**6.3. Single mode non-linear identification**

For the single mode non-linear identification the polynomial functions of Eqs. (40) and (41) with the modal displacements and velocities of the aileron mode 60 *r* are employed. The powers 60 *q* and 60 *q* are increased from max *i* 1 to max *i* 3 . The selection of terms is performed in the way that several analysis runs with different terms on a trial and error basis are performed. The goal is to minimize the deviations between the measured and recalculated restoring forces and the measured and recalculated modal signals. Terms are included if they appear necessary to model the non-linear behaviour. They are excluded if they do not reduce the deviations. It turns out that the curve fit on the basis of Eqs. (40) and (41) is not completely successful. Thus, additional anti-symmetric terms on a trial and error basis are introduced. A stiffness term with the second power of 60 *q* , namely 60 60 *q q* reduces clearly the deviations and is therefore additionally included.

Table 1 shows the identified parameters which contribute clearly. The low value of the linear damping parameter *<sup>i</sup>* constitutes only small changes with respect to the linear term, which is identified with the Phase Resonance Method and is a priori included in Eq. (34). No non-linear damping terms are detected.

Inserting the identified parameters in Eq. (37), the restoring force is calculated and displayed in Figure 10 together with the measured restoring force. It shows that the measured und recalculated restoring force match really well. Nevertheless, some deviations occur at the minima and maxima of the functions. The RMS-value of the deviation amounts to *0.90 %*.


**Table 1.** Parameters for single mode identification

identification for the two modes 60 *r* and 71 *r* .

**Figure 9.** Measured restoring forces of the aileron mode

**6.3. Single mode non-linear identification** 

powers 60 *q* and 60 *q* are increased from max

linear damping parameter *<sup>i</sup>*

non-linear damping terms are detected.

reduces clearly the deviations and is therefore additionally included.

Under consideration of the observed mode coupling it makes sense to perform two types of non-linear identification: single mode identification for mode 60 *r* and coupled mode

For the single mode non-linear identification the polynomial functions of Eqs. (40) and (41) with the modal displacements and velocities of the aileron mode 60 *r* are employed. The

performed in the way that several analysis runs with different terms on a trial and error basis are performed. The goal is to minimize the deviations between the measured and recalculated restoring forces and the measured and recalculated modal signals. Terms are included if they appear necessary to model the non-linear behaviour. They are excluded if they do not reduce the deviations. It turns out that the curve fit on the basis of Eqs. (40) and (41) is not completely successful. Thus, additional anti-symmetric terms on a trial and error basis are introduced. A stiffness term with the second power of 60 *q* , namely 60 60 *q q*

Table 1 shows the identified parameters which contribute clearly. The low value of the

which is identified with the Phase Resonance Method and is a priori included in Eq. (34). No

Inserting the identified parameters in Eq. (37), the restoring force is calculated and displayed in Figure 10 together with the measured restoring force. It shows that the measured und recalculated restoring force match really well. Nevertheless, some deviations occur at the minima and maxima of the functions. The RMS-value of the deviation amounts to *0.90 %*.

constitutes only small changes with respect to the linear term,

*i* 1 to max

*i* 3 . The selection of terms is

**Figure 10.** Measured and recalculated restoring forces (without coupling)

### **6.4. Coupled mode non-linear identification**

For the coupled mode identification the polynomial functions of Eqs. (42) and (43) with the modal displacements and velocities of the aileron mode 60 *r* and the winglet mode 71 *r* are employed. The powers of 60 *q* , 71 *q* and 60 *q* , 71 *q* are increased from max max *i j* 1 to max max *i j* 3 . The selection of terms is performed in the same way as above. Several analysis runs with different terms on a trial and error basis are performed. Terms are only included if they contribute clearly to reduce the deviations between measured and recalculated modal signals.

Table 2 shows the identified parameters which contribute clearly. For mode 60 *r* itself three stiffness and one damping parameter are identified again. In addition, two coupled stiffness and five coupled damping terms are identified. A significant difference to single mode identification for the four identified parameters of mode 60 *r* is observed. The reason is that the analytical model has changed and that the coupled mode identification requires additional terms until the measured restoring forces are fitted with good accuracy. The different terms in the stiffness series compensate partly for each other. Thus, the physical meaning of the polynomial coefficients is limited. The polynomial coefficients may be considered rather as 'numbers' which enable a good fit to the measured data. The main criterion are the restoring functions.


Non-Linearity in Structural Dynamics and Experimental Modal Analysis 197

depends clearly from both modal coordinates 60 *q*

**Figure 12.** Restoring surface of single mode identification (black) and coupled mode identification (red)

the coupled mode identification is computed at the grid points from Eq. (42) with the parameters of Table 2. This surface is depicted as red mesh. Both surfaces exhibit a clear

and 71 *q* . Single mode identification with the black mesh restoring surface is not able to

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 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

difference. Thus, the restoring force 60

**7. Conclusion** 

describe the complete non-linear behaviour.

the application of the approach in practice.

196 Nonlinearity, Bifurcation and Chaos – Theory and Applications

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

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

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 terms, the restoring stiffness forces 60 are visualized as surfaces in Figure 12.

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

**Figure 12.** Restoring surface of single mode identification (black) and coupled mode identification (red)

the coupled mode identification is computed at the grid points from Eq. (42) with the parameters of Table 2. This surface is depicted as red mesh. Both surfaces exhibit a clear difference. Thus, the restoring force 60 depends clearly from both modal coordinates 60 *q* and 71 *q* . Single mode identification with the black mesh restoring surface is not able to describe the complete non-linear behaviour.
