SPECTRAL METHOD FOR VIBRATION ANALYSIS OF CRACKED BEAM SUBJECTED TO MOVING LOAD

INTRODUCTION VIET NAM ACADEMY OF SCIENCE AND TECHNOLOGY GRADUATE UNIVERSITY OF SCIENCE AND TECHNOLOGY ---o0o--- PHI THI HANG SPECTRAL METHOD FOR VIBRATION ANALYSIS OF CRACKED BEAM S

INTRODUCTION

VIET NAM ACADEMY OF SCIENCE AND TECHNOLOGY

GRADUATE UNIVERSITY OF SCIENCE AND

TECHNOLOGY -o0o -

PHI THI HANG

SPECTRAL METHOD FOR VIBRATION ANALYSIS OF CRACKED BEAM

SUBJECTED TO MOVING LOAD

Specialized in: Engineering Mechanics

Code: 62 52 01 01

SUMMARY OF PhD THESIS

Hanoi, 2016

1 Prof.DrSc Nguyen Tien Khiem

on , 2016

Hardcopy of the thesis be found at Vietnam National Library and Library of Graduate university of Science and Technology

1 Necessity of the theme

Dynamic analysis of structure subjected to moving load is

an important problem in the practice of engineering, especially, for the bridge and railway engineerings This problem was investigated very early, in the 19th Century However, it is studying at present by the following reasons: (1) moving load models need to be improved to describe more accurately the moving vehicle-structure interaction; (2) structures subjected to moving loads become more complicated so that a lot of new problems in dynamic analysis

of such the structures has been posed; (3) more exact methods

of dynamic analysis need also to be developed for solving the problems

The most popular method used for dynamic analysis of a structure under moving load is the Bubnov-Galerkin method that is based on the eigenfunctions of the structure and therefore is called also superposition or modal method This method is difficult to apply for the structures eigenfunctions of which are unavailable In that case, the Finite Element Method (FEM), the most powerful technique for structure analysis, is employed Finite element model of a structure is conducted basically on the specific shape functions that are static solution

of a finite element of the structure Therefore, high frequency dynamic response of a structure could be investigated by the finite element model with very large number of elements Recently, the dynamic stiffness method that uses the dynamic shape functions instead of the static one for constructing a

frequency dependent matrix called dynamic stiffness matrix is developed Such development of the FEM enables to study dynamic response of arbitrary frequency for a structure as a distributed system This method called Dynamic Stiffness Method (DSM) is then formulated as a method used for dynamic analysis of structure in the frequency domain and termed Spectral Element Method (SEM)

2 Objective of the thesis

This thesis aimed to apply the SEM for dynamic analysis of cracked beam subjected moving harmonic force in the frequency domain Namely, the frequency response of a cracked beam subjected to moving harmonic force is obtained explicitly and examined in dependence upon the load and crack parameters This task is acknowledged herein spectral analysis of cracked beam subjected to moving load

3 Subject of research

Subject of this study is a multiple cracked beam-like structure under loading of a concentrated force moving with constant speed The Euler-Bernoulli theory of beam is used and crack is modeled by an equivalent spring of stiffness calculated from its depth accordingly to the fracture mechanics theory

4 Methodology of research

Method used in this study is mostly analytical method that

is illustrated by numerical results obtained by MATLAB

5 Thesis’s content

Thesis consists of introduction, 4 chapters and a conclusion

Chapter 1 describes an overview of the moving load

problem and conventional methods used for solving the

problem; the crack detection problem is also presented in this

chapter

Chapter 2 presents the methodology for spectral analysis of

cracked beam subjected to moving force

Chapter 3 provides an exact solution in frequency domain

of the moving load problem for intact beam and frequency

response is thoroughly examined

Chapter 4 studies cracked beam subjected to moving force

and proposes a method for calculating natural frequencies of

continuous multispan cracked beam A procedure for crack

detection by using frequency response is developed

Conclusion chapter summaries major results obtained in the

thesis and some problems for further investigation

Chapter 1 OVERVIEW 1.1 The moving load problem

Consider a beam subjected to the load produced by a

moving mass as shown in Fig 1.1 Equations of motion for the

system are

)]

( [ ) ) , ( )

, ( )

,

(

0 2

2

4

4

t x x t P t

t x w F t

t x w F

P        ;

],)([)

;)]

()[)(

;)()

In the latter equations w ( t x, ) is the transverse displacement

of beam, y (t)- vertical displacement of mass; x0( t )is

position of mass on the beam measured from the left end;

)

(t

 is delta Dirac function From the given system the

following problems can be obtained for dynamic analysis of beam:

1 The moving force problem, when the force P (t) is known, for instance, P(t)P0exp{t0};

2 The moving mass problem if P(t)m[gw0(t)] ;

3 The moving vehicle problem when Eq (1.1) are solved for both the beam and vehicle

Fig 1.1 Model of beam under moving load

1.2 Conventional methods for moving load problem

a) The Bubnov-Galerkin method is based on an expansion of

time domain response of a structure in a series of its eigenfunctions and, as result, a system of ordinary differential equations is obtained and solved by using the well-developed methods Most important results in the moving load problem have been obtained for simple beam-like structures by using the method However, this method is difficult to apply for

complicate structures such as cracked ones, eigenfunctions of which are unavailable

b) The finite element method is the most powerfull technique

that may be applied for arbitrary complicate structures due to involved specific shape functions being static solution of a finite element Nevertheless, since the static shape functions have been used the finite element method is unable to apply

for studying high frequency response of a structure

c) The dynamic stiffness method gets to be advanced in

comparison with the finite element method by that allows one

to investigate dynamic response of arbitrary frequency This is due to frequency-dependent shape functions are employed instead of the static ones However, applying the dynamic stiffness method for the moving load problem leads the Gibb’s phenomena to appear when shear force is converted from the frequency domain to the time domain So, the frequency response obtained by the dynamic stiffness method should be analyzed directly rather in the frequency domain than in the time domain This leads to spectral analysis of frequency response of beam subjected to a moving load that is subject of

the present thesis

1.3 Crack detection problem

The problem of crack detection in structures has attracted

a great attention of researchers and engineers because of its vital importance to safely employ a structure and avoid serious catastrophe might be caused from not early recognized cracked members The methods developed for solving the problem can be categorized as follows:

(1) Frequency-based method means crack location and depth

being predicted by using only measured natural frequencies

(2) Mode shape-based method proposes to evaluate the crack

parameters from measurements of mode shapes of structures under consideration

(3) Time domain method is that uses time history response

measured in-situ of a structure for its crack detection

(4) Frequency response function method proposes to carry

out the crack detection task based on the Frequency Response Function (FRF) measured by the dynamic testing technique Though all of the aforementioned methods are helpful in solving various specific problems of crack detection, they are all faced with either insensitivity of chosen diagnostic criterion

to crack or noisy measured signal used for the crack detection Among the diagnostic indicators the frequency response function is most accurately measured by the dynamic testing method However, the FRF-based method is limited by the following facts First, measurement of FRF needs the testing load measured at a large number of positions on structure and, secondly, the presence of crack may be hidden by the interaction of vibration modes predominated in the measured FRF The shortcomings of the FRF-based method in crack detection may be avoided by using frequency response of a testing structure subjected to controlled moving load

1.4 Determination of thesis’s subject

The short overview allows one to conclude that, firstly, the most efficient approach to the moving load problem is the dynamic stiffness method but it must be used directly for

dynamic analysis of a structure in the frequency domain Secondly, the frequency response of a structure subjected to a well-controlled moving load provides a constructive signal for crack detection, especially, in beam-like structures

So, subject of the present thesis is to further develop the frequency response method proposed by N.T Khiem et al to spectral analysis of cracked beam under moving force and to use that method for multi-crack detection from measured frequency response

,()

,()

,

(

2 2 2

4 5

1 4

4

t x p t

t x w t

t x w F t

x

t x w x

E, I, F, ρ, L - the beam’s material and geometry constants and



;),(),(

;),(),(

;),

w

x

)1/(

)/(

);

1(/

2 1 2

1

The so-called frequency response W(x,) determined from

Eq (2.1.1) must satisfy boundary conditions The frequency

response is complex function of frequency ,

),(),

,()

,

S w   w  w , (2.1.2)

is the frequency-amplitude characteristic of beam subjected to

arbitrary loadp ( t x, ) The functionS w(x,)considered with

respect to frequency  for fixed x is called herein response

spectrum of beam at the section x The function (2.1.2) of

variable x with fixed frequency 0 is called deflection

diagram of frequency0 Content of the frequency response

method applied for moving load problem is first to solve Eq

(2.1.2) for a given moving load p ( t x, )

2.2 Frequency response method in the moving load

problem

As well-known, load produced by a moving force P(t)

expressed in the form p(x,t)P(t)(xvt) has the

frequency-amplitude characteristic

v/)/()

v()

v x P dt e t x t

,

W   (2.2.2)

0),(/

),

,

with L1(x),L2(x) being the independent particular solutions of

homogeneous equation (2.1.1) and satisfying boundary

conditions at the left end of beam Therefore, constants C, D

can be determined from the boundary conditions at the right

end as

)(),()(),(

) ( 2 ) ( 1 )

( 2 ) (

1

) ( 2 ) ( 1 )

( 2 )

(

1

1 1

1 1

1 1

1 1

q p

q p

p q

L L

L L

L L

)(),()(),(

) ( 2 ) ( 1 )

( 2 ) (

1

) ( 1 ) ( 1 )

( 1 )

(

1

1 1

1 1

1 1

1 1

q p

p q

q p

L L

L L

L L

D





2.3 Tikhonov regularization method

A lot of problems in science and engineering leads to

solve the equation

,

b

Ax (2.3.1) (2.3.1)

where A is a matrix of arbitrary dimension and singularity and

b is a vector that is known as an approximation of vector b .

The conventional methods are inapplicable for such the

system A N Tikhonov proposed the so-called regularization

method that suggests regularizing the Eq (2.3.1) by

0

LxLbAx)LLA

A

( T  T  T α T (2.3.2)

with a prior solution x0 and regularizing matrix L and factor

α Finally, regularized solution is calculated by

.vxv

bux

xˆ

1 0

0

k n r k k k

r

T k k k

Concluding remarks for Chapter 2

In this Chapter, the concept of frequency response of a

structure subjected to a moving load is defined that provides

basic instrument for developing the so-called frequency

response method to spectral analysis of beam under moving

force Also, the Tikhonov’s regularization method is shortly

described with the aim to use for solving the crack detection

by measurement of frequency response to moving force

Chapter 3

FREQUENCY RESPONSE OF BEAM SUBJECTED TO

MOVING HARMONIC FORCE

3.1 Vibration of beam under constant moving force

For convenience, the following dimensionless parameters are used:  v/V c v/1- speed parameter (that is ratio of actual speed to the critical speed V c 1/);

]2

 - frequency parameter (the ratio of frequency

to the fundamental frequency of beam); v v/ is called driving frequency

Fig 3.1 Response spectrum in

dependence on the load speed

Fig 3.2 Eigenmode amplitude

in dependence on load speed

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 0

at the fundamental frequency implies predomination of eigenmode of response and it is observed for speed greater than 1/3 critical one Fig 3.2 shows that there exist values of the load speed that may cancelate amplitude of eigenmode response This is approved by graphs given in Fig 3.3 that were plotted for so-called anti-resonant speeds

3.2 Frequency response to harmonic moving force

Fig 3.4 shows response spectrum in the case of moving harmonic force of frequency   0 4 1 The peak attains at load frequency for load speed less than 0.1vc This means predomination of forced mode of response However, the peak

is rapidly reduced and completely disappears when load speed reaches 0.3vc For the speed exceeding 0.3vc it is observed only peak at fundamental frequency Similarly, we can find the anti-speeds for the moving harmonic load as shown in Fig 3.5 and 3.6

Concluding remark for Chapter 3

The obtained numerical results allow one to make the following concluding remarks for Chapter 3:

(a) Response spectrum enable one to identify various vibration modes that are predominated in dependence on the load speed Namely, for the load speed less than 0.1vc response behaviors as vibration mode of load frequency and eigenmode of the response becomes governed if load speed exceeds 1/3vc

(b) There exist speeds of load that may concelate the vibration mode of natural frequencies and such speeds are called anti-resonant ones Antiresonant speeds are elementarily calculated from given natural and load frequencies

(c) Action of combined harmonic forces with different frequencies is also investigated Namely, the constant load is predominate for low speeds and for high speed the load with frequency more closed to the natural one has more effect on the response of beam The loads with frequencies symmetrical about the fundament frequency are equally affecting on the beam vibration

Chapter 4 VIBRATION OF CRACKED BEAM SUBJECTED TO

MOVING FORCE 4.1 Free vibration of cracked beam