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

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

The equations of motion for an elastomechanical system with linear and non-linear restoring forces are given according to Eq. (12) by

$$\left\{ \left[ M \right] \middle| \left\{ \ddot{u} \right\} + \left[ \mathbf{C} \right] \middle| \left\{ \dot{u} \right\} + \left[ K \right] \middle| \left\{ u \right\} + \left\{ F\_{nl} \left( \left\{ u \right\}, \left\{ \dot{u} \right\} \right) \right\} = \left\{ F\_{ext} \right\}, \tag{27}$$

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 external excitation forces.

The equations of motion Eq. (27) can be transformed from physical to modal space by using the modal matrix of the associated linear undamped system

$$\left\{\boldsymbol{\mu}\left(\boldsymbol{t}\right)\right\} = \sum\_{r=1}^{n} \left\{\boldsymbol{\phi}\right\}\_{r} q\_{r}(\boldsymbol{t}) = \left[\boldsymbol{\phi}\right] \left\{q(\boldsymbol{t})\right\} \tag{28}$$

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

$$
\begin{bmatrix} \begin{bmatrix} \boldsymbol{\rho} \end{bmatrix}^{\mathsf{T}} \begin{bmatrix} \boldsymbol{M} \end{bmatrix} \begin{bmatrix} \boldsymbol{\rho} \end{bmatrix} \begin{bmatrix} \boldsymbol{\rho} \end{bmatrix} \begin{bmatrix} \boldsymbol{\epsilon} \end{bmatrix} + \begin{bmatrix} \boldsymbol{\rho} \end{bmatrix}^{\mathsf{T}} \begin{bmatrix} \boldsymbol{C} \end{bmatrix} \begin{bmatrix} \boldsymbol{\epsilon} \end{bmatrix} \begin{bmatrix} \boldsymbol{\epsilon} \end{bmatrix} + \begin{bmatrix} \boldsymbol{\rho} \end{bmatrix}^{\mathsf{T}} \begin{bmatrix} \boldsymbol{K} \end{bmatrix} \begin{bmatrix} \boldsymbol{\rho} \end{bmatrix} \begin{bmatrix} \boldsymbol{q} \end{bmatrix} + \begin{bmatrix} \boldsymbol{\rho} \end{bmatrix}^{\mathsf{T}} \begin{bmatrix} \boldsymbol{F}\_{\boldsymbol{n}\boldsymbol{l}} \left( \begin{Bmatrix} \boldsymbol{u} \end{Bmatrix}, \begin{Bmatrix} \boldsymbol{i} \end{Bmatrix} \right) \end{bmatrix} = \begin{bmatrix} \boldsymbol{\rho} \end{bmatrix}^{\mathsf{T}} \begin{Bmatrix} \boldsymbol{F}\_{\boldsymbol{c}\boldsymbol{t}\boldsymbol{l}} \end{Bmatrix}. \tag{29}
$$

This equation can be rewritten as

$$
\begin{bmatrix} m \\ \end{bmatrix} \begin{Bmatrix} \ddot{q} \\ \end{Bmatrix} + \begin{bmatrix} c \\ c \end{bmatrix} \begin{Bmatrix} \dot{q} \\ \end{Bmatrix} + \begin{bmatrix} k \\ \end{bmatrix} \begin{Bmatrix} q \\ \end{Bmatrix} + \begin{Bmatrix} \delta \\ \end{Bmatrix} = \begin{Bmatrix} f\_{ext} \\ \end{Bmatrix}, \tag{30}
$$

Non-Linearity in Structural Dynamics and Experimental Modal Analysis 185

by solving Eq. (28) e.g. with least squares:

*u t* (35)

*f t* is calculated from measured eigenvectors and

*t qt qt qt* (37)

5

 

(36)

 <sup>3</sup> , <sup>5</sup> ,.

(38)

Here, *mr* , *<sup>r</sup> c* and *<sup>r</sup> k* are experimentally identified e.g. from vector polar plot curve fit, evaluation of real part slopes or from the complex power method, see (Niedbal & Klusowski, 1989). The modal coordinate ( ) *<sup>r</sup> q t* is calculated from the physical acceleration

<sup>1</sup> ( ) (). *T T*

 in Eq. (35) represents the experimental modal matrix. This modal matrix contains the eigenvectors in the frequency band of interest, which were previously determined from linear modal analysis. If significant modal responses for other than the investigated mode of vibration are observable, coupling terms between the investigated modes and other modes

The modal velocities ( ) *<sup>r</sup> q t* and modal displacement responses ( ) *<sup>r</sup> q t* of the mode can be obtained by an integration of the modal acceleration responses. Prior to the integration, a band-pass filtering of the data is required in order to avoid a drift of the time domain

> () (). *<sup>T</sup> <sup>r</sup> <sup>r</sup> <sup>f</sup> t Ft*

With the purpose of identifying the non-linear parameters it is required to use an analytical expression which is able to describe the non-linear behaviour. If the modal DoF *r* is nonlinear in the stiffness and depends only on the modal coordinate ( ) *<sup>r</sup> q t* , a polynomial

13 5 () () () () *rrrr*

 

( ) () () ()

*t qt qt qt t qt qt qt*

*r rr r r rr r*

*r rr r*

( ) () () ()

*t qt qt qt*

characterize the cubic and higher polynomial parts of the stiffness.

3 5

3 5 1 111 1 3 5 2 222 3

( ) () () () .

The vector on the left hand side of the equation can be computed from Eq. (34) by inserting values for the modal parameters, *mr* , *<sup>r</sup> c* , *<sup>r</sup> k* and time domain data at time steps *<sup>j</sup> t* with *j* 1,2, , of the modal coordinates ( ), ( ), ( ) *rj rj rj qt qt qt* , and the modal force ( ) *r j f t* . The matrix on the right hand side is formed by time domain data of the modal coordinate ( ) *r j q t* .

3 5

 

1 describes the linear part of the stiffness and

*<sup>i</sup>* of the function can be computed by writing Eq. (37) for several time

 

*q t*

responses *u t* ( ) and the modal matrix

signals. The modal excitation force ( ) *<sup>r</sup>*

excitation forces according to

can be used. The coefficient

 

exist.

function like

The coefficients

steps *<sup>j</sup> t*

where *m* , *c* and *k* are the (generalized) modal mass, damping and stiffness matrices. The modal mass and stiffness matrices *m* , *k* are diagonal since the real normal modes *<sup>r</sup>* of the associated undamped system are orthogonal with respect to the physical mass and stiffness matrices *M* and *K* . In case of so-called proportional damping also the modal damping matrix *c* is diagonal. ( )*t* is the vector of modal nonlinear restoring forces, which includes stiffness and damping non-linearities, and *f ext*( )*t* is the vector of (generalized) modal excitation forces.

If the damping is proportional Eq. (30) simplifies to

$$k m\_r \ddot{q}\_r + c\_r \dot{q}\_r + k\_r q\_r + \delta\_r = f\_r, \quad r = 1, 2, \dots, n. \tag{31}$$

In case of ( ) 0 *<sup>r</sup> t* , the dynamic equation for mode *r* is in the form of a single degree of freedom system. When the Phase Resonance Method according to the above described test concept is used, the excitation forces are appropriated to the specific mode and the whole structure vibrates in the linear case as a single DoF system. However, if non-linearities are present the modal DoF *r* may be coupled with other modal DoF. This is because the vector of the non-linear modal restoring forces ( )*t* is, according to Eqs. (30) and (29), a function of all physical displacements and velocities

$$\left\{\mathcal{S}(\mathbf{t})\right\} = \left[\boldsymbol{\phi}\right]^{\mathrm{T}} \left\{F\_{nl}\left(\{\boldsymbol{u}\}, \{\dot{\boldsymbol{u}}\}\right)\right\}.\tag{32}$$

And thus, in the general case, the non-linear modal restoring forces ( ) *<sup>r</sup> t* can be a function of all modal coordinates

$$
\delta\_r \delta\_r = \delta\_r (q\_{1'} q\_{2'}, \dots, q\_n; \dot{q}\_{1'} \dot{q}\_{2'}, \dots, \dot{q}\_n). \tag{33}
$$

The basic idea of the non-linear modal identification is to use time domain data of the modal DoF and to perform a so-called direct parameter estimation (DPE) in the modal space (Worden & Tomlinson, 2001) as well as to apply ideas of the non-linear 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).

When the excitation forces are appropriated the whole structure vibrates in the linear case as a single DoF system. Thus, the analysis in modal space offers an effective way of identifying the non-linear damping and stiffness properties. Such a non-linear identification requires the previous identification of the linear modal parameters mass *mr* , damping *<sup>r</sup> c* and stiffness *<sup>r</sup> k* . Also, it is required to determine the time histories of the modal coordinates ( ) *<sup>r</sup> q t* and the modal forces ( ) *<sup>r</sup> f t* .

The rearrangement of Eq. (31) delivers

$$
\delta\_r \delta\_r(t) = -m\_r \ddot{q}\_r(t) - c\_r \dot{q}\_r(t) - k\_r q\_r(t) + f\_r(t). \tag{34}
$$

Here, *mr* , *<sup>r</sup> c* and *<sup>r</sup> k* are experimentally identified e.g. from vector polar plot curve fit, evaluation of real part slopes or from the complex power method, see (Niedbal & Klusowski, 1989). The modal coordinate ( ) *<sup>r</sup> q t* is calculated from the physical acceleration responses *u t* ( ) and the modal matrix by solving Eq. (28) e.g. with least squares:

184 Nonlinearity, Bifurcation and Chaos – Theory and Applications

damping also the modal damping matrix *c* is diagonal.

the vector of (generalized) modal excitation forces. If the damping is proportional Eq. (30) simplifies to

of the non-linear modal restoring forces

of all physical displacements and velocities

where

normal modes *<sup>r</sup>*

In case of ( ) 0 *<sup>r</sup>* 

of all modal coordinates

( ) *<sup>r</sup> q t* and the modal forces ( ) *<sup>r</sup>*

The rearrangement of Eq. (31) delivers

, *mq cq kq fext*

matrices. The modal mass and stiffness matrices *m* , *k* are diagonal since the real

physical mass and stiffness matrices *M* and *K* . In case of so-called proportional

, 1,2, , . *mq cq kq f r n rr rr rr r r* 

freedom system. When the Phase Resonance Method according to the above described test concept is used, the excitation forces are appropriated to the specific mode and the whole structure vibrates in the linear case as a single DoF system. However, if non-linearities are present the modal DoF *r* may be coupled with other modal DoF. This is because the vector

> ( ) , . *<sup>T</sup> nl*

12 12 ( , , , ; , , , ). *rr n n*

The basic idea of the non-linear modal identification is to use time domain data of the modal DoF and to perform a so-called direct parameter estimation (DPE) in the modal space (Worden & Tomlinson, 2001) as well as to apply ideas of the non-linear resonant decay method (NLRDM) (Wright, Platten, Cooper, & Sarmast, 2001), (Platten, Wright, Cooper, & Sarmast, 2002), (Wrigth, Platten, Cooper, & Sarmast, 2003), (Platten, Wrigth, Worden,

When the excitation forces are appropriated the whole structure vibrates in the linear case as a single DoF system. Thus, the analysis in modal space offers an effective way of identifying the non-linear damping and stiffness properties. Such a non-linear identification requires the previous identification of the linear modal parameters mass *mr* , damping *<sup>r</sup> c* and stiffness *<sup>r</sup> k* . Also, it is required to determine the time histories of the modal coordinates

( ) ( ) ( ) ( ) ( ). *r rr rr rr r*

*t mq t cq t kq t f t* (34)

 *t F uu*

 

*f t* .

And thus, in the general case, the non-linear modal restoring forces ( ) *<sup>r</sup>*

Cooper, & Dimitriadis, 2005), (Platten, Wrigth, Dimitriadis, & Cooper, 2009).

*t* , the dynamic equation for mode *r* is in the form of a single degree of

linear restoring forces, which includes stiffness and damping non-linearities, and *f*

*m* , *c* and *k* are the (generalized) modal mass, damping and stiffness

of the associated undamped system are orthogonal with respect to the

(30)

( )*t* is the vector of modal non-

(31)

( )*t* is, according to Eqs. (30) and (29), a function

(32)

*qq qqq q* (33)

*t* can be a function

*ext*( )*t* is

$$\left\{ \ddot{\boldsymbol{\eta}}(t) \right\} = \left( \left[ \boldsymbol{\phi} \right]^{T} \left[ \boldsymbol{\phi} \right]^{-1} \right) \left[ \boldsymbol{\phi} \right]^{T} \left\{ \ddot{\boldsymbol{u}}(t) \right\}. \tag{35}$$

 in Eq. (35) represents the experimental modal matrix. This modal matrix contains the eigenvectors in the frequency band of interest, which were previously determined from linear modal analysis. If significant modal responses for other than the investigated mode of vibration are observable, coupling terms between the investigated modes and other modes exist.

The modal velocities ( ) *<sup>r</sup> q t* and modal displacement responses ( ) *<sup>r</sup> q t* of the mode can be obtained by an integration of the modal acceleration responses. Prior to the integration, a band-pass filtering of the data is required in order to avoid a drift of the time domain signals. The modal excitation force ( ) *<sup>r</sup> f t* is calculated from measured eigenvectors and excitation forces according to

$$\{f\_r(t) = \left\{\phi\right\}\_r^T \left\{ F(t) \right\}. \tag{36}$$

With the purpose of identifying the non-linear parameters it is required to use an analytical expression which is able to describe the non-linear behaviour. If the modal DoF *r* is nonlinear in the stiffness and depends only on the modal coordinate ( ) *<sup>r</sup> q t* , a polynomial function like

$$
\delta\_r \delta\_r(t) = \alpha\_1 q\_r(t) + \alpha\_3 q\_r^3(t) + \alpha\_5 q\_r^5(t) + \dots \tag{37}
$$

can be used. The coefficient 1 describes the linear part of the stiffness and <sup>3</sup> , <sup>5</sup> ,. characterize the cubic and higher polynomial parts of the stiffness.

The coefficients *<sup>i</sup>* of the function can be computed by writing Eq. (37) for several time steps *<sup>j</sup> t*

$$\begin{Bmatrix} \delta\_r(t\_1) \\ \delta\_r(t\_2) \\ \vdots \\ \delta\_r(t\_\ell) \end{Bmatrix} = \begin{vmatrix} q\_r(t\_1) & q\_r^3(t\_1) & q\_r^5(t\_1) & \cdots \\ q\_r(t\_2) & q\_r^3(t\_2) & q\_r^5(t\_2) & \cdots \\ \vdots & \vdots & \vdots & \cdots \\ q\_r(t\_\ell) & q\_r^3(t\_\ell) & q\_r^5(t\_\ell) & \cdots \end{vmatrix} \begin{Bmatrix} a\_1 \\ a\_3 \\ a\_5 \\ \vdots \end{Bmatrix} . \tag{38}$$

The vector on the left hand side of the equation can be computed from Eq. (34) by inserting values for the modal parameters, *mr* , *<sup>r</sup> c* , *<sup>r</sup> k* and time domain data at time steps *<sup>j</sup> t* with *j* 1,2, , of the modal coordinates ( ), ( ), ( ) *rj rj rj qt qt qt* , and the modal force ( ) *r j f t* . The matrix on the right hand side is formed by time domain data of the modal coordinate ( ) *r j q t* . The solution of Eq. (38) with least squares or any other appropriate method delivers the coefficients 135 , , , . Care is needed for the appropriate number of time steps in Eq. (38) because too few or too many time steps can cause problems.

The quality of the non-linear identification can be checked by comparing the restoring force ( ) *<sup>r</sup> t* of Eq. (34), which is based on measured data, and the recalculated restoring force, which is computed from Eq. (37) with the identified coefficients *<sup>i</sup>* . However, in cases of 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 compare the modal accelerations of the measurement ( ) *<sup>r</sup> q t* with the recalculated modal accelerations ( ) *<sup>r</sup> q t* , which are computed from the rearranged Eq. (34)

$$
\ddot{q}\_r(t) = \frac{1}{m\_r} \left( f\_r(t) - c\_r \dot{q}\_r(t) - k\_r q\_r(t) - \delta\_r(t) \right),
\tag{39}
$$

Non-Linearity in Structural Dynamics and Experimental Modal Analysis 187

*<sup>i</sup>* and *<sup>i</sup>* 

versus the excitation level can be calculated and

(42)

(43)

*i*

the dependency of

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

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

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

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

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

max max

max max

*i j*

0 0

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

The estimation of the coefficients of the functions in Eq. (42) or Eq. (43) always leads to the

*c r ij r s i j*

 *t q tq t* 

*i j*

0 0

*k r ij r s i j*

 *t q tq t* 

can be used. As above, the involvement of terms with even powers in Eq. (42) allows for

() ()

() ()

*i j*

*i j*

,

,

possible non-symmetric characteristics of the restoring forces.

To model damping non-linearities the polynomial function

data by using a suitable identification equation.

solution of an over-determined set of linear equations like

. The computation of the coefficients is in all cases based on an equation like Eq. (38).

employed, the overall restoring force would be completely anti-symmetric.

used, and on the basis of the identified non-linear parameters

visualised in graphs, the so-called modal characterizing functions.

*<sup>r</sup>* and damping *<sup>r</sup>*

displacements and velocities.

**4.4. Coupled mode identification** 

stiffness, the polynomial function

and *<sup>i</sup>* 

eigenfrequency

where ( ) *<sup>r</sup> t* is computed from Eq. (37). A qualitative comparison can be performed by visualizing the time histories of ( ) *<sup>r</sup> q t* and ( ) *<sup>r</sup> q t* . In addition, a quantitative comparison can be obtained by the root mean square (RMS) values of the measured acceleration signal and the deviation between measured and recalculated signals.
