**2. Equations of motion for structures**

Most large and complex technical structures, or at least large parts of them, can be considered as elastomechanical systems. That means, the dynamic behaviour and the vibration characteristics are determined by the quantity and distribution of masses, stiffness and damping. In principle, all of these structures are assembled by continuous parts. However, an analysis of continuous structures is only possible if the geometry is rather simple. Beams, plates and shells can be analysed by using ordinary or partial differential equations. However, the coupling of the basic elements, which are described by differential equations, becomes difficult and impossible due to complicated boundary conditions if the number of elements is high. Under practical considerations it is appropriate to discretize the structures. Discrete points have to be defined at all suitable locations and the dynamic motions are described by motions of these discrete points. If a computational analysis with e.g. the Finite Element Method (FEM) is performed the nodal points are such discrete points. If an experimental analysis is carried out, suitable points have to be defined. Here, it is essential to select all structural points which are required to describe the dynamic motions with sufficient accuracy. The displacements, velocities and accelerations of the selected discrete points can then be assembled in the vectors *u u* , and *u* , the so-called displacement, velocity and acceleration vectors.

The equations of motion can be setup with different methods. As most general method, *Hamilton's* principle of least action (Williams, 1996), (Szabo, 1956), (Landau & Lifschitz, 1976) can be utilized. *Hamilton's* principle states that the time integral

$$\mathfrak{T} = \bigwedge\_{t\_1}^{t\_2} (L + \mathcal{W})dt,\tag{1}$$

Non-Linearity in Structural Dynamics and Experimental Modal Analysis 177

*kin E u Mu* (4)

*pot E u Ku* (5)

. *<sup>T</sup> W uF ext ext* (6)

, *<sup>T</sup> W u Cu <sup>c</sup>* (7)

, . *nl nl F F uu* (8)

*W F u u du* (9)

Let us now separate notionally the structure in a complete linear part and some non-linear elements. In this case the kinetic energy of the linear part can be written with the physical

> <sup>1</sup> . <sup>2</sup> *T*

 <sup>1</sup> . <sup>2</sup> *T*

In a similar way the potential energy of the structure's linear part is given by the physical

The work of the non-conservative forces consists firstly of the work of the external forces

The damping of the elastomechanical structure can be taken into account by assuming discrete dampers, separating notionally the damping elements from the structure and considering the damping forces as external forces. Following this, the work of the damping forces is given for the structure's linear part by the physical damping matrix *C* and the

where the minus sign indicates that the damping forces act into the opposite direction of the

At next, the non-linear part of the structure has to be taken into account. Here, all non-linear elements are considered as discrete elements, are notionally separated from the linear structure and it is assumed that the forces between the structure and the non-linear elements

Thus, the non-linearties of the structure can be considered as the effect of external forces.

where the minus sign indicates that the non-linear forces act into the opposite direction of

,

depend only from the deformations and velocities at the connection points

1

<sup>2</sup>

*u t nl nl u t*

Following this, the work of the non-linear forces is given by

The work of the non-conservative forces can now be written as

the related deformations and velocities.

mass matrix *M* and the velocities *u* as follows

stiffness matrix *K* and the elastic deformations *u* to

*Fext* and the related deformations *u*

velocities *u* to

related velocities.

which contains *Lagrange's* function *L* and the work of non-conservative forces *W* , reaches a stationary value for the actual dynamic motions of the structure. The meaning of *Hamilton's* principle is that from all possible dynamic motions between two fixed states at points in time 1*t* and 2*t* the actual dynamic motions are those which cause a stationary value of the time integral of Eq. (1). Thus, arbitrary variations of Eq. (1) have to vanish and this leads to a method for setting up equations of motion (Williams, 1996).

*Lagrange's L* function consists of the kinetic and potential energy of the structure and can be written as

$$L = E\_{kin} - E\_{pot} \,. \tag{2}$$

The work of non-conservative forces *F* can be computed with the displacements at discrete points *u* at two fixed states at points in time 1*t* and 2*t* to

$$\mathcal{W} = \bigcap\_{\{u(t\_1)\}}^{\{u(t\_2)\}} \left\{ F \right\}^T d\left\{ u \right\}. \tag{3}$$

Let us now separate notionally the structure in a complete linear part and some non-linear elements. In this case the kinetic energy of the linear part can be written with the physical mass matrix *M* and the velocities *u* as follows

176 Nonlinearity, Bifurcation and Chaos – Theory and Applications

**2. Equations of motion for structures** 

displacement, velocity and acceleration vectors.

can be utilized. *Hamilton's* principle states that the time integral

method for setting up equations of motion (Williams, 1996).

discrete points *u* at two fixed states at points in time 1*t* and 2*t* to

written as

Most large and complex technical structures, or at least large parts of them, can be considered as elastomechanical systems. That means, the dynamic behaviour and the vibration characteristics are determined by the quantity and distribution of masses, stiffness and damping. In principle, all of these structures are assembled by continuous parts. However, an analysis of continuous structures is only possible if the geometry is rather simple. Beams, plates and shells can be analysed by using ordinary or partial differential equations. However, the coupling of the basic elements, which are described by differential equations, becomes difficult and impossible due to complicated boundary conditions if the number of elements is high. Under practical considerations it is appropriate to discretize the structures. Discrete points have to be defined at all suitable locations and the dynamic motions are described by motions of these discrete points. If a computational analysis with e.g. the Finite Element Method (FEM) is performed the nodal points are such discrete points. If an experimental analysis is carried out, suitable points have to be defined. Here, it is essential to select all structural points which are required to describe the dynamic motions with sufficient accuracy. The displacements, velocities and accelerations of the selected discrete points can then be assembled in the vectors *u u* , and *u* , the so-called

The equations of motion can be setup with different methods. As most general method, *Hamilton's* principle of least action (Williams, 1996), (Szabo, 1956), (Landau & Lifschitz, 1976)

which contains *Lagrange's* function *L* and the work of non-conservative forces *W* , reaches a stationary value for the actual dynamic motions of the structure. The meaning of *Hamilton's* principle is that from all possible dynamic motions between two fixed states at points in time 1*t* and 2*t* the actual dynamic motions are those which cause a stationary value of the time integral of Eq. (1). Thus, arbitrary variations of Eq. (1) have to vanish and this leads to a

*Lagrange's L* function consists of the kinetic and potential energy of the structure and can be

The work of non-conservative forces *F* can be computed with the displacements at

*u t*

1

<sup>2</sup>

*u t*

*T*

.

,

*L W dt* (1)

. *kin pot LE E* (2)

*W F du* (3)

2

*t*

1

*t*

$$E\_{\rm kin} = \frac{1}{2} \{\dot{\boldsymbol{\mu}}\}^T \begin{bmatrix} \boldsymbol{M} \end{bmatrix} \begin{bmatrix} \dot{\boldsymbol{\mu}} \end{bmatrix}. \tag{4}$$

In a similar way the potential energy of the structure's linear part is given by the physical stiffness matrix *K* and the elastic deformations *u* to

$$E\_{pot} = \frac{1}{2} \{ \boldsymbol{u} \}^T \begin{bmatrix} \boldsymbol{K} \end{bmatrix} \{ \boldsymbol{u} \}. \tag{5}$$

The work of the non-conservative forces consists firstly of the work of the external forces *Fext* and the related deformations *u*

$$\mathcal{W}\_{ext} = \begin{Bmatrix} \boldsymbol{\mu} \end{Bmatrix}^T \begin{Bmatrix} F\_{ext} \end{Bmatrix}. \tag{6}$$

The damping of the elastomechanical structure can be taken into account by assuming discrete dampers, separating notionally the damping elements from the structure and considering the damping forces as external forces. Following this, the work of the damping forces is given for the structure's linear part by the physical damping matrix *C* and the velocities *u* to

$$\mathcal{W}\_c = -\{\boldsymbol{\mu}\}^T \left[\boldsymbol{\mathbb{C}}\right] \{\boldsymbol{\upmu}\}\_{\prime} \tag{7}$$

where the minus sign indicates that the damping forces act into the opposite direction of the related velocities.

At next, the non-linear part of the structure has to be taken into account. Here, all non-linear elements are considered as discrete elements, are notionally separated from the linear structure and it is assumed that the forces between the structure and the non-linear elements depend only from the deformations and velocities at the connection points

$$\left\{F\_{nl}\right\} = \left\{F\_{nl}\left(\left\{\mu\right\}, \left\{\dot{\mu}\right\}\right)\right\}.\tag{8}$$

Thus, the non-linearties of the structure can be considered as the effect of external forces. Following this, the work of the non-linear forces is given by

$$\mathcal{W}\_{nl} = -\int\_{\{u(t\_1)\}}^{\{u(t\_2)\}} \left\{ F\_{nl}\left(\{u\}, \{\dot{u}\}\right) \right\} d\left\{u\right\} \tag{9}$$

where the minus sign indicates that the non-linear forces act into the opposite direction of the related deformations and velocities.

The work of the non-conservative forces can now be written as

$$\mathcal{W} = \mathcal{W}\_{\text{ext}} + \mathcal{W}\_c + \mathcal{W}\_{nl} = \begin{Bmatrix} \boldsymbol{\upmu} \end{Bmatrix}^T \left\{ F\_{\text{ext}} \right\} - \begin{Bmatrix} \boldsymbol{\upmu} \end{Bmatrix}^T \left\{ \mathbf{C} \right\} \begin{Bmatrix} \boldsymbol{\upmu} \end{Bmatrix} - \begin{Bmatrix} \boldsymbol{\upmu}(\boldsymbol{t}\_2) \\ \boldsymbol{\upmu}(\boldsymbol{t}\_1) \end{Bmatrix} \left\{ F\_{\text{nl}} \begin{Bmatrix} \{\boldsymbol{\upmu} \}, \{\boldsymbol{\upmu}\} \end{Bmatrix} \right\} \boldsymbol{d} \begin{Bmatrix} \boldsymbol{\upmu} \end{Bmatrix} . \tag{10}$$

The variation of Eq. (1)

$$\delta \mathcal{S} \mathfrak{I} = \delta \oint\_{t\_1} (L + \mathcal{W}) dt = \int\_{t\_1}^{t\_2} (\delta E\_{kin} - \delta E\_{pot} + \delta \mathcal{W}) dt = 0 \tag{11}$$

Non-Linearity in Structural Dynamics and Experimental Modal Analysis 179

*m*

*k*

000

0 00 , 00 0

000

0 00 . 00 0

*k*

000

(18)

(19)

(20)

(21)

 (22)

(23)

*k*

*m*

0 00

*m*

 , *<sup>T</sup> <sup>r</sup> r r m M* 

 . *<sup>T</sup> <sup>r</sup> r r k K* 

> <sup>0</sup> , *<sup>r</sup> r*

*r*

 

structure (13) due to an impulse or a release from any initial condition are

1

*r k m*

, <sup>2</sup> *r*

 1 2 . *T T <sup>r</sup> r rr <sup>r</sup> c C MK*

> *rr r*

Using the above modal parameters it can be shown that the dynamic responses of a

*ut A t B te*

*<sup>n</sup> <sup>t</sup> r rr r <sup>r</sup> <sup>r</sup>*

 

2 <sup>0</sup> 1 .

sin cos . *r r*

  

(24)

 

 

*c k m*

*r r*

1 2

<sup>1</sup> <sup>1</sup> 2 2

1 2

2 2

*n*

*n*

(17)

 

(16)

*T n*

*T n*

 

<sup>1</sup> <sup>1</sup>

*T*

*T*

*n*

*T*

*T*

*n*

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

*M M*

 

> 

*T*

*T*

In addition it is valid

where

and

*K K*

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

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

$$\left\{ \left[ \boldsymbol{M} \right] \middle| \{ \dot{\boldsymbol{u}} \} + \left[ \boldsymbol{C} \right] \middle| \{ \dot{\boldsymbol{u}} \} + \left[ \boldsymbol{K} \right] \middle| \{ \boldsymbol{u} \} + \left\{ F\_{nl} \left( \{ \boldsymbol{u} \}, \{ \dot{\boldsymbol{u}} \} \right) \right\} = \left\{ F \right\}\_{\prime} \tag{12}$$

which is the well-known basic equation of linear structural dynamics extended by a term accounting for non-linearities.
