Luận văn thạc sĩ ảnh hưởng của khối lượng nền lên ứng xử động của kết cấu dầm và tấm

THE UNIVERSITY OF DA NANG
UNIVERSITY OF SCIENCE AND TECHNOLOGY

PHAM DINH TRUNG

THE INFLUENCE OF FOUNDATION MASS ON DYNAMIC
RESPONSES OF BEAM AND PLATE STRUCTURES

MAJOR : ENGINEERING MECHANICS
CODE : 9520101 (62520101)

SUMMARY OF DOCTOR OF ENGINEERING DISSERTATION

Da Nang - 2018


The work was finished at
University of Science and Technology – The University of Da Nang

Advisors:
1. Assoc. Prof. Dr. HOANG PHUONG HOA
2. Assoc. Prof. Dr. NGUYEN TRONG PHUOC

Reviewer 1st: Prof. Dr. Pham Duy Huu
Reviewer 2nd: Assoc. Prof. Dr. Pham Hoang Anh
Reviewer 3rd: Assoc. Prof. Dr. Dang Cong Thuat

This dissertation was defended at The Doctor of Engineering
Committee at University of Science and Technology-The University
of Da Nang on 29th of July, 2018.

For the detail of the dissertation, please contact:
- Information and Library Center of University of Science
and Technology - The University of Da Nang.
- National Library of Vietnam.


1

INTRODUCTION
1. Motivation of study
The problem model of plate and beam structures on foundation
subjected to moving vehicles as transportation, runway, railway, etc,
has very important meaning both in theory and practical application.
In almost the studies, the foundation model applied to analyze
dynamic response of structures was described by various foundation
models. The first is the one-parameter foundation model Winkler [87]
and it was developed into the parameters foundation models as
Filonenko-Borodich [30], Hetényi [36], Pasternak [70], Reissner [77],
Kerr [44], Vlasov [83]. The common character of the foundation
models used elastic spring without density to describe the behavior of
foundation.
However, the nature of foundation has density and it does not
have meaning in the problem of static analysis but it can have the
influence on the problem of dynamic analysis. When the above
structure is vibrating under dynamic loads, the foundation mass also
causes vertical inertia force as external load subjected to the structures.
The force depends on the value of mass and acceleration of the
foundation and it completely participates in the response to the above
structures.
It is seen that the foundation mass has an influence on the

dynamic response of the structures and do not consider any researches
presented clearly the influence of foundation mass. The dissertation
studies “The influence of foundation mass on dynamic responses of
beam and plate structures” to describe more exactly the problem of
structures on foundation under moving load. It can be seen that this
idea agrees with developing tendency, it has inheritability from the last


2
foundation models and a new problem is foundation mass, it has
meaning science and more agreement with reality.
2. Purpose of study
The purpose of dissertation proposes a new foundation model
and establishes theory to describe characteristic parameters of the
influence of foundation mass. Then, the experimental model is used to
determine the parameters of the influence of foundation mass on the
dynamic response of the structures.
3. Object and scope of the study
The subject of study: The dissertation studies the influence of
foundation mass on the dynamic response of the beam and plate
subjected to moving load.
The scope of study: The material properties of the structural
model and foundation are assumed as homogeneous, continuous,
isotropic and linear elastic based on small deformation theory.
4. Research content
Theory research: The dissertation studies overview of the
characteristics of foundation models applied to the problem of the
structures on the foundation. And then, the dissertation proposes a new
foundation

model

and

establishes

basic

theory

to

describe

characteristic parameters of the influence of foundation in the problem
of dynamic analysis response of beam and plate structures.
Experimental

research:

The

dissertation

establishes

the

experimental models to verify and establish the relation of
characteristic parameters of the influence of foundation mass. And

then, the parameter of the influence of foundation mass is determined
based on a comparison between experimental and theoretical results.


3
Computer program: The dissertation also establishes a program
for the above research problems based on finite element method and
dynamic structures.
5. Methodology of study
With above purpose and content of research, the study
methodology of the dissertation is a combination between theory
research such as proposition new foundation model, numerical
simulations and experimental research to determine the parameter of
the influence of foundation mass on the dynamic response of the
structures.
6. Dissertation layout
In addition to the introduction and conclusion parts, the
dissertation is organized into four chapters as:
Chapter 1: Overview
Chapter 2: Dynamic foundation model
Chapter 3: Numerical simulations
Chapter 4: Experimental research.
7. Contributions of the dissertation
The dissertation proposed a new foundation and established
basic theory to describe the characteristic parameters of the influence
of foundation mass.
The experimental model is established and the characteristic
parameters of the influence of foundation mass are determined.
The computer program of the problem of the dynamic response
of the beam and plate is also established.

So, the dissertation “The influence of foundation mass on the
dynamic response of the beam and plate” has certain contributions and
practical significance in the problem of dynamic analysis response of
the structures on foundation subjected to moving load. The results are


4
also meaning in the problem of structures such as road-vehicle,
runway, foundation-track-train interaction.

Chapter 1
OVERVIEW
1.1. Introduction
The purpose of the chapter presents an overview of foundation
models and the applied of the foundations in the problem of dynamic
analysis response of the structures on the foundation are systematically
studied.
1.2. Overview of the foundation models
1.2.1. The one-parameter foundation model
The Winkler foundation model is suggested in 1867 [87]; called
one-parameter foundation model. But, one of the most important
deficiencies of the model is that it appears a displacement discontinuity
between the loaded and unloaded part of the foundation surface.
1.2.2. The many-parameter foundation models
One of the methods to overcome the deficiencies of Winkler
model adds to the upper surface of linear springs a layer without mass
density, the parameter of this layer is called the second parameter of
foundation models.
1.3. The application of foundation models
1.3.1. The studies outside the country

In many last decades, the above foundation models have been
quite applied in many problems of dynamic analysis response of the
structures on the foundation.
1.3.2. The studies inside the country
In many last years, the dynamic analysis response of structures
on foundation under moving load models has been attracted many
Vietnam researchers.


5
1.4. The studies of the influence of foundation mass
The previous works also considered the influence of foundation
mass and the results showed that the foundation mass had a significant
effect on the dynamic characteristic of the structures system. This
study does not propose any foundation models or parameters
describing the influence of foundation mass [54-57, 6].
1.5. The problem of analysis response of structures on foundations
Recently, the moving oscillator model is one of the moving load
models described nearly as real nature of moving vehicles has been
applied in many studies [12], [26], [43], [50], [64], [75].
1.6. Conclusion
The above overview of foundation models shows that the
problem of the dynamic response of structures on the foundation is
always topic that attracts many attentions and researches in recent
years. One of the most common of almost above studies is foundation
model described by parameters without mass density, as same as, it
overlooks the influence of foundation mass on the dynamic response of
the structures.
But, the true nature of foundation has mass density, so, the mass
density of foundation has certainly effect on the character and dynamic

response of the above structures.
And then, the research and promotion of a new foundation
model used to analyze the influence of foundation mass on dynamic
characteristic and response of the structure on the foundation are really
necessary, meaningful science and relevant practice.

Chapter 2
DYNAMIC FOUNDATION MODEL
2.1. Introduction
This chapter proposes a new foundation model, called Dynamic


6
foundation model, and establishes the theory to describe parameter of
the influence of foundation mass on responses of the structures.
2.2. The dynamic foundation model
2.2.1. The basic theory of the model
The study proposes the new foundation model considering fully
foundation parameters such as stiffness elastic, shear layer, viscous
damping and especially consideration characteristic parameters of the
influence of foundation mass, called dynamic foundation model,
plotted in Fig. 2.1.
The pressure-deflection relationship at the time t is determined
based on balance principle of shear layer, see in Fig. 2.2. The pressuredeflection relationship can be expressed mathematically as follows
w(x, y,t)
2w(x, y,t)
q(x, y,t) = kw(x, y,t)  c
m
 ks2w(x, y,t)
(2.1)

t
t 2

Fig. 2.1. The dynamic foundation model
(a)

(b)

Fig. 2.2. The mechanical model of the dynamic foundation:
(a) Stress in shear layer, (b) Force acting on shear layer
2.2.2. The parameter of foundation mass
The lumped mass m is determined based on the principle


7
balance kinetic energy of the elastic spring element shown in Fig. 2.3,
can be expressed as
m = aF F H F

(2.2)

where a F is a parameter of the influence of foundation mass.
(a)

(b)

Fig. 2.3. The lumped mass model: (a) The elastic spring,
(b) The straight rod.
2.2.3. Comment
The Dynamic foundation model describes nearly identical real

nature of the soil, at the same time it is also overall the above
foundation models in both the static and dynamic problem of the
structures on the foundation.
2.3. The beam on the foundation
2.3.1. The beam model
Consider the Euler-Bernoulli beam element on the dynamic
foundation model, plotted in Fig. 2.4.

Fig. 2.4. The beam element on the dynamic foundation model
2.3.2. The properties matrices of the beam element
The stiffness, mass and damping matrices of the beam element
can be expressed as
b

w

s

K e, B =  K e, B   K e, B  K e, B

(2.3)


8
b

F

M e, B =  M e, B   M e, B

(2.4)

l
F

T

Ce, B =   N e, B c  N w, B dx

(2.5)

0
b

w

s

where  K e, B ,  K e, B and  K e, B are stiffness matrix of the beam
b

element, elastic foundation, and shear layer, respectively and  M e, B
F

M e, B

are mass matrix of the beam element and mass matrix of the

foundation, respectively.
2.4. The plate on the foundation

2.4.1 The plate model
Consider the Reissner-Mindlin plate element on the dynamic
foundation model, shown in Fig. 2.5.

Fig. 2.5. The plate element on the dynamic foundation model
2.4.2 The matrices of the plate element
The dynamic properties matrices of the plate element can be
expressed as
b, s

w

s

K e, P =  K e, P   K e, P  K e, P
b
F
M e, P =  M e, P   M e, P
F
T
Ce, P =   N w, P c  N w, P dAe

(2.6)
(2.7)
(2.8)

Ae

b,s

w

s

where  K e, P ,  K e, P and  K e, P are stiffness matrix of the plate


9
b

element, elastic foundation, and shear layer, respectively and  M e, P
F

M e, P

are mass matrix of the plate element and foundation,

respectively.
2.5. The governing equation of motion
2.5.1. The moving oscillator model
Consider a moving oscillator model [51], shown in Fig. 2.6.

Fig. 2.6. The moving oscillator model on the dynamic foundation
The equation of motion of the oscillator can be expressed as
0
Mv 0 zv   cv cv zv   kv kv zv  

(2.9)
 0 m z c c z k k z =f  M m g
 c  v w 

w  w  v
v  w   v
v  w 

where f c is contact force between the moving oscillator and structure.
2.5.2. The equation of motion of the system
The governing equation of motion is defined as follows
M U   CU    K U = F

(2.10)

So, the dynamic characteristic of the structure is given by
det  K   w 2  M  = 0
(2.11)
2.6. The time integration method
One of the numerical methods applied in many types of research
is Newmark method having a good accuracy of the result. The steps of
the algorithm of Newmark method is shown in Fig. 2.7.
2.7. The computer program
The general algorithm flowchart in Fig. 2.7 analyzed response of
the structures on the dynamic foundation subjected to the moving load


10
is set into computer program based on Matlab code, shown in Fig. 2.8.

Fig. 2.7. Flowchart for analyzing the response of the structures

Fig. 2.8. Detail background of the computer program

11
2.8. Conclusion
This chapter proposed a new foundation model, called dynamic
foundation model. And then, the force-deflection relationship of the
foundation has been established based on the characteristic parameters of
the dynamic foundation model. The comments and estimations of the new
foundation model show practice and general than another models.
The governing equation of motion of the structures subjected to
moving load is also formulated based on dynamic balance principle
and the Newmark method is chosen and described by algorithm
flowchart; a computer program based on Matlab language is also
established to automatic calculation in this problems.
The above analysis contents are basic for next analysis
numerical to investigate the influence of foundation mass on the
dynamic response of the structures, at the same time it is also content
of theory research of the dissertation.

Chapter 3
NUMERICAL SIMULATIONS
3.1. Introduction
This chapter presents the results of numerical simulations to
describe the influence of parameter of foundation mass on dynamic
responses of the beam and plate structures.
3.2. Verified computer program
The calculation program is reliable in both the problem of beam
and plate.
3.3. The beam on the foundation
The dimensionless parameters of the system are defined as


k L2
A
kL4
 = F , b = a F H F , K1 =
, K 2 = 2s , l = wL2
(3.1)
EI
EI
 EI


 v = ( M v  mw ) / M ,  v = wv / w = kv / M v / w

(3.2)


12
The influence of parameter of the foundation mass on free
vibration of the beam is shown in Fig. 3.1.
120
60

120

(b)

b=0,2
b=0,6
b=1

90

l1

b=0
b=0,4
b=0,8

90

l1

(a)

60

10

100
K1

0

0
1

10

100
K1

b=0
b=0.4
b=0.8

b=0.2
b=0.6
b=1

1000

60

(d)

90

30

1000

10000

1000

10000

120

l1

90

1

10000

120

l1

b=0.2
b=0.6
b=1

30

30

(c)

b=0
b=0.4
b=0.8

60

b=0
b=0.4
b=0.8

b=0.2
b=0.6
b=1

10

100
K1

30

0

0
1

10

100
K1

1000

10000

1

Fig. 3.1 The dimensionless frequencies l1 of the beam with
K 2 = 1 ,  = 0.75 : (a) S-S, (b) C-C, (c) CF, (d) C-S

The parameters of the model are given by: L = 5 m, L / h = 50 ,

 = 7860 kg/m3, E = 206.109 N/m2,  v = 0.5 and  v = 0.5 .
(a)

1.7

1.7

(b)

1.5
DMF

DMF

1.5
1.3
b=0
b=0.25
b=0.5
b=0.75

1.1

1.3
b=0
b=0.25
b=0.5

b=0.75

1.1
0.9

0.9
0

20

40
60
80
Velocity (m/s)

0

100

20

40
60
80
Velocity (m/s)

100

Fig. 3.2. The DMF of the beam with various values of elastic
spring: ( K 2 = 1 , c = 103 ,  = 0.5 ) : (a) K1 = 75 , (b) K1 = 150

1.7

(a)

1.7

(b)

1.3
b=0
b=0.25
b=0.5
b=0.75

1.1
0.9
0

20

40
60
80
Velocity (m/s)

100

DMF

DMF

b=0
b=0.25
b=0.5
b=0.75

1.5

1.5

1.3
1.1
0.9
0

20

40
60
80
Velocity (m/s)

100

Fig. 3.3. The DMF of the beam with various values of shear
layer: ( K1 = 100 , c = 103 ,  = 0.5 ) : (a) K 2 = 2 , (b) K 2 = 5


13
1.7

(a)

1.7

(b)

1.5

DMF

DMF

1.5
1.3
b=0
b=0.25
b=0.5
b=0.75

1.1

1.3
b=0
b=0.25
b=0.5
b=0.75

1.1

0.9

0.9
0

20

40
60
80
Velocity (m/s)

100

0

20

40
60
80
Velocity (m/s)

100

Fig. 3.4. The DMF of beam with various values of viscous damping:
( K1 = 100 , K 2 = 1 , c = 103 ,  = 0.5 ): (a) c f = 102 , (b) c f = 104
1.7

(a)

1.7

(b)

1.5

DMF

DMF

1.5
1.3
b=0
b=0.25
b=0.5
b=0.75

1.1

1.3
b=0
b=0.25
b=0.5
b=0.75

1.1

0.9

0.9
0

20

40
60
80
Velocity (m/s)

100

0

20

40
60
80
Velocity (m/s)

100

Fig. 3.5. The DMF of the beam with various values of  v :
( K1 = 100 , K 2 = 1 , c = 103 ,  = 0.5 ): (a)  = 0.25 , (b)  = 1 .
1.6

(a)

1.6

(b)

1.4
DMF

DMF

1.4
1.2

1.2

b=0
b=0.25
b=0.5
b=0.75

1

b=0
b=0.25
b=0.5
b=0.75

1
0.8

0.8
0

20

40
60
80
Velocity (m/s)

0

100

20

40
60
80
Velocity (m/s)

100

Fig. 3.6. The DMF of the beam with various values of  v :
( K1 = 100 , K 2 = 1 , c = 103 ,  = 0.5 ): (a)  v = 0.75 , (b)  v = 1.5 .
1.6

(a)

1.6

(b)

1.4
DMF

DMF

1.4
1.2
b=0
b=0.25
b=0.5
b=0.75

1

1.2
b=0
b=0.25
b=0.5
b=0.75

1
0.8

0.8
0

20

40

60
80
Velocity (m/s)

100

0

20

40
60
80
Velocity (m/s)

100

Fig. 3.7. The DMF of the beam with various values of cv :
( K1 = 100 , K 2 = 1 , c = 103 ,  = 0.5 ): (a)  v = 5% , (b)  v = 10%


14
3.4. The plate on the foundation
3.4.1. The parameters of the plate model
k B2
wa2
kB 4
K '1 =
, K '2 = s ,  = 2


D
D

h

(3.3)

D

3.4.2. Free vibration of the plate
The effects of characteristic parameters of the foundation on free
vibration of the square plate for  = 0.5 are shown in Table 3.1.
Table 3.1. The dimensionless frequencies of the plate on foundation
K’1

102

50

SSSS (=0.2, h/B=0.01) CCCC (=0.2, h/B=0.01)

b

K’2

1

2

3

1

2

3

0

3.8957

7.2044

10.334

5.1522

9.3363

13.058

0.25

1.0604

1.9612

2.8134

1.4024

2.5416

3.5551

0.5

0.7641

1.4132

2.0273

1.0105

1.8314

2.5618

0.75

0.6279

1.1613

1.6660

0.8304

1.505

2.1052

3.4.3. Dynamic response of the plate
The parameters of the plate on the dynamic foundation are given
by B = 10 m, L = 20 m, h = 0.3 m,  = 2500 kg/m3, E = 3.1x1010 N/m2,

 = 0.2 ,  v = 0.5 ,  v = 0.5 , mw = 0 , K '1 = 50 , K '2 = 5 , c = 102 Ns/m2
and  = 0.75 , with simple support along the short two sides.
(a)

1.7

b=0

b=0.25

b=0.5

b=1

b=0

(b)

1.5

b=1

DMF

DMF

1.3

1.1

1.1

0.9

0.9

0

20

1.9

b=0

40
60
`
Velocity
(m/s)
b=0.25

80

b=0.5

100

0

(d)

1.15

40
60
Velocity (m/s)
b=0.25

80

b=0.5

100
b=1

1.65
DMF

1.4

20
b=0

1.9

b=1

1.65

DMF

b=0.5

1.5

1.3

(c)

b=0.25

1.7

1.4

1.15

0.9

0.9
0

20

40
60
80
Velocity (m/s)

100

0

20

40
60
80
Velocity (m/s)

100


15
Fig. 3.8. The DMF of the beam with various values of elastic
spring: (a) K1' = 25 , (b) K1' = 50 , (c) K1' = 75 , (d) K1' = 100
(a)

b=0

1.7

b=0.25

b=0.5

(b)

b=1

1.5

b=0.25

b=0.5

b=1

DMF

DMF

1.5

1.3

1.3

1.1

1.1

0.9

0.9

0

(c)

b=0

1.7

20
b=0

1.7

40
60
Velocity (m/s)
b=0.25

80

b=0.5

100
b=1

0

(d) 1.7

40
60
Velocity (m/s)
b=0.25

80

b=0.5

100
b=1

DMF

1.5

DMF

1.5

20
b=0

1.3

1.3

1.1

1.1

0.9

0.9
0

20

40
60
Velocity (m/s)

80

100

0

20

40
60
Velocity (m/s)

80

100

Fig. 3.9. The DMF of the beam with various values of shear
layer: (a) K 2' = 1 , (b) K 2' = 5 , (c) K 2' = 25 , (d) K 2' = 50
(a)

b=0

1.7

b=0.25

b=0.5

b=1

b=0.5

b=1

DMF

1.3

1.3

1.1

1.1

0.9

0.9
0

20
b=0

1.5

40
60
Velocity (m/s)
b=0.25

80

b=0.5

100
b=1

0

(d)

20
b=0

1.5

40

60
Velocity (m/s)
b=0.25

80

b=0.5

100
b=1

1.35
DMF

1.35
DMF

b=0.25

1.5

DMF

1.5

(c)

b=0

(b)1.7

1.2

1.05

1.2

1.05
0.9

0.9
0

20

40
60
80
Velocity (m/s)

100

0

20

40
60
80
Velocity (m/s)

100

Fig. 3.10. The DMF of the beam with various values of viscous
damping: (a) c = 102 , (b) c = 103 , (c) c = 5 x103 , (d) c = 104


16
b=0

1.7

(a)

b=0.25

b=0.5

b=1

b=0

(b)1.7

1.5

b=0.25

b=0.5

b=1

DMF

DMF

1.5
1.3

1.3

1.1

1.1

0.9

0.9
0

20
b=0

(c) 1.7

40
60
Velocity (m/s)
b=0.25

80

b=0.5

0

100
b=1

20
b=0

(d) 1.7

40
60
Velocity (m/s)
b=0.25

80

b=0.5

100
b=1

1.5

DMF

DMF

1.5

1.3

1.3

1.1

1.1

0.9

0.9
0

20

40
60
Velocity (m/s)

80

0

100

20

40
60
Velocity (m/s)

80

100

Fig. 3.11. The DMF of the beam with various values of  v :
(a)  v = 0.25 , (b)  v = 0.5 , (c)  v = 1 , (d)  v = 2
b=0

(a) 1.7

b=0.25

b=0.5

b=1

1.5

b=0

(b)1.7

b=0.25

b=0.5

b=1

DMF

DMF

1.5

1.3

1.3

1.1

1.1

0.9

0.9

0

20
b=0

(c) 1.7

40
60

Velocity (m/s)
b=0.25

80

b=0.5

100
b=1

0

20
b=0

(d)1.7

40
60
Velocity (m/s)
b=0.25

80

b=0.5

100
b=1

1.5

DMF

DMF

1.5

1.3

1.3

1.1

1.1

0.9

0.9
0

20

40
60
Velocity (m/s)

80

100

0

20

40
60
Velocity (m/s)

80

Fig. 3.12. The DMF of the beam with various values of  v :
(a)  v = 0.25 , (b)  v = 0.5 , (c)  v = 1 , (d)  v = 2

100


17
b=0

(a)1.7

b=0.25

b=0.5

b=1

b=0

(b)1.7

b=0.5

b=1

DMF

1.5

DMF

1.5

b=0.25

1.3

1.3

1.1

1.1

0.9

0.9
0

20
b=0

(c) 1.7

40
60
Velocity (m/s)
b=0.5

b=1

80

100

b=1.5

0

20
b=0

(d)1.7

40
60
Velocity (m/s)
b=0.25

80

b=0.5

100
b=1

1.5

DMF

DMF

1.5

1.3

1.3

1.1

1.1

0.9

0.9
0

20

40
60

Velocity (m/s)

80

100

0

20

40
60
Velocity (m/s)

80

100

Fig. 3.13. The DMF of the beam with various values of  v (a)

 v = 0.01 , (b)  v = 0.1 , (c)  v = 0.15 , (d)  v = 0.2
3.4. Conclusion
This chapter realized numerical results based on basic theory
and the computer program of chapter 2. With many investigation
results of input circumstance (ngữ pháp) show that the influence of
foundation mass is significant than without foundation mass. It almost
increases the dynamic response of structures and causes more
unfavorable for the above structures.
The results are important in the content of theory research of
dissertation, it presented quantitative results of the influence of

foundation mass on the dynamic response of the above structures and it
is basic to establish an experimental model in the next chapter.

Chapter 4
THE EXPERIMENTAL STUDY FOR DETERMINING
THE INFLUENCE OF FOUNDATION MASS


18
4.1. Introduction
This chapter uses the experimental model to determine
characteristic parameters of the influence of the foundation mass on
the dynamic response of the structures.
4.2. An experimental model of single degree of freedom
4.2.1. The model
The experimental model is present in Fig. 4.1.

Fig. 4.1. An experimental model of single degree of freedom
The parameter of foundation mass is determined
m  mS
a F = eff
eff H F

(4.1)

where mS is mass of structure and meff is vibration mass.
4.2.2. The results
The efficiency stiffness of foundation is determined based on the
relationship between force-displacement and the characteristic
parameters of the foundation are drawn in Table 4.1.

Table 4.1. The character parameters of experimental models
Symbol

M1

M2

M3

M4

keff (kN/mm)
HF (mm)
ρeff (kg/m)
mS (kg)

2.558
102.675
48.503
1.939

1.140
203.500
48.872
1.968

0.758
303.475
48.923
1.989

0.586
404.775
48.714
1.938

The results of natural frequency in experimental models are the
average value of many times, shown in Table 4.2 and the characteristic
parameter of the influence of foundation mass is plotted in Table 4.3.


19
Table 4.2. The natural frequency of experimental models
The natural frequency at sensors

Model
M1
M2
M3
M4

Value of natural

A47490

A47491

A47492

frequency wF (rad/s)

694.711
456.159
354.372
296.776

689.684
451.342
353.534
302.640

682.144
453.646
353.743
295.729

688.847
453.716
353.883
298.381

Table 4.3. The parameter of influence of foundation mass
Model

keff
wF
(kN/mm) (rad/s)

mS
(kg)

mF
(kg)

meff
(kg)

HF
(mm)

aF

M1

2.558

688.847

1.939 3.451

5.391

102.675

0.693

M2
M3
M4

1.140
0.758
0.586

453.716
353.883
298.381

1.968 3.569
1.989 4.064
1.938 4.644

5.538
6.053
6.582

203.500
303.475
404.775

0.359
0.274
0.260

4.2.3. Comment and estimate
The analysis results are shown from Fig. 4.2 to Fig. 4.6.
5.00

1200

4.50

1000

w F (rad/s)

m F.

Experiment

4.00
3.50
3.00

600
400
200

50

150

250
H F (mm)

350

450

Fig.4.2. The relationship mF - H F

50

5.00

0.65

4.50

0.50

150

250
H F (mm)

350

450

Fig. 4.3. The frequency results

0.80

mF

aF

Without foundation mass

800

4.00
3.50

0.35

3.00

0.20
50

150 250 350
H F (mm)

450

0.5 1.0 1.5 2.0 2.5 3.0
k eff (kN/mm)

Fig. 4.4. The relationship aF - HF Fig. 4.5. The relationship mF - keff


20
0.80

aF

0.65
0.50

2

0.35

R = 0.9963

0.20
0.5 1.0 1.5 2.0 2.5 3.0
k eff (kN/mm)

Fig. 4.6. The relationship a F - keff
4.3. An experimental model of the beam on the foundation
4.3.1. The model
The model of the beam on the dynamic foundation is described
by a steel beam on rubber layer in the Fig. 4.7.

Fig. 4.7. The experimental model
The characteristic parameters of the beam are determined based
on experiment, shown in Table 4.4 and Table 4.5.
Table 4.4. The character parameters of the steel beam.
Model

L

b

h

ρ (kg/m3)

E (N/m2)

Steel beam

500.00

40.00

2.80

7691.267

1.808x1011

Table 4.5. The characteristic of the rubber layer
Model

HF mm)

D1

105.98

D2

211.96

D3

317.75

D4

423.64

ρF (kg/m3)

kS (N/m)

k (N/m3)
6.367x107

1206.690

1.773x105

2.807x107
1.874x107
1.452x107


21
Table 4.6. The experimental results of natural frequency of beam
Model

The natural frequency at sensors

Value of natural

A47490

A47491

A47492

frequency wF (rad/s)

D1

490.298

494.696

487.366

490.787

D2

428.723

427.885

428.094

428.234

D3

335.522

336.150

337.407

336.360

D4

313.950

314.997

314.369

314.439

Table 4.7. The parameter of influence of foundation mass
Model

ρF
3

(kg/m )

k (N/m3)

kS (N)

HF

wF

(mm)

(rad/s)

aF

6.367x107 105.98 490.787 2.253

D1
D2

2.807x107 211.96 428.234 0.748

1206.690 1.773x105

D3

1.874x107 317.75 336.360 0.592
1.452x107 423.64 314.439 0.461

D4
4.3.2. The experimental results

The results of natural frequency in experimental models of the
beam are shown in Table 4.6 and the parameter of the influence of
foundation mass is determined, plotted in Table 4.7.
4.3.3.

Comment and estimate
The analysis results are shown from Fig. 4.8 to Fig. 4.10.

2000

1.80

aF

1550

w F (rad/s)

2.40

Experiment
Without foundation mass

1100

1.20
0.60

650

0.00
200
50

150

250
H F (mm)

350

450

50

150

250 350
H F (mm)

450

Fig. 4.8. The frequency results Fig. 4.9. The relationship a F - H F


22
2.40

aF

1.80
1.20
0.60

2

R = 0.9858

0.00
1.E+07 3.E+07 5.E+07 7.E+07
3

k (N/m )

Fig. 4.10. The relationship mF - k
4.4. Conclusion
This chapter realized the content of experimental study:
- Choosing material, designing model and using experimental
equipment agreed with the object of study in the dissertation are
presented. The main material is rubber having the homogeneous
property quite well and ideal elastic; the beam structure uses steel
material having homogeneous property quite well and characteristic
mechanic clearly. The simple experiment model as a single degree of
freedom is described by a steel plate on rubber layer and the beam
structure model on rubber layer are also designed. The recorder is used
to determine natural frequency based on many data and various sizes
of rubber layer in an experimental model of free vibration.
- The experiment models are realized to determine the influence
of foundation mass on the structures. The results show that the
parameter a F has effect significantly on the dynamic response of the
structures; it causes foundation mass to participate in vibration mF and
it increases the general mass of the structures. At the same time, the
experimental results also show that the relation between foundation
mass with the depth of foundation layer and stiffness of foundation

agree with judgment based on characteristic physics of the structures.
- The analysis of the relation between the depth of foundation
H F and the parameter of the influence of foundation mass a F : with

an increase of depth of foundation causes increase vibration foundation


23
mass mF , and at the same time the parameter of foundation mass

a F decreases with the increase of depth of foundation; the results agree
in both experimental models.
- The analysis of the relationship between stiffness of foundation
k and the parameter of the influence of foundation mass a F : the
relation is linear; with an increase of the stiffness of foundation cause
increase value of the parameter of the influence of foundation mass.
From the above comments, it can be seen that the contents of
experimental research quite agree with the dynamic foundation model
in the theoretical research of chapter 2 and 3 of the dissertation; at the
same time, the experimental results also determined relation between
the parameter of the influence of foundation mass a F with depth H F
and stiffness of foundation k , and especially value of a F in the above
experiments.
CONCLUSIONS
Conclusion
The following conclusions are summarized as follow:
1. The study proposed a new foundation model, called the
dynamic foundation model and established basic theory to describe
parameter of the influence of foundation mass on the dynamic
response of beam and plate structures.

2. The process of finite element method to describe the influence
of foundation mass on the dynamic response of the structures subjected
to moving load is established clearly, it was described by flowchart
algorithm and established the computer program.
3. The analysis results show that the foundation mass effects
significantly on dynamic properties of the structures, it increases
general vibration mass of the structures.
4. The study determined relationship between the parameter of