**7. Conclusion**

196 Nonlinearity, Bifurcation and Chaos – Theory and Applications

3

2

2

2

2 3

3 3

**Table 2.** Parameters for coupled mode identification

**Figure 11.** Measured and recalculated restoring forces (with coupling)

terms, the restoring stiffness forces 60

Term of non-linearity Coupled mode identification

1 

2 

3 

1 

4 

5 

2 

3 

4 

5 

6 

Figure 11 shows the measured and recalculated restoring forces. A nearly perfect agreement can be seen, even at the minima and maxima of the functions. The quantitative assessment via RMS-value delivers a deviation of *0.15 %*. In order to show the influence of the coupling

The restoring surfaces are computed at a grid of data points which is spanned by the minimum and maximum values of 60 *q* and 71 *q* . The restoring force surface of the single mode nonlinear identification is computed at the grid points from Eq. (41) with the parameters of Table 1. This surface is depicted as black mesh. The restoring force surface of

are visualized as surfaces in Figure 12.

2.264 10

6.425 10

4.001 10

1.190 10

1.397 10

2.041 10

1.806 10

1.458 10

1.247 10

2.323 10

5.266 10

<sup>60</sup> *<sup>q</sup>* <sup>6</sup>

60 60 *q q* <sup>9</sup>

<sup>60</sup> *<sup>q</sup>* <sup>13</sup>

<sup>60</sup> *<sup>q</sup>* <sup>2</sup>

60 71 *q q* <sup>9</sup>

60 71 *q q* <sup>13</sup>

60 71 *q q* <sup>4</sup>

60 71 *q q* <sup>7</sup>

60 71 *q q* <sup>6</sup>

60 71 *q q* <sup>11</sup>

60 71 *q q* <sup>11</sup>

This book chapter derives first the basic dynamic equations of structures with nonlinearities and considers the experimental modal identification. Then the theoretical basis for non-linear identification is explained and a test strategy for non-linear modal identification, which can be used within a test concept for modal testing, is described. The basic idea is to use modal force appropriation, to employ equations in modal space and to identify the modal non-linear restoring forces. This is realized by computing the coefficients of applicable functions for the restoring forces from time domain data. The required steps for single mode and coupled mode non-linear identification are developed and discussed in detail. The identification is then illustrated by an analytical example where it could be shown that the method is able to identify the non-linear coupled modes of vibration. A second example taken from a modal identification test on a large transport aircraft shows the application of the approach in practice.

The non-linear identification may be further developed by using other functions for the restoring forces or to extend it to a higher number of modal DoF. Also, it can be elaborated whether and how it would be possible to derive the required linear modal parameters from applying Phase Separation Techniques. Thus, the experimental effort of applying the Phase Resonance Method could be avoided leading to a reduced test duration.

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