MINISTRY OF EDUCATION AND TRAINING
NATIONAL UNIVERSITY OF CIVIL ENGINEERING

Ngo Trong Duc

VIBRATION ANALYSIS AND CRACKS
IDENTIFICATION ON MULTIPLE CRACKED BEAMLIKE STRUCTURES MADE OF FUNCTIONALLY
GRADED MATERIALS

Speciality: Engineering Mechanics
Code: 9520101

SUMMARY OF DOCTORAL DISSERTATION

ACADEMIC ADVISOR
PROF. DR TRAN VAN LIEN

Ha Noi - 2019


1

The dissertation was completed in National University of Civil Engineering

Academic advisor:
Prof. Dr Tran Van Lien
National University of Civil Engineering

Reviewer 1: Prof. Sci Dr. Nguyen Dong Anh
Institute of Mechanics, Vietnam Academy of Science and
Technology.

engineers can repair and maintenance structures to prevent seriously damages.
Recently, domestic and world researchers started to study the effects of
cracks and cracks identification problem in FGM structures through NonDestructive Testing - NDT using dynamic characteristics such as: frequencies,
mode shapes, vibration displacements,.... However, authors often study simple
beam structure with limited number of cracks, the problem of complicated beam
structures like multiple cracked continuous-beam hasn’t been studied yet.
2. Purpose of the research
Contributing vibration modal of multiple cracked beam-like structures
made of FGM using dynamic stiffness method (DSM). Therefore, offer some
identification method to determine crack parameters in beam-like structures
based on measured frequencies, mode shapes and vibration displacements.
3. Objects and scopes of the research
Objects of the research: Beam-like structures are simple beams and
continuous beams made of FGM that have one-side open cracks.
Scope of the research:
- Beam-like structures made of FGM which mechanical characteristics varies
though beam thickness (P-FGM) with predestination material, geometric
parameters and boundary conditions.
- Cracks are one-side open cracks that perpendicular to beam axis. We don’t
consider the reasons and procedures of the formation and development of


3

cracks. Cracks at special positions like boundaries or joints are also out of
the range of the research.
5. Methodologies of the research
Theatrical methodology and numerical simulation calculation.
6. Scientific basis of the research
Based on elastic theory, destructive mechanics, structural dynamic



4

related to the field of the research such as: Structural health monitoring;
functionally graded material (FGM); structural damage model; crack models
that are normally used in vibration analysis of frame structures; researches of
the multiple cracked FGM beam-like structures; dynamic stiffness method. The
dissertation also review crack identification methods based on dynamic
parameters; crack detection methods using wavelet analysis and artificial
neutral network. Besides, elastic spring model of cracks were presented to
clarify multiple cracked beam element model which is used in the research.
Chapter conclusion evaluated the unsolved problem and set the research goals.
CHAPTER 2. MULTIPLE CRACKED FGM BEAM VIBRATION
MODEL
2.1. Vibration of intact Timoshenko FGM beam
Considered the beam with the length L, rectangular section with
dimensions A=b×h made of FGM (Fig 2.1). The FGM characteristic function is
component style (P-FGM)
z
x
u
u0

w0



z
w

(2.1)

  2 I11
0
  2 I12
0 
0 0
 A11  A12 0 

A   A12 A22 0  ; Π  0 0 A33  ; D()   2 I12 2 I 22  A33 0 
 0
 0
0  A33 0 
0
A33 
0
 2 I11  (2.23)


z  {U, ,W}T ; q  {P,0, Q}T


5

Based on Hamilton principle, motion equation in frequency domain was

Az  Πz  Dz  q

(2.24)


x

 H ( x   ,  ) q ( ,  )d

(2.34)

0

where [H(x,)] is matrix of transfer functions that satisfied equations

A. H  Π. H  D. H  0

(2.35)

With the left boundary conditions

H(0)  [0] ; H (0)  A1

(2.36)

Hence, general solution forced vibration differential equations were
~z c ( x ,  )  z c ( x ,  )  z q ( x ,  )
(2.39)
2.2. Continuous condition at crack positions. Two equivalent springs model
keY

a
h
a)


 E
 

RE  1  3(3  n)
2n
1 n
 RE  11  n 

f1 ( z )  s 2 (0.6272  0.17248s  5.92134 s 2  10.7054 s 3  31.5685s 4  67.47 s 5 
 139.123s 6  146.682 s 7  92.3552 s 8 )
f 2 ( z )  s 2 (0.6272  1.04533s  4.5948s 2  9.9736 s 3  20.2948s 4  33.0351s 5 
 47.1063s 6  40.7556 s 7  19.6 s 8 )

2.3. Vibrations of multiple cracked FGM Timoshenko beam





2.3.1. Crack function matrix G(x) and displacement expression z c ( x, )
Propose 3×3 matrix [Gc(x,)] as bellowed
[G c ( x,  )]  [L( x,  )][ Σ]
(2.58)
where
 1 0 0
1 cosh k1x 2 cosh k21x 3 cosh k3x 11 12 13 
[L(x,)]   cosh k1x
cosh k21x
cosh k3x   21 22 23  , Σ   0  2 0
 0  2 0

j

j

k

k

(2.63)

k 1

2.3.2. Frequencies and mode shapes of multiple cracked Timoshenko beam
For single span beam, boundary conditions on beam ends can be written
B0 z c  x0   0 ; B L z c  xL   0
(2.64)
Apply boundary conditions (2.64), general solutions of beam with n cracks were
n


z
(
x
,

)

G
(
x

j

L

(2.80)

Solutions of equations (2.24) can be written in the form of
n


ɶ
z
(
x
,

)
G
(
x
,

)


c
  0
  G( x  e j , ). χ j  CL   zq ( x, )
j 1


2.4. Dynamic stiffness matrix and nodal load vector of multiple cracked
FGM Timoshenko beam element :
2.4.1. Dynamic stiffness matrix and nodal load vector
Considered beam element made of FGM which is bended and compressed
at the same time. Defined node coordinates and nodal forces as Fig 2.4. We
obtained  Kˆ e  and Fˆ e  respectively are dynamic stiffness matrix and nodal
load vector of multiple cracked FGM beam element.
z
Q2
Q1
L
N1
x N2
j
i
M
M1 W
W2
U1
U2
1
2
Fig 2.4 Node coordinates, nodal load of beam element

 

 

(2.97)


B F Ψ
    Ψ

  ( L,  ) 
xL  
 
T

e

1

1

1

2

2

2

và

Fˆ  là
e

 
 



ɶ ( L,  )   z q ( L)
    Ψ



 
xL  

(2.98)

(2.99)

with BF is free boundary condition matrix operator. ej is position of jth crack and


8

n

ɶ  x,     G( x, )  G( x  e ). χɶ 
Ψ

j   j

j 1

j 1

χɶ j   G(e j )   G(e j  ek ) . χɶ k  ; j  1, 2,3,..., n

(2.105)
Where natural frequencies j are obtained from equations
det Kˆ ( )  0
(2.106)
Mode shape  j  corresponded to natural frequency j



   









 (x)  C Ψˆ  x, Uˆ 
j

0
j

j

(2.107)

j


2.6. Conclusions of Chapter 2
1. Establishing vibration differential equations of Timoshenko FGM beam in
frequency domain taking into account real position of neutral. Using 2 spring
model of cracks, the dissertation obtained frequency equations, mode shape
expressions and dynamic displacement of multiple cracked Timoshenko
FGM with different boundary conditions through DSM.
2. Establishing dynamic stiffness matrix and nodal load vector of multiple
cracked Timoshenko FGM beam element which is bended and compressed
at the same time through dynamic stiffness method. Therefore, the


9

dissertation obtained frequency equations, mode shape expressions and
dynamic displacement of multiple cracked beam-like structure for structural
analysis using dynamic stiffness method.
3. Establishing block diagram and algorithms of the programs that can
determine frequencies, mode shapes and dynamic displacements of multiple
cracked FGM structures.
The obtained results allowed us to study the effects of crack parameters
(number, position, depth), geometric parameter, FGM material parameters and
boundary conditions to dynamic characteristics of FGM beam-like structures
(direct problem of damage structure analysis). These results also are basic data
to solve inverse problem to identify crack parameters on FGM beam-like
structures using dynamic characteristic (inverse problem of cracks identification
on beam).
CHAPTER 3. VIBRATION ANALYSIS OF MULTIPLE CRACKED
FGM BEAM-LIKE STRUCTURES
3.1. Result verification of proposed programs
3.1.1. Verifying natural frequency results

Fig 3.6. Variation of 1 calculated
with NA and MA

3.2.3. The effects of FGM material parameters to natural frequencies
Analyze the variation of the first 3 non-dimensional frequencies i of
simple supported FGM Timoshenko beam with different ratio L/h , index n. We
saw that all frequencies decreased when n increased from 0 with 3 boundary
conditions, when n

0.9

0.9

1

0.95
0.94

0.92

1-ah=0.1
2-ah=0.2
3-ah=0.3

0.91

a)

0.96

0.93

0.92

1-ah=0.1
2-ah=0.2
3-ah=0.3


3-ah=0.3

0.91
0.9

1

b)

0

The relation of ratios of frequency No2 and the location of the last cracks

The relation of ratios of frequency No1 and the location of the last cracks

1

0.99

0.99

0.99

0.98

0.98

0.98

0.96

0.7

0.8

0.9

1

c)

0.97
0.96
0.95
0.94

1-n=0.5
2-n=5
3-n=10

0.93
0.92

0.1

The relation of ratios of frequency No4 and the location of the last cracks

1

omega2/omega02



0.3

0.4
0.5
0.6
Crack positions(m)

0.7

0.8

0.9

1

0

0.1

0.2

0.3

0.4
0.5
0.6
Crack positions(m)

0.7


Consider FGM beam with geometric parameters: L=1.0m, b=0.1m,
h=0.1m and material parameters: n=0.5, Et=70GPa, Eb/Et=5, t=2780kg/m3,
b=7800kg/m3, t=b=0.3. Fig 3.10 show the variation of the first 3 frequency
ratios with simple supported FGM beam with and without a crack when the
crack depth (a-c) and volume index n (d-f) changed. We saw that:
a) When the number and depth of cracks increased, the frequencies of the
beam decreased significantly.
b) There are some positions on beam that the cracks there don’t effect
individual frequency.
c) When n or Eb/ Et decreased, the beam were mores sensitive to cracks.
3.3.2. Mode shapes of the multiple cracked Timoshenko FGM beam:
-3

Shape mode :1

x 10

0.1

0.04
0.03

-4

0.02

-6

0.01

2-ah=20%
3-ah=30%

-16
-18

Shape mode :4

Shape mode :2
0.05

phi2-phi02

phi1-phi01

2

0

0.1

0.2

0.3

0.4
0.5
0.6
0.7
1.Simply supported beam

3-ah=30%

-0.04

0

0.1

0.2

0.3

0.4
0.5
0.6
0.7
1.Simply supported beam
Shape mode :2

0.8

0.9

b)
1

-0.15

0



-0.02

0

phi4-phi04

0

phi2-phi02

phi1-phi01

0.05

-0.05

0

-0.05

-0.1

-0.04

-0.1

1-1crack
2-2cracks
3-3cracks

c) The effects of symmetric cracks through beam middle axis were the same
when the beam has symmetric boundary conditions.
3.3.3. Forced vibration of the multiple cracked Timoshenko FGM beam
Considered clamped FGM Timoshenko beam with concentrate load
P=-3000N at position x=0.4m. Fig 3.27 is diagram of displacement, rotation of
clamped FGM beam with a crack at x=0.6m and crack depth are 10%, 20%,
0

0.1

0.2

0.3

0.4
0.5
0.6
0.7
1.Simply supported beam

0.8

0.9

1

0

0.1


0.8

0.9

1


12

30%. We had some remarks:
a) When the crack depths increased, the displacement and rotation of the beam
increased while the moment and shear force has small changes.
b) There were the breaks at crack positions on displacement diagram, the
jumps on rotation diagram of the beam.
c) Displacement of beam increased with more number of crack, there were the
breaks on displacement diagram while the moment of the beam decreased
significantly.
-5

0.5

-4

1.Chuyen vi

x 10

1.5

2.Goc xoay

1-ah=0%
2-ah=10%
3-ah=20%
4-ah=30%

-1

-1.5

a)
b)
Fig 3.27: Displacement (a), rotation (b) of the claimped FGM beam with a
crack, crack depths a/h=0%-30%, ω=200rad/s
3.4. Vibration analysis of multiple cracked FGM continuous beam
0.1

0.2

0.3
0.4
0.5
0.6
0.7
2.Beam with clamped ends

0.8

0.9

1

Eb=350GPa;
3
Fig 3.29: Continuous FGM beam
t=2780kg/m ;
b=7800kg/m3;
t=b=0.33; n=0.5 (Fig 3.29). Fig
3.32 showed the variation of the first 3 frequency ratio of FGM continuous
beam with a crack when the crack depth (a-c) index n (d-f), ratio Eb/Et (g-i)
changed. We have remarks:
a) Cracks on different beam span effected different individual frequencies.
b) There are some positions on beam that the cracks there don’t effect
individual frequency same as simple beam.
c) When index n increased or Eb/Et decreased, the beam was more sensitive to
cracks. With n<1 (or Eb/Et>1), natural frequencies varied much more than
they did with n>1 (hoặc Eb/Et
1-ah=10%
2-ah=20%
3-ah=30%
0

0.5

1
1.5
Crack positions(m)

2

2.5

0.965

a)

0.97

0.96

1-ah=10%
2-ah=20%
3-ah=30%

0.97

0.965

b)

0

0.5

1
1.5
Crack positions(m)

2

2.5

c)

d)
e)
f)
Fig 3.32: Variation of frequency ratios of the continuous FGM beam with a
crack when crack depth ah, and volume index n changed
Mode shape: 1

Mode shape: 2

0.2

0

0.05


0.2
1
2
3
4

0.1

-0.1

-0.05

0.05

-0.1
1
2
3
4

-0.4

-0.5

Mode shape: 3

0.15

0.1

0.5

1
1.5
Three-Span(m)

2

1
2
3
4

0.015

Crack
Cracks
Cracks
Cracks

Amplitude

Amplitude

0
-0.005
-0.01

-0.02
1

2
3
4

0.01

0.01

-0.015

-0.05

Crack
Cracks
Cracks
Cracks

0.005

0.005

-0.01

-0.03

b)

Comparision of the eigenmodes: 2

0

f)
Fig 3.39: Mode shapes and mode shape deviation of continuous FGM
beam which has from 1 to 4 equidistant cracks on the second span
0.5

1
1.5
Three-Span(m)

2

2.5

0

0.5

1
1.5
Three-Span(m)

2

2.5

0

0.5

1

Displacement of beam:

x 10

4

0

0

Displacement of beam:

x 10

2

-2

0

-4

Displacement

Displacement

Displacement

-2
-4

2.5

a)

Displacement variation:

x 10

1
2
3
4

3

Crack
Cracks
Cracks
Cracks

Amplitude

Amplitude

2

1

0


0.5

-7

8

6

-2.5

1
1.5
Three-Span(m)

1
1.5
Three-Span(m)

1
2
3
4

-8

0

-2

0.5

-10

Amplitude

1
2
3
4

-8

1
1.5
Three-Span(m)

2

Displacement variation:

x 10

1
2
3
4

Crack
Cracks
Cracks
Cracks


-4

2.5

-6

0

0.5

1
1.5
Three-Span(m)

2

2.5

d)
e)
f)
Fig 3.53: Dynamic displacement, displacement deviation of continous
FGM beam which has from 1 to 4 cracks on each span, crack depth =20%
3.5. Conclusions of Chapter 3
1. Obtaining the programs to determine natural frequencies, mode shapes and
forced displacements of uncracked and cracked beam-like structures using
dynamic stiffness method taking into account real neutral axis position.
Verifying calculated results to announced one to prove the reliability of
obtained programs.







 cD j  k  j ,k  x  

k 



J

k 

j J

 cAJ  k  j ,k  x    D j  x   Aj  x  (4.10)

where Dj(x) and Aj(x) are detail and approximate functions at J level
Dj x 





 cD  k   x  ; A  x    cA  k   x 

k  


The advantage of SWT is the dimention of datas after transformation is not
reduced, so the detail coefficients of SWT have more imfomation about basic
signal, hence the signal identification such as highlights or interrupts of the
signal,.... is better than DWT.
4.1.2. Popular wavelet style: Daubechies, Haar, Morlet, Mexican hat.
4.1.3. Measured noise and denoise: Actually, mode shape data of the structures
include 3 components: y  yin  ynoise  ycrack . De-noise procedure in wavelet is
operated through preset thresholding.
4.1.4. Wavelet toolbox of MatLab
Commands of Stationary wavelet transformation SWT in MatLab is
[SWA,SWD] = swt (X,N, 'wname')
[SWA,SWD] = swt (X,N, Lo_D, Hi_D)
(4.26)

a)

b)

c)

d)
f)
e)
Fig 4.6: SWT detail coefficients of the first 2 mode shapes of FGM beam
which has 4 cracks, noise level is 75, 80dB and 90dB


16


more sensitive with the presence of cracks.
4.1.6.3. Crack identification on continuous FGM beam using SWT of
dynamic displacements


17

Considered continuous FGM beam with geometric and material parameters
same as task 3.4.1 (Fig 3.29) subjected to distribution load q0=1000N/on the
second span and forced frequency is ω.
Fig 4.12 is detail coefficients SWT of dynamic displacement of the
continuous FGM beam when the number and depth of cracks varied on the
second span. We easily detail coefficient SWT paragraph change suddenly at
crack positions. Its amplitude increase with the raise of crack depth.

b)

a)

Fig 4.12: SWT detail coefficient of dynamic displacement on the second
span of continuous FGM beam when the number and depth of cracks varied
4.2. Crack identification using artificial neural network
4.2.1. Artificial neurons:
Artificial Neural Network - ANN is the data processing model consist
artificial neurons that has operation procedure similar to biological neurons of
the human brain (Fig 4.1). ANN is formed from neurons linked together by
layer structures.
x w
x


4.2.4. ANN toolbox of MatLab
Commands of MATLAB that are used to create the network are newff,
train, and sim [116]. Commands to creat a MLP network is net as bellowed
net = newff( PR , [ S1 S2 … SNl ], [ TF1 TF2 … TFNl ], BTF ) (4.29)
4.2.5. Block diagram crack identification on beam using ANN (diagram 4.2)
4.2.6. Numerical results of crack identification on beam using ANN
4.2.6.1. Crack identification on FGM cantilever beam using ANN of the
natural frequencies
Considered cantilevel FGM beam that has 2 cracks at 0.4m and 0.8m from
the left end, crack depth respectively are 20% and 30%. Table 4.3 is detection
results, which is the best when we used 4 natural frequencies.
Table 4.3: Crack parameters detection result based on 3,4 natural frequencies
Position,
Frequency
Frequency
depth of 2
number 1,2,3
number 2,3,4
cracks
(1)
(2)
(1)
(2)
0.4m 0.8m 0.3243 0.7903 0.3185 0.7138
0.20 0.30 0.1631 0.2676 0.1621 0.2895
Error
Li
18.9% 1.21% 20.4% 10.8%
ahi 18.5% 10.8% 18.9% 3.50%


Sai số Position
Depth
Coefficient R

Mode shape #1
(1)
(2)
0.3944 0.8027
0.1980 0.3031
1.40%
0.34%
1.00%
1.03%
0.9998

Mode shape #2
(1)
(2)
0.4037
0.7896
0.2021
0.3052
0.93%
1.30%
1.05%
1.73%
0.9997

Mode shape #3
(1)


Result of ANN
(1)
(2)
0.3946
0.8145
0.1992
0.3005
1.35%
1.81%
0.4%
0.16%
0.9994

Position, depth of
the 2 cracks
0.3m
0.6m
0.20
0.30
Error Position
Depth
Coefficient R

Result of ANN
(1)
(2)
0.2931
0.5906
0.2015

subjected to load same as task 4.4.1, we identify cracks on FGM beam by ANN
using input datas of SWT detail coefficients of dynamic displacements. Crack
positions are determined from SWT analysis, and crack depths are detected by
ANN using input datas of SWT detail coefficients of dynamic displacements
(Fig 4.24c-d).
4.5. Conclusions of Chapter 4
1. Using artificial neural network (ANN) with measured natural frequencies.
The identification results are reliable when number of measured frequencies
equals to crack parameters on beam.
2. Using stationary wavelet transformation (SWT) of the mode shapes or
dynamic displacements to determine the number and position of cracks on
FGM beam-like structures. The advantage of this method is ability to detect
crack positions on beam with the noise level >=75dB. The dissertation also
survey the effects of material parameters (index n and ratio Et/Eb), crack
parameters (position, depth), and noise level SNR to detail coefficients of
SWT.
3. Using ANN with input datas of the mode shapes or dynamic displacements
to determine the number, position and depth of cracks on beam. The
detection results are highly accurate.
4. To decrease variable numbers of inverse problem, the dissertation combines
ANN and SWT of the mode shapes and dynamic displacements to determine
the number, position and depth of cracks on beam. The results proved that
this method has higher accuration with lower operation time.
GENERAL CONCLUSIONS
The main new achieved results of the dissertation are:
1. Modeling multiple cracked FGM beam as a single beam element by DSM
with spring model of cracks. Therefore, the dissertation established dynamic
stiffness matrix and nodal load vector of multiple cracked Timoshenko FGM
beam element that is compressed and bended at the same time while
previous researches only established the dynamic stiffness model of intact

on the structures.
These are new results of crack identification on cracked FGM beam-like
structures when we have mesured datas of natural frequencies, mode shapes
or dynamic displacements. These results offer a crack identification method
on FGM beam-like structures.
SCIENCE PAPERS RELATED TO DISSERTATION HAVE BEEN
PUBLISHED
1. Trần Văn Liên, Ngô Trọng Đức, Nguyễn Tiến Khiêm (2016). Phân tích dao động
tự do của dầm Timoshenko làm bằng vật liệu chức năng có nhiều vết nứt. Tuyển
tập Hội nghị khoa học toàn quốc Vật liệu và Kết cấu Composite – Cơ học, công
nghệ và ứng dụng, Trường Đại học Nha Trang, 391-399, 28-29/7/2016.
2. Tran Van Lien, Nguyen Tien Khiem, Ngo Trong Duc (2016). Free vibration
analysis of functionally graded Timoshenko beam using dynamic stiffness method.
Journal of Science and Technology in Civil Engineering, National University of
Civil Engineering, 31 (10/2016), 19-28.
3. Tran Van Lien, Ngo Trong Duc, Nguyen Tien Khiem (2016). Mode shape
analysis of multiple cracked functionally graded Timoshenko beam. Proceeding of
the International Conference on Sustainable Developement in Civil Engineering,
Hanoi, 15-16/11/2016, 213-223.
4. Tran Van Lien, Ngo Trong Duc, Nguyen Tien Khiem (2017). Mode shape
analysis of functionally graded Timoshenko beams. Latin American Journal of


22

Solids and Structures, 14 (7), 1327-1344 (ISI Journal list).
5. Trần Văn Liên, Nguyễn Tiến Khiêm, Ngô Trọng Đức (2017). Phân tích dao động
cưỡng bức của dầm Timoshenko bằng vật liệu FGM có nhiều vết nứt. Tạp chí
khoa học công nghệ xây dựng, Trường Đại học Xây dựng, 11(3), 10-19.
6. Trần Văn Liên, Ngô Trọng Đức, Nguyễn Tiến Khiêm (2017). Xây dựng ma trận

e157 (ISI Journal list).