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The additive loading includes among others the product of the harmonic components cos(wt) and cose( t),see Eq.(B.1).It turns out that the response period is determined from the interference of the response caused by the circular frequencies w0+ and w0- .The corresponding periods become T=2pi/ w0+ ,T=2pi/ w0- . In order to find the combined period of the response T the following ratios are evaluated T/w=n(1+2/w). The factor n is found as the minimum value at which both T tand T1 attain integer values.Poincare¡ämaps of 2000 excitation periods are plotted in Fig.3 for various ratios of w0/ .=0.8.The amplitude is u0=0:3 m, the damping ratios c1=c2=1 and the frequency ratio w0/w1=2 Tindicate the phase value ,every excitation period T0,andeTindicates the phase value at every response period T.As seen the period tends towards infinity as w0/ .becomes irrational.As an example w0/=3:14159 results in n=314 159. For an irrational frequency ratio a so-called almost periodic response is achieved in which case a continuous closed curve is obtained in the phase plane for the Poincare¡ämap.As seen from Fig.3 the amplitude of q1 increases as w0/ . is increased.The reason is that the fundamental blade circular eigenfrequency for the considered example is given as w0/ .=3:2125.Hence,the simulations tend towards resonance in the fundamental eigenmode as w0/ . is increased. |
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