**6. Experimental example**

190 Nonlinearity, Bifurcation and Chaos – Theory and Applications

**Figure 4.** Mode participation

max max

mean square (RMS) values. From the figures it can be seen that the above sine sweep excites

With the purpose to investigate the influence of measurement noise and errors in the data, a random signal with an RMS-value of *5 %* is added to the clean signals of excitation forces and responses prior to the non-linear identification. For the modal parameters , *m c r r* and

used. A careful modal analysis at an appropriate excitation level should be able to deliver

In the following the simulated *20 s* time histories of 1 2 *qt qt* ( ), ( ) and 3 *q t*( ) are used for the non-linear modal identification. First, single mode identification on a trial basis is

employed. The result is always the same: the deviations between the 'measured' and recalculated signals remain high. Also, it shows that there are effects which cannot be accounted for with single mode non-linear identification. Figure 5 shows as an example the

Since the single mode non-linear identification is not sufficient, as next step coupled mode identification is performed. For the coupled mode identification the polynomial function of

The deviations between the 'measured' and recalculated signals disappear completely for

*i j* 3 and if the clean signals (without the additional random noise) are utilized.

apparently unnecessary coefficients are computed to *0*. The analysis of noisy signals leads to deviations. However, the deviations are not much higher than the noise itself. E.g. in the

*i* increasing from *1* to *5* is

*<sup>i</sup>* causes no problems. The

*<sup>i</sup>* causes no problems. The

*i j* 1 to max max

*t* according

the correct data are

*i* 5 . By the way,

*i j* 3 .

*<sup>r</sup> k* which are required for the computation of the non-linear restoring forces ( ) *<sup>r</sup>*

to Eq. (34), the correct values are used. Also, for the eigenvectors *<sup>r</sup>*

measured and recalculated restoring force of the modal DoF 1 *r* for max

Eq. (42) is used. The number of terms is increased from max max

performed. The polynomial function of Eq. (40) with max

it is interesting that the usage of too much coefficients

Again, it shows that the usage of too many coefficients

apparently unnecessary coefficients are computed to *0*.

such accurate data of the underlying linear system.

clearly the modal DoF *1* and *3*, whereas DoF *2* responds only very weakly.

In this section an example of the application of the method in practice is shown. The method is exemplarily applied to an aileron mode of a large transport aircraft (Goege, Fuellekrug, Sinapius, Link, & Gaul, 2005), (Goege & Fuellekrug, 2004) (Goege, 2004).

### **6.1. Test structure and test performance**

A modal identification test is performed as a Ground Vibration Test on an aircraft using the modal identification concept described above. The test duration was about two weeks and the aircraft was tested in two configurations. A total number of 352 accelerometers was employed to measure the mode shapes of the structure with a sufficient spatial resolution.

Non-Linearity in Structural Dynamics and Experimental Modal Analysis 193

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

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

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

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

and integrated to obtain velocities and displacements as described above.

changing over the load level by approximately *27 %*.

other modes is comparatively small.

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

production tolerances of the aileron/wing attachment.

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

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 mode.
