Cap 2 Structural Dynamics and Modal Analysis

8/9/2019 Cap 2 Structural Dynamics and Modal Analysis

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D. A. Rade and V. Steffen, Jr

Federal University of Uberlandia, School of Mechanical

Engineering, Brazil

STRUCTURAL DYNAMICS AND MODAL ANALYSIS



Rade and Steffen Jr. Structural Dynamics and Modal Analysis

1

D. A. Rade and V. Steffen, Jr Federal University of Uberlandia, School of Mechanical Engineering, Brazil

Key Words: mechanical vibrations, finite elements, vibration testing, modal analysis, structural

dynamics.

Contents

1. Introduction

1.1. Basic Definitions

2. Theoretical Foundations of Structural Dynamics

2.1. Equations of Motion for Discrete Systems

2.2. Undamped Free-Vibrations. Eigenvalues and Eigenvectors

2.3. Expansion Theorem

2.4. Free Responses of Undamped Systems to Initial Conditions

2.4.1. Systems with Rigid Body Modes

2.5. Harmonic Vibrations of Undamped Systems

2.6. General Forced Responses of Undamped Systems

2.7. Free Responses of Viscously Damped

2.7.1. The Condition 1 1

C M K K M C

is satisfied


C M K K M C

is satisfied

2.8. Harmonic Responses of Viscously Damaped Systems

2.9. Forced Responses of Viscously Damped Systems

2.10. Examples

3. Continuous Systems

3.1. Equations of Motion

3.2. Eigenvalue Problem


3.4. Free Response to Initial Conditions

3.5. General Forced Responses

3.6. Examples

3.6.1. Natural Frequencies and Eigenfunctions

3.6.2. Free Response to Initial Conditions

4. Finite Element Modeling in Structural Dynamics4.1. Finite Element Equations of Motion of Euler-Bernoulli Beams

4.1.1. Element-level Equations of Motion

4.1.2. Assembling of Global Equations of Motion

4.2. Enforcing Boundary Conditions

4.3. Performing Dynamics Analysis

5. Introduction to Experimental Modal Analysis

5.1. Setup for Experimental for Modal Analysis

5.2. Estimation of Modal Parameters

6. Conclusions

STRUCTURAL DYNAMICS AND MODAL ANALYSIS




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Summary

This contribution is devoted to two inter-related topics in the field of Structural Mechanics, namely,

Structural Dynamics and Modal Analysis. It has been conceived aiming at providing the reader with

the knowledge about the essentials of numerical and experimental techniques developed for

characterizing the dynamic behavior of structural systems. In this context, “structural systems” broadly

encompass a large range of engineered products such as civil engineering structures (buildings, towers,

bridges, etc.), vehicles (airplanes, automobiles, trains, ships, spaceships, etc.), industrial equipment

(pipes, off-shore platforms, electric power lines, etc.) and machines (compressors, turbines, internal

combustion engines, etc.) and household appliances (refrigerators, washing machines, air-conditioners,

etc.).

Structural dynamics is the subject of continuously increasing interest in engineering, especially due to

the fact that, in many applications, dynamic behavior is influential upon operational effectiveness,

comfort and safety of the products mentioned above. It should not be forgotten that environmental and

energy-related aspects, which have become subject of concern, are frequently in relation with structural

dynamics

Given the broadness of the topics addressed, this presentation is considered to be introductory. To learn

about more advanced aspects, the reader should refer to a number of excellent text-books available,

some of which are included in the bibliography list provided.

This chapter comprises five sections, which are devoted to the following aspects of structural

dynamics: Section 1: Introduction; Section 2: Vibrations of discrete systems; Section 3: Vibrations of

continuous systems; Section 4: Finite element modeling in structural dynamics; Section 5: Introduction

to experimental modal analysis.

1. Introduction

In the context of the present contribution, Structural Dynamics, is related to the study of the vibratory

behavior of mechanical systems (machines, vehicles, industrial equipment and civil engineering

structures, etc.) when subjected either to sustained or impulsive external forces, which are known as

excitations. Vibrations are considered as oscillations about the equilibrium position of the system and

result from a continuous exchange between kinetic and potential energy. Kinetic energy is related to

the system’s mass or inertia, while the potential energy is associated to the system’s flexibility.

Modal Analysis is understood as the ensemble of analytical and experimental techniques intended for

the modeling of the dynamic behavior of vibrating systems that derive from the fact that, under certain

conditions, the dynamic response can be represented as a superposition of the dynamic responses of

elementary mechanical systems, in terms of the so-called modal characteristics. From the mathematicalstandpoint, modal analysis can be interpreted as a set of techniques intended for solving partial linear

differential equations or systems of linear ordinary differential equations by performing

transformations from the physical coordinates to the so-called modal space.

Modal analysis encompasses two main types of techniques, which will be addressed in the remainder

of this chapter, namely:

Analytical modal analysis, which is primarily related to the modeling of the dynamic behavior in the

modal space, making use of the modal characteristics. Hence, analytical modal analysis is intended to

solve a class of direct problems;




3

Experimental modal analysis, which consists in determining a set of modal characteristics of a given

structural system from a set of measured responses. Thus, experimental modal analysis deals with

inverse or identification problems.

For the sake of better understanding, it becomes necessary to present some basic definitions regardingthe vibratory behavior of dynamic systems.

1.1. Basic Definitions

Inertia: is understood as the system property that quantifies its resistance to being accelerated when

acted upon by external efforts. In this sense, the system mass quantifies the resistance to translational

acceleration, whilst the moment of inertia quantifies the resistance to rotational acceleration. The

inertia is responsible for the storage of kinetic energy.

Flexibility: or, inversely, stiffness, is the property related to the system’s inherent capacity to deformunder the action of external efforts and recover the deformation upon the removal of such efforts. The

flexibility is the property that makes mechanical systems capable of storing strain energy.

Damping: is understood as any energy dissipation mechanism found in mechanical systems, such as

friction and viscous resistance. When damping exists, the total mechanical energy of the system is not

conserved. On the contrary, it is reduced gradually with respect to time. Structural vibration can be

understood as the result of a continuous interchanging between kinetic and strain energy. Part of this

energy can be dissipated by damping mechanisms.

Free vibrations: are related to the dynamic response of the system when subjected to a set of initial

conditions (displacements or velocities) without no sustained external excitations (forces or torques).

Forced vibrations: are related to the dynamic response of the system when subjected to sustained

external excitations. These latter can be of various types such as harmonic, periodic, transient, or

random.

Number of degrees-of-freedom: is the smallest number of independent coordinates that are necessary

to express the position of the mechanical system, completely. Figure 1.1(a) shows that the simple

pendulum is a single degree-of-freedom system, associated to the angular coordinate , while the

double pendulum, shown in Figure 1.1(b) is a two-degree-of-freedom systems, represented to

coordinates 1 and 2.




4

Figure 1.1 - Examples of single-degree-of-freedom and two-degree-of-freedom systems

Discrete or lumped-parameter systems: are those whose physical parameters (inertia, stiffness, and

damping) are concentrated in a finite number of fixed points in space. Consequently, their number of

degrees of freedom is also finite. It will be shown that these systems are mathematically represented by

a set of ordinary differential equations having the time as independent variable. Figure 1.2 illustrates a

three- degree-of-freedom (3 d.o.f) discrete system, in which x1 , x2, x3 are the displacement coordinates

and f 1 , f 2, f 3 are the excitation forces applied to the masses, respectively

Figure 1.2 - Three-degree-of-freedom discrete system

(a) (b)




5

Continuous or distributed-parameter systems: are those whose physical parameters are distributed

in a given volume in space. As an infinite number of points exist in this volume, an infinite number of

degrees-of-freedom are necessary to express mathematically the dynamic behavior of the system. It

will be shown that partial differential equations having time and space coordinates as independent

variables are used to represent continuous systems. Figure 1.3 shows a tout string in transversevibrations as an example of a continuous system. It should be pointed-out that, in reality, all structural

systems are continuous. However, by assuming some hypotheses, then can modeled as discrete

systems.

Figure 1.3 - Vibrating string as an example of continuous system

2. Theoretical Foundations of Structural Dynamics

In the section, the theoretical foundations of structural dynamics and modal analysis are presentedunder the assumption of linear behavior.

2.1. Equations of Motion for Discrete Systems

We consider herein structural systems modeled as the one depicted in Figure 1.2, which are assumed to

exhibit linear behavior. This means that the resilient elements (springs) establish proportionality

between displacements and restoring forces: kx f e and the viscous dashpots (dampers) establish

proportionality between velocities and damping forces: xc f d .

From Newton’s Second Law or, alternatively, from the so-called Lagrange’s equations, it is possible to

obtain the set of coupled second-order differential equations of motion that represent the dynamic

behavior of a viscously damped N d.o.f. mechanical system of, in the following matrix form:

M x t C x t K x t f t (2.1)

where:




6

n

n

R

t x

t x

t x

1

is the vector of time responses;

n

n

R

t f

t f

t f

1

is the vector of excitation forces;

Matrices M (positive-definite), C and nn R K , (positive definite or semi-positive definite). are

the symmetric matrices of mass, damping and stiffness, respectively.

2.2. Undamped Free-Vibrations. Eigenvalues and Eigenvectors

For the undamped system, without any excitation force, the equations of motion given by Eq. (2.1)

become:

0M x t K x t , (2.2)

whose general solution is searched in the form:

t ie xt x (2.3)

This type of solution means that we are assuming harmonic free responses with circular frequency .

By introducing Eq. (2.3) into Eq. (2.2), the following eigenvalue problem is obtained:

0 x M K (2.4)

where 2 .

The problem expressed by Eq. (2.4) admits N pairs of non-trivial solutions r r x, , r =1,2,…,n, the

so-called eigensolutions, where:

Rr are the eigenvalues. The natural frequencies are given by: r r , r = 1, ..., n;

nr R x are the eigenvectors or vibration natural mode-shapes.

The eigenvalues are obtained from the following condition:

det 0K M (2.5)




7

whose development leads to the following characteristic polynomial :

001

1 aa n

nn

(2.6)

For each one of the n eigenvalues, Eq. (2.4) gives the corresponding eigenvector r X . It is usual to

regroup the eigensolutions in the following matrices:

Modal Matrix: nn

n R x x x X ,21

(2.7)

Spectral Matrix: nn

n Rdiag ,21

(2.8)

In general, it is assumed that the diagonal entries of the spectral matrix and the columns in the modal

matrix are ordered according to the increasing magnitudes of the eigenvalues r .

Matrix K can be either positive-definite or positive semi-definite, according to the boundary

conditions (kinematic constraints) of the system: when the constraints are sufficient to prevent any

motion without deformation of the flexible elements K will be positive-definite; on the contrary, K

will be positive semi-definite. In the first case, all the eigenvalues of K will be positive and the

corresponding vibration modes will be named as elastic modes. In the second case, the system will

have a number p of null eigenvalues 6p , to which correspond the so-called rigid body modes and a

set of n p elastic modes that are associated with positive eigenvalues.

Examples: The 3 d.o.f. system represented by Fig 2.1(a) presents a null eigenvalue that corresponds to

the rigid body mode represented by the set of displacements 1 2 3x x x that may occur without

deformation of the springs k1 and k2. On the other hand, the system represented by Fig. 2.1(b) does not

exhibit any null eigenvalue because any arbitrary set of displacements 1 2 3, ,x x x satisfying the

boundary conditions leads to the deformation of at least one of the elastic elements.

Figure 2.1 - Three d.o.f. system under two different boundary conditions

One of the most important property of the eigenvectors, is the so-called Orthogonality Property, with

respect to mass and stiffness matrices, which are demonstrated in the following.

For a given pair of eigensolutions r r x, , Eq. (2.4) is rewritten as:

r r r X M X K

(2.9)




8

For another pair s s x, , one has:

s s s X M X K

(2.10)

Pre-multiplying Eq. (2.9) by T s x and Eq. (2.10) by T

r x , successively, subtracting one resulting

equation from the other, and accounting for the symmetry of matrices [K] and [M], one writes:

0 sT

r r s x M x (2.11)

Assuming distinct eigenvalues, namely r s , Eq. (2.15) is satisfied only if:

0 s

T

r

x M x (2.12)

0 sT

r x K x , (2.13)

for each and every pair of eigenvectors with r s.

Altogether, equations (2.12) and (2.13) express the orthogonality properties of the eigenvectors, with

respect to the mass and stiffness matrices, respectively. This is a fundamental property that is the basis

of many of the vibration analysis methods, as will be seen in the remainder.

From Eq. (2.4) it is possible to see that if r x is an eigenvector, any collinear vector r x , with 0

is also an eigenvector. This means that the norms of the eigenvectors are not determined uniquely and

can be chosen arbitrarily, so that they satisfy:

, 1,2, ,

T

r r r

T

r r r r

x M xr n

x K x

(2.14)

where , 1,2, ,r r n are the so-called generalized masses, whose values must be arbitrarily

prescribed. Usually the eigenvectors are normalized in such a way to have unit generalized masses, i.e.:

1, 1,2, ,

Tr r

T

r r r

x M xr n

x K x

(2.15)

Based on the definitions given by equations (2.7) and (2.8), the orthogonality relations (2.12) and

(2.13), and the normalization equations (2.15) can be grouped in the following matrix equations, in

terms of the spectral and modal matrices:

T

X M X N (2.16)

T

X K X N (2.17)




9

where:

1 2, , , r N diag (2.18)

is the matrix of generalized masses.


The orthogonality property of the eigenvectors demonstrated in the previous section ensure that, under

the assumption of distinct eigenvalues, the eigenvectors of a n d.o.f. vibratory system form a set of n

linearly independent vectors. In the context of Linear Algebra, this set is known to constitute a vector

basis of the n-dimensional space which comprises the vectors that represent all the possible forms of

motion of the system satisfying the prescribed boundary conditions. This means that any response of

the system (free or forced motions) can be expressed uniquely as a linear combination of the n

eigenvectors, as follows:

1

n

r r

r

x t x c t X c t

(2.19)

where r c t are the coefficients of linear combination, which are grouped in the vector

T n t ct ct ct c 11 .

Equation (2.19) expresses the Expansion Theorem or Principle of Modal Superposition, which represents the

basis of all Modal Analysis procedures for linear mechanical systems.

2.4. Free Responses of Undamped Systems to Initial Conditions

For an n d.o.f. undamped system subjected to a set of prescribed initial conditions, the equations of

motion are obtained from Eq. (2.1), resulting:

0M x t K x t (2.20)

satisfying:

0 00 , 0x x x x (2.21)

By using the Expansion Theorem, the solution of eq. (2.20) is written as:

1

n

r r

r

x t X c t x c t

(2.22)




10

By introducing Eq. (2.22) into Eq. (2.20) and pre-multiplying the resulting equation by T

X , one has:

0T T

X M X c t X K X c t (2.23)

Taking into account the orthogonality/norm relations (2.16) and (2.17), Eq. (2.23) becomes:

0N c t N c t (2.24)

As N and are diagonal matrices, Eq. (2.24) is comprised by n uncoupled second order differential

equations of the type:

2 0, 1,2, ,r r c t t r n (2.25)

Each equation in (2.25) is similar to the equation of motion of a single d.o.f. undamped system, the

solution of which is given below:

t Dt C t c r r r r r sincos (2.26)

Then, by introducing Eq. (2.26) into Eq. (2.22), one writes:

r

n

r

r r r r xt Dt C t x

1

sincos (2.27)

The 2n constants C r and Dr , r = 1,..., n appearing in Eq. (2.27) are obtained by imposing the initial

conditions, as follows:

0

1

0

n

r r

r

x x C x

(2.28)

By pre-multiplying Eq. (2.28) by T

sX M , accounting for the orthogonality relations, one has:

s sr T

s

n

r s

T s C x M xC x M x 1

0 (2.29)

Then:

0

1, 1, ,

T

s ss

C x M x s n

(2.30)

By deriving Eq. (2.27) with respect to time, one obtains:




11

r

n

r

r r r r r xt Dt C t x

1

cossin (2.31)

from which:

n

r

r r r x D x1

0 (2.32)

By following a procedure similar to that presented above for dealing with initial displacements, one

has:

s sr r T

s

n

r

r r T

s D x M x D x M x 1

0 (2.33)

from which:

0

1 x M x D

T s

s s s

, s=1,2,...,n (2.34)

By associating equations (2.27) and (2.30) and (2.34) one obtains a general expression for the system’s

response to an arbitrary set of initial conditions. It is worth mentioning that Eq. (2.27) states that the

system’s time response is a linear combination of the eigenvectors, in which the coefficients of linear

combination are harmonic functions whose frequencies correspond to the natural frequencies of the

system ωr , r = 1,…n.

2.4.1. Systems with Rigid Body Modes

As discussed before, in the cases in which the kinematic constraints do not preclude the existence of

rigid-body modes, the systems present a number p of null eigenvalues 6p . In such cases, for

convenience, matrices X , and N are partitioned according to:




12

where the index 1 holds for the rigid body modes and the index 2 refers to the elastic modes.

By using the orthogonality relations (2.16) and (2.17), one writes:

1 1 1

1 2

2 2 2

0

T

T T

T

X M X N

X M X N X M X

X M X N

(2.35)

1 1 1

1 2

2 2 2 2

0 0

0

T

T T

T

X K X K X

X K X N X K X

X K X N

(2.36)

For each rigid-body mode pr r r ,...,2,1,02 corresponding modal equation of motion (2.25)

simplifies to:

0, 1, ,r c t r p (2.37)

whose solution is expressed as follows:

, 1, ,r r r c t C D t r p (2.38)

As seen before, for the elastic modes, the solution is given by:

n p pr t Dt C t c r r r r r ,...2,1,sincos (2.39)

Returning to Eq. (2.27), the complete solution is obtained:

p

r

n

pr

r r r r r r r r xt Dt C xt DC t x1 1

sincos (2.40)

By introducing the initial conditions, and following the procedure described previously in Section 2.5,one obtains the following expressions for the constants C r and Dr :

0

1, 1, ,

T

r r r

C x M x r n

(2.41)

0

0

1, 1, ,

1, 1, ,

T

r r

r T

r r r

x M x r p

D

x M x r p nw

(2.42)




13

The association of (2.41), (2.42) and (2.27) provides the complete expression for the response to initial

conditions of a n d.o.f system containing p rigid-body modes.

2.5. Harmonic Vibrations of Undamped Systems

We are interested here in the vibration response due to permanent harmonic excitations. should be

remembered that the complete response of the system is obtained as the superposition of the free

response due to the initial conditions and the forced responses, engendered by the excitation forces. In

this section, only these latter are considered and the former have been addressed in section 2.5.

Assuming harmonic excitation with a forcing frequency , the equations of motion of a n d.o.f.

undamped mechanical system under harmonic excitation are given by:

t ie F t x K t x M

(2.43)

where n R F is the vector of the amplitudes of the external forces.

The steady-state harmonic solution of Eq. (2.43) is written as(1)

:

t ie X t x (2.44)

where n R X is the vector of the amplitudes of the harmonic responses.

By substituting Eq. (2.44) into Eq. (2.43) one obtains the following system of frequency-dependent

linear equations:

F X M K 2 (2.45)

or:

F X Z (2.46)

where nn R M K Z ,2 is the dynamic stiffness matrix.

By multiplying Eq. (2.46) by 1

Z one obtains the vector of harmonic response amplitudes:

F H X (2.47)

where:

nn R M K H ,12

(2.48)




14

is the so-called dynamic flexibility matrix, frequency response function (FRF) matrix or receptance

matrix.

For better understanding, equation (2.47) is expanded according to:

nnnnn

n

n

n F

F

F

H H H

H H H

H H H

X

X

X

2

1

21

22121

11211

2

1

(2.49)

It can be seen, from Eq. (2.49), that the response amplitudes are linear combinations of the elements of

the dynamic flexibility matrix. It should be remembered that, according to Eqs. (2.43) and (2.44), it is

assumed that all the excitation forces have same excitation frequencies. The cases in which the system

is excited by forces with various frequencies can be dealt with by using the superposition principle.

It is convenient to express the matrix H in terms of the system’s eigensolutions. With this aim,

starting from Eq. (2.48), the following equation is written:

1 2T T T

X H X X K X X M X

(2.50)

By using the orthogonality relations (2.16) and (2.17), after some manipulations, the equation above

leads to:

1

12 TH X I N X

(2.51)

It is possible to write (2.51) in the following form:

T r r

n

r r r

x x H

1

22

1

(2.52)

The general element of the matrix H takes the form:

jr ir

n

r r r

ij x x H

1

22

1

(2.53)

where ir x represents the i-th component of the eigenvector r x .

If the system has p rigid-body modes pr r r ,...,2,1,02 , Eq. (2.53) can be cast as follows:

jr ir

n

pr r r

jr ir

p

r r

ij x x x x H

1

22

1

2

11

(2.54)




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2.6. General Forced Responses of Undamped Systems

We are now interested in the solution of the forced vibration problem, which is mathematicallymodeled by the following non-homogenous system of differential equations:

M x t K x t f t , (2.55)

subjected to the following initial conditions:

0

0

0

0

x x

x x

(2.56)

where nf t R is the vector of external forces applied to the system. These forces can have an

arbitrary variation with time (harmonic, periodic, transient, random, etc.)

Once again, the Expansion Theorem is used, so that the solution of Eq. (2.55) is written as:

1

n

r r

r

x t X c t x c t

(2.57)

Equation (2.57) is introduced into Eq. (2.55). The resulting equation is pre-multiplied by T

X and the

orthogonality relations (2.16) and (2.17) are used, leading to:

N c t N c t g t (2.58)

where:

T

g t X f t (2.59)

The system (2.58) is comprised of n uncoupled, non-homogeneous differential equations of the form:

nr t g t ct c r r

r r r ,...,2,1,12

(2.60)

From the theory of linear differential equations, it is known that the general solution of Eq. (2.60) is

given by:

h pr r r c t c t c t , (2.61)

where hr c t is the solution of the associated homogenous differential and p

r c t is a particular solution

of Eq. (2.67).




16

The solution hr c t is given by Eq. (2.26), rewritten here:

t Dt C t c r r r r hr sincos (2.62)

The particular solution pr c t

, obtained by the Method of Variation of Parameters, is :

0

1, 1, ,

t

pr r r

r

c t g h t d r n

(2.63)

where:

t t h r r

r

sin1 (2.64)

is recognized as the impulse response function associated to the r -th vibration mode. This function

gives the contribution of the r -th mode to the response of an ideal excitation represented by (0)

(Dirac’s delta function) applied at t = 0.

Association of equations (2.57), (2.61), (2.62) and (2.63) yields:

n

r

r

t

r r r r

r r r r xd t g t Dt C t x1 0

sin1sincos

(2.65)

As shown in the previous sections, the constants C r and Dr , r = 1, ..., n are calculated by taking into

account the initial conditions, making use of the orthogonality properties of the eigenvectors:

0

1 x M xC

T r

r r

(2.66)

0

1 x M x D

T

r r r r

, r =1,2,...,n (2.67)

Once more it is possible to observe, in Eq. (2.65), that the system’s general response results from the

sum of the contributions of the n vibration modes, being expressed by a linear combination of the

eigenvectors, in which the coefficients of linear combination are functions of time which depend on the

corresponding natural frequencies.

The response of systems having rigid-body modes can be easily obtained by following a procedure

similar to the one presented in Section 2.5.1.




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2.7. Free Responses of Viscously Damped Systems

Taking a particular case of the more general equations of motion (2.1) the following mathematical

model of the structural system, accounting for viscous damping, is considered:

0M x t C x t K x t (2.68)

with the initial conditions:

0

0

0

0

x x

x x

(2.69)

For obtaining the responses of viscously damped systems, two different cases regarding the damping

matrix should be considered separately, as follows.


C M K K M C

is satisfied

It can be demonstrated that this is a sufficient condition for generalized damping matrix

[β]= T

X C X to be diagonal. It can be easily verified that the widely known model of proportional

damping or Rayleigh damping, in which the damping matrix is expressed as a linear combination of

mass and stiffness matrices, ( K b M aC ) is a particular case of this more general condition.

Under this condition, it becomes still convenient to express the response of the damped system as alinear combination of the eigenvectors of the associated undamped system (solutions of the eigenvalue

problem (2.4)), as follows:

1

n

r r

r

x t X c t x c t

(2.70)

By replacing Eq. (2.70) into Eq. (2.68), pre-multiplying the resulting equation by T

X , and making

use of the orthogonality equations given by (2.16) and (2.17), one obtains:

0N c t c t N c t (2.71)

where nT

diag X C X 11 .

The system given by Eq. (2.71) is formed by set of n uncoupled differential equations of the type:

02 2 t ct ct c r r r r r r (2.72)

where:




18

r r

r r

2 (2.73)

are the so-called modal damping factors.

The nature of the system’s response depends strongly on the amount of damping. For the cases in

which damping is small ( 1, 1, ,r r n , i.e., sub-critical damping), the general solution to Eq. (2.72)

takes the form:

t Dt C et cr d r r d r

t r r r

sincos (2.74)

where:

21 r r d (2.75)

is known as damped natural frequency.

Replacing Eq. (2.74) into Eq. (2.70) yields:

r

n

r r d r r d r

t r r xt Dt C et x

1

sincos (2.76)

By enforcing the initial conditions to the solution (2.76) the following expressions for the constants C r ,

Dr , r = 1, ..., n are obtained:

0

1 x M xC

T r

r r

(2.77)

02

02 11

1 x M x x M x D

T r

r r

r T r

r r r

r

(2.78)


C M K K M C

is satisfied

For this case, which is known as general viscous damping , it can be shown that, as opposed to the case

treated in Section 2.8.1, the modal transformation based on the eigenvectors of the associated

undamped system cannot provide the simultaneous diagonalization of matrices [ M ], [C ] and [ K ]. As a

result, the modal transformation must be made in terms of the modal basis of the damped system. With

this aim, the system has to be modeled in the state-space 2nR , as follows:

0M x t C x t K x t

M x t M x t

(2.79)




19

or:

U y t A y t (2.80.a)

with the following initial conditions:

N R y y 00 (2.80.b)

where:

n N R M

K A R

M

M C U N N N N 2

0

0,

0

,,

(2.81)

N

x ty t

x t

R (2.82)

The solutions of (2.80) are searched in the form:

sty t y e (2.83)

Substituting Eq. (2.83) into Eq. (2.80), the following eigenvalue probem is obtained:

0 A s U y (2.84)

For small damping, the solutions of Eq. (2.84) are given by complex quantities:

Eigenvalues:

d r

d r r r i s , d

r d r r r i s nr ,...,2,1, (2.85)

Eigenvectors:

, N

d r r

d r

r C x s

x y

nr C x s

x y N

d r r

d r

r ,...,2,1,

(2.86)

where the overscore indicates complex conjugates.

It can be easily demonstrated that the complex eigenvectors satisfy the following orthogonality

relations:

N sr

s y A y

yU y

s

d

srs s

T

r

d srs s

T r ,2,1,

(2.87)




20

where

sr

sr rs

,1

,0

The following matrices are defined:

N N n C s sdiag S ,

1 : spectral matrix (2.89)

N N n C y yY ,

1 : modal matrix (2.90)

N N d N

d d C diag N ,1 : matrix of generalized masses (2.91)

Based on these definitions, the two sets of orthogonality relations, Eq. (2.87), are expressed by:

T dY U Y N (2.92)

T dY A Y N S (2.93)

Once the eigensolutions are obtained, the Theorem of Expansion is used so that the solution of the

initial-value problem (2.80) is written as a linear combination of the N complex eigenvectors:

N

r

r r t cY yt ct y1

(2.94)

Substituting Eq. (2.94) into Eq. (2.80.a) and pre-multiplying the resulting equation by T

Y , accounting

for the orthogonality relations given by equations (2.92), the following equation is obtained:

t cS t c , (2.95)

which encompasses N uncoupled first order ordinary differential equations of the form:

, 1, ,r r r c t s c t r N (2.96)

whose solution is given by:

r s tr r c t A e (2.97)

Introducing Eq. (2.97) into Eq. (2.94), one writes:

1

r

Ns t

r r

r

y t A y e

(2.98)




21

By enforcing the initial conditions to the solution (2.98), using the orthogonality relations (2.92-2.93),

the constants , 1, ,r A r N are obtained as follows:

0

1 T

r r dr

A y U y , (2.99)

so that the free response of the system is finally expressed as:

0

1

1r

NT s t

r r dr r

y t y U y e y

(2.100)

2.8. Harmonic Responses of Viscously Damped Systems

Consider the following system of equations of motion:

M x t C x t K x t f t (2.101)

where:

i tf t f e , (2.102)

being the frequency of the harmonic excitation.

We are interested in obtaining the steady state harmonic responses, which are due to the excitation

forces, exclusively. For computing the total response of the vibrating system, theses responses must be

superimposed to those associated to the initial conditions, which have been addressed in the preceding

Section.

Only the case of general viscous damping ( 1 1

C M K K M C

) will be considered.

In this case, the equations of motion in state-space representation, previous introduced in Section 2.8,

are used:

i tU y t A y t g e (2.103)

where:

0

Nf g

R (2.104)

The solution to Eq. (2.103) is searched in the form:




22

t iecY t y (2.105)

Introducing Eq. (2.105) into Eq. (2.103), pre-multiplying the resulting equation by T

Y and making use

of the orthogonality equations (2.92), one obtains:

1 Td dc i N S N Y g

(2.106)

By introducing (2.106) into Eq. (2.105), the following relation is obtained between the amplitudes of

the responses and the amplitudes of the excitation forces:

Y H g (2.107)

where:

1 1 TdH Y N i I S Y

(2.108)

is the complex receptance matrix.

For the case of light damping, in which the eigensolutions appear in complex conjugate pairs, the

receptance matrix can be expressed as:

1 1

TTn n

r r r r

d dr r r r r r

y yy y

H i s i s

(2.109)

and the general element i jH of the receptance matrix is given by:

n

r r d

r

jr ir

r d r

jr ir ij

si

y y

si

y y H

1 (2.110)

2.9. Forced Responses of Viscously Damped Systems

We are interested in the systems exhibiting general viscous damping ( 1 1

C M K K M C

),

represented by the following equations of motion in the state-space:

U y t A y t g t (2.111)

with the initial conditions:

00 y y




23

The excitation vector

0

Nf t

g t

R

is assumed to be formed by forces having arbitrary variations with time.

Once again, the modal transformation (2.94) and the orthogonality relations (2.92-2.93) are associated

to (2.103), leading to:

Td dN c t N S c t Y g t

(2.112)

The system above is represented by a set of N uncoupled differential equations of the type:

N r t g yt pt pt c st cT

r r r d r

r r r ,,2,1,,1

(2.113)

Accounting for the initial conditions, the solutions of the non-homogeneous differential equations

(2.113) can be obtained by using the Method of Varying Parameter, as follows:

N r d g yee yU yt c

t T

r t r st r sT

r d r

r ,,2,1,1

0

0

(2.114)

Finally, by introducing Eq. (2.114) into (2.94), the system’s response, including the response to the

initial conditions and the response to the excitation forces is obtained as follows:

2

0

1 0

1r r

tnT Ts ts t

r r r dr r

y t y U y e e y g d y

(2.115)

2.10. Examples

In this section, numerical examples are presented aiming at illustrating some of the main aspects of the

theory presented in the previous sections.

Example 1: Eigensolutions of undamped systems. For the three d.o.f. systems depicted in Fig. 2.1(a)

and 2.1(b) the following values of the physical parameters are assumed:

1 2 3

5 5 510 12 23

10 12 23

1.0 kg 0.5kg 1.0kg

1.0 10 N/m 2.0 10 N/m 1.0 10 N/m

10.0 N.s/m 20.0 N.s/m 40.0 N.s/m

m m m

k k k

c c c




24

The equations of motion for both systems, in the general form (2.1) can be obtained either by applying

Newton’s laws or energy principles (Hamilton Variational Principle or Lagrange Equations). For each

system, the mass and stiffness matrices take the following forms:

For the system illustrated in Fig. 2.1(a)

For the system shown in Fig 2.1(b)

1

2

3

10 12 125

12 12 23 23

23 23

10 12 12

12 12 23 23

23 23

0 0 1.0 0 00 0 0 0.5 0 kg

0 0 0 0 1.0

0 3.0 2.0 0

2.0 3.0 1.0 10 N/m

0 0 1.0 1.0

0 30 20 02

0

m M m

m

k k k

K k k k k

k k

c c cC c c c c

c c

0 60 40 Ns/m

0 40 40

1

2

3

12 12

512 12 23 23

23 23

10 12 12

12 12 23 23

23 23

0 0 1.0 0 0

0 0 0 0.5 0 kg

0 0 0 0 1.0

0 2.0 2.0 0

2.0 3.0 1.0 10 N/m

0 0 1.0 1.0

0 20 20 0

20 60

0

m

M m

m

k k

K k k k k

k k

c c c

C c c c c

c c

40 Ns/m

0 40 40




25

The eigensolutions of the associated undamped systems (with damping neglected) are obtained bysolving numerically the eigenproblem (2.4) and are shown in Table 2.1, while the complex

eigensolutions of the viscously damped systems, calculated by solving the eigenvalue problem (2.84)

are given in Table 2.2. It can be verified that both systems are included in the case of general viscous

damping, in which the condition 1 1

C M K K M C

is satisfied (refer to Section 2.8.2)

Table 2.1 – Eigensolutions of the associated undamped systems

System illustrated in Fig. 2.1(a) System illustrated in Fig. 2.1(b)

Natural frequencies Mode shapes Natural frequencies Mode shapes

1 0.0 rad/s

633.0

633.0

633.0

1 X

1 167.38 rad/s

806.0

580.0

427.0

1 X

2 360.34 rad/s

752.0

224.0

640.0

2 X

2 424.67 rad/s

566.0

454.0

760.0

2 X

3 877.59 rad/s

186.0

245.1

437.0

3 X

3 889.74 rad/s

175.0

207.1

491.0

3 X

Table 2.2 – Eigensolutions of the damped systems

System illustrated in Fig. 2.1(a) System illustrated in Fig. 2.1(b)

Complex eigenvalues Damped natural

frequencies and

damping factors

Complex eigenvalues Damped natural

frequencies and

damping factors

1 s 0.0 (rad/s)

1 s 0.0 (rad/s) ---1 s -2.2 + 167.41i (rad/s)

1 s -2.2 -167.41i (rad/s)

1 167.41(rad/s)

2 0.0129

i360.2+-20.82 s (rad/s)

i360.2--20.82 s (rad/s)

360.22 (rad/s)

2 0.0578

i424.5+-24.72 s (rad/s)

i424.5--24.72 s (rad/s)

2 424.5 (rad/s)

2 0.0581

i873.6+-69.23 s (rad/s)

i873.6--69.23 s (rad/s)

873.63 (rad/s)

3 0.0792

i885.8+-68.23 s (rad/s)

i885.8--68.23 s (rad/s)

8.8583 (rad/s)

3 0.0770




26

The results above demonstrate the existence of a rigid-body mode for the system of Fig. 2.1(a),

associated to a null natural frequency. The eigenvectors have been normalized to unit generalized

masses according to Eq.(2.14).

Example 2: Free responses to initial conditions of undamped systems . Here one considers the same

three d.o.f. systems considered in Example 1, submitted to the following sets of initial conditions:

The free responses are computed according to the formulation presented in Section 2.5. Figures 2.2 and

2.3 below depict the variations of the coefficients , 1,2,3r

c t r , appearing in Eq. (2.22), as well as

displacement responses of each mass of the system, in the interval [0; 5s]. As can be seen, each the

coefficient r c t has harmonic variation in time and its period is determined by the value of the

corresponding natural frequency.

0 0.01 0. 02 0. 03 0.04 0.05 0. 06 0.07 0.08 0.09 0. 1-0.015

-0.01

-0.005

0

0.005

0.01

0.015

time (s)

c1(t)

c2(t)

c3(t)

0 0. 01 0. 02 0. 03 0. 04 0. 05 0. 06 0. 07 0. 08 0.09 0. 1-8

-6

-4

-2

0

2

4

6

8

10

12x 10

-3

time (s)

d i s p l a c e m e n t s

( m )

x1(t)

x2(t)

x3(t)

Figure 2.2 - Responses to initial conditions for the system illustrated in Fig. 1(a)

0 0

0.005 -2.00

0.008 (m) 0.00 (m/s)

-0.005 2.00

x x




27

0 0. 01 0. 02 0. 03 0. 04 0.05 0.06 0.07 0.08 0.09 0. 1-0.015

-0.01

-0.005

0

0.005

0.01

0.015

time (s)

c1(t)

c2(t)

c3(t)

0 0. 01 0. 02 0. 03 0. 04 0. 05 0. 06 0. 07 0. 08 0. 09 0. 1-0.015

-0.01

-0.005

0

0.005

0.01

0.015

time (s)

d i s p l a c e m e n t s ( m )

x1(t)

x2(t)

x3(t)

Figure 2.3 - Responses to initial conditions for the system illustrated in Fig. 1(b)

Example 3: Harmonic responses of undamped systems. Here one considers the same three d.o.f.

systems considered in Example 1, submitted to harmonic excitations. The elements of the first column

of the receptance matrix H are calculated according to Eq. (2.48) in the interval [0; 200 Hz]. In

the figures 2.4 and 2.5, are shown, for each system, the contributions of the individual vibration modes

as well as the combined contribution of the three modes.




28

0 20 40 60 80 100 120 140 160 180 200

-160

-140

-120

-100

-80

-60

-40H(1,1)

frequency (rad/s)

A m p l i t u d e ( d B ) - r e f . = 1

C(1)C(2)

C(3)C(1)+C(2)+C(3)

0 20 40 60 80 100 120 140 160 180 200

-160

-140

-120

-100

-80

-60

-40H(2,1)

frequency (rad/s)

A m p l i t u d e ( d B ) - r e f . = 1

C(1)

C(2)

C(3)

C(1)+C(2)+C(3)

0 20 40 60 80 100 120 140 160 180 200

-160

-140

-120

-100

-80

-60

-40H(3,1)

frequency (rad/s)

A m p l i t u d e ( d B ) - r e f . = 1

C(1)

C(2)

C(3)

C(1)+C(2)+C(3)

Figure 2.4 - Amplitudes of the FRFs for the system illustrated in Fig. 1(a)

0 20 40 60 80 100 120 140 160 180 200-160

-140

-120

-100

-80

-60

-40H(1,1)

frequency (rad/s)

A m p l i t u d e ( d B ) - r e f . = 1

C(1)

C(2)

C(3)

C(1)+C(2)+C(3)

0 20 40 60 80 100 120 140 160 180 200-160

-140

-120

-100

-80

-60

-40H(2,1)

frequency (rad/s)

A m p l i t u d e ( d B ) - r e f . = 1

C(1)

C(2)

C(3)

C(1)+C(2)+C(3)




29

0 20 40 60 80 100 120 140 160 180 200-160

-140

-120

-100

-80

-60

-40H(3,1)

frequency (rad/s)

A m p l i t u d e ( d B ) - r e f . = 1

C(1)

C(2)

C(3)

C(1)+C(2)+C(3)

Figure 2.5 - Amplitudes of the FRFs for the system illustrated in Fig. 1(b)

3. Continuous Systems

We consider in this section continuous, or distributed parameter systems, which have been defined

previously in Section 1. In the realm of structural dynamics, various types of structural components,

such as cables, beams, plates and shells can be modeled as continuous vibrating systems. For each of

them, specific hypotheses in terms of the space dimension and kinematics must be considered. This fact

makes it difficult to develop a general theory encompassing the different types of structural elements of

interest. As a result, for the purpose of illustrating the main theoretical aspects involved in the

mathematical modeling and modal analysis of continuous systems, we shall consider, in this Section,taut cables in transverse vibrations, which are considered to be an adequate example.

3.1. Equations of Motion

Consider the flexible string of length L that vibrates under the action of a distributed load ,f x t and a

tension force T , as illustrated by Fig. 3.1. The amplitudes of the deflections ,y x t are assumed to be

small, which enables to model the system as linear.




30

Figure 3.1 - Vibrating String

In Figure 3.1, one has:

,f x t : distributed force per unit length [N/m]

x : material linear density (kg/m)

T x : tension (N)

,y x t : lateral displacement (m)

By applying Newton’s second law to the differential element depicted in Figure 3.1(b) for the

components in the direction y, one has:

dx

x

t x y

x

t x ydx

x

xT xT

x

t x y xT dxt x f dx

t

t x y x

2

2

2

2 ,,,,

, (3.1)

Neglecting the second-order terms with respect to dx, Eq. (3.1) becomes:

L x x

t x y xT

xt x f

t

t x y x

0

,,

,2

2

(3.2)

The boundary conditions and initial conditions of the problem are expressed as:

0, , 0y t y L t , t (3.3.a)

dx x

y

x

y2

2




31

L x x y x y x y x y 0,0,,0, 00 (3.3.b)

3.2. Eigenvalue Problem

Consider the case of free vibrations in which , 0f x t . For this case, Eq. (3.2) is written as:

L x x

t x y xT

xt

t x y x

0

,,2

2

(3.4)

The solution to Eq. (3.4) is searched under the form, which can be interpreted as a separation of

variables:

,y x t Y x C t (3.5)

By introducing Eq. (3.5) into (3.4) one obtains:

L x

dx

xdY xT

dx

d

xY xdt

t C d

t C

0

112

2

(3.6)

It can be demonstrated that Eq. (3.6) holds only if both sides take a same constant value, which will be

for convenience, noted as – ω2. Then:

02

2

2

t C dt

t C d (3.7)

L xdx

xdY xT

dx

d xY x

02 (3.8)

Equation (3.8) is a linear ordinary second order differential equation whose solution is known to be of

the form:

t C t C t C sincos 21 (3.9)

where 1C and 2C are constant. It becomes now clear that ω is the circular frequency of the transverse

motion. Together, this frequency and the function xY , which characterize the spatial distribution of

the amplitudes of motion, must be determined to fully characterize the string’s free motion.




32

Equation (3.8), complemented with the boundary conditions (3.3.a) constitutes an eigenvalue problem

which is similar in nature to the eigenvalue problem pertaining discrete systems, expressed by Eq.

(2.4). The solutions of this problem are the following:

2r , ,,2,1 r eigenvalues, which correspond to the square of the natural frequencies;

,,2,1, r xY r eigenfunctions, which correspond to the vibration mode shapes.

The eigenfunctions satisfy the orthogonality relations as demonstrated in the following.

Equation (3.8) is written for two different sets of eigensolutions xY xY s sr r ,,, :

2r r r

dY xdT x x Y x

dx dx

(3.10.a)

2ss s

dY xdT x x Y x

dx dx (3.10.b)

Multiplying (3.10.a) by sy x , (3.10.b) by r y x and integrating the resulting equations by parts in

the interval 0 < x < L , ones gets, respectively:

2

0 0 0

L L L

r r ss r s r

dY x dY x dYY x T x T x dx x Y x Y x dx

dx dx dx (3.11.a)

2

0 0 0

L L L

s s r r s s r

dY x dY x dYY x T x T x dx x Y x Y x dx

dx dx dx (3.11.b)

By subtracting (3.11.a) from (3.11.b) and taking into account the boundary conditions (3.3.a), one

obtains:

2 2 0

L

r s r s

o

x Y x Y x dx

Assuming that the eigenvalues are distinct, i.e., ωr ≠ ω s, the equation above leads to:

0

0

L

r sx Y x Y x dx (3.12.a)

The property of the eigenfunctions, expressed by Eq. (3.12), in which the density operator x is

involved, is equivalent to the orthogonality property of the eigenvectors of undamped discrete systems,

expressed by (2.12).




33

Introducing (3.12.a) into (3.11), the following orthogonality property satisfied by the derivatives of the

eigenfunctions which respect to x is obtained:

0

0

L

r sdY x dY xT x dxdx dx

(3.12.b)

It is convenient to normalize the eigenfunctions so that:

2

0

, 1,2,

L

r r x Y x dx r (3.13.a)

2

2

0

, 1,2,

L

r r r

dY xT x dx r

dx

(3.13.b)

where ,2,1, r r are normalization parameters known as generalized masses, whose values are

arbitrarily chosen. Most frequently, unit values are adopted, i.e., 1r , 1,2,r


The orthogonality properties described in the previous section ensure that the eigenfunctions r Y x ,

1,2,r form a complete basis of the vector space of containing all the displacement fields ,y x t of

the string compatible with the prescribed boundary conditions. As a result, the Expansion Theorem

can be announced as follows: the free or forced response of the continuous system to arbitrary sets of

initial conditions and forcing functions can be expressed uniquely as a linear combination of the

eigenfunctions, as follows:

t C xY t x y r

r

r

1

, (3.14)

By introducing (3.14) into (3.4), multiplying the resulting equation by sY x , integrating in the interval

0 < x < L , and taking into account of the orthogonality relations (3.10) one gets:

,2,1,02 r t C t C r r r (3.15)

Equations (3.15) can be considered as representing the equations of motion of the continuous system in

the modal coordinates t C r . Their solutions take the form:

,2,1,sincos r t Bt At C r r r r r (3.16)

It should be pointed-out that (3.15) is totally in accordance with (3.7).




34

Finally, the solution of (3.4) is expressed in terms of physical coordinates by inserting (3.16) in (3.16):

1

, cosr r r r r

r

y x t Y x A t B sen t (3.17)

where the coefficients Ar and Br are determined by the initial conditions.

3.4. Free Response to Initial Conditions

The interest here is to obtain the vibration response of the cable to an arbitrary set of initial conditions

expressed by (3.3.b), based on the solution (3.17). With this aim, the constants Ar and Br must be

determined by enforcing the initial conditions as follows:

r

r

r A xY x y

1

0 (3.18.a)

r r

r r B xY x y

1

0 (3.18.b)

Pre-multiplying each equation above by a specific weigthed eigenfunction xY x s , integrating both

sides of the resulting equations in the interval L x 0 , and making use of the orthogonality and norm

relations (3.12.a) and (3.13.a), one obtains the following expressions for the coefficients r A and r B :

dx xY x y x A

L

r r

r 0

0

1

(3.19.a)

dx xY x y x B

L

r r r

r 0

0

1

(3.19.b)

By associating equations (3.19) and (3.17), the complete expression for the vibration response is

obtained.

3.5. General Forced Responses

We shall consider the equation of motion considering a general force function t x f , , obtained from

(3.2):

L x

x

t x yT t x f

t

t x y

0

,,

,2

2

2

2

(3.20)




35

By introducing (3.14) into (3.20), multiplying the resulting equation by a specific weigthed

eigenfunction xY x s , integrating in the interval 0 < x < L , and making use of the orthogonality and

norm relations (3.12) and (3.13), one obtains the following uncoupled equations of motion in terms ofmodal coordinates:

,2,1,12 r t f t C t C r r

r r r

(3.21.a)

with:

,2,1,,

0

r dxt x f xY t f

L

r r (3.21.b)

It can be seen that (3.21.a) has the same for as (2.60). As a result, we can follow the same resolution

procedure discussed in Section 2.7, which is leads to:

xY d t f t Dt C t x yr

r

t

r r r r

r r r r

1 0

sin1

sincos,

(3.22.a)

with:

dx x y xY C r

L

r r 0

0

1

(3.22.b)

dx x y xY D r

L

r r r 0

0

1

, r =1,2,..., (3.22.c)

3.6. Examples

In the examples presented in this section, it will be considered a uniform taut cable subjected to a

constant tension force, i.e., ,T constant (independent from x).

3.6.1. Natural Frequencies and Eigenfunctions

Under these assumptions above, the eigenvalue problem (3.8) becomes:




36

2

2 0, 0

d Y xY x x L

dx (3.23)

with

0 0Y Y L (3.24)

22

T (3.25)

The general solution of (3.23) is:

cosY x Asen x B x (3.26)

By taking into account the boundary conditions (3.21) into (3.23), one gets:

0B (3.27)

0 Asen L (3.28)

As we are looking for non trivial solutions, we must have A 0 and:

0sen L (3.29)

Equation (3.29) is called characteristic equation or frequency equation, whose solutions are given by:

,2,1, r r Lr (3.30)

By associating Eqs. (3.22) and (3.26) one obtains the expressions for the natural frequencies of the

string:

,2,1,2

r L

T r r

(3.31)

By introducing (3.25) into (3.30), the following expressions for the eigenfunctions are obtained:

,2,1,sin r L

xr A xY r r

(3.32)

The first three eigenfunctions of the string are illustrated in Fig. 3.2.




37

Figure 3.2 - Eigenfunctions of the string

We shall normalize the eigenfunctions (2.26) to unit generalized masses, according (3.13.a):

2

0

1L

r Y x dx ,1sin

0

2

L

r dx L

xr A

,2,1,

2 r

L Ar

The normalized eigenfunctions are given by:

,2,1,sin2

r L

xr

L xY r

(3.33)

3.6.2. Free Response to Initial Conditions

Let us assume that the string is submitted to a short-time stimulus which is represented by a initial

velocity2

0 L

V applied at the mid-point of the string as shown in Fig. 3.3.

20 L

V

2 L 2 L




38

Figure 3.3 – Scheme of a string subjected to an impulsive load

The initial conditions are then represented as:

L x L

xV y y L

o

0,

2,0

200

where x is the Dirac’s Delta function.

From (3.19), one gets:

,2,1,0 r Ar

oddfor 21

pair for ,0

2sin

212

0

220

2

r V T

L

r

r r

V T

L

r B

L L

r

From (3.17), the response of the string is found to be expressed according to:

,5,3,12

220 sin

12,

r

L

t L

T r

r T

LV t x y

(3.34)

4. Finite Element Modeling in Structural Dynamics

As shown in Chapter 3, the problems involving the dynamic behavior of continuous systems are

represented by partial differential equations in which the independent variables are spatial coordinates

and time. In this case, the problems present an infinite number of unknowns that are the values of

displacement for each considered time instant for every one of the infinite number of points of the

continuum.

In most cases the exact analytic solution of the equations of motion is impossible to be obtained due to

various factors, such as complex boundary conditions, non-homogeneity and anisotropy of the

materials. Consequently, approximate numerical techniques should be used to overcome these




39

difficulties. Among the existing alternative techniques, the finite element method is undoubtedly the

most popular one.

The finite element method (FEM), applied to problems of structural dynamics, consists in dividing the

system domain in sub-domains (meshing) that are called as “elements”. Points are defined either in the boundaries or in the interior of these elements, which are called as “nodes”. The set of elements and

nodes form the “mesh” of the finite element model. The primary variable of the problem under study

(the displacement field, for example) is expressed as a linear combination of the values that this

variable assumes at the nodes. This way, the unknowns of the problem are these latter values. The

finite number of the unknowns is the so-called “number of degrees of freedom” of the problem. The

resulting mathematical model is represented by a set of second order ordinary differential equations

with respect to time, which take the same form as the equations of motion of discrete vibrating systems,

considered in Chapter 2. The number of differential equations is equal to the number of degrees-of-

freedom. It can be said that the finite element method performs a transformation of a continuous system

into an equivalent discrete system. For this reason, the finite element method is frequently associated to

the notion of discretization that is understood as the transformation of infinite-dimensional problems

into finite-dimensional ones.

4.1. Finite Element Equations of Motion of Euler-Bernoulli

For the sake of simplification, the fundamentals of the finite element method are presented in this

chapter for the case of straight beams under the hypotheses assumed by the Euler-Bernoulli theory.

Clearly, the basic procedures can be extended to other types of structural elements, such as cables,

bars, plates and shells.

The following hypotheses are admitted:

1st. Only the effects of the transverse bending are taken into account (the effects due to the longitudinal

motion are neglected);

2nd. The simplifications of the Euler-Bernoulli theory of beams are adopted: the effects of the

transverse shear and the rotary inertial of the cross sections are neglected;

3rd

. The displacements and rotations are considered to be small and the material exhibit isotropic linear

elastic behavior;

4th. Energy dissipation is neglected, i.e., the system is considered to be undamped.

4.1.1 Element-level Equations of Motion

Consider the beam of length L illustrated on Figure 4.1, for which the relevant properties are the linear

mass density m x (kg/m), the Young’s modulus E ( x) (N.m-2

), the second momento of area of the cross

sections with respect to the centroidal axis z , I x (m4). Moreover, t xq , (N.m-1) is the external

transverse distributed load, and t xv , (m) represents the lateral displacement field of the beam.

q(x,t)




40

Figure 4.1 - Illustration of a Euler-Bernoulli beam

The beam shown in Figure 4.1 is considered to be divided according to an arbitrary number of elements

each of each is delimited by two nodes. Figure 4.2 shows one of these elements that is generically

identified by the index i , containing two nodes, denoted by i L and i R . For this element, the relevant

properties are indicated by i , im x , i E x , and i I x . Besides, ,iq x t and ,iv x t represent the

external load and the transverse displacements field in the interior of the element ( 0 i x ),

respectively. Also degrees-of-freedom (or nodal coordinates) defined for the element are indicated:

they represent by the values of the transverse displacements t v Li and t v R

i and rotations of the cross

sections at the nodes, Li t and R

i t . These degrees of freedom are organized in a vector as given by

Eq. (4.1):

T

e L L R Ri i i i iu t v t t v t t (4.1)

where the superscript(e)

indicates the degrees-of-freedom in the element level.

It should be observed that, according to the theory of Euler-Bernoulli, the following relation between

the rotations of the cross sections and the transverse displacements of the beam holds:

,

,i

i

v x t x t

(4.2)

Li

Figure 4.2 - Generic Euler-Bernoulli beam element

Li

Ri

R

iv t

R

i t L

i t

L

iv t

i ,iv x t




41

The transverse displacements are interpolated by the following cubic polynomial:

2 3

1 2 3 4, 0

i iv x t c t c t x c t x c t x x (4.3)

Consequently, by taking into account Eq. (4.2), the rotations of the cross sections can be approximated

by the following quadratic function:

2

2 3 4, 2 3 0i i x t c t c t x c t x x (4.4)

The Equations (4.3) and (4.4) have to satisfy the following kinematic conditions:

0, L

i iv t v t , R

i i iv t v t (4.5.a)

0

Lii

x

v t x

i

Rii

x

v t x

(4.5.b)

By imposing the conditions (4.5) to the approximations (4.3) and (4.4), four linear algebraic equations

are obtained. Once these equations are solved, the expressions corresponding to the constants ic t are

obtained as functions of the nodal displacements and rotations as follows:

1 2 3 4, L L R R

i i i i iv x t x v t x t x v t x t

(4.6.a)

or:

,e

i iv x t x u t

(4.6.b)

where:

1 2 3 4 x x x x

(4.7)

2 3

1 1 3 2 x x

xl l

(4.8.a)

2 3

2 2 x x

x x l l l l

(4.8.b)

2 3

3 3 2 x x

xl l

(4.8.c)

2 3

4

x x x l l

l l

(4.8.d)




42

The functions 4,3,2,1i xi are known as shape functions or interpolation functions.

Equations (4.6) clearly show that the transverse displacements field ant the rotations of the cross

sections are expressed as linear combinations of the nodal displacements and rotations, in which the

coefficients of linear combinations are given by the shape functions.

Aiming at obtaining the equations of motion for the beam elements by using Lagrange’s equations, it is

necessary to formulate both the kinetic and potential energies together with the virtual work of the

external load, as follows:

Kinetic Energy: The kinetic energy of the element is given by the expression:

2

0

,1

2

i

i

i i

v x t T m x dx

t

(4.9)

By introducing (4.6b) into (4.9) and deriving iv t with respect to the time, results:

1

2

T e e e

i i i iT u t M u t

(4.10)

where:

4 40

iT e

i im x x x dx

(4.11)

is the so-called mass matrix (or inertia matrix) of the beam element, whose general element is

calculated according to:

0

i

e

i i m nmn

m x x x dx

, 4,3,2,1, nm (4.12)

For the case in which im x is constant, equations (4.11) and (4.12) lead to the following matrix:

2 2

2

156 22 54 13

4 13 3

156 22420

4

i i

e i i ii ii

i

i

m M

simétrica

(4.13)

Potential Energy: For the Euler-Bernoulli beam, the strain energy is expressed as:

symmetric




43

22

2

0

,1

2

i

i

i i i

v x t V E x I x dx

x

(4.14)

By introducing (4.6b) into (4.14) and deriving twice iv t with respect to x it is possible to write:

1

2

T e e e

i i i iV u t K u t (4.15)

where:

4 4

0

iT e

i i i K E x I x x x dx

, , 1a 4m n (4.16)

is the so-called stiffness matrix of the Euler-Bernoulli beam element, whose general element is

calculated according to:

0

i

e

i i i m nmn

K E x I x x x dx

(4.17)

For the case in which the product i i E x I x is constant, the equations (4.16) and (4.17) lead to the

following matrix:

2 2

3

2

12 6 12 6

4 6 2

12 6

4

i i

e i i ii ii

i i

i

E I K

simétrica

(4.18)

Virtual work of the distributed load: The work of the distributed external load associated to a virtual

displacements field iv x is given by:

0 , ,

i

i i iW t v x t q x t dx

(4.19)

so that the virtual displacements are obtained from (4.6) according to:

,e

i iv x t x u t (4.20)

Combining (4.19) and (4.20) results:

T

e e

i i iW t u t Q t (4.21)

symmetric




44

where the vector of generalized efforts (forces and moments applied at the nodes) is given by:

4 1

0

,il

T e

i iQ t q x t x dx

(4.22)

being the general element expressed as:

0

, , 1a 4il

e

i i mm

Q t q x t x dx m (4.23)

In the case for which the distributed load is constant along the element, the Eqs. (4.22) and (4.23) lead

to the following vector of generalized efforts:

2

2

2

12

2

12

ii

ii

e

i

ii

ii

q t

q t

Q t

q t

q t

(4.24)

By considering the degrees of freedom as generalized coordinates, the Lagrange’s equations are written

as:

ei iie e

i i

L Ld Q

dt u u

(4.25)

where i i i L T V is defined as the Lagrangian.

Using the equations (4.25), (4.21), (4.15) and (4.10) the following equations of motion can be obtained

for the beam element:

( ) ( )e e ee e

i i i i iu t K u t Q t (4.26)

4.1.2. Assembling of Global Equations of Motion

In the preceding section, the equations of motion have been obtained for a generic beam element.

However, it should be considered that the beam is discretized in various elements so that neighboring

elements share nodes, as illustrated in the Figure 4.3 for two contiguous elements i and j.




45

T e L L R R

i i i i iu t v t t v t t T e L L R R

j j j j ju t v t t v t t

1 1 2 2 3 3

T g u t v t t v t t v t t

Figure 4.3 - Assembling scheme of two beam elements

The equations of motion of the two elements are combined so that the dynamic balance and the

continuity of displacements and rotations of sharing nodes of two neighboring elements are assured.

This procedure will be illustrated in the following for the two elements depicted in the Figure 4.3.

For each element linear transformations relating the elementary degrees of freedom and the global ones

are written according to:

4 64 1 6 1

e g

i iu t L u t

(4.27.a)

4 64 1 6 1

e g

j ju t L u t

(4.27.b)

The scheme shown in Figure 4.3 enables to construct easily the matrices i L and j L as explained

below. It should be noted that these are Boolean matrices (formed by zeroes and ones) in which the

positions of the ones permit to localize the elementary degrees of freedom among the global ones.

Element i

3

3v t

3 t 1

2

2v t

2 t 1 t

1v t

L j R j

R

jv t

R

j t L

j t

L

jv t

Li Ri

R

iv t

R

i t L

i t

L

iv t




46

1

1

2

2

3

3

1 0 0 0 0 0

0 1 0 0 0 0

0 0 1 0 0 0

0 0 0 1 0 0

L

i

L

i

Ri

R

i

v t

v t t

t v t

v t t

t v t

t

(4.28.a)

Element j

1

1

2

2

3

3

0 0 1 0 0 0

0 0 0 1 0 0

0 0 0 0 1 0

0 0 0 0 0 1

L

j

L

j

R

j

R

j

v t

v t t

t v t

v t t

t v t

t

(4.28.b)

For the system formed by the elements i and j the total kinetic energy is written as:

i jT T T (4.29)

Utilizing Eq. (4.10) for both the elements i and j together with the equations (4.27) it is possible toobtain the total kinetic energy as expressed in terms of the global degrees of freedom:

1

2

T g g g

T u t M u t

(4.30)

where the global mass matrix is written as:

6 6

T T g e e

i i i j j j L M L L M L

(4.31)

Similarly, by manipulating the total potential energy and the virtual work of the external load, results:

i jV V V = 1

2

T g g g

u t K u t (4.32)

with:

6 6

T T g e e

i i i j j j K L K L L K L

(4.33)




47

i jW W W =

T g g

u t Q t (4.34)

where:

6 1

T T g e e

i i j jQ L Q L Q

(4.35)

By applying again the Lagrange’s equations considering the global degrees of freedom as the

generalized coordinates, one can write:

g

g g

i

L d LQ

dt u u

com L T V (4.36)

Associating the equations (4.30), (4.32), (4.34) and (4.36), the global equations of motion are obtained

for the two connected elements:

g g g g g u t K u t Q t (4.37)

The equations (4.26) forms a system of six ordinary differential equations in the time domain that

exhibits the same form of the equations of motion of undamped discrete systems as presented in

Chapter 2. Consequently, the methods presented before for discrete systems can be used to calculate

the different types of dynamic responses of the system for different excitation forms, given thefollowing initial conditions:

00 g g

u u 00 g g

u u (4.38)

The procedure presented above for the system represent for only two beam elements can be applied

successively for systems containing an arbitrary number of elements.

4.2. Enforcing Boundary Conditions

As opposed to static problems, the problems of dynamic analysis can be easily solved when the

structure is free in the space without any physical connection with the fixed reference frame. This is

indeed the case of aeronautical and space structures. In such cases, the system of differential equations

(4.37) can be processed directly. However, if kinematic constraints, as represented by prescribed

displacements at some points of the structure exist, the equations of motion have to be modified to take

into account such constraints. In the following, a straightforward procedure is presented to perform this

task.




48

The matrices and vectors that appear in (4.37) are partitioned according to the “free” degrees of

freedom (those that are not affected by the kinematic constraints) and the “imposed” degrees of

freedom (those whose values are prescribed). The first are indicated by the index “L” and the last ones

are denoted by the index “I” in the equations of motions that are written as:

g g g g g g g

L L L LL LI LL LI

g g g g g g g

IL II IL II I I I

u t u t Q t M M K K

M M K K u t u t Q t

(4.39)

The two blocks of equations resulting from the partitioning indicated in (4.37) can be organized as

follows:

g g g g g g g g g

LL L LL L L LI I LI I u t K u t Q t M u t K u t

(4.40)

g g g g g g g g g

I IL L II L IL L II I Q t M u t M u t K u t K u t (4.41)

The solution of the equations (4.40) and (4.41) enables to determine the dynamic response of the “free”

coordinates and, the reaction efforts associated to the “imposed” degrees of freedom, respectively.

4.3. Performing Dynamics Analysis

At this point, it should be pointed-out that the equations of motion of the original continuous systems

obtained after space discretization according to the FE method take exactly the same form as the

equations of motion of discrete vibrating systems addressed in Chapter 2. As a result, all the different

types of dynamic analysis previous discussed as related to discrete systems can be performed, as

summarized in the following.

Modal Analysis: This type of dynamic analysis consists in determining the natural frequencies and

corresponding mode shapes of the discretized structure by solving numerically the eigenvalue problem

associated to the differential equation (4.40), expressed as follows:

02

g L

g LL

g LL U M K

Harmonic analysis: The goal of harmonic analysis is to evaluate the amplitudes of vibration when the

structure is submitted to harmonic forces t i g

L g

L eQt Q . This is made by solving the following

linear set of algebraic equations, by varying the excitation frequency in a previously prescribed

frequency band:

g L

g LL

g LL U M K 2

U L

g LQ




49

Transient analysis: Transient analysis is carried-out to obtain the time-response of the structure to

arbitrary set of initial conditions or time varying excitation forces in a given time interval. This is made

by solving the following set of coupled second-order ordinary differential equations.

t Qt u K t u M g

L g

L g

LL g

L g

LL T t 0

g L

g L

g L

g L ut uuu 00 ;0

It should be pointed-out that, when performing dynamic analysis based on finite element models, a

large number of simultaneous equations, which is determined by the particular type of finite element

used as well as the discretization mesh, must be solved. The number of equations can reach the order of75 10or 10 , which makes computation effort a primary concern. As a result, the dynamic analyses

described above are generally performed by solving efficient numerical methods.

5. Introduction to Experimental Modal Analysis

In the preceding sections, analytical and numerical techniques for modeling structural systems based on

the principle of modal superposition were presented. According to this principle, the dynamic response

of any linear structure (either in time domain or in frequency domain) to an arbitrary set of initial

conditions or excitation forces can be expressed uniquely as a linear combination of the natural

vibration modes of the structure. A major consequence of this fact is that the set of modal parameters

(natural frequencies, modal damping factors, vibration modes and generalized masses) constitute a

reduced set of properties which fully characterize the dynamic behavior of structural systems. In otherwords, the modal characteristics constitute a convenient “dynamic signature” of linear vibrating

systems.

Analytical and numerical techniques for modeling of dynamic systems have been developed in the last

decades. However, in practical applications of industrial interest, it is inevitable that theoretical models

involve inaccuracies and uncertainties that can jeopardize the correct prediction of the dynamical

behavior of the systems.

The drawbacks of theoretical modeling can be circumvented by performing vibration tests, and

experimental results can be used for the various purposes, such as: 1) validation of theoretical models

and model updating, if found necessary; 2) obtaining information regarding the dynamic behavior thatwould be very difficult to obtain through theoretical modeling, such as the level of damping of the

structure; 3) monitoring the structure under operating conditions for maintenance purposes (condition

health monitoring).

In the modern literature, the term Experimental Modal Analysis (EMA) is generally understood as the

set of hybrid numerical-experimental procedures that encompass, sequentially: 1) the realization of

vibration testing that include the application of adequate excitation forces, followed by the

measurement and conditioning of the dynamic responses; 2) the numerical processing of the responses

for extracting the desired dynamic characteristics.




50

Regarding the first step, different types of excitation can be considered, namely: a) impact excitation,

generally using an impact hammer that is instrumented with a load cell; b) periodic or random

excitation by using electrodynamic, hydraulic or piezoelectric shakers. The response signals are

conditioned according to the objectives of the dynamic testing that may include amplification, filtering,

windowing, average calculation, Fourier transform, and estimation of frequency spectra.

The second step is dedicated to the determination of modal parameters (natural frequencies, modal

damping factors, vibration modes and generalized masses) from the processed dynamic responses. In

general, this task is performed by adjusting theoretical expressions of the dynamic responses (in which

the modal parameters intervene as a result of the principle of modal superposition) to the experimental

dynamic responses. This process is also known as modal parameter estimation or modal parameter

identification.

For the sake of conciseness, this chapter presents briefly an approach regarding some of the essential

aspects of EMA. More details can be found in a number of text books, namely Ewins (2000); Maia e

Silva (1997), McConnel and Varoto (2008).

5.1. Setup for Experimental for Modal Analysis

Figure 5.1 illustrates a typical setup of vibration testing for EMA in which the essential hardware is

presented. The functions of setup components are the following: the signal generator (1) is used to

provide voltage signals, which, once amplified by the power amplified (2), will drive the

electrodynamic shaker (3) to apply a time-varying force to the test-structure (4). Usually, the signal

generator has the capability of providing various types of output voltage signals (sine-waves, sine-

sweep, burst random, white noise, etc.). A piezoelectric load cell, integrated into the impedance head

(5), is placed between the shaker and the test-structure to measure the actual force applied to this latter.A piezoelectric accelerometer (also integrated in the impedance head) is used to measure the vibratory

response of the test-structure.

The signal conditioners (6) (also called charge amplifiers) enable to convert the electric chargers

generated within the piezoelectric sensors – accelerometer and load cell, into voltage signals which are

proportional to the measured acceleration and applied force, respectively. They can also perform other

types of signal treatments such as integration (from acceleration to velocity and displacement,

successively), low-pass and high-pass filtering and amplification.

The spectrum analyzer or frequency analyzer (7) is the equipment which performs the numerical

processing of the time responses. One of the most useful capabilities of this equipment is thecomputation of the frequency response functions of the structure from frequency spectra of the forces

and responses, obtained by Fourier transforming of the corresponding time signals. Those FRFs can be

used for modal parameter estimation, as discussed in the next section.




51

Figure 5.1 - Essential elements of a typical setup for EMA. (1) Signal Generator; (2)Signal Amplifier;

(3) Electrodynamic shaker; (4) Test specimen; (5) detail of the impedance head (load cell and

accelerometer combined); (6) Signal amplifiers for force and acceleration signals; (7) Spectrum

Analyzer; (8) PC for data acquisition

5.2. Estimation of Modal Parameters

According to Craig Jr. and Kurdila (2006), the techniques of modal parameter estimation can be

classified as below.

Modal parameters versus structural parameters. The methods that operate on the so-called

structural parameters aim at constructing models formed by matrices of mass, stiffness and damping,

associated to a second order model such as given Eq. (2.68) or the state matrices in a state-space

model, as represented by Eq. (2.111). The modal parameters are then obtained by solving the

associated eigenvalue problems.

Frequency domain versus time domain techniques. In the frequency domain methods the modal

parameters are estimated from the frequency response functions (FRFs) or auto-spectra and crossed

frequency spectra. On the other hand, time domain techniques use free or forced time responses to

perform the identification.

Single degree of freedom methods (SDOF) versus multiple-degree- of-freedom (MDOF) methods.

SDOF methods intend to estimate the modal parameters of each individual mode, separately, while

MDOF methods are intended to estimate the modal parameters corresponding various vibration modes,

simultaneously.

Single-input methods versus multi-input methods. The former use, for modal parameter estimation,

the vibratory responses associated to a single excitation applied at a given point of the test-structure,

while the latter operate on an ensemble of responses obtained by applying various excitations at

different points of the test-structure, successively.




52

The text books by Ewins (2000) and Maia e Silva (1997) describe some of the most important modal

parameter estimation methods available. As an example, a MDOF/SIMO (single input, multiple

outputs) frequency domain method is presented in the following.

For the application of the parameter estimation technique, it is necessary that an external force i be

applied at a given point i of the structure and the vibration responses be measured at a set of points j ( j

= 1,2,3,..r, including j=i). This scenario is illustrated in Figure 5.2, in which the external force f 1 is

applied at point 1 and the responses (lateral displacements 1 2 3 4 , y , y , y ) are measured at points 1 to 4. It

should be observed that, in the general case of three-dimensional structures, measurements at a given

point of the structure should be referred to the specific directions along which the sensor used is

oriented. This way, it is possible that a same measurement point be related to dynamic responses along

more than one direction.

Figure 5.2 - Scheme of the testing scenario

For this example, it is considered that a set of four complex FRFs 11 1 2 3 4exp

ii H y f , i , , , have

been obtained from the experimental dynamic responses, after processing the excitation and response

signals.

A general damping model is assumed 1 1

C M K K M C

and damping is considered to be

subcritical. The FRFs expressed by Eq. (2.137), assuming unit generalized masses, are rewritten below:

n

r r

r r n

r r

r r teo

ij si

y y

si

y y

H

ji ji

11 (5.1)

where the exponent teoindicates that (5.1) corresponds to the theoretical expression of the FRF.

In Eq. (5.1), the modal parameters to be estimated are: the complex eigenvalues, which can be

expressed as 21r r r r r s . They contain the natural frequencies r and the modal damping

factors r ; and the complex components of the n eigenvectors, at the c points of the structure in which

the responses have been measured, indicated byir ( r =1,2,...n; i=1,2,...,c).

y

x

f 1

y1 y2 y3 y4




53

As a continuous structure exhibits an infinite number of vibration modes, the parameter estimation

should be limited to a reduced finite number of p vibration modes within a previously selected

frequency band. These modes are considered to be the dominant ones (those that contribute the most to

the dynamic response) in the frequency band of interest. Thus, the FRF are approximated by a limited

series as in Eq. (5.2):

ijij

p

r r

r r teo

ij ba si

y y H ji

1

(5.2)

where the terms ij ija b are included to account for the influence of the truncated modes (including

the complex-conjugated modes) in the interest frequency band. The coefficients andij ija b are also

considered as parameters to be identified together with the modal parameters.

Now, for the example considered above, for which the experimental FRFs 11 1 2 3 4exp

ii H y f , i , , , are available for q excitation frequency values ( 1 2 q , , , ) in the

interest frequency band, the goal is to adjust the theoretical model (5.2) to the experimental values of

the FRFs. In this sense, the modal identification procedure can be understood as a curve-fitting

procedure in a function containing some unknown parameters must be adjusted to interpolate a set of

experimental points.

For each measurement point, the theoretical counterparts of the experimental FRFs are written as

follows:

11

1

11

ik i

p

r r k

r r

k

teo

i ba si

y y

H i

, k =1,2,…q, i=1,2,…c , (5.2)

It can be observed that, for each FRF, there are 2 p+2 unknowns to be determine: p values of the

products1r r y y

i, also called residuals; p complex eigenvalues r s , and the coefficients 1ia and 1ib .

To determine the unknowns the theoretical expressions of the FRFs are equated to the corresponding

experimental values:

11

1

exp

11

ik i

p

r r k

r r

k i ba

si

y y H i

, k =1,2,…q, i=1,2,…c , (5.2)

As the number of equations q is determined by the number of frequency points within the frequency

band of interest, generally it is preferred to solve an over-determined set of equations by choosing

q>2 p+2. It should be noted that the equations to be solved are nonlinear since the unknowns r s appear

in the denominators of the partial fractions in (5.2). The resolution of the equations must be performed

iteratively by using appropriate numerical algorithms.

The procedure is carried-out c time for each FRF available. At the end of this process one has: p×c

values of the complex eigenvalues r s , r =1,2,…, p. As the eigenvalues are global parameters, which do

not depend from the measurement point, their values, for a fixed r , should be theoretically equal.




54

However, dispersions can occur due to experimental and numerical errors. In practice, for each r , the c

values of the complex eigenvalues are averaged; p×c values of the residuals rjr y yi

, i=1,2,...c.

In the equations above a total number of 4q equations with complex coefficients and 6 p complexunknowns are found. From these, 4 p represent the product of the mode components such asm nr r y , and

p complex eigenvalues r s and p modal masses d r are identified. Admitting that 4q6 p, the system of

equations can be solved numerically for the determination of the unknowns. However, the difficulty in

the solution of the system of equations comes from the fact that these equations are nonlinear

(unknowns appear in the denominator), which requires the use of iterative procedures. Besides, it

should be noted that: a) once the product1 1r r y is obtained,

1 1 1r r r y y can be calculated and, in the

sequence,2 1 1 1 3 3 1 1 4 4 1 1

1 2r r r r r r r r r r r r y y y ; y y y y ; y y y y , r , , p are also determined; b) once the

complex eigenvalues r s are obtained, the natural frequencies and the modal damping factors can be

calculated.

Very frequently, the modal parameters identified from experimental modal analysis are intended to be

used for validation or adjustment of finite element models. Most frequently, damping is neglected in

these latter. As a result, one is faced with the problem of comparing complex eigenvectors, obtained

from experimental modal analysis, to real eigenvectors provided by a finite element model. A number

of numerical procedures have been developed for this goal and are described in a number of books that

are found in the literature.

6. Conclusion

In the previous sections of this contribution, material was presented aiming at providing theunderstanding of some of the main aspects of Structural Dynamics and Modal Analysis. An effort was

made to keep a strong connection between these two sub-topics, by exploring the fundamental

principle of modal superposition, expressed by the Expansion Theorem. Such principle, which can be

seen as a technique for solving linear ordinary or partial differential equations from the Mathematics

point of view, provides a sound interpretation of the dynamic responses as sums of contributions of

single-degree-of-freedom systems. It constitutes the basics of most experimental modal analysis

procedures and enables the use of computer-effective numerical procedures.

As the scope of this contribution was limited, a number of relevant, more advanced points were not

addressed, such as: nonlinear behavior, damping models, random excitations, rotating systems(accounting for gyroscopic effects), signal processing and instrumentation for vibration testing. To

learn about these subjects, the reader is encouraged to refer to text-books available, some of which are

included in the bibliography list that follows.

It should be pointed-out that, in modern Engineering applications, highly complex structures must be

dealt with. In such cases, to ensure the desired predictive capacity, the numerical models (most

frequently generated by finite element discretization) are characterized by a large number of degrees-

of-freedom (which can reach the order of 105-107). Such problems can be treated by using various

software packages available in the market, which enable to perform dynamic analysis by using very

effective numerical routines developed lately. Clearly this has become possible due to the great




55

increase in computation power observed in the last two decades.

As for vibration testing, technological achievements in electronics and signal processing have enabled

to perform vibration measurements at moderate costs. As a result, typical measurement chains include

hundreds of sensors and multi-channel signal processors. Measurement capabilities have been greatly

improved by the development of laser-based techniques, such as laser vibrometry and holography.

The advances mentioned above contributed to the incorporation of structural dynamics analyses in the

engineering routine in many industrial sectors. At the same time research efforts are carried-out in

universities and research centers aiming at improving the effectiveness and applicability range of

existing techniques.

It should also be mentioned the efforts of professional associations for the development of modal

analysis and structural dynamics within the engineering community. A number of international

conferences are held in various countries on a regular basis, having those subjects among their topics.

Acknowledgements

The authors wish to acknowledge the Brazilian research agencies CNPq, CAPES and FAPEMIG for

the continued support to their research activities. They are grateful to the collaborators in the Structural

Mechanics Laboratory “Prof. José Eduardo Tannús Reis” of the School of Mechanical Engineering of

the Federal University of Uberlândia.

Bibliography

Craig Jr., R. R., and Kurdila, A. J. (2006), Fundamentals of Structural Dynamics. 2nd Edition, John

Wiley & Sons, Inc., ISBN-13: 978-0-471-430445. [This is an updated version of the book which

brings a comprehensive coverage of structural dynamics fundamentals, finite-element-based

computational methods, dynamic testing methods, and an introductory chapter to active

structures(adapted from the publisher).]

Ewins, D.J., (2000), Modal Testing, Theory, Practice and Application. Research Studies Press Ltd.,

ISBN 0 86380 2184. [ In this book, all the steps involved in planning, executing, interpreting and

applying the results from modal tests are described. In the second edition, new topics have been

included, such as modal testing of rotating machinery and the use of laser vibrometry (adapted from

the publisher).]

Geradin, M, Rixen, D., (1997), Mechanical Vibrations: Theory and Applications to Structural

Dynamics, 2nd Edition, ISBN-10: 047197546X. [This book provides a comprehensive and unified

approach to problems encountered in the field of mechanical vibrations and structural dynamics.

The mathematical and phenomenological aspects underlying structural dynamics are addressed,

with emphasis put on computational methods (adapted from the publisher).]

McConnell, K., Varoto, P.S., (2008), Vibration Testing – Theory and Practice, 2nd Edition, Wiley &

Sons, Ltd., ISBN-10: 0471666513. [This book contains a step-by-step guide that shows how to




obtain meaningful experimental results by the use of modern equipment, with emphasis put on

signal processing operations performed with a frequency analyzer(adapted from the publisher).]

Inman, D. J., (2007), Engineering Vibrations, 3rd Edition. Prentice-Hall, ISBN-10: 0132281732 [This

book combines the study of traditional introductory vibration with the use of vibration design,analysis and testing in engineering practice. This text represents an integration of traditional topics

with a design emphasis, the introduction of modal analysis, and the inclusion of a professional

quality software package (adapted from the publisher).]

Maia, N.M.M., Montalvão e Silva, J. M., Montalvão e Silva, J. M., (1997). Theoretical and

Experimental Modal Analysis. John Wiley & Sons, Ltd., ISBN-10: 0471666513. ISBN-10:

0471970670.[This book address various aspects of theoretical and experimental modal analysis,

including the theory of mechanical vibrations and signal processing, identification techniques,

substructure coupling, structural modification, finite element model updating and nonlinear

analysis.]

Cap 2 Structural Dynamics and Modal Analysis

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