**5. Illustrative analytical example**

In this section the non-linear identification is applied to an analytical vibration system with 3 DoF. The purpose is to illustrate the principles of and to demonstrate the applicability.

The vibration system is shown in Figure 2. The non-linearity consists of a non-linear spring with a cubic characteristic ( <sup>3</sup> *nl nolin* <sup>2</sup> *Fk u* ). The non-linear spring is attached parallel to the medial spring 2 *k* . The eigenfrequencies of the associated linear undamped system are located at *2.845 Hz*, *3.774 Hz* and *8.954 Hz*. The modal matrix of the associated linear undamped vibration system is

$$
\begin{bmatrix} \phi \end{bmatrix} = \begin{bmatrix} \begin{Bmatrix} \phi \end{Bmatrix}\_1 & \begin{Bmatrix} \phi \end{Bmatrix}\_2 & \begin{Bmatrix} \phi \end{Bmatrix}\_3 \end{bmatrix} = \begin{bmatrix} 1 & -1 & -0.216 \\ 0.432 & 0 & 1 \\ 1 & 1 & -0.216 \end{bmatrix} \tag{45}
$$

where the columns of the modal matrix are the eigenvectors , 1,2,3 *<sup>r</sup> r* with components at the 3 masses <sup>123</sup> *mmm* , , . Due to the position of the non-linear spring, only modes 1 and 3 behave non-linear while mode 2 is completely linear. The reason is that mode 2 has no deformation at the attachment point of the non-linear spring *nolin k* .

Non-Linearity in Structural Dynamics and Experimental Modal Analysis 189

**Figure 2.** Vibration system with 3 Dof

188 Nonlinearity, Bifurcation and Chaos – Theory and Applications

coordinates ( ), ( ), ( ), ( ) *rrss qtqtqtqt* and vector

Detect the modes that behave non-linear.

**5. Illustrative analytical example** 

with a cubic characteristic ( <sup>3</sup>

undamped vibration system is

can be summarized as follows:

modes are coupled.

modal signals.

, *ij ij* 

where contains the values of the non-linear restoring forces

**4.5. Summarization of steps for non-linear modal identification** 

vibrations alone, or signals of forced and free decay vibrations.

 Perform single mode non-linear identification for the uncoupled modes. Perform coupled non-linear mode identification for the coupled modes.

<sup>123</sup>

2 has no deformation at the attachment point of the non-linear spring *nolin k* .

appropriate parameter estimation methods can be applied.

Resonance Method or Phase Separation Techniques.

*Q*

(computed according to Eq. (34)), *Q* is comprised by time domain data of the modal

The steps for performing a non-linear modal identification according to the above theory

Identify the linear modal characteristics of the tested structure with the Phase

 Excite the non-linear modes with appropriated exciter forces at different force levels and use harmonic or sine sweep excitation. Measure time domain signals of forced

Compute the participation of the modal coordinates according to Eq. (35) and check if

Check the quality of the identification by comparing the measured and recalculated

In this section the non-linear identification is applied to an analytical vibration system with 3 DoF. The purpose is to illustrate the principles of and to demonstrate the applicability.

The vibration system is shown in Figure 2. The non-linearity consists of a non-linear spring

medial spring 2 *k* . The eigenfrequencies of the associated linear undamped system are located at *2.845 Hz*, *3.774 Hz* and *8.954 Hz*. The modal matrix of the associated linear

components at the 3 masses <sup>123</sup> *mmm* , , . Due to the position of the non-linear spring, only modes 1 and 3 behave non-linear while mode 2 is completely linear. The reason is that mode

where the columns of the modal matrix are the eigenvectors , 1,2,3 *<sup>r</sup>*

*nl nolin* <sup>2</sup> *Fk u* ). The non-linear spring is attached parallel to the

1 1 0.216 0.432 0 1 , 1 1 0.216

(45)

*r* with

. The solution of Eq. (44) can be obtained by using least squares. However, also other

, (44)

is assembled by the unknown coefficients

( )*t* at discrete time steps

For the simulation of 'measured' data the vibration system is excited with two single forces at mass 2 and mass 3. As excitation signal a sine sweep is used, which runs in *10 s* linearly from *2 Hz* to *12 Hz*.

For the non-linear analysis the *10 s* of the sine sweep excitation and *10 s* of the following free decay vibrations are used. The time domain integration of the 'measured' acceleration signals is realized by applying a digital band-pass filter to the accelerations and by integrating them once. The resulting velocities are also digitally band-pass filtered and then integrated to obtain displacements. Thus, no drift occurs during time domain integration. The force signals are also digitally high-pass filtered twice with the purpose to retain the correct phase relationship between the input and the output of the system.

Figure 3 shows the structural displacement responses following the above sine sweep excitation. *20 s* of the time histories of the modal coordinates 1 2 *qt qt* ( ), ( ) and 3 *q t*( ) are displayed. Figure 4 shows the mode participation of the modal coordinates as scaled root

**Figure 3.** Time histories of the modal displacements

mean square (RMS) values. From the figures it can be seen that the above sine sweep excites clearly the modal DoF *1* and *3*, whereas DoF *2* responds only very weakly.

Non-Linearity in Structural Dynamics and Experimental Modal Analysis 191

(, ) *q q* identified from signals with *5 %*

*ij* and

**Figure 5.** Non-linear restoring force of modal DoF 1 for single mode identification

Figure 6 shows as an example the restoring force 11 3

identification.

restoring force 11 3

**6. Experimental example** 

describe the complete non-linear behaviour.

**6.1. Test structure and test performance** 

case of *5 %* noise the deviations between the 'measured' and recalculated modal coordinates <sup>1</sup> *q t*( ) and 3 *q t*( ) amount to *7.4 %* and *7.7 %* respectively. It is apparent that no smaller deviation than *5 %* will be possible. Thus, the deviations are acceptable and indicate a good

noise for mode 1 *r* . In the figure the 'measured' and recalculated restoring forces at all time steps are plotted as points and crosses. Also, the interpolated restoring surface is depicted. The interpolated restoring surface is computed at a grid of 25 10 data points. The grid is spanned between the minimum and maximum values of 1 *q* and 3 *q* . The values

inserting the values at the grid points for 1 *q* and 3 *q* . The figure shows clearly that the

would be set to zero (or any other fixed value), which is assumed during single mode identification, just a trim curve would be identified. However, this trim curve is not able to

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,

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.

Sinapius, Link, & Gaul, 2005), (Goege & Fuellekrug, 2004) (Goege, 2004).

of the restoring surface are obtained by using Eq. (42) with the identified coefficients

(, ) *q q* depends on both modal coordinates. If modal coordinate 3 *q*

**Figure 4.** Mode participation

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 *<sup>r</sup> k* which are required for the computation of the non-linear restoring forces ( ) *<sup>r</sup> t* according to Eq. (34), the correct values are used. Also, for the eigenvectors *<sup>r</sup>* the correct data are used. A careful modal analysis at an appropriate excitation level should be able to deliver such accurate data of the underlying linear system.

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 performed. The polynomial function of Eq. (40) with max *i* increasing from *1* to *5* 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 measured and recalculated restoring force of the modal DoF 1 *r* for max *i* 5 . By the way, it is interesting that the usage of too much coefficients *<sup>i</sup>* causes no problems. The apparently unnecessary coefficients are computed to *0*.

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 Eq. (42) is used. The number of terms is increased from max max *i j* 1 to max max *i j* 3 . The deviations between the 'measured' and recalculated signals disappear completely for max max *i j* 3 and if the clean signals (without the additional random noise) are utilized. Again, it shows that the usage of too many coefficients *<sup>i</sup>* causes no problems. The 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

**Figure 5.** Non-linear restoring force of modal DoF 1 for single mode identification

case of *5 %* noise the deviations between the 'measured' and recalculated modal coordinates <sup>1</sup> *q t*( ) and 3 *q t*( ) amount to *7.4 %* and *7.7 %* respectively. It is apparent that no smaller deviation than *5 %* will be possible. Thus, the deviations are acceptable and indicate a good identification.

Figure 6 shows as an example the restoring force 11 3 (, ) *q q* identified from signals with *5 %* noise for mode 1 *r* . In the figure the 'measured' and recalculated restoring forces at all time steps are plotted as points and crosses. Also, the interpolated restoring surface is depicted. The interpolated restoring surface is computed at a grid of 25 10 data points. The grid is spanned between the minimum and maximum values of 1 *q* and 3 *q* . The values of the restoring surface are obtained by using Eq. (42) with the identified coefficients *ij* and inserting the values at the grid points for 1 *q* and 3 *q* . The figure shows clearly that the restoring force 11 3 (, ) *q q* depends on both modal coordinates. If modal coordinate 3 *q* would be set to zero (or any other fixed value), which is assumed during single mode identification, just a trim curve would be identified. However, this trim curve is not able to describe the complete non-linear behaviour.
