**4.3. Single mode identification**

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 single DoF problem.

To model stiffness non-linearities a polynomial with even and odd powers of the displacements ( ) *<sup>r</sup> q t* can be used

$$\mathcal{S}\_{k,r}\left(t\right) = \sum\_{i=0}^{i\_{\text{max}}} \alpha\_i q\_r^{-i}(t). \tag{40}$$

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 terms *i* and the associated coefficients *<sup>i</sup>* determine whether the overall force , ( ) *k r t* always acts into the opposite direction of the respective displacements and has really the physical meaning of a restoring force.

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

$$\mathcal{S}\_{c,r}\left(t\right) = \sum\_{i=0}^{i\_{\text{max}}} \mathcal{Y}\_i \dot{q}\_r^{\;i}(t). \tag{41}$$

Here as well, the involvement of terms with even powers in Eq. (41) allows for possible nonsymmetric characteristics of the restoring forces. If only terms with odd powers would be employed, the overall restoring force would be completely anti-symmetric.

If stiffness and damping non-linearities occur together the functions of Eqs. (40) and (41) can be combined. In some cases it may also be appropriate to use mixed terms with displacements and velocities.

By modelling the non-linearities with functions of Eq. (40), Eq. (41) or an appropriate combination the non-linear identification is reduced to the estimation of the coefficients *i* and *<sup>i</sup>* . The computation of the coefficients is in all cases based on an equation like Eq. (38).

The article (Goege, Fuellekrug, Sinapius, Link, & Gaul, 2005) describes in detail the identification of the non-linear parameters for a single mode of vibration. In addition, the paper shows a way of characterizing the identified non-linearities. The Harmonic Balance is used, and on the basis of the identified non-linear parameters *<sup>i</sup>* and *<sup>i</sup>* the dependency of eigenfrequency *<sup>r</sup>* and damping *<sup>r</sup>* versus the excitation level can be calculated and visualised in graphs, the so-called modal characterizing functions.
