Finite Element Model Updating using Rotational Response

저 시-비 리- 경 지 2.0 한민

는 아래 조건 르는 경 에 한하여 게

l 저 물 복제, 포, 전송, 전시, 공연 송할 수 습니다.

다 과 같 조건 라야 합니다:

l 하는, 저 물 나 포 경 , 저 물에 적 된 허락조건 명확하게 나타내어야 합니다.

l 저 터 허가를 면 러한 조건들 적 되지 않습니다.

저 에 른 리는 내 에 하여 향 지 않습니다.

것 허락규약(Legal Code) 해하 쉽게 약한 것 니다.

Disclaimer

저 시. 하는 원저 를 시하여야 합니다.

비 리. 하는 저 물 리 목적 할 수 없습니다.

경 지. 하는 저 물 개 , 형 또는 가공할 수 없습니다.

http://creativecommons.org/licenses/by-nc-nd/2.0/kr/legalcode

http://creativecommons.org/licenses/by-nc-nd/2.0/kr/

Finite Element Model Updating

using Rotational Response

Myunghwan Bae

Department of Urban Environmental Engineering

(Urban Infrastructure Engineering)

Graduate school of UNIST

2015



Myunghwan Bae

Department of Urban Environmental Engineering

(Urban Infrastructure Engineering)

Graduate school of UNIST



A thesis

submitted to the Graduate School of UNIST

in partial fulfillment of the

requirements for the degree of

Master of Science

Myunghwan Bae

December 18th, 2014

Approved by

Advisor

Sung-Han Sim

Finite Element Model Updating using Rotational Response

Myunghwan Bae

This certifies that the thesis/dissertation of Myunghwan Bae is approved

December 18th, 2014

Sung-Han Sim

Assistant Professor of Urban and Environmental Engineering

Ulsan National Institute of Science and Technology

Advisor

Myoungsu Shin

Associate Professor of Urban and Environmental Engineering


Marco Torbol

Assistant Professor of Urban and Environmental Engineering


i

ABSTRACT

The Finite Element (FE) models have been used in civil, mechanical and aerospace engineering field

for system identification and response simulation under various unexperienced loadings. However, the

original FE models are difficult to use directly in simulation because the original FE model is different

with the present state of actual structure. Therefore the FE models are updated to minimize the

differences with real structure. In the existing FE model updating method, generally translational

responses measured by accelerometers have been employed to identify the structural properties and

update numerical model controlling parameter. This paper proposes FE model updating method only

uses rotational response such as angular velocity measurement because the rotational response is more

sensitive to damage than translational response in numerical analysis on simply supported beam. First

sensitivity of translational and rotational response is investigated by sensitivity analysis on a

numerical simply-supported beam. The FE model updating is carried out for a numerical simply-

supported beam using an optimization algorithm which minimizes the gap between responses from

the actual structure and the FE model reduce. The used responses in FE model updating are natural

frequencies and rotational mode shape obtained from angular velocities measured by gyroscope

sensor. The sensitivity analysis shows that rotational responses have higher sensitivity than

translational responses to the structural and boundary condition changes. Then, the updated model

using existing translational response from the experiment two FE models updated using translational

and rotational responses are compared to validate he improvement by proposed FE model updating

method. From the experiment, using rotational responses is a good enough in FE model updating

compared to existing method, using translational responses.

ii

iii

TABLE OF CONTESTS

ABSTRACT ...................................................................................................................................... i

TABLE OF CONTENTS ............................................................................................................... iii

LIST OF FIGURES ........................................................................................................................ iv

LIST OF TABLES.......................................................................................................................... vi

CHAPTER 1. INTRODUCTION ......................................................................................................................................... 1

CHAPTER 2. BACKGROUNDS ........................................................................................................................................ 3

CHAPTER 3. PRELIMINARIES ....................................................................................................................................... 7

3.1 Modeling for Numerical Analysis & System Identification of Model ........................................ 7

3.2 Sensitivity Analysis of Translational and Rotational Mode Shapes Using MAC value .... 10

3.3 FE Model Updating Process Approached by Using Rotational Response .............................. 16

CHAPTER 4. NUMERICAL SIMULATION ............................................................................................................... 17

4.1 Selecting Parameter ........................................................................................................................................ 17

4.2 Numerical Simulation using Exact Values ............................................................................................ 19

4.3 Numerical Simulation using Excitation ................................................................................................. 22

4.4 Validation of FE Model Updating using Rotational Response..................................................... 34

CHAPTER 5. EXPERIMENT ............................................................................................................................................. 35

5.1 Experimental Set up ....................................................................................................................................... 35

5.2 Measurement Data from Experiment ...................................................................................................... 36

5.3 Comparison of FE Model Updating Performance ............................................................................. 38

CHPATER 6. CONCLUSION ............................................................................................................................................. 42

REFERENCE............................................................................................................................................................ 43

iv

List of Figures

Figure 2.1: The concept of FE model updating

Figure 2.2: Iteration process of Simplex method

Figure 3.1.1: Simply supported beam model with 10 elements Figure 3.1.2: Theoretical beam with 11 node and 22 DOFs

Figure 3.2.1: The 1st, 2nd, 3rd Translational Mode Shape

Figure 3.2.2: The 1st, 2nd, 3rd Rotational Mode Shape

Figure 3.3.1: The MAC sensitivity analysis of Material properties

Figure 3.3.2: The sensitivity analysis of Sectional properties

Figure 3.3.3: The sensitivity analysis of rotational stiffness of boundary condition and damage by

adding mass

Figure 4.1: Compared MAC sensitivity of beam Thickness – 2nd elements and 6th elements

Figure 4.2: Height decreases in 2nd, 4th, 6th element to select damage location

Figure 4.3: The sensor location to get simulation data for each case

Figure 4.4.1: The PSD of simulated for Baseline beam

Figure 4.4.2: The PSD of simulated angular velocity for Case 1


Figure 4.5.1: Comparison of Mode Shape for Base

Figure 4.5.2: Comparison of Mode Shape for Case 1


Figure 4.6.1: natural frequency by updating in Baseline

Figure 4.6.2: natural frequency by updating inCase1

Figure 4.6.3 : natural frequency by updating in Case2

Figure 4.7.1: Change of Modeshape by updating of Baseline is expressed as MAC value.

Figure 4.7.2: Change of Modeshape by updating of Case1 is expressed as MAC value.


Figure 4.8: The updated heights of each element by FE model updating for three cases

Figure 4.9: updated parameters and error percentage

Figure 5.1: Experimental set up for measurement

Figure 5.2.1 : Angular velocity measurement

Figure 5.2.2 : Enlarged angular velocity

Figure 5.2.3 : Acceleration measurement

v

Figure 5.2.4 : Enlarged acceleration

Figure 5.3.1 : The frequency domain, PSD of angular velocity and acceleration of before-damaged

Figure 5.3.2 : The frequency domain, PSD of angular velocity and acceleration of after-damaged

Figure 5.4 Comparison of the natural frequencies

Figure 5.5.1 : Comparision MAC value of 1st mode shape for each case

Figure 5.5.2 comparision MAC value of 2nd mode shape for each case

Figure 5.6 : The updated heights of each element by FE model updating

Figure 5.7 : The error percentage of updated parameter for each case

vi

List of Tables

Table 3.1: The spec of theoretical Beam model

Table 4.1.1: RMSD value of updated parameter in damage of 2nd element

Table 4.1.2: RMSD value of updated parameter in damage of 4th element


Table 4.1.4: RMSD value of updated parameter in multi damage

Table 4.2: Selected cases for numerical simulation using excitation

Table 4.3.1: Natural Frequency of Baseline from Numerical Simulation

Table 4.3.2: Natural Frequency of Case1 from Numerical Simulation


Table 4.4: Compare RMSD value for each damage case

Table 5.1: The accelerometer and gyroscope for measuring response

Table 5.2: Natural Frequencies of from experiment

Table 5.3.1: Comparison of FE model updating result – before damaged beam

Table 5.3.2: Comparison of FE model updating result – after damaged beam

Table 5.4: Compare RMSD value to select location of damage

1

Chapter 1

1. Introduction

In the engineering and mathematics field, the finite element method is used to numerically find

approximate solution for given conditions. The FE models are made and used for system

identification and response simulation under various unexperienced conditions. Because the finite

element method finds solutions for each element, the FE model of complex structures which have

many elements have many solutions to be solved. So the simple FE model is solved manually but for

these complex structures, there are many softwares to model finite element and to solve the response

of FE model such as ABAQUS, ANSYS, MIDAS, SAP200 and etc. To simulate using FE model

under conditions, generally initial FE model should be updated to respond the existing structure. The

FE model updating is a method to update the initial model by changing parameters. In the civil

engineering, the initial FE model is updated using the field measurement data to simulate the real

existing structure. In civil engineering, the structures are mainly made of concrete which has time

dependence feature such as aging, creep and shrinkage, FE models different from the real existing

structures need to updating to be used for simulation. Without the time dependence feature of

structures, the initial FE models need to be updated due to deterioration, construction error and so on.

By updating FE model, we could make a reference after construction and many year of operation and

assess the safety of structure to various events (e.g., earthquake, flood, typhoon, tsunami etc.) for

maintenance. [1-5]. Especially, the urban infrastructures like nuclear power plant needs the perfect

modeling and model sensitive updating skill to achieve safety by simulating to various conditions. To

updated initial FE model, by the optimization updating algorithm, parameter are changed to minimize

the gap between the response from FE model and from experiment until the objective function defined

as the gap of responses meets given value. In general, in the FE model updating process, the

compared responses are mainly natural frequency and mode shape. To improve the performance of FE

model updating, the researches related with each part are ongoing to solve problem in updating

2

process. The key issues in FE model updating are field measurement, updating parameter, objective

function, optimization techniques, and interpretation of updated parameters.

When it comes to field measurement, for measuring responses from the real structure, the

accelerometers are most widely used by measuring the translational response because to measure

translational response using accelerometer is accurate and clear even easy to measure. In the most of

existing researches, only translational responses are adopted to assess or evaluate natural mode. For

instance, [6] shows the change of natural frequency and mode shape of Bernoulli-Euler beam after

crack growth. [7] is representative research which uses FE model updating method to detect damage

of simply supported beam. [8] is research of FE model updating using natural frequency, damping

ratio, and mode shape measured by accelerometer and FBG strain sensors. [6-8] and most of existing

researches adopted only translational response to assess natural frequency and mode shape. By the

result of [6], the change of mode shape due to crack growth is little. In the [6], the aim is to detect the

damage of beam structure by checking the change of mode shape after crack growth but it is hard to

detect damage because the change of mode shape is too small to distinct. Actually, in the FE model

updating, because the changes of natural frequencies and mode shapes is not sensitive as the structure

changes, to update initial model as precise FE model which has similar response with real structure is

difficult. For simply supported beam, the translational mode shapes are sine functions whereas the

rotational mode shape are cosine functions. Therefore, on the both end, the translational mode shape is

the minimum and the rotational mode shape is maximum. Based on the [6] the damaged parts on the

continuous beam have similar boundary conditions with hinge so the rotational mode shape could be

more sensitive for the damaged structures. In other words, because the change of rotational mode

shape is expected to be more sensitive than change of translational mode shape for the non-continuous

state of beam like crack or damage, using rotational mode shape in FE model updating is also

expected more effective.

So in this paper, the FE model updating using rotational response expected to be more sensitive will

be carried out to improve the performance of FE model updating process and also the result will be

compared to the result by existing method. In the chapter 2, the basic theory related with FE model

updating will be briefly covered. In the chapter 3, the sensitivity of rotational response to change of

parameter will be analyzed and based on the result of sensitivity analysis, the FE model updating to

compare the approaches by using rotational response and translational response will be carried out by

numerical simulation in the chapter 4. And in the Chanter 5, the result of lab scale experiment to

verify the FE model updating using rotational response will be reported and the conclusion will be

followed with the possibility of using rotational response in FE model updating in the chapter 6.

3

Chapter 2

2. Backgrounds

The basic concepts for the FE model updating are reviewed briefly considering overall process.

Figure 2.1 illustrates the concept and process of FE model updating. To updated initial FE model, by

the optimization updating algorithm, parameter are changed to minimize the gap between the response

from FE model and from experiment until the objective function defined as the gap of responses

meets given value. In general, in the FE model updating process, the compared responses are mainly

natural frequency and mode shape. In case of comparing natural frequency, the response by FEM

analysis is directly compared with experimental response with the natural frequency value itself.

However, in case of comparing mode shape, the responses by finite element method analysis and

experimental result are compared by MAC value which means the mode assurance criteria. So in this

chapter, fist, the mode shape of simply supported beam and MAC value are review. Next, the portion

of natural frequency and mode shape decided by objective function is reviewed. Finally, the review

about the simplex method which controls the overall updating process is followed.

Figure 2.1: The concept of FE model updating

4

2.1. Mode shape of Bernoulli-Euler beam

From the Newton’s Second Law, the equation of motion of transverse vibrating beam is derived with

in assumption of elementary beam theory. Below summary of Bernoulli-Euler assumptions of

elementary beam theory are explained in chapter 6.3 of [9].

Bernoulli-Euler assumptions of elementary beam theory

1. When a beam deforms the principle plane remains plane.

2. When a beam undergoes bending due to deformation, an axis of the beam is not extended or

contracted to axial direction called neutral axis. The neutral axis is included by neutral surface

perpendicular to principle plane.

3. The cross sections, which are parallel to principle planes and perpendicular to neutral axis in

unbending remain plane and remain perpendicular to the bended neutral axis.

4. The beam is uniform because of linearly elastic material.

+

= ( , ), 0 < < (1)

is displacement of transverse vibration. ρ is mass density, E is elastic modulus, I is moment of

inertia and L is length of beam. A is area of cross section. denotes external excitation. In chapter

12.2 of [10] the above equation is derived called the differential equation of motion governing

transverse vibration. And above assumptions of elementary beam theory are applied in equation (1).

Mode Shape of Bernoulli-Euler beam

In case of free vibration, the right side of equation (1) is zero. Then the equation could be solved as;

( ) = ℎ + ℎ + ℎ + ℎ (2)

( , ) = ( ) ( − ) (3)

where V is displacement by assumption of harmonic motion and is the eigenvalue solution and four

are amplitude solution. On simply supported at both ends, the boundary condition is like below

=

= 0 (4)

The equation (2) could be solved using boundary condition expressed in equation (4). When solving

5

equation (2), ‘ =

, =

, ⋯ =

and = = =0’ is obtained. Then V is expressed as

multiple of arbitrary amplitude factor and sin function. The equation for mode shapes becomes

(x) = sin

(5)

where the mode has been normalized by C=1 and renamed as The mode shape of a uniform

simply supported beam is illustrated in Figure. The mode shape from translational DOF is sinusoidal

function based on sine function while the modes shape from rotational DOF which is based on cosine

function.

2.2. MAC value

Because the mode shape is not value but it is shape, to compare the mode shape, the MAC value has

been adopted in FE model updating. The Modal Assurance Criteria, MAC is index for correlation

between mode shape vectors introduced by Allemang and Brown [11]. (1982). Close to 1 of MAC

means the mode shape vectors have high similarity and close to 0 of MAC means the mode shape

vector has no similarity. As shown in equation (6), actually, the MAC value is cosine of angle

between two vectors - the mode for comparison and the mode for reference. In FE model updating,

the MAC value employed to compare mode shape will be updated close to 1.

= (

)

(( )

)(( )

) (6)

The MAC value employed in FEMU is used in comparing the mode shape from finite element method

of initial model or updating model and mode shape of experimental result.

2.3. Objective function

In the FE model updating process, the responses of initial model or updating model and actual model

such as natural frequencies and mode shapes are compared and updated minimize the gap of

responses by changing parameters. In this optimization process, the objective function is used. In

other words, the Fe model updating method changes the parameters which affect the modal properties

of initial FE model and minimize the difference of modal properties from initial FE mode and real

structure. On the way, frequencies and mode shapes of initial FE model are updated and the optimal

6

parameters are found. To find optimal minimization of difference, the objective function is adopted in

FE model updating algorithm. Möller and Friberg [12] presented the best objective function for civil

engineering scale as;

= ∑ , , + , , (7)

where , is normalized residual function of natural frequency and , is normalized residual

function of mode shape. , and , denote the portion of each residual function.

, = , ,

,

(8)

, =( )

(9)

where , and , are the i-th natural frequency from the real structure and FE model. The

denotes modal assurance criterion value which is covered in previous chapter 2.2 between i-th

mode shape from the real structure and FE model.

2.4. Simplex method

Simplex method is an algorithm to find optimum

solution where the gap of response of reference

and response of evaluation is smallest by iteration

as shown in figure 2.2. The objective function is

improved on each step until the objective function

cannot be improved. The entire FE model updating

process in this research is carried out Nelder-Mead

simplex method. The Nelder-Mead simplex

method [13] is widely used in FE model updating.

The principle of algorithm is simple and this

algorithm doesn’t use differential value to objective function. Comparing with Genetic Algorithm

which is also used for FE model updating the speed of calculation is prominent. Also Simplex method

is more efficient than Genetic Algorithm in estimating the system parameters.

Figure 2.2: Iteration process of Simplex method

0 20 40 60 80 100 120 1400

0.5

1

1.5x 10

-6 현재 함수 값 : 5.31465e-09

7

Chapter 3

3. Preliminary

The goal of this research is comparison of the performance of FE model updating using translational

response and the rotational response. Because the translational mode shape has limitation of

sensitivity, using translational response is difficulty in distinguishing the transformation of structure

from original shape of structure. However, using rotational mode shape is expected to be more

sensitive than using translational mode shape because rotational mode shape is type of half of cosine

function which has no zero value on the boundary even though no one tried FE model updating using

rotational mode shape ever. So, before the FE model updating is carried out, the sensitivity analysis of

translational modeshape and rotational modeshape from numerical simply supported beam model is

carried out first in this chapter following the process of Jaishi et al. [14] where the structural FE model

updating using vibration test result. In this chapter, the numerical beam model for basic system

identification is made and the sensitivity analysis to various parameters is carried out. To compare the

sensitivity, the modeshape is estimated by MAC value covered in previous chapter 2.2.

3.1. Modeling for Numerical Analysis & System Identification of Model

In theoretical beam model, the elements could be considered as a thick cross section. Then the divided

elements also satisfy the beam theory of 1st and 3rd assumption based on Newton’s Law.

Table 3.1: The spec of theoretical Beam model

Number of Elements 10

Width of Section 0.08 m

Height of Section 0.01 m

Elastic Modulus 200 GPa

Density of Steel 7850 kg/m

Boundary condition 0 N ∙ M/rad

8

Figure 3.1.1: Simply supported beam model with 10 elements.

Figure 3.1.2: Theoretical beam with 11 node and 22 DOFs.

A simply-supported beam with 2m length following the assumptions of Euler-Bernoulli beam theory

is modeled in MATLAB as shown in figure 3.1.1. The beam is composed of 10 elements with 0.2m

and rectangular section. The detailed specs of beam are on the Table 3.1. Each element has same

rectangular uniform section and same material properties such as elastic modulus and mass density.

The 10 elements beam model has 11 nodes and each node is named such as ‘N1’, ‘N2’, ‘N3’… as

shown in figure 3.1.2. Because each node has 2 degree of freedom by translational displacement and

rotational displacement, the beam has 11 translational-DOFs and 11 rotational-DOFs. Note that

because the vertical displacement is 0 on the boundary node, the translational response at first node

and last node is zero. In this case, the number of total DOF is 20 not 22. To analyze the Modal

properties of translational response and rotational response, the natural frequencies and mode shapes

are obtained by solving Eigen-value problem. The first three natural frequencies of beam model are

5.792Hz, 24.015Hz, and 57.197Hz. The first three mode shapes by translational response are shown

in figure 3.2.1 and the first three mode shapes by rotational response are shown in figure 3.2.2. The

obtained 1st translational mode shape is half of sine function and the obtained 1st rotational mode

shape is half of cosine function like theory. By calculating modal properties, the system identification

is done and using this natural frequencies and mode shapes of base line, the next step, the sensitivity

analysis will be carried out.

9

Figure 3.2.1: The 1st, 2nd, 3rd Translational Mode Shape

Figure 3.2.2: The 1st, 2nd, 3rd Rotational Mode Shape

2 4 6 8 10

Node

Translational Mode shape

1st

2nd

3rd

2 4 6 8 10

Node

Rotational Mode shape

1st

2nd

3rd

10

3.2. Sensitivity Analysis of Translational and Rotational mode shapes

The transformation of structure such as damage or deterioration affects the parameters such as height

of section, elastic modulus. Because the changes of these parameters make the changes of modal

properties, inversely it is possible to detect transformation by analyzing and comparing modal

properties using MAC value which indicates the similarity of mode shapes vectors as covered in

chapter 2. Though the natural frequency from translational responses is same with the natural

frequency from rotational responses, the translational mode shapes are different from rotational mode

shapes. Therefore, the change of MAC values caused by the change of parameters is different as type

of responses. In this chapter, the analysis is carried out to compare the sensitivity of parameters vs.

MAC value in two types of response - translational mode shape and rotational mode shape. The

parameters used in this analysis are like below.

3.2.1. Material Property

The material properties are character of the materials (such as stone, steel and concrete) consisting

structures. Mass of density or elastic modulus is included in material property. In civil engineering

field, the change of material properties is caused by inner damage or change of inner state. For

example, in case of concrete structure, because of the time dependent phenomena of concrete, the

changes of external conditions and deformation by cyclic applied load, the change of material

properties is caused in concrete structure. So recently the NDE methods are adopted to detect the

inner state of concrete or damaged. In this chapter, the MAC sensitivity analysis is done by

controlling the elastic modulus and mass density.

(1) Elastic Modulus

The sensitivity analysis to elastic modulus is carried out by controlling the elastic modulus of the 6th

elements on the modeled beam whose original elastic modulus is 200 GPA and the result with in the

first three modes is shown in Figure 3.3.1. The change of MAC value from rotational mode shape is

more sensitive than the change of MAC value of translational mode shape as change of elastic

modulus. In addition, when the elastic modulus decreases, the MAC value decreases more sensitive.

(2) Mass density

The sensitivity analysis to mass density is carried out by controlling density of the 6th elements on

the modeled beam whose original mass density is 7850 kg/m and the result with in the first three

modes is shown in figure 3.3.1. In only 1st mode, the change of MAC value from rotational mode

11

shape is more sensitive than the change of MAC value of translational mode shape as change of mass

density. However from the next modes, the change of MAC value from rotational mode shape is less

sensitive than the change of MAC value of translational mode shape as change of mass density. The

difference of MAC sensitivity in 1st mode is much smaller than the difference of MAC sensitivity in

next modes.

Figure 3.3.1: The MAC sensitivity analysis of Material properties

150 200 250 30099.7

99.75

99.8

99.85

99.9

99.95

100

Elastic Modulus, E [GPa]

MA

C v

alu

e (

100%

)

MAC value of mode #1

translational

rotational

7.2 7.4 7.6 7.8 8 8.2 8.4 8.6

x 105

99.95

99.96

99.97

99.98

99.99

100

100.01

mass density, rho

MA

C v

alu

e (

100%

)


translational

rotational

150 200 250 30099.7

99.75

99.8

99.85

99.9

99.95

100


MA

C v

alu

e (

100%

)


translational

rotational

7.2 7.4 7.6 7.8 8 8.2 8.4 8.6

x 105

99.95

99.96

99.97

99.98

99.99

100

100.01

mass density, rho

MA

C v

alu

e (

100%

)


translational

rotational

150 200 250 30099.7

99.75

99.8

99.85

99.9

99.95

100


MA

C v

alu

e (

100%

)


translational

rotational

7.2 7.4 7.6 7.8 8 8.2 8.4 8.6

x 105

99.95

99.96

99.97

99.98

99.99

100

100.01

mass density, rho

MA

C v

alu

e (

100%

)


translational

rotational

12

3.2.2. Sectional Property

Sectional properties are factors to be determined by the shape of section. From the sectional

properties, the area and moment of inertia are calculated to analyze the bending in beam structure. The

sectional properties are changed when the external load is applied on structure to make

transformations or scratches. In addition, the section of concrete based structure could be changed due

to creep and shrinkage and the effective section could be changed by crack growth.

(1) Height of beam

The sensitivity analysis to height of section is carried out by controlling height of the 6th elements on

the modeled beam whose original height is 10 cm and the result with in first three modes is shown in

figure 3.3.2. The changed eight of beam is from 8cm to 12cm. The change of MAC value from

rotational mode shape is more sensitive than the change of MAC value of translational mode shape as

change of height of beam. In addition, when height of the 6th elements decreases, the MAC value

decreases more drastic.

(2) Width of beam

The sensitivity analysis to width of section is carried out by controlling width of the 6th elements on

the modeled beam whose original height is 8cm and the result with in the first three modes is shown

in figure 3.3.2. The tendency is similar with case of height of beam as expectation because both height

and width of section is related with 2nd section moment and area of section. Because the 2nd section

moment is calculated by multiplying width and cube of height, the height is more sensitive parameter

than width to MAC value. The change of MAC value from rotational mode shape is more sensitive

than the change of MAC value of translational mode shape as change of width of beam. In addition,

when width of the 6th elements decreases, the MAC value decreases more drastic.

13

Figure 3.3.2: The sensitivity analysis of Sectional properties

0.8 0.85 0.9 0.95 1 1.05 1.1 1.1598

98.5

99

99.5

100

Heigh of Beam, h [cm]

MA

C v

alu

e (

100%

)MAC value of mode #1

translational

rotational

6.5 7 7.5 8 8.5 9 9.5

x 10-4

99.7

99.75

99.8

99.85

99.9

99.95

100

Width of Beam, b [cm]

MA

C v

alu

e (

100%

)


translational

rotational

0.8 0.85 0.9 0.95 1 1.05 1.1 1.1598

98.5

99

99.5

100


MA

C v

alu

e (

100%

)


translational

rotational

6.5 7 7.5 8 8.5 9 9.5

x 10-4

99.7

99.75

99.8

99.85

99.9

99.95

100


MA

C v

alu

e (

100%

)


translational

rotational

0.8 0.85 0.9 0.95 1 1.05 1.1 1.1598

98.5

99

99.5

100


MA

C v

alu

e (

100%

)


translational

rotational

6.5 7 7.5 8 8.5 9 9.5

x 10-4

99.7

99.75

99.8

99.85

99.9

99.95

100


MA

C v

alu

e (

100%

)


translational

rotational

14

3.2.3. Boundary Conditions – end of beam

The boundary condition of both end of simply supported beam is mentioned in previous chapter 2.1.

As equation (6), the transverse displacement of beam is zero and the moment is also zero. However

when the damage is detected on boundary the boundary conditions are considered as changed. In this

study, the rotational change of boundary is considered. The sensitivity analysis to boundary condition

is carried out by controlling rotational of stiffness of boundaries on the modeled simply supported

beam whose original rotational stiffness is 0 and the result with in the first three modes is shown in

figure 3.3.3. The change of MAC value from rotational mode shape is more sensitive than the change

of MAC value of translational mode shape as increase of rotational stiffness of both boundaries.

3.2.4. Added Mass

The sensitivity analysis to add mass is carried out by adding mass on the 6th elements on the modeled

beam and the result with in first three modes is shown in figure 3.3.3. In only 1st mode, the change of

MAC value from rotational mode shape is more sensitive than the change of MAC value of

translational mode shape as the mass added on 6th elements. However from the next modes, the

change of MAC value from rotational mode shape is less sensitive than the change of MAC value of

translational mode shape as the mass added on 6th elements. The difference of MAC sensitivity in 1st

mode is much smaller than the difference of MAC sensitivity in next modes.

15

Figure 3.3.3: The sensitivity analysis of rotational stiffness of boundary condition and damage by

adding mass

1 2 3 4 5 6 7 8 9 10

x 104

100

100

100

100

100

Rotational Stiffness(Boundary) [N*m/rad]

MA

C v

alu

e (

100%


translational

rotational

0.2 0.4 0.6 0.8 199.85

99.9

99.95

100

100.05

Added mass on 2th Elements [kg]

MA

C v

alu

e (

100%

)


translational

rotational

1 2 3 4 5 6 7 8 9 10

x 104

100

100

100

100

100

100


MA

C v

alu

e (

100%

)


translational

rotational

0.2 0.4 0.6 0.8 199.85

99.9

99.95

100

100.05


MA

C v

alu

e (

100%

)


translational

rotational

1 2 3 4 5 6 7 8 9 10

x 104

100

100

100

100

100

100


MA

C v

alu

e (

100%

)


translational

rotational

0.2 0.4 0.6 0.8 199.85

99.9

99.95

100

100.05


MA

C v

alu

e (

100%

)


translational

rotational

16

3.3. FE model updating process approached by using rotational response

So far before simulation or experiment, the MAC sensitivity is analyzed as damage parameters based

on the theoretical modeshape. As the result, among the parameters used in sensitivity analysis, elastic

modulus, the size of section and rotational stiffness of boundary are more sensitive when using

rotational response than when using translational response. Therefore in case of damage of elastic

modulus, the size of cross section or rotational stiffness of boundary using gyroscope to measure

angular velocity is recommended in FE model updating process. In case of mass density change and

change of mass by adding, those are not appropriate parameters to employ in FE model updating

using rotational mode because using rotational response is more sensitivity only at the first mode.

Among the factors which decide the section, the height of section is more sensitive than the width of

section. The change of the height and the width affect the area of section and inertia of section

because the inertial of moment is proportion to cube of height. In summary, measuring the angular

velocity using gyroscope sensors is expected to improve the performance of the FE model updating in

several damage cases. From the next chapters, the FE model updating process is carried out based on

the responses of numerical simulation or experiment measurement. In each step, the performance and

error of updated modal properties and parameters are compared as which sensors are used.

17

Chapter 4

4. Numerical Simulation

From the numerical simulation, the transformation of original beam is defined as damage of beam.

Before verifying the FE model updating using rotational modeshapes from numerical simulation

result, appropriate damaged parameter was selected by considering the possibility of realization in

laboratory.

4.1. Selecting parameter

In the sensitivity analysis of the previous chapter 3, rotational modeshape has more sensitive

changes than translational modeshape especially to elastic modulus, size of section and boundary

rotational stiffness. Sung et al. [15] proposed new sensing method by data fusion of acceleration and

angular velocity based on that to the rotational stiffness of boundary changes such as hinge, using

angular velocity could be supplement the disadvantage of using accelerometer especially on edge or

hinge of beam. The sensitivity analysis result completed in the previous chapter 3.2.3 is accord with

the result of Sung et al.[15].

In civil engineering field most of structure is composed of concrete or steel e.g., tall buildings using

reinforced concretes, prestressed concrete bridges and tower using steel truss. The most widely used

material, concrete is not homogeneous material and for high external stress, concrete has nonlinear

behavior. In addition concrete has time dependent characteristics such as strain due to relaxation,

creep and shrinkage. The damage of concrete structure is highly related with material property –

elastic modulus. Therefore covering elastic modulus as parameter in FE model updating process is

highly recommended to show the superiority of using rotational modeshape. However for simply

supported beam made of steel in laboratory making damage cases similar to real reduces of elastic

modulus is difficult. In this paper, height of section is selected as parameter FE model updating

18

process to compare the efficiency of FE model updating processes by using rotational responses and

translational responses. In real field, the increase or decrease of height of section could be depicted as

local damage by scratch or surface deformation by cyclic load. Considering the size of section

determines the area and moment of inertia change of sectional properties suggests damage of rebar

and connection parts near the rebar in reinforced concrete structure because the section of RC is

expressed as virtual section by amount of concrete and rebar. For example, the decrease of connection

force between rebar and concrete or decreases of ratio of reinforcement bring the decrease size of

section. From the sensitivity analysis of section height in previous chapter 3.2.2, selecting the location

of damage is started by comparing the MAC sensitivity for each damage case as shown in figure 4.1.

On the next page, for the selected damage parameter, height of section, the details such as magnitude

of damage and location of damage is analyzed.

Figure 4.1: Compared MAC sensitivity of beam Thickness – 2nd elements and 6th elements

0.8 0.85 0.9 0.95 1 1.05 1.1 1.1598

98.5

99

99.5

100


MA

C v

alu

e (

100%

)


translational

rotational

0.8 0.85 0.9 0.95 1 1.05 1.1 1.1598

98.5

99

99.5

100


MA

C v

alu

e (

100%


translational

rotational

0.8 0.85 0.9 0.95 1 1.05 1.1 1.1598

98.5

99

99.5

100


MA

C v

alu

e (

100%

)


translational

rotational

0.8 0.85 0.9 0.95 1 1.05 1.1 1.1598

98.5

99

99.5

100


MA

C v

alu

e (

100%

)


translational

rotational

0.8 0.85 0.9 0.95 1 1.05 1.1 1.1598

98.5

99

99.5

100


MA

C v

alu

e (

100%

)


translational

rotational

0.8 0.85 0.9 0.95 1 1.05 1.1 1.1598

98.5

99

99.5

100


MA

C v

alu

e (

100%

)


translational

rotational

19

4.2. Numerical Simulation using Exact Values

The figure 4.1 compares the MAC sensitivity of 2 case of damage – damaged on 2nd element and

damaged on 4th element. As the height of 2nd element or 4th decreases, the MAC value also decreases.

On the first mode, the sensitivity is almost same but on 2nd and 3rd mode, case of damaged 2nd element

is more sensitive to change of height. Based on the difference of sensitivity of MAC, the location of

damage is selected by dividing the cases into near the boundary and near the middle of beam. As

mentioned in chapter 3.1, the natural frequency and modeshape of theoretical simply supported beam

are provided by solving Eigen value problem. In same way, the theoretical natural frequency and

modeshape of damaged beam are calculated first. The height of section is reduced from initial value,

from 1cm to 0.8cm in damaged elements. The FE model updating process using exact value found by

theoretical model is carried out to decide the locations of damage. As shown in figure 4.2, the

decreased height of 2nd element, 4th element and 6th element is assumed in each case.

Figure 4.2: Height decreases in 2nd, 4th, 6th element to select damage location

For each case, the FE model updating using theoretical frequencies and modeshape is carried out

twice. First, translational frequency and modeshapes are used in process and second rotational

frequency and modeshapes are used in process. In this analysis, the magnitude of damage and the

portion of objective function covered in chapter 2.3 are determined as scenario of damage case.

20

2nd element Magnitude of Damage

RMSD [%] / Error [%] J 1mm 2mm 3mm 4mm

Rotational

High

M.S

1.5043 /

0.46

2.5298 /

0.15

5.4771 /

0.07

7.1460 /

1.5

Low

M.S

2.8360 /

2.44

9.8396 /

2.47

10.8099 /

1.09

10.8400 /

7.32

Translational

High

M.S

2.7668 /

1.10

7.7739 /

5.44

10.0795 /

5.67

9.5180 /

2.08

Low

M.S

3.9560 /

7.06

9.8424 /

8.03

15.4028 /

22.33

20.5333 /

9.56

Table 4.1.1: RMSD value of updated parameter in damage of 2nd element

4th element Magnitude of Damage


Rotational

High

M.S

3.3690 /

2.09

4.5219 /

1.11

6.9117 /

0.81

8.8759 /

1.12

Low

M.S

3.4628 /

6.16

7.4129 /

11.32

15.3223 /

33.05

35.5033 /

104.41

Translational

High

M.S

3.5997 /

3.29

6.2896 /

1.61

15.3841 /

19.96

16.4815 /

0.06

Low

M.S

4.4829 /

9.09

7.7796 /

13.52

12.9439 /

27.92

14.3720 /

8.35


6th element Magnitude of Damage


Rotational

High

M.S

2.4806 /

0.54

5.9875 /

1.82

6.6780 /

0.06

7.3096 /

0.02

Low

M.S

2.4818 /

0.56

5.9874 /

1.82

6.8917 /

0.43

8.8569 /

0.40

Translational

High

M.S

1.5813 /

0.66

4.5987 /

0.74

7.1237 /

1.98

6.9570 /

0.26

Low

M.S

2.3257 /

0.46

4.1500 /

0.47

5.9910 /

0.04

10.4440 /

1.74


21

2nd & 5th elements Magnitude of Damage

RMSD [%] / Error [%] J 1mm 2mm 3mm

Rotational

High

M.S

3.8236 /

1.45

11.4634 /

23.23

16.0623 /

0.10

Low

M.S

4.0408 /

54.33

12.9180 /

12.85

12.4720 /

14.15

Translational

High

M.S

4.4349 /

29.26

8.0078 /

34.65

27.8197 /

13.32

Low

M.S

3.5909 /

12.42

6.9711 /

13.44

33.1946 /

156.55

Table 4.1.4: RMSD value of updated parameter in multi damage

The table 4.1.1~4.1.4 shows the percentage RMSD value of updated parameters comparing with the

assumed value of height depend on magnitude of damage, type of used modeshape in updating, the

portion of objective function to respond modeshape and the location of damage. A percentage RMSD

(root mean square deviation) method is used to quantify the accuracy of the updated heights of section:

(10)

where ℎ denotes updated height of element and ℎ denotes initial value of assumed height of

each element. The Error [%] is calculated by comparing updated parameter of damaged element and

assumed value of damaged element. The J represents objective function to inform the degree of

weight on modeshapes. High M.S means the higher weight on modeshape and Low M.S means the

higher weight on natural frequency. In the above table 4.1.1~4.1.4, the cases whose RMSD or Error

value is more than 10(%) are marked as red color because this updating process adopts exact values

from theoretical modeshapes and natural frequencies. As sensitivity analysis, using rotational

responses in model updating is more efficient than using translational responses in all analyzed

location. Considering the accuracy of updated parameters, using rotational responses is efficient for

detecting the damage of boundary of beam and using translational responses is efficient for detecting

the damage of middle of beam. The magnitude of damage is also related with the performance of

model updating because the simplex method is especially dependent on the initial point. If the height

loss is more than 4mm, adjusting the initial value from 1.0cm to 0.9, 0.8 or 0.7 or changing the

portion of objective function to respond mode shape part more is recommended. For damage on 2nd

element, the 4mm damage could be also discovered in the process of updating parameter by using the

(%) = ∑ (ℎ − ℎ )

∑ (ℎ )

× 100

22

objective function which has high weight on modeshape as shown in table 4.1.1. For damaged on 6th

element case, using rotational response is more efficient for precise updating of parameters than using

rotational response as shown in table 4.1.3. For the multiple damages as shown in table 4.1.4, though

the exact values from theoretical modal properties are employed, it is hard to update the parameters as

assumed height.

4.3. Numerical Simulation using Excitation

In the analysis of previous chapter 4.2, the overall FE model updating result is good because exact

value, the theoretical solution of initial beam, was used. Theoretical natural frequency and the

theoretical modeshape are used in this updating process to predict theoretical beam. Even though

overall updated result is good, the advantage and disadvantage of using rotational response and using

translational response are turned out. From the comparing the updated parameters, height, the

updating process based on the rotational mode has superior performance than the updating process

based on the translational mode especially in damage of 2nd element case as shown in table 4.1.1. The

FE model updating approached by rotational modeshape using high weight on modeshape sensitive to

height loss is successful for all damage case – height decrease of 2nd, 4th and 6th element. Rotational

modeshape based FE model updating is predominant to detect damage near the end of beam. Because

the goal of this research is to verify the possibility of using rotational response in FEMU, the FE

model updating using rotational response will be carried out and be compared with the translational

response for the three cases like below table 4.2.

Table 4.2: Selected cases for numerical simulation using excitation

Scenario

Base No damaged on the beam

Case1 2mm Damaged on 2nd

element

Case2 4mm damaged on 2nd element

23

Figure 4.3: The sensor location to get simulation data for each case

4.3.1 Getting numerical simulation data

Before using the modal properties measured in experiment, the FE model updating using the modal

properties from Numerical simulation with Matlab Simulink was carried out to verify the FE model

updating using rotational modeshape. From numerical simulation in this chapter, the acceleration is

measured for translational response and angular velocity is measured for rotational response. By the

Matlab Simulink, a random input excitation is created and the position of excited input is illustrated in

figure 4.3. Also Figure 4.3 shows the location of sensors to measure acceleration and angular velocity.

In each case, the position of sensors and input is same with base line. To avoid symmetric placement

of sensor, the distance between sensors are different e.g., the distance between 1st location and 2nd

location is 40cm but the distance between 2nd location and 3rd location is 60cm. In this simulation, the

24

measured responses have 5% of noise. Sampling rate of simulation is about 2.56× 70Hz ⋍180Hz and

the simulated responses are cut off within 70Hz because in this research, only the first three modes are

considered. The responses of acceleration and angular velocity by random excitation are shown in

figure 4.4.1~4.4.3. The first mode of simply supported beam is half of sine function approached by

translational-DOF while when approached by rotational-DOF the first mode is half of cosine function.

Therefore the 2nd location of sensor has high PSD in 1st mode in acceleration measurement and the 3rd

location of sensor has high PSD in 1st mode in angular velocity measurement. The Frequency

Response Function of Acceleration is distinguished from the FRF of Angular velocity. The result

gained from chapter 4. 2 could be also related to it.

Figure 4.4.1: The PSD of simulated for Baseline beam



0 10 20 30 40 50 60 70 80 9010

-10

10-8

10-6

10-4

10-2

freqeuncy [Hz]

PS

D

Angualar velocity measurement

1st location

2nd location

3rd location

0 10 20 30 40 50 60 70 80 9010

-6

10-4

10-2

100

102

freqeuncy [Hz]

PS

D

Acceleration measurement

1st location

2nd location

3rd location

0 10 20 30 40 50 60 70 80 9010

-8

10-6

10-4

10-2

freqeuncy [Hz]

PS

D


1st location

2nd location

3rd location

0 10 20 30 40 50 60 70 80 9010

-6

10-4

10-2

100

102

freqeuncy [Hz]

PS

D


1st location

2nd location

3rd location

0 10 20 30 40 50 60 70 80 9010

-8

10-6

10-4

10-2

freqeuncy [Hz]

PS

D


1st location

2nd location

3rd location

0 10 20 30 40 50 60 70 80 9010

-6

10-4

10-2

100

102

freqeuncy [Hz]

PS

D


1st location

2nd location

3rd location

25

The natural frequencies from simulation result are shown in table 4.3.1~3. Though the natural

frequencies of first three modes are not perfectly coincide with the theoretical value, the over all

measured natural frequencies from numerical simulartion are appropirate considering the erorr of

measurement. The natural frequency of beam is changed as damage which adjust the height of section.

In FE model updating process, the initial model is updated by changing the parameters to update the

natural frequency for reducing the gap between the natural frequcy from measurement and updated

natural frequency. The modeshape is also updated following the algorithm which reduces the gap

between MAC value of modeshape vectors from measurment and mode shape vectors of updated

model. Likewise natural frequency, the mode shape is also changed as decrease of the height of

section defined as damage in this paper. In figure 4.5.1~4.5.3, the greenline depicts the the modeshape

of initial beam and the redline depicts the theoretically transformed modeshape of damaged beam.

The transfromation of modeshape comparing with the initial modeshape is outstading in case of

rotational mode as expected with the result of sensitiviry analysis in the previous chapter 3.

Table 4.3.1: Natural Frequency of Baseline from Numerical Simulation

[Hz] 1st mode 2nd mode 3rd mode

Rotational mode 5.781 24.028 57.283

Translational mode 5.780 23.997 57.078

Theoretical 5.792 24.015 57.197





Theoretical 5.705 23.012 54.271





Theoretical 5.447 20.760 49.842

26

Figure 4.5.1: Comparison of Mode Shape for Base

2 4 6 8 10

Node

Rotational Mode Shape #1 base

initial

damaged

measure

updated

2 4 6 8 10

Node

Translational Mode Shape #1 base

initial

damaged

measure

updated

2 4 6 8 10

Node


initial

damaged

measure

updated

2 4 6 8 10

Node


initial

damaged

measure

updated

2 4 6 8 10

Node


initial

damaged

measure

updated

2 4 6 8 10

Node


initial

damaged

measure

updated

27


2 4 6 8 10

Node

Rotational Mode Shape #1 case1

initial

damaged

measure

updated

2 4 6 8 10

Node

Translational Mode Shape #1 case1

initial

damaged

measure

updated

2 4 6 8 10

Node


initial

damaged

measure

updated

2 4 6 8 10

Node


initial

damaged

measure

updated

2 4 6 8 10

Node


initial

damaged

measure

updated

2 4 6 8 10

Node


initial

damaged

measure

updated

28


2 4 6 8 10

Node


initial

damaged

measure

updated

2 4 6 8 10

Node


initial

damaged

measure

updated

2 4 6 8 10

Node


initial

damaged

measure

updated

2 4 6 8 10

Node


initial

damaged

measure

updated

2 4 6 8 10

Node


initial

damaged

measure

updated

2 4 6 8 10

Node


initial

damaged

measure

updated

29

4.3.2 Comparison of FEMU Performance using excitation

To achieve goal of this research, the performance of FE model updating using rotational response

from simulation response is evaluated. To compare the performance of FE model updating using

rotational response and translational response, the damage is on 2nd element as result of FE model

updating using exact value as chapter 4.2. Though the performance of FE model updating using

translational response could be improved by changing objective function to consider natural

frequency more, the adjusting the objective function to consider natural frequency more is excluded

because, in this research, based on the MAC value sensitivity of rotational response to damage, the

objective function is designed to consider modeshape. By changing the parameters, the initial FE

model is updated to have modal property proximate with the modal property from simulation. To

reduce the gap between modal property of FE model and modal property from simulation the FE

model is repeatedly updated following simplex method algorithm mentioned in previous chapter 2.4.

The updated natural frequencies optimizing the objective function by simplex method are shown in

figure 4.6. The natural frequencies are updated with measured natural frequencies for all case. The

natural frequencies in both translational response and rotational response are updated from initial

value to almost same value of natural frequency from measurement. The updated mode shape for each

case is marked as shown figure 4.5.1~4.5.3 and the figure 4.7 shows the updated mode shapes for

each case using MAC value. Initial and updated MAC value is calculated by comparing with the

modeshape vector from simulation measurement. The initial mode shapes are updated to have MAC

value near 1 for all case whether using rotational response or translationa response. In the case 1 and

case2, when using rotational response, the MAC value is lower than when using translational response

for same damage before updating but after updating the MAC value get close to 1 in all of cases –

updated to measurement. The figure 4.8 shows the updated parameter, height for each case. For all

cases, the updated parameters has less error using rotational response the updated parameters than

using translational response as shown in table 4.4 and figure 4.9. For baseline, updating process using

rotational response updated parameters within 5% error but using translational response updated with

large error especially to one parameter-9th element. For case 1 and case 2, both methods could detect

the location of damage and magnitude of damage in the damaged element, but using rotational

responses has less error than using translational responses also.

30

Figure 4.6.1: natural frequency by updating in Baseline

Figure 4.6.2: natural frequency by updating inCase1

Figure 4.6.3 : natural frequency by updating in Case2

1 2 30

10

20

30

40

50

60frequency updated in baseline

mode

Natu

ral

Fre

quen

cy

initial

t-measurment

t-updated

r-measurment

r-updated

theoretical

1 2 30

10

20

30

40

50

60frequency updated in case1

mode

Natu

ral F

requ

en

cy

initial

t-measurment

t-updated

r-measurment

r-updated

theoretical

1 2 30

10

20

30

40

50

60frequency updated in case2

mode

Natu

ral

Fre

quen

cy

initial

t-measurment

t-updated

r-measurment

r-updated

theoretical

31

Figure 4.7.1: Change of Modeshape by updating of Baseline is expressed as MAC value.



1 2 30.7

0.75

0.8

0.85

0.9

0.95

1MAC value of Baseline

mode

MA

C v

alu

e

t-initial

t-updated

r-initial

r-updated

1 2 30.7

0.75

0.8

0.85

0.9

0.95

1MAC value of case1

mode

MA

C v

alu

e

t-initial

t-updated

r-initial

r-updated

1 2 30.7

0.75

0.8

0.85

0.9

0.95

1MAC value of case2

mode

MA

C v

alu

e

t-initial

t-updated

r-initial

r-updated

32

Figure 4.8: The updated heights of each element by FE model updating for three cases

2 4 6 8 100

0.5

1

1.5

heig

ht

[cm

]

Updated height - Baseline

assumed acc gyro

2 4 6 8 100

0.5

1

1.5

he

ight

[cm

]

Updated height - Case1

assumed acc gyro

2 4 6 8 100

0.5

1

1.5

heig

ht

[cm

]

Updated height - Case2

assumed acc gyro

33

Table 4.4: Compare RMSD value for each damage case

RMSD(%) Baseline Case1 Case2



Figure 4.9: updated parameters and error percentage

2 4 6 8 100

10

20

30

40

50

perc

enta

ge

[%

]

Error of updated height - Baseline

acc gyro

2 4 6 8 100

10

20

30

40

50

pe

rce

nta

ge

[%

]

Error of updated height - Case1

acc gyro

2 4 6 8 100

10

20

30

40

50

pe

rcen

tag

e [

%]

Error of updated height - Case2

acc gyro

34

4.4 Validation of FE Model Updating using Rotational Response

The FE model updating to find the damaged location and the damaged height using rotational

modeshape and translational modeshape from the simulation using matlab Simulink is carried out. In

this process, location of damaged is confined as 2nd element of beam and the objective function is

escaped from only focusing on the natural frequency. The magnitude of damage is only controlled as

case. As a result, the FE model updating performance using rotational responses is more efficient than

the FE model updating performance using translational responses. The percentage RMSD of updated

parameter is lower, overall error percentage of each element is lower and the modeshape is also

updated closer to modeshape from measurement when using rotational response. From numerical

simulation result, the validation of using rotational response in FE model updating is completed.

Actually the accelerometer is most widely used sensor to measure structure because of sensitive to all

of responses and the response is clear and accurate, the validation using rotational sensor is

challenging. In next chapter, The FE model updating is carried out from the experimental

measurement to verify FE model updating using rotational responses. Because using gyroscope sensor

to measure angular velocity is more sensitive to damage of each parameter, the result and performance

of model updating could be also improved as simulation result.

35

Chapter 5

5. Experiment

In this chapter, the FE model updating is carried out using modal properties from the experimental

measurement to verify the FE model updating using rotational response. The transverse responses of

before-damaged beam and after-damaged beam are measured by accelerometers and gyroscopes

sensors.

5.1 Experimental Set up

As shown in figure 4.3, the experiment to get responses of transverse vibration is set up. Before-

damaged beam is made following the baseline and after-damaged beam is made following the case2

in figure 4.3. A steel beam, an impact hammer, 3 accelerometers, 3 gyroscope sensors and DAQ

system are prepared as shown in figure 5.1. The 2m steel beam is manufactured as numerical model

made in chapter 3 and to depict the decrease of height (damage), the part of steel beam is cut off as

shown in figure 5.1. Table 5.1 shows the spec of accelerometers and gyroscopes sensors to measure

the acceleration and angular velocity. The sensors are installed on the three location 40cm, 100cm and

Table 5.1: The accelerometer and gyroscope for measuring response

Accelerometer Gyroscope

Model PCB 353B33 ADXRS646

Sensitivity 100mV/g 9mV/°/sec

Measurement range ±50 g pk ±300°/sec

Figures

36

Figure 5.1: Experimental set up for measurement

190cm far from the left boundary of beam. PXIE-1082 of National instrument is used for data

acquisition. The sampling rate is 2048Hz and the duration time of each measurement is 180s. Each

case is measured 3times repeatedly.

5.2 Measurement data from experiment

The measured transverse responses using accelerometer and gyroscope sensor are shown as below

table 5.2 and figure 5.2 ~ 5.3. Because it is hard to find optimization solution using first three modes,

the first mode and second mode (third mode is not included unlike the simulation) are used in the FE

Table 5.2: Natural Frequencies of from experiment

Before-damaged 1st mode 2nd mode

Rotational mode 5.647 21.844

Translational mode 5.594 21.843

Theoretical 5.792 24.015

After-damaged 1st mode 2nd mode

Rotational mode 5.333 18.360

Translational mode 5.240 18.360

Theoretical 5.447 20.760

37

Figure 5.2.1 : Angular velocity measurement Figure 5.2.2 : Enlarged angular velocity

Figure 5.2.3 : Acceleration measurement Figure 5.2.4 : Enlarged acceleration

Figure 5.3.1 : The frequency domain, PSD of angular velocity and acceleration of before-damaged

Figure 5.3.2 : The frequency domain, PSD of angular velocity and acceleration of after-damaged

0 20 40 60 80 10010

-10

10-8

10-6

10-4

10-2

freqeuncy [Hz]

PS

D


1st location

2nd location

3rd location

0 20 40 60 80 10010

-6

10-4

10-2

100

102

freqeuncy [Hz]

PS

D


1st location

2nd location

3rd location

0 20 40 60 80 10010

-10

10-8

10-6

10-4

10-2

freqeuncy [Hz]

PS

D


1st location

2nd location

3rd location

0 20 40 60 80 10010

-6

10-4

10-2

100

102

freqeuncy [Hz]

PS

D


1st location

2nd location

3rd location

0 1 2 3 4 5 6 7 8

x 104

-0.05

-0.04

-0.03

-0.02

-0.01

0

0.01

0.02

0.03

0.04

0.05

time[s]

an

gula

r ve

loc

ity [

theta

/s]

1st location 2nd lodation 3rd location

1.004 1.006 1.008 1.01 1.012 1.014 1.016 1.018 1.02

x 104

-0.2

-0.15

-0.1

-0.05

0

0.05

0.1

0.15

0.2

0.25

time[s]

accele

rati

on [

m/s

2]


3.04 3.041 3.042 3.043 3.044 3.045 3.046 3.047 3.048 3.049 3.05

x 104

-2

-1.5

-1

-0.5

0

0.5

1

1.5

2

2.5

x 10-3

time[s]

an

gula

r ve

loc

ity [

theta

/s]


0 1 2 3 4 5 6 7 8

x 104

-10

-8

-6

-4

-2

0

2

4

6

8

10

time[s]

accele

rati

on [

m/s

2]


38

FE model updating process from measurement in this experiment. Figure 5.2 is time domain

measurement data and figure 5.3 is frequency domain data. To imitate stationary signal, the impulse is

repeated on input point by hammer as shown in figure 5.2.

5.3 Comparison of FE Model Updating Performance

From the measurement, the translational response and rotational response are updated as shown in

table 5.3 and figure 5.4~5. By the FE model updating, the natural frequency is updated close to

measured natural frequency reduced by damage (decrease of height of section). The mode shape is

also updated by FEMU process using responses from experiment as shown in figure 5.5 where the

MAC values are compared to measurement data – translational modeshape of initial state, updated

translational modeshape, theoretical translational modeshape, rotational modeshape of initial state,

updated rotational modeshape, and theoretical rotational modeshape. In figure 5.5.1 shows the

comparison data of 1st modeshape. The initial values are updated to have the similar response of

measurement. In figure 5.5.2 shows the comparison data of 2nd modeshape. The initial values of 2nd

mode are also updated. The MAC between measured mode shape and initial mode shape by rotational

response is 0.7321 which means quite different vectors but in this case, the mode shape is updated by

using rotational mode shape verified that has more sensitivity than translational response. Figure 5.6

illustrates the updated heights with much errors compared to the result of using simulation responses.

Even though the updated heights have much error than simulation result, using rotational response has

lower RMSD than using translational response as shown in table 5.4 and figure 5.7. For the non-

damaged beam, the RMSD (%) error of parameters by FE model updating using rotational response is

11.072% and the RMSD (%) error of parameters by FE model updating using translational response is

23.161%. For the non-damaged beam, the RMSD (%) error of parameters by FE model updating

using rotational response is 17.169% and the RMSD (%) error of parameters by FE model updating

using translational response is 27.344%. By the FE model updating using response from experiment,

using rotational response is turned out to be more efficient in model updating.

For the before damaged beam, actually the initial state of response is same with the theoretical

response but there is difference between initial response and measured response. It could be caused by

not considered conditions such as weight of sensors, the exact position of sensors. In this experiment,

the first mode and second mode are used to find the solution of simplex method because of

optimization problem. To decide the not considered conditions previous to FE model updating is one

of method to use 1st, 2nd and 3rd mode and to get the more accurate updated result.

39

Table 5.3.1: Comparison of FE model updating result – before damaged beam

Translational Rotational

Initial Updated Initial Updated

Natural

frequency

1st freq [Hz] 5.792 5.594 5.792 5.647

2nd freq [Hz] 24.015 21.844 24.015 21.844

Mode

shape

1st MAC 0.9885 0.9890 0.9991 0.9994

2nd MAC 0.9987 0.9993 0.9993 0.9999

Table 5.3.2: Comparison of FE model updating result – after damaged beam

Translational Rotational

Initial Updated Initial Updated

Natural

frequency

1st freq [Hz] 5.447 5.240 5.447 5.332

2nd freq [Hz] 20.760 18.360 20.760 18.364

Mode

shape

1st MAC 0.9890 0.9902 0.9749 0.9869

2nd MAC 0.9421 0.9927 0.7321 0.9948

40

Figure 5.4 Comparison of the natural frequencies

Figure 5.5.1 : Comparision MAC value of 1st mode shape for each case

Figure 5.5.2 comparision MAC value of 2nd mode shape for each case

0

1

2

3

4

5

6frequency updated - 1st mode

before damaged after damaged

Natu

ral

Fre

qu

enc

y

initial

t-measurment

t-updated

r-measurment

r-updated

0

5

10

15

20

25frequency updated - 2nd mode

before damaged after damaged

Natu

ral

Fre

qu

enc

y

initial

t-measurment

t-updated

r-measurment

r-updated

10.9

0.92

0.94

0.96

0.98

1MAC value of before damaged

mode

MA

C v

alu

e

t-initial

t-updated

t-theo

r-initial

r-updated

r-theo

10.7

0.75

0.8

0.85

0.9

0.95

1MAC value of after damaged

mode

MA

C v

alu

e

t-initial

t-updated

t-theo

r-initial

r-updated

r-theo

20.9

0.92

0.94

0.96

0.98

1MAC value of before damaged

mode

MA

C v

alu

e

t-initial

t-updated

t-theo

r-initial

r-updated

r-theo

20.7

0.75

0.8

0.85

0.9

0.95

1MAC value of after damaged

mode

MA

C v

alu

e

t-initial

t-updated

t-theo

r-initial

r-updated

r-theo

41

Figure 5.6 : The updated heights of each element by FE model updating

Figure 5.7 : The error percentage of updated parameter for each case

Table 5.4: Compare RMSD value to select location of damage

RMSD(%) Translational Rotational

Before damaged 23.161 11.072

After damaged 27.344 17.169

2 4 6 8 100

0.5

1

1.5

he

ight

[cm

]

Updated height - before damaged

assumed gyro acc

2 4 6 8 100

0.5

1

1.5

heig

ht

[cm

]

Updated height - after damaged

assumed gyro acc

2 4 6 8 100

20

40

60

80

100

perc

enta

ge

[%

]

Error of updated height - before damaged

acc

gyro

2 4 6 8 100

20

40

60

80

100

perc

enta

ge

[%

]

Error of updated height - after damaged

acc

gyro

42

Chapter 6

6. Conclusion

So far, the Finite Element model updating using rotational response and translational response is

carried out to retain validity of FE model updating using rotational responses. Started from the

analysis the MAC sensitivity of rotational response to damage, FE model updating using rotational

response is verified by controlling the thickness of beam in numerical simulation and in lab-scale

experiment. In the preliminary analysis, by using the rotational response in FE model updating, the

updated parameter is expected to have more accurate than using translational response especially

when the damage is occurred near the boundary and high mode-shape responded objective function is

useful in FEMU when the updated model is far from the initial. In numerical simulation, using

rotational response is also evaluated to have more efficient performance in FE model updating

compared to using translational response. In the numerical simulation, the thicknesses of beam of

each element were accurately assessed with error less than 8% when using rotational modeshape. In

the experiment, however, the performance in FE model updating when using rotational responses was

short of expectation from preliminary work and simulation result although the validity of using

rotational response is achieved by comparing the performance of using translational response in

experiment. In the experiment, natural frequencies and modeshapes are updated close to measured

data and the RMSD error of updated thicknesses is less than 18%. Comparing the error is about 28%

when using translational response, using rotational response is considerable to use in finite element

model updating. To reduce error of experiment first using the high performance gyroscope is needed.

And not only thickness of each element but also the mass of each element with sensors is selected as

parameters to be updated in finite element model updating.

43

Reference

[1] C. P. Fritzen, D. Jennewein, and T Kiefer, “Damage detection based on model updating

methods,” Mechanical Systems and Signal Processing, vol. 12, no. 1, pp. 163-186,

1998.

[2] J. M. W. Brownjohn, and P. Q. Xia, “Dynamic assessment of a curved cable-stayed

bridge by model updating,” Journal of Structural Engineering-ASCE, vol. 126, no. 2, pp.

252-260, 2000.

[3] A. Teughels, J. Maeck, and G. De Roeck, “Damage assessment by FE model updating

using damage functions,” Computers & Structures, vol. 80, no. 25, pp. 1869-1879, 2002.

[4] J. M. W. Brownjohn, P. Moyo, P. Omenzetter, and Y. Lu, “Assessment of highway bridge

upgrading by dynamic testing and finite-element model updating,” Journal of Bridge

Engineering-ASCE, vol. 8, no. 3, pp. 162-172, 2003.

[5] B. Jaishi, and W. X. Ren, “Damage detection by finite element model updating using

modal flexibility residual,” Journal of Sound and Vibration, vol. 290, no. 1, pp. 369-387,

2006.

[6] Shen, M. H., & Pierre, C. (1990). Natural modes of Bernoulli-Euler beams with

symmetric cracks. Journal of sound and vibration, 138(1), 115-134.

[7] Jaishi, B., & Ren, W. X. (2005). Structural finite element model updating using ambient

vibration test results. Journal of Structural Engineering, 131(4), 617-628.

[8] Moaveni, B., He, X., Conte, J. P., & De Callafon, R. A. (2008). Damage identification of

a composite beam using finite element model updating.Computer‐Aided Civil and

Infrastructure Engineering, 23(5), 339-359.

[9] R.R. Craig, Jr.,(2000) Mechanics of of Materials, 2nd ed., New York, Wiley.

[10] R.R. Craig, Jr., A. J. Kurdila, (2006). Fundamentals of Structural Dynamics, 2nd ed.,

New York ,Wiley.

[11] Allemang, R. J. (2003). The modal assurance criterion–twenty years of use and

abuse. Sound and vibration, 37(8), 14-23.

44

[12] P. W. Möller and O. Friberg, “Updating large finite element models in structural

dynamics,” AIAA Journal, vol. 36, no. 10, pp. 1861-1868, 1998

[13] J. A. Nelder, and R. Mead, “A simplex method for function minimization,” The

Computer Journal, vol. 7, no. 4, pp. 308-313, 1965.

[14] Jaishi, B., & Ren, W. X. (2005). Structural finite element model updating using ambient

vibration test results. Journal of Structural Engineering, 131(4), 617-628.

[15] Sung, S. H., Park, J. W., Nagayama, T., & Jung, H. J. (2014). A multi-scale sensing

and diagnosis system combining accelerometers and gyroscopes for bridge health

monitoring. Smart Materials and Structures, 23(1), 015005.

45

Acknowledgement

제일 먼저, 부족한 제게 UEE에서 공부할 수 있는 기회를 주신 윤정 수님께 감사

드립니다. 리고 졸업논문을 비롯해 석사과정 동안 도해주신 조수 사님께도 감사

의 말씀을 전합니다. 난 2년간 완 히 이해 못하는 부족한 학생에게도 친절하게 설명

을 해주신 김재홍 수님과 이영주 수님께도 감사 드립니다. 졸업논문 표 평가를 해

주신 신명수 수님, Torbol 수님께도 감사 드립니다. 요 자주 찾아 뵙 는 못했 만

정신적인 멘토가 되어주신 문수현 수님, 윤새라 수님께도 늦게나마 감사 드립니다.

2년 동안 같은 수업을 많이 들어 수업동료라고 할 수 있는 민영이와 성우 리고 Fritz

에게도 많은 도움을 았던 것 같아 고맙고 항상 많이 도와 던 연 실 동료인 현 이,

화, 은 이, R.P.에게 나는 잘해 주 못한 것 같아 고마움과 미안한 마음이 듭니다.

작년 여름 ANCRISST 학회 행 스태프로 일할 때 많이 도와주신 금은 학 에 안 계시

는 조해영 선생님께도 감사의 말씀 드립니다. 또 작년 여름, 엄청 고생했던 하얼빈

에서 만나 이것저것 도움 많이 주신 임승현 사님께도 감사의 말을 전합니다. 대학원생

활을 힘들게 한 것은 아니 만 나고 나니 때때로 스트레스를 거나, 운동을 하고 싶

을 때 같이 축 를 했던 위대 도시락 리고 참석율이 높 는 않았 만 말만

폭탄주 전문 모임인 급형에도 고맙다는 말을 전합니다. 위대 노땅들만 모인

경로당 멤 , 특히 6년넘게 함께 해온 09학 동기들에게도 고맙다는 말을 전

하고 싶습니다.

남들에게는 쉬울 도 모르는 석사졸업이 만 다리를 다쳐 수술하는 개인사정도 있고

여러 가 로 힘든 점이 있어 개인적으로 힘든 과정이었다고 생각하는데 무사히 끝마쳐서

기쁘게 생각하고 있고, 앞으로 어떤 일이 있어도 열심히 도전해봐야겠다는 생각이 듭니

다. 끝으로, 학문 연 뿐만 아니라 생활 전 에 걸쳐서 저를 2년동안 도해주신

도 수님, 심성한 수님께 깊은 감사의 뜻을 표합니다.

2015.01.16

명환드림

Documents

Finite Element Model Updating using Rotational Response