**3. Dynamic identification and modal analysis**

With the purpose to validate analytical models of complex technical structures it is required to perform measurements on components or prototypes and to identify the dynamic properties. The most important dynamic properties are the modal parameters. Their identification is the essential goal of experimental modal analysis (Maia & Silva, 1997), (Ewins, 2000).

### **3.1. Modal parameters**

To explain the basic ideas, let us first assume that the structure undergoing a modal identification test is linear and that the damping matrix is proportional to the mass and stiffness matrix. In this case Eq. (12) simplifies to

$$
\begin{bmatrix} \mathbf{M} \\ \mathbf{\ulcorner} \end{bmatrix} \begin{Bmatrix} \ddot{\boldsymbol{u}} \end{Bmatrix} + \begin{bmatrix} \mathbf{C} \\ \mathbf{\ulcorner} \end{Bmatrix} \begin{Bmatrix} \dot{\boldsymbol{u}} \end{Bmatrix} + \begin{bmatrix} \mathbf{K} \\ \mathbf{\ulcorner} \end{bmatrix} \begin{Bmatrix} \boldsymbol{u} \end{Bmatrix} = \begin{Bmatrix} \mathbf{F} \end{Bmatrix}, \tag{13}
$$

where it is assumed

$$
\begin{bmatrix} \mathbb{C} \ \end{bmatrix} = \beta\_1 \begin{bmatrix} M \ \end{bmatrix} + \beta\_2 \begin{bmatrix} K \ \end{bmatrix}. \tag{14}
$$

The eigenvalues und eigenvectors of the undamped structure are determined by the eigenvalue problem

$$\left(\alpha\_{0r}^2 \left[\begin{matrix} M \\ \end{matrix}\right] + \left[\begin{matrix} K \\ \end{matrix}\right]\right) \left\{\phi\right\}\_r = \left\{\begin{matrix} 0 \\ \end{matrix}\right\} \tag{15}$$

and are of great practical importance. The values <sup>0</sup>*r* are the so-called eigenfrequencies and *<sup>r</sup>* are the eigenvectors of the undamped structure. A fundamental property of the eigenvectors *<sup>r</sup>* is the fact that the matrix of eigenvectors, the so-called modal matrix, diagonalises the mass and stiffness matrices

### Non-Linearity in Structural Dynamics and Experimental Modal Analysis 179

$$
\begin{bmatrix} \boldsymbol{\phi} \end{bmatrix}^{T} \begin{bmatrix} \boldsymbol{M} \end{bmatrix} \begin{bmatrix} \boldsymbol{\phi} \end{bmatrix} = \begin{bmatrix} \begin{bmatrix} \boldsymbol{\phi} \end{bmatrix}\_{1}^{T} \\\\ \begin{bmatrix} \boldsymbol{\phi} \end{bmatrix}\_{2}^{T} \\\\ \begin{bmatrix} \boldsymbol{M} \end{bmatrix} \begin{bmatrix} \boldsymbol{M} \end{bmatrix} \begin{bmatrix} \begin{bmatrix} \boldsymbol{\phi} \end{bmatrix}\_{1} & \begin{bmatrix} \boldsymbol{\phi} \end{bmatrix}\_{2} & \cdots & \begin{bmatrix} \boldsymbol{\phi} \end{bmatrix}\_{n} \end{bmatrix} = \begin{bmatrix} m\_{1} & \boldsymbol{0} & \boldsymbol{0} & \boldsymbol{0} \\\ \boldsymbol{0} & m\_{2} & \boldsymbol{0} & \boldsymbol{0} \\\ \boldsymbol{0} & \boldsymbol{0} & \ddots & \boldsymbol{0} \\\ \boldsymbol{0} & \boldsymbol{0} & \boldsymbol{0} & m\_{n} \end{bmatrix}, \tag{16}
$$
 
$$
\begin{bmatrix} \boldsymbol{\phi} \end{bmatrix}^{T} \end{bmatrix} \tag{17}
$$

$$
\begin{bmatrix} \left[\boldsymbol{\phi}\right]^{T} \left[\boldsymbol{K}\right] \left[\boldsymbol{\phi}\right] = \begin{bmatrix} \left\{\boldsymbol{\phi}\right\}\_{1} \\ \left\{\boldsymbol{\phi}\right\}\_{2}^{T} \\ \vdots \\ \left\{\boldsymbol{\phi}\right\}\_{n}^{T} \end{bmatrix} \begin{bmatrix} \left\{\boldsymbol{\phi}\right\}\_{1} & \left\{\boldsymbol{\phi}\right\}\_{2} & \cdots & \left\{\boldsymbol{\phi}\right\}\_{n} \end{bmatrix} = \begin{bmatrix} k\_{1} & 0 & 0 & 0 \\ 0 & k\_{2} & 0 & 0 \\ 0 & 0 & \ddots & 0 \\ 0 & 0 & 0 & k\_{n} \end{bmatrix} . \tag{17}
$$

The terms 1 2 , , , *mm m <sup>n</sup>* are the so-called modal mass

$$m\_r = \left\{\phi\right\}\_r^T \left[\!\!\!\!\!M\right] \left\{\!\!\phi\right\}\_{r\;\!\prime} \tag{18}$$

and in analogy, the terms 1 2 , , , *<sup>n</sup> kk k* are the so-called modal stiffness

$$\boldsymbol{k}\_r = \left\{ \boldsymbol{\phi} \right\}\_r^T \left[ \boldsymbol{K} \right] \left\{ \boldsymbol{\phi} \right\}\_r. \tag{19}$$

In addition it is valid

$$
\alpha\_{0r} = \sqrt{\frac{k\_r}{m\_r}}\,\,\,\,\,\tag{20}
$$

$$
\zeta\_r = \frac{c\_r}{2\sqrt{k\_r m\_r}} \tag{21}
$$

where

178 Nonlinearity, Bifurcation and Chaos – Theory and Applications

 

**3. Dynamic identification and modal analysis** 

stiffness matrix. In this case Eq. (12) simplifies to

and are of great practical importance. The values

diagonalises the mass and stiffness matrices

The variation of Eq. (1)

leads with Eqs. (4), (5) and (10) to

accounting for non-linearities.

(Ewins, 2000).

**3.1. Modal parameters** 

where it is assumed

eigenvalue problem

eigenvectors *<sup>r</sup>*

 *<sup>r</sup>* 

 

, , *M nl u Cu Ku F u u F* (12)

*T T ext c nl ext nl*

1 1

*t t*

*t t*

*W W W W u F u C u F u u du*

2 2

*L W dt E E W dt*

which is the well-known basic equation of linear structural dynamics extended by a term

With the purpose to validate analytical models of complex technical structures it is required to perform measurements on components or prototypes and to identify the dynamic properties. The most important dynamic properties are the modal parameters. Their identification is the essential goal of experimental modal analysis (Maia & Silva, 1997),

To explain the basic ideas, let us first assume that the structure undergoing a modal identification test is linear and that the damping matrix is proportional to the mass and

> 1 2 *CMK*

The eigenvalues und eigenvectors of the undamped structure are determined by the

 <sup>2</sup> <sup>0</sup> <sup>0</sup> *<sup>r</sup> <sup>r</sup>*

 

 

is the fact that the matrix of eigenvectors, the so-called modal matrix,

are the eigenvectors of the undamped structure. A fundamental property of the

*kin pot*

<sup>2</sup>

*u t*

*u t*

1

(11)

*<sup>M</sup> u Cu Ku F* , (13)

. (14)

*M K* (15)

<sup>0</sup>*r* are the so-called eigenfrequencies and

, .

(10)

0

$$\mathcal{L}\_r = \left\{ \boldsymbol{\phi} \right\}\_r^T \left[ \boldsymbol{\mathbb{C}} \right] \left\{ \boldsymbol{\phi} \right\}\_r = \left\{ \boldsymbol{\phi} \right\}\_r^T \left( \boldsymbol{\gamma}\_1 \left[ \boldsymbol{M} \right] + \boldsymbol{\gamma}\_2 \left[ \boldsymbol{K} \right] \right) \left\{ \boldsymbol{\phi} \right\}\_r. \tag{22}$$

and

$$
\alpha\_r = \alpha\_{0r} \sqrt{1 - \zeta\_r^{'2}} \,. \tag{23}
$$

Using the above modal parameters it can be shown that the dynamic responses of a structure (13) due to an impulse or a release from any initial condition are

$$\left\{\boldsymbol{u}\left(t\right)\right\} = \sum\_{r=1}^{n} \left(\boldsymbol{A}\_{r}\sin\alpha\_{r}\boldsymbol{t} + \boldsymbol{B}\_{r}\cos\alpha\_{r}\boldsymbol{t}\right)e^{-\boldsymbol{\zeta}\_{r}\alpha\_{r}^{\alpha\_{r}}t} \left\{\boldsymbol{\phi}\right\}\_{r}.\tag{24}$$

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 *t* 0 . The time history of the vibrations is determined by the eigenfrequency *<sup>r</sup>* for the harmonic part and by the modal damping value *<sup>r</sup>* as well as the eigenfrequency *<sup>r</sup>* for the decay part.

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

$$\left\{ F\left(t\right) \right\} = \left\{ \hat{F} \right\} e^{i\alpha t} \tag{25}$$

Non-Linearity in Structural Dynamics and Experimental Modal Analysis 181

assured that all modes in the requested frequency range are identified and that the accuracy

All these demands lead to the fact that a highly sophisticated concept for the modal identification is required (Gloth, et al., 2001). During the modal identification testing of large complex structures also the possible non-linear behaviour has to be investigated. Usually, linear dynamic behaviour of the structure is assumed in the applied modal identification methods. However, in practice most of the investigated and tested structures exhibit some non-linear behaviour. Such non-linear behaviour can occur for example as a result of free play and different connection categories (e.g. welded, bolted) within joints or e.g. from

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

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,

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

of the modal parameters is as high as possible.

hydraulic systems in control surfaces of aircraft.

Method (PhRM) with Phase Separation Techniques (PhST).

**4. Non-linear modal identification** 

Fuellekrug, & Link, 2005).

sufficient.

is

$$\left\{\boldsymbol{u}\left(t\right)\right\} = \left\{\hat{\boldsymbol{u}}\right\}e^{i\boldsymbol{\alpha}t} = \sum\_{r=1}^{n} \left\{\boldsymbol{\phi}\right\}\_{r} \frac{\left\{\boldsymbol{\phi}\right\}\_{r}^{T} \left\{\hat{F}\right\}}{m\_{r} \left(\boldsymbol{\alpha}\_{0r}^{2} - \boldsymbol{\alpha}^{2} + i2\zeta\_{r}\boldsymbol{\alpha}\_{0r}\boldsymbol{\alpha}\right)} e^{i\boldsymbol{\alpha}t}.\tag{26}$$

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> r F* , the modal mass *mr* and the relationship of the excitation frequency to the respective eigenfrequency <sup>0</sup>*<sup>r</sup>* . Near the resonance frequencies, where approaches <sup>0</sup>*<sup>r</sup>* , i.e. <sup>0</sup>*<sup>r</sup>* , the modal damping *<sup>r</sup>* becomes important and limits the vibration amplitudes to finite values.

Considering Eqs. (24) and (26) shows that the complete dynamic behaviour of a complex structure is determined by a set of modal parameters 0 , , , *r rr <sup>r</sup> m* . Thus, the experimental identification of these parameters is of great practical importance and allows a detailed insight into the dynamic behaviour.
