10 0 10 2 31.0358 31.0358 31.0359 31.036 31.036 Energy Frequency (Hz) 0 0.01 0.02 0.03 −4 −2 0 2 4 x 10 −4 Time (s) Displacement (m) 0 0.01 0.02 0.03 −5 0 5 x 10 −4 Time (s) Displacement (m) 0 0.01 0.02 0.03 −0.01 −0.005 0 0.005 0.01 Time (s) Displacement (m) 1 3 5 −2 0 2 4 x 10 −4 Harmonic i A i 1 3 5 −2 0 2 4 6 x 10 −4 Harmonic i A i 1 3 5 −5 0 5 10 x 10 −3 Harmonic i A i (a) (b) (c) a b c Figure 12: 3:1 internal resonance between the first wing torsional mode and a higher tail mode of the Paris aircraft. Top plot: close-up in the FEP of the 3:1 tongue of Figure 10. Bottom plots: NNM motions at (a) beginning of the tongue (in the vicinity of the backbone of the the first wing torsional mode); (b) middle of the tongue; and (c) extremity of the tongue. From top to bottom: NNM shapes; time series of the vertical displacements at the rear tip of the right tank (——) and at the right side of the horizontal tail (−−−); Fourier coefficients of both displacements (in grey and black, respectively). 240
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