**4. Non-linear modal identification**

180 Nonlinearity, Bifurcation and Chaos – Theory and Applications

harmonic part and by the modal damping value *<sup>r</sup>*

decay part.

eigenfrequency

values.

the modal damping *<sup>r</sup>*

detailed insight into the dynamic behaviour.

**3.2. Experimental modal analysis** 

is

excitation with frequency

This equation reveals that the free decay vibrations are determined by a superposition of eigenvectors with damped harmonic vibrations at the respective eigenfrequencies. The contribution of each eigenvector depends on *Ar* and *<sup>r</sup> B* , i.e. the initial conditions at time

Also it can be shown that the steady state dynamic responses of a structure to a harmonic

<sup>ˆ</sup> *i t Ft Fe*

*ut ue e*

This equation shows that the steady state harmonic vibrations are defined by a superposition of eigenvectors with frequency dependent amplification or attenuation factors. The contribution of each eigenvector depends on the so-called modal force <sup>ˆ</sup> *<sup>T</sup>*

Considering Eqs. (24) and (26) shows that the complete dynamic behaviour of a complex

experimental identification of these parameters is of great practical importance and allows a

Since the 1970s numerous methods for experimental modal analysis have been developed (Maia & Silva, 1997), (Ewins, 2000), (Fuellekrug, 1988). In addition to the classical Phase Resonance Method (PhRM) a large number of Phase Separation Techniques (PhST) operating in the time or frequency domain has been developed and can be applied

For the practical performance of high quality modal identification tests several concerns have to be accounted for. First, in many cases several hundred sensors are required to achieve a sufficient resolution of the spatial motions of all structural parts. Second, the excitation requires several large exciters which have to be operated simultaneously in order to excite all vibration modes. Third, the results have to be of high quality and accuracy since they are used for the verification and validation of analytical models. Therefore it has to be

1 0 0

*m i*

*<sup>T</sup> <sup>n</sup> i t <sup>r</sup> i t <sup>r</sup> <sup>r</sup> r r r r*

ˆ .

 

2 2

2

 

becomes important and limits the vibration amplitudes to finite

ˆ

*F*

as well as the eigenfrequency

(25)

<sup>0</sup>*<sup>r</sup>* , i.e. <sup>0</sup>*<sup>r</sup>* ,

 

approaches

(26)

*<sup>r</sup>* for the

*<sup>r</sup>* for the

*r F* ,

to the respective

*m* . Thus, the

*t* 0 . The time history of the vibrations is determined by the eigenfrequency

the modal mass *mr* and the relationship of the excitation frequency

nowadays as a matter of routine during modal identification tests.

<sup>0</sup>*<sup>r</sup>* . Near the resonance frequencies, where

structure is determined by a set of modal parameters 0 , , , *r rr <sup>r</sup>*

The classical procedure for the modal identification is to perform normal-mode force appropriation with the Phase Resonance Method (PhRM). The structure is harmonically excited by means of an excitation force pattern appropriated to a single mode of vibration. However, the exclusive application of the Phase Resonance Method (PhRM) is timeconsuming. Thus, an improved test concept is required which combines Phase Resonance Method (PhRM) with Phase Separation Techniques (PhST).

The core of such an optimized test concept applied e.g. to aircraft as Ground Vibrations Tests (GVT) is to combine consistently Phase Separation Techniques and the Phase Resonance Method with their particular advantages (Gloth, et al., 2001), see Figure 1. After the setup the GVT starts with the measurement of Frequency Response Functions (FRFs) in optimized exciter configurations. Second, the FRFs are analysed with Phase Separation Techniques. Hereafter the Phase Resonance Method is applied for selected vibration modes, e.g. for modes that indicate significant deviations from linearity, for modes known to be important for flutter calculations (if an aircraft is tested), or for modes which significantly differ from the prediction of the finite element analysis. Optimal exciter locations and amplitudes can be calculated from the already measured FRFs in order to accelerate the time-consuming appropriation of the force vector. The calculated force vector is applied and the corresponding eigenvector is tuned. Once a mode is identified, the classical methods for identifying modal damping and modal mass are applied. Also, a linearity check by simply increasing the excitation level is performed. During this linearity check, a possible change of the modal parameters with the force level can be investigated, see (Goege, Sinapius, Fuellekrug, & Link, 2005).

The identified eigenvectors are compared with the prediction of the finite element model and by themselves during the measurement in order to check the completeness of the data and its reliability. Multiply identified modes are sorted out. Additional exciter configurations have to be used and certain frequency ranges need to be investigated if not all expected modes are experimentally identified or if the quality of the results is not sufficient.

Non-Linearity in Structural Dynamics and Experimental Modal Analysis 183

 Uncoupled modes, which are influenced by non-linear effects. Coupled modes, which are influenced by non-linear effects.

**4.2. Basic equations for non-linear modal identification** 

non-linear modes and coupled non-linear modes is described.

forces are given according to Eq. (12) by

external excitation forces.

This equation can be rewritten as

the modal matrix

structures.

Most of the modes of real structures behave linear so that an identification using the classical linear methods and the test concept described above is still possible. Nevertheless, some modes show significant non-linear behaviour, which makes it impossible to adopt linear theory. A solution to this problem is a non-linear identification which can be based on the Masri-Caughey approach (Masri & Caughey, 1979), the force-state mapping (Crawley & Aubert, 1986) and a variant of it (Al\_Hadid & Wright, 1989). The idea and basics of the nonlinear resonant decay method (NLRDM) (Wright, Platten, Cooper, & Sarmast, 2001), (Platten, Wright, Cooper, & Sarmast, 2002), (Wrigth, Platten, Cooper, & Sarmast, 2003), (Platten, Wrigth, Worden, Cooper, & Dimitriadis, 2005), (Platten, Wrigth, Dimitriadis, & Cooper, 2009) appear to be an appropriated method for applying it to large and complex

In this section the theoretical background of the non-linear analysis of structures is outlined. The basic equations are established and a way for the modal identification in case of single

The equations of motion for an elastomechanical system with linear and non-linear restoring

where, as above, *M* , *C* and *K* are the mass, damping and stiffness matrices, and *u* , *u* and *u* are the vectors of physical displacements, velocities and accelerations. The non-linear restoring forces are given by *F uu nl* , , and *Fext* is the vector of the

The equations of motion Eq. (27) can be transformed from physical to modal space by using

where *q t*( ) is the vector of (generalized) modal coordinates, which represent modal degrees of freedom (DoF). Substituting the modal expansion of Eq. (28) into the equations of

() ()

(28)

yields

 

of the associated linear undamped system

1

motion and pre-multiplying by the transposed of the modal matrix *<sup>T</sup>*

*<sup>r</sup> <sup>r</sup> <sup>r</sup> ut q t qt* 

, . *T TTT T M q C q K q F uu F nl ext*

(29)

*n*

, , *Mu Cu Ku F u u F nl ext* (27)
