and computational cost. In some cases applying a Fast Fourier or Wavelet Transformation to the time signal helps to find dominant harmonics. By having the ansatz for the response from Eq. (12.4) the nonlinear forces Fnl.Px;x; t/ in Eq. (12.1) can be developed in a Fourier series Fnl.Px;x; t/ DA0 C 1 X D1 1 X D1 A = cos !t CB = sin !t ; (12.5) where the Fourier coefficients are determined by the following integrals A0 D 1 T Z T 0 Fnl.Px;x; t/ dt; A = D 2 T Z T 0 Fnl.Px;x; t/cos !t dt; B = D 2 T Z T 0 Fnl.Px;x; t/sin !t dt: (12.6) For the calculation of these coefficients, the regarded period length T D2 =! has to have at least the length of one full period, also for the subharmonic terms which have -times the period length of the excitation. Using an infinite number of these coefficients, the nonlinear force is approximated by Fnl.Px;x; t/ A0 C h X D1 h X D1 A = cos !t CB = sin !t : (12.7) Introducing complex values the nonlinear force can be written as complex valued amplitudes denoted by the hat symbol, Fnl.Px;x; t/ Re OFnl;0 C h X D1 h X D1 OFnl; = e i h h !t DRe A0 C h X D1 h X D1 .A h= h i B h= h/ e i h h !t : (12.8) Considering steady state behavior, the time dependency can be canceled and the system of equations can be arranged in matrix form in the frequency domain as
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