**4.4. Coupled mode identification**

186 Nonlinearity, Bifurcation and Chaos – Theory and Applications

because too few or too many time steps can cause problems.

which is computed from Eq. (37) with the identified coefficients

*r*

compare the modal accelerations of the measurement ( ) *<sup>r</sup>*

the deviation between measured and recalculated signals.

coefficients 135

accelerations ( ) *<sup>r</sup> q t*

where ( ) *<sup>r</sup>* 

( ) *<sup>r</sup>* 

visualizing the time histories of ( ) *<sup>r</sup>*

**4.3. Single mode identification** 

displacements ( ) *<sup>r</sup> q t* can be used

terms *i* and the associated coefficients

physical meaning of a restoring force.

single DoF problem.

The solution of Eq. (38) with least squares or any other appropriate method delivers the

The quality of the non-linear identification can be checked by comparing the restoring force

*t* of Eq. (34), which is based on measured data, and the recalculated restoring force,

weak non-linearities (small non-linear restoring forces) the deviations may be high, although the agreement for the modal coordinates is very good. For this reason it is better to

<sup>1</sup> () () () () (), *<sup>r</sup> r rr rr r*

be obtained by the root mean square (RMS) values of the measured acceleration signal and

If the non-linearity in the modal DoF *r* is solely caused by displacements ( ) *<sup>r</sup> q t* and velocities ( ) *<sup>r</sup> q t* of the same DoF *r* , the problem of non-linear identification is reduced to a

To model stiffness non-linearities a polynomial with even and odd powers of the

0

The involvement of terms with even powers in Eq. (40) allows for possible non-symmetric characteristics of the overall restoring force. If only terms with odd powers were employed, the overall restoring force would be completely anti-symmetric. Of course, the number of

always acts into the opposite direction of the respective displacements and has really the

 *t qt* 

( ).

*i*

max

In a quite similar way, the damping non-linearities can be modelled by the function

,

max

*c r i r i*

*i*

0

 *t qt* 

( ).

*i*

*k r i r i*

*i*

,

*t* is computed from Eq. (37). A qualitative comparison can be performed by

*q t f t cq t kq t t <sup>m</sup>*

, which are computed from the rearranged Eq. (34)

*q t* and ( ) *<sup>r</sup> q t*

, , , . Care is needed for the appropriate number of time steps in Eq. (38)

. In addition, a quantitative comparison can

(40)

*<sup>i</sup>* determine whether the overall force , ( ) *k r*

(41)

*t*

*<sup>i</sup>* . However, in cases of

(39)

*q t* with the recalculated modal

In the case of coupled modes the function of Eq. (40) has to be extended by the contribution of other modal coordinates ( ) *<sup>s</sup> q t* . If two modes *r* and *s* are coupled with respect to the stiffness, the polynomial function

$$\mathcal{S}\_{k,r}\left(t\right) = \sum\_{i=0}^{i\_{\text{max}}} \sum\_{j=0}^{j\_{\text{max}}} \alpha\_{ij} q\_r^{\ i}(t) q\_s^{\ j}(t) \tag{42}$$

can be used. As above, the involvement of terms with even powers in Eq. (42) allows for possible non-symmetric characteristics of the restoring forces.

To model damping non-linearities the polynomial function

$$\delta \mathcal{S}\_{c,r}\left(t\right) = \sum\_{i=0}^{i\_{\text{max}}} \sum\_{j=0}^{j\_{\text{max}}} \mathcal{Y}\_{ij} \dot{q}\_r^{\ i}(t) \dot{q}\_s^{\ j}(t) \tag{43}$$

can be used. For more general cases the functions of Eqs. (42) and (43) can be combined. In some cases it may also be appropriate to use mixed terms with displacements and velocities. If three or more modes are non-linearly coupled the functions of Eqs. (42) and (43) can be extended accordingly. Also, the identification is not generally restricted to polynomial functions. Any other function may be used where it is appropriate. The important fact is that the function has to contain parameter coefficients, which can be computed from measured data by using a suitable identification equation.

The estimation of the coefficients of the functions in Eq. (42) or Eq. (43) always leads to the solution of an over-determined set of linear equations like

$$\{\Delta\} = \begin{bmatrix} Q \end{bmatrix} \{a\} \tag{44}$$

Non-Linearity in Structural Dynamics and Experimental Modal Analysis 189

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

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

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

correct phase relationship between the input and the output of the system.

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

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

from *2 Hz* to *12 Hz*.

where contains the values of the non-linear restoring forces ( )*t* at discrete time steps (computed according to Eq. (34)), *Q* is comprised by time domain data of the modal coordinates ( ), ( ), ( ), ( ) *rrss qtqtqtqt* and vector is assembled by the unknown coefficients , *ij ij* . The solution of Eq. (44) can be obtained by using least squares. However, also other appropriate parameter estimation methods can be applied.
