**6.2. Non-linear analysis**

At first, the modal characteristics of the aileron mode were identified with the Phase Resonance Method at a level of the modal force of *10 N*. The aileron mode was excited with one single exciter, which was located at the aileron. Figure 7 displays schematically the test setup for the excitation of the aileron.

**Figure 7.** Excitation of the aileron mode of a large transport aircraft

Next, the level of the modal force was increased in several steps up to *121 N*. At each force level the aileron mode was measured with the Phase Resonance Method. Significant nonlinear characteristics were observed: The resonance frequency of the aileron mode was changing over the load level by approximately *27 %*.

For the detailed non-linear analysis short parts of the time domain signals with harmonic steady-state excitation at the linear resonance frequency were measured. About 16 cycles of vibration were recorded. The modal accelerations were computed from the measured signals of the 352 accelerometers according to Eq. (35). The acceleration signals were filtered and integrated to obtain velocities and displacements as described above.

The analysis of the modal displacements was performed in the same way as for the simulated example. Figure 8 shows the RMS-values of the modal displacements for the lowest and highest excitation level. It shows that for the highest excitation level only the aileron mode 60 *r* itself responds. However, for the lowest excitation level, a significant response of the bending mode of one winglet (mode 71 *r* ) is also observed. This is not surprising because the motions of an aileron are in principle capable of exciting wing bending modes and thus motions of a winglet. The coupling of the aileron mode with all other modes is comparatively small.

**Figure 8.** Mode participation for two different force levels

192 Nonlinearity, Bifurcation and Chaos – Theory and Applications

**Figure 6.** Restoring force of modal DoF 1 for coupled mode identification

**Figure 7.** Excitation of the aileron mode of a large transport aircraft

mode.

**6.2. Non-linear analysis** 

setup for the excitation of the aileron.

The transport aircraft is dynamically characterized by a high modal density. During the GVT about 73 modes were identified. Most of the modes were linear. Only few modes exhibit non-linear behaviour. One mode with significant non-linear behaviour is the aileron

At first, the modal characteristics of the aileron mode were identified with the Phase Resonance Method at a level of the modal force of *10 N*. The aileron mode was excited with one single exciter, which was located at the aileron. Figure 7 displays schematically the test

> Figure 9 displays the restoring force of mode 60 *r* . The restoring force was calculated according to Eq. (34). In this equation the modal parameters, which were identified with the Phase Resonance Method on the highest level, are inserted together with the measured modal displacement 60 *q* and the modal force 60 *f* . The restoring function shows a hysteresis behaviour and may indicate a clearance non-linearity. This is imaginable because the structure vibrates at the lowest force level with only small amplitudes, which are close to the production tolerances of the aileron/wing attachment.

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

Non-Linearity in Structural Dynamics and Experimental Modal Analysis 195

Term of non-linearity Single mode identification

1 

2 

3 

7.0

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

3.204 10

4.662 10

3.814 10

<sup>60</sup> *<sup>q</sup>* <sup>5</sup>

60 60 *q q* <sup>10</sup>

<sup>60</sup> *<sup>q</sup>* <sup>14</sup>

<sup>60</sup> *q* <sup>1</sup>

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

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

criterion are the restoring functions.

3

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

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