Modelling and Simulation of a bridge interacting with a moving vehicle system

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Master's Degree Thesis ISRN: BTH-AMT-EX--2013/D11--SE

Department of Mechanical Engineering Blekinge Institute of Technology

Karlskrona, Sweden 2013

Jingrui Yang Rui Duan

Modelling and Simulation of a Bridge interacting with a moving

Vehicle System

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Modelling and Simulation of a bridge interacting with a moving

vehicle system

Jingrui Yang

Rui Duan

Department of Mechanical Engineering Blekinge Institute of Technology

Karlskrona, Sweden 2012

Thesis submitted for completion of Master of Science in Mechanical

Engineering with emphasis on Structural Mechanics at the Department of

Mechanical Engineering, Blekinge Institute of Technology, Karlskrona,

Sweden.

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Abstract:

The problem about the vibration of vehicles and bridge is mainly the problem of coupling vibration between the vehicles dynamic system and bridge dynamic system. The vehicles running on the bridge at some speed can make the bridge vibrate, at the same time the vibration of the bridge will also affect the vibration of the vehicles. This problem with interaction and mutual effects is the problem of coupling vibration between vehicles and bridge.

Because its close relation to practical application it is paid extensive attention by more and more people. In this thesis we construct the kinetic model of the bridge subsystem and the vehicle subsystem separately. Then according to the Euler–Bernoulli beam theory [1], mode superposition method and D’Alambert principle [2] we get the functions of the two models. In addition we refer to the relationship of the geometric on the contact point and the static equilibrium conditions to construct the relation between the equation of the vehicle vibration and the equation of bridge vibration. Finally we use MATLAB to write the code according to the Newmark-beta method [3][4] to handle the coupled equation.

Keywords:

Vehicles-bridge coupling vibration, Dynamic system, D’Alambert

principle, Euler–Bernoulli beam theory, Mode superposition method,

Newmark-beta method.

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Acknowledgements

From the selection of topic to the finalization of this thesis, our supervisor Mr.Andreas Josefsson devoted all his attention to each process. Every detail of the paper was finished with Mr. Andreas Josefsson’s painstaking care.

We would like to express our grateful acknowledgement to our responsible and humble supervisor of this thesis, Mr. Andreas Josefsson, for helping, guiding and encouraging us to complete this thesis. We wish to express our sincere appreciation to him for helping us check out the thesis and suppose many constructive suggestions. We also would like to thank our friends providing technical knowledge and valuable advice whenever we needed during our research work.

At last we would like to thank all of family members for their unconditional

love and understanding. We are also grateful to all of our friends who

support and motivate us to work hard on the thesis.

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Acknowledgements 2

Contents 3

1 Notations 4

2 Introduction 6

2.1 Background 6

2.2 Aim and Scope 7

2.3 Structure of the thesis 7

3 Theoretical modelling 9

3.1 Overview 9

3.2 The kinetic model of the vehicle subsystem 10

3.3 The dynamic model of the bridge subsystem 15

3.4 The compatibility conditions 25

3.5 Construction of complete model 28

4 MATLAB simulation 37

4.1 The influence of the speed 40

4.2 The influence of the bridge’s span 44

4.3 The influence of mass of the vehicle 49

4.4 Verification 54

5 Conclusion and future work 58

5.1 Conclusion 58

5.2 Future work 58

5.2.1 Models and calculation method 58

5.2.2 Numerical method 59

6 References 60

Appendices 61

Appendix A: MATLAB scripts 61

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1 Notations

  M

v

Mass matrix of vehicle model

  C

v

Damping matrix of vehicle model

  K

v

Stiffness matrix of vehicle model

  y

v

Matrix of the degree of freedom of the vehicle

  F

v

Matrix of exciting force of vibration of the vehicle E Young's modulus

I Moment of inertia

 Mass of the beam per unit length

 Damping coefficient per unit length

  ^,

F x t Coupled force on the beam

  ^,

y

b

x t Beam’s displacement at the contact point

 Dirac function

 

i

x

  Mode of vibration

 

i

t

 Modal Coordinates

y

ci

Displacement of the i’th wheel at the contact point

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 

i

^,

r x t Irregularity of the bridge’s surface at the i’th point

W Static load of the wheel and the vehicle body gravity of the

i

wheel undertake

K Stiffness of the tire

ti

C Damping of the tire

ti

y

ti

Displacement the wheel’s middle point in the vertical direction y _

ti

Velocity of the wheel’s middle point in the vertical direction y Displacement in the vertical direction at the contact point

ci

between the tire and the bridge

y _

ci

Velocity in the vertical direction at the contact point between

the tire and the bridge

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2 Introduction

2.1 Background

In recent decades, with the rapid development of highway communication around the world, the increasing number of bridges, the emergence of new bridge’s structure, the increasing bridge span and a significant increase in vehicle speed, the research of dynamic interaction between vehicles and the bridge is more and more intensive. The study of vehicle-bridge coupling vibration focuses on two aspects generally: accurate modeling of coupled vehicle-bridge system and efficient methods and algorithms for simulating the dynamic response.

Figure 2.1. A car is running on a bridge

When the vehicles run across the bridge, the pocket weight of the vehicles,

irregularities of the bridge and the acceleration and braking of the vehicle

can lead to excessive vibrations. The vibration will be transferred to the

bridge by tires, which can make the bridge vibrate forcefully. The bridge

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lead to damage in the bridge structure. This problem with interaction and mutual effects is related to coupled vibrations between the vehicle and the bridge.

Running vehicles on the bridge at high speed can lead to damage to the bridge structure. At the same time, the vibration of the bridge can affect the stability, the comfortableness and the safety of the running vehicles.

2.2 Aim and Scope

A major goal of this study is to create a simplified model of a vehicle-bridge system in MATLAB. The model will then be used to study the influence of relevant parameters to vehicle-bridge vibrations.

Due to the extensive amount of parameters which affect the vehicle-bridge coupling vibration, we cannot analyze the influence of all the parameters comprehensively. Because of a variety of bridges and vehicles, the theoretical model cannot represent all the bridges and vehicles. Hence, the following assumptions are made in this thesis:

1) The bridge is modelled as a simply supported beam using Euler–

Bernoulli beam theory.

2) The vehicle is modelled as a lumped parameter system with four degrees of freedom.

3) Only one vehicle on the bridge is considered in the model.

2.3 Structure of the thesis

A short description of every chapter is presented below to get an overview

of the structure of the thesis.

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In chapter 3, the main structure of the theoretical model will be illustrated.

Firstly, a simply supported beam is going to be taken as the study project.

According to the Euler–Bernoulli beam theory and mode superposition method we will build the kinetic model of the bridge subsystem.

Secondly, we will build the double-axle-2D kinetic model of vehicle subsystem and get the dynamic function referring to the D’Alambert principle.

Then, we consider these two systems which interact and affect each other as one system. Base on the relationship of the geometry on the contact point and the static equilibrium conditions we will construct the relation between the equation of the vehicle vibration and the equation of bridge vibration.

In chapter 4, the well-known software MATLAB is used to simulate the model using the Newmark-β method. Simulation results are demonstrated using different set of parameters.

Finally, chapter 5 summarizes the results and gives some suggestion for future work.

Appendix contains MATLAB scripts.

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3 Theoretical modelling

3.1 Overview

The structural form of vehicles is generally complex. From the point of view of vibration, a vehicle may be seen as complex multi-variant system consisting of "mass, stiffness and damping". Then different simplified analysis models are formed. Three kinds of simplified models are commonly used. They are two degrees of freedom vehicle model with single axle, double-axle-2D model of vehicle with four degrees of freedom and three-dimensional model with seven degrees of freedom. In this thesis, we take the double-axle-2D model of vehicle as an example to carry out the analysis.

Figure 3.1. The complex structural form of vehicles

As for the bridge we consider it as a simply-supported beam. It is assumed

that the vibration of the bridge is mainly made up by the basic modes of

vibration with lower frequencies. [5] That is to say that the accuracy is high

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enough to use the lower modes of vibration to describe the vibration of bridge’s structure. That is the mode superposition method which can decrease the amount of degrees of freedom of the bridge’s structure. So we will use this method combined with the Euler-Bernoulli beam theory to get the kinetic equation in this thesis.

Figure 3.2. A simply-supported bridge

3.2 The kinetic model of the vehicle subsystem

Newton’s law is used to build the 2D model of the double-axle vehicle.

Before building the differential equations we must get some basic assumptions as follows:

1) The vehicle’s speed is a constant and the vehicle is in linear motion.

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occurs.

3) Wheels and body of the vehicle are vibrating with small displacement.

4) Wheels and body are assumed to be rigid bodies with mass, and there is no internal elastic deformation. They are connected by spring and dampers to the main structure.

5) Road surface displacement input function impacts on the points of contact between the tires and the road.

When the double axles vehicle symmetries for the longitudinal vertical plane altogether and the change of the load roughness below the left and right wheel is the same, we can consider the vehicle vibrates in the vertical direction only.

Then we can get the model as follows:

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Figure 3.3. The model of the double axles vehicle with four degrees of freedom.

This model has 4 degrees of freedom. The body of the vehicle has two degrees of the freedom; they are ups and downs of freedom and nod degrees of freedom  separately. The front and back wheel set have a vertical displacement degree of freedom and separately. Then we build the kinetic equilibrium function of the vehicle about every degree of freedom according to the Newton’s law.

Body vibrates up and down:

1 1 1 2 2 2

1 1 1 1 2 2

y ( ) ( )

( ) ( ) 0

s s s s t s s t

s s t s s t

m c y y a c y y a

k y y a k y y a

 

     

      

 

    

(3.2.1) Body nodes vibration:

( ) ( )

J    k a y  y   a  k a y  y   a

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Front axle vibrates in vertical motion:

1 1 1 1 1 1 1 1

1 1 1 1 1 1

( ) ( )

( ) ( ) 0

t t s s t s s t

t t c t t c

m y k y y a c y y a

k y y c y y

 

     

    

   

  (3.2.3)

Back axle vibrates in vertical motion:

2 2 2 1 2 2 2 2

2 2 2 2 2 2

( ) ( )

( ) ( ) 0

t t s s t s s t

t t c t t c

m y k y y a c y y a

k y y c y y

 

     

    

   

  (3.2.4)

Where,

m is the mass of the body and the frame of the vehicle.

s 1

m ,

t

m are the mass of the axle between the front and back wheel

_t₂

set and the tires

1

k ,

s

k ,

^s²

c ,

^s¹

c are the stiffness and damping between wheel set

^s²

and the body of the vehicle.

1

k ,

t

k ,

^t²

c ,

^t¹

c are the stiffness and damping of the tires

^t²

a ,

1

a are the displacement from the center of gravity to the back

²

wheel set or to the front wheel set. The total length is a   a

¹

a

²

.

1

y ,

c

y are the displacement on the point which the bridge contacts

^c²

with the front and back wheel set in the vertical.

1

y _

c

, y _

^c²

are the velocity at the point which the bridge contacts with

the front and back wheel set.

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Transforming the equations into matrix we can get:

  M

v

   y

v

   C

v

  y 

v

   K

v

    y

v

 F

v

(3.2.5) Where,

  M

v

is the mass matrix,

  C is the damping matrix,

v

  K is the stiffness matrix separately of the model,

v

  y is the matrix of the degree of freedom of the vehicle,

v

  F is the matrix of exciting force of vibration of the vehicle.

v

These matrices are given in Eq. (3.2.6) to Eq. (3.2.10).

 

1 2

0 0 0

s

v

t t

m M J

m m

 

 

  

 

  (3.2.6)

 

1 2 s

v t t

y

y y

y



 

 

  

 

  (3.2.7)

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 

1 1 1 1

2 2 2 2

0 0

v

t c t c

F k y c y

k y c y

 

 

     

  

 



 (3.2.8)

 

1 2 1 1 2 2 1 2

2 2

1 1 2 2 1 1 2 2 1 1 2 2

1 1 1 1 1

2 2 2 2 2

0 0

s s s s s s

v

s s s t

k k k a k a k k

k a k a k a k a k a k a

K k k a k k

k k a k k

   

 

    

 

     

   

  (3.2.9)

 

1 2 1 1 2 2 1 2

2 2

1 1 2 2 1 1 2 2 1 1 2 2

1 1 1 1 1

2 2 2 2 2

0 0

s s s s s s

v

s s s t

c c c a c a c c

c a c a c a c a c a c a

C c c a c c

c c a c c

   

 

    

 

     

   

  (3.2.10)

3.3 The dynamic model of the bridge subsystem

The dynamic model of the bridge is considered next. To begin with, we

construct the half of the vehicle model running on the bridge.

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Figure 3.4. The half of the vehicle model in figure 3.3.

Figure 3.5. The model of simply supported beam with the function of running vehicle.

According to the Euler-Bernoulli beam theory, the governing equation for flexural vibrations can be written as:

         

4 2

, , ,

,

b b b

y x t y x t y x t

EI F x t x vt

x  t  t 

  

    

   (3.3.1)

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E is Young's modulus of the beam,

I is the moment of inertia of the cross-section, EI is the flexural rigidity of the beam,

 is the mass of the beam per unit length,

 is the damping coefficient per unit length,

  ^,

F x t is the coupled force on the beam,

  ^,

y

b

x t is the beam’s displacement on the contact point in the vertical at time t.

 is the function of Dirac.[6]

The Dirac function has the following characteristics:

     

   

 

0, , 0,

b

a

t a f x x t dx f t a t b

t b





 

    

 





(3.3.2)

The function ^{f x}   is continuous in the closed interval   ^{a b} ^, ^.

According to the mode superposition method we make,

     

1

b

,

i i

i

y x t

^

 x  t



  ^

(3.3.3) Where,

 

i

x

  is the function of beam mode of vibration,

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 

i

t

 is modal Coordinates.

Because the respond of the bridge’s vibration is mainly dominated by some lower order modes, we need to choose some lower order basic vibration modes to analyze the response. In this way, we can reduce the number of degrees of freedom. Hence, we choose the number of modes N to perform the analysis.

So we can get,

   

1

( , )

N

b i i

i

y x t  x  t



  ^

(3.3.4) According to the condition of normalization of the simply supported beam, we can get,

  ^sin

i

x A i x l

   

,

    ^sin ^,  ^{1, 2,3, ,} 

i i

x x A i x i N

l

        

(3.3.5)

 

2 0

1

l

i

x dx

 

 (3.3.6)

Then we rewrite it as,

 

2 2 2 2

0 0

sin 1 cos 2

l l

i

l

x dx A i x dx

l i x

l

   







 



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Where, A 2

  l

  

(3.3.8) So the function of mode of vibration will become,

  ² ^sin

i

x i x

l l

 

 

(3.3.9) Absolute coordinates can be written as a function of modal coordinates with

     

1

,

^N

t

b i i

i

y x t  x 



 

(3.3.10) to make the superposition analysis.

Then we can get,

 

_{ }

   

4

4 1

,

^N

b

i i

i

y x t

x t

x  



 

 

;

     

2

2 1

,

^N

b

i i

i

y x t

x t

t  



 

  ^

;

     

1

,

^N

b

i i

i

y x t

x t

t  



 

  ^

(3.3.11)

Substitute equation (3.3.11) into the function of bridge’s flexural

vibration (3.3.1) we can get that:

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 

           

   

4

1 1 1

,

N N N

i i i i i i

i i i

EI x t x t x t

F x t x vt

       



  

 

  

  ^  ^

(3.3.12) The next step is to multiply the function of mode of vibration

  

n

x n N

  on both sides and integrate the equation from 0 to l .

According to the orthogonality of the function of mode of vibration, we can get:

 

1

   

²

   

0

0 l

0

l

n i i

i n n

i n

x x t dx

x dx t i n

  

 





 

     



  

(3.3.13) So we can deduce that,

 

^{ }

         

     

4

1 1

0 0

0 1

0

,

l N l N

n i i n i i

i i

l N

n i i

i l

n

x EI x t dx x x t dx

x x t dx

x F x t x vt dx

      

   

 

 





 

 

 

 

 







(3.3.14)

After simplification, it follows that:

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 

^{ }

       

   

4 2

0 0

2

0

,

l l

n n n n n

l

n n

n

EI x x dx t x dx t

x dx t vt F vt t

     

  



  

 

 

 







(3.3.15) Since,

 

^{ }⁴

   

²

0 0

l l

n

x

n

x dx

n

x dx

   ^  ^ ^

 

  (3.3.16)

The proof of Eq. (3.3.16) is as follows:

 

^{ }⁴

   

^{ }³

 

0 ^{ }³

   

0 0

|

l l

l

n

x

n

x dx

n

x

n

x

n

x

n

x dx

   ^    ^     ^

 

 

²

   

0 ^{ }³

   

0 0

|

l l

l

n

x dx

n

x

n

x

n

x

n

x dx

    

         

   

 

If we want to prove

 

^{ }⁴

   

²

0 0

l l

n

x

n

x dx

n

x dx

   ^   ^ ^ 

 

We must prove that ^  

_n

  x 

_n^{ }³

  x    |

0^l

^  

_n

^     x 

_n

^ x ^  |

0^l

As we know that

n

  ² ^sin

x n x

l l

 

 

,

So we can get,

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  ² ^cos

n

n n x

x l l l

 

 ^ ^ 

 

²

² ^sin

n

n n x

x l l l

 

 

 

   

 

 ³

 

³

² ^cos

n

n n x

x l l l

 

 

 

  

 

 

^{ }

 

3 3

0 0

3 4 0

2 2

| sin cos |

2 1 2

sin | 0

2

l l

n n

l

n x n n x

x x

l l l l l

n n x

l l

  

 

 

 



   

        

       

 

     

 

 

   

 

2

0 0

3 4 0

2 2

| sin cos |

2 1 2

sin | 0

2

l l

n n

l

n n x n x

x x n

l l l l l

n n x

l l

  

  

 

 



   

          

       

 

     

 

 

Hence, we get:

 

^{ }⁴

   

²

0 0

l l

n

x

n

x dx

n

x dx

   ^   ^ ^ 

 

)

Then we can change equation (3.3.15) into

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       

   

2

0 0

2

0

,

l l

n n n n

l

n n

n

EI x dx t x dx t

x dx t vt F vt t

    

  



     

 

 

 

 







(3.3.17) Substitute equation (3.3.6) into equation (3.3.17) we can get

 

²

     

0

+ ,

l

n n n n n

EI  x dx  t     vt F vt t

      

 

 ^ ^

(3.3.18) And,

 

   

2 2 2

0 0

4 4 4

5 5

0

2 sin 1 cos 2

2 2 1

2 2

l l

n

l

n n x

x dx dx

l l l

n x

n l dx n l n

l l l

 

 

   

  

   

         

       

        

 

 (3.3.19)

Then we assume that 2  

_n _n

 ;

 

(3.3.20)

 

² ⁴

2 0 l

n n

EI x dx EI n

l

  



 

  

        

(3.3.21) So, the mode equation for the bridge is,

   

2

,

n n n n n n

F vt t

n

vt

             (3.3.22)

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As for the model illustrated figure 3.3.

Because of the different number of contact points, the force will be different. For the model in the figure3.1, there will be two forces on the contact points. So the vibration equation of the bridge can be written as

     

   

4 2

2 1

, , ,

,

b b b

i i i i

i

y x t y x t y x t

EI x t t

F x t x vt

 





  

 

  

   

(3.3.23) Then, the mode equation for the bridge will be,

   

2

1 1 1 2 2 2

n

2

n n n n n

F t

n

F t

n

                 (3.3.24)

Where

 

1

2 sin

n n

vt n vt

l l

  

  

(3.3.25)

   

2

2 sin

n n

n vt a vt a

l l

  



   

(3.3.26)

 

1

1 0 0

t l

t v

else

 _{ } ^ ^ ^{ }

 (3.3.27)

 

2

1 0

a l a

t v t v

else

 _{ } ^ ^ ^{ } ^



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3.4 The compatibility conditions

The compatibility condition of the displacement is as follows:

    ^, ^,

ci b i i

y  y x t  r x t

(3.4.1) Where y is the displacement of the i’th wheel on the contact point in the

_ci

vertical direction, y

b

  x t

i

^, is the displacement of bridge on the i’th contact point in the vertical direction, r x t  

i

^, is the irregularity from the bridge’s surface on the i’th point.

And there are two contact points, so equation (3.4.1) will become:

   

       

1 1 1

1 1

| , | |

,

c x vt b x vt x vt

N

b i i

i

y y x t r x

y vt t r vt  vt  t r

  



  

   

(3.4.2)

   

2 2 2

2 1

| , | |

c x vt a b x vt a x vt a

N

i i

i

y y x t r x

vt a t r

 

     



  

 

 (3.4.3)

Where,

1

 

r  r vt

; r

2

 r vt   a 

.

Then we can get

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       

1 1 1

1

|

^N

c x vt i i i i

i

y

_

 vt  t  vt  t r



 

      ^  ^     

 

(3.4.4)

       

2 2 2

1

|

^N

c x vt a i i i i

i

y

_{ }

 vt a  t  vt a  t r



 

      ^    ^     

 

(3.4.5) When the vehicle is running on the bridge, the force to the bridge from the i’th wheel is made up of two parts. One is the static load of the wheel and the vehicle body gravity that wheel undertakes; the other one is the elasticity from the transformation of the wheel and the damping force generate from the viscous damping. So the function is

  ^,   ⁽ ⁾

i i ti ti ci ti ti ci

F x t  W  K y  y  C y   y  (3.4.6) Where,

W

i

is the static load of the wheel and the vehicle body gravity of the wheel undertakes,

K ,

ti

C is the stiffness and the damping of the tire separately;

^ti

y ,

ti

y _

^ti

is the displacement and the velocity of the wheel’s middle in the vertical direction separately;

y ,

ci

y _

^ci

is respectively the displacement and velocity in the vertical direction on the contact point between the tire and the bridge.

So we can get the function of force as follows:

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   

1 1

,

1

k

_t1 _t1 _c1 _t1

(

_t1 _c1

)

F x t  W  y  y  c y   y  (3.4.7)

   

2 2

,

2

k

_t2 _t2 _c2 _t2

(

_t2 _c2

)

F x t  W  y  y  c y   y  (3.4.8) Where

2

1 s t1

W m a m g

a

 

      (3.4.9)

1

2 s t2

W m a m g

a

 

      (3.4.10)

According to the equations (3.2.1), (3.2.2), (3.2.3) and (3.2.4) we can get:

   

²

1 1 1 1 1 1 1 1 s s

t t c t t c t t

a m y J

c y y k y y m y

a a

      ^   ^

  

(3.4.11)

   

¹

2 2 2 2 2 2 2 2 s s

t t c t t c t t

a m y J

c y y k y y m y

a a

      ^   ^

  

(3.4.12) Substitute equations (3.4.11) and (3.4.12) into equations (3.4.7) and (3.4.8), we can get that:

    ^{ } _{ }

2 2

1 1 1

1

2 1 1

2 2 2

,

s t t t s s

a a J

m m g m y m y

F t a a a

F x t

F t a a J

m m g m y m y

a a a



      

  

  

   

 

       

   



  

(3.4.13)

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3.5 Construction of complete model

Figure 3.6. The model of double axles vehicle-bridge.

We analyze the situations that the two wheels of vehicle run on and run down the bridge in consideration of the headway. In our thesis, we will analyze passing the bridge from three time ranges.

The vehicle starts to run on the bridge, the time range is 0, a v

 

   ; The vehicle runs on the bridge, the time range is a l ,

v v

 

   ;

The vehicle runs down the bridge just on the right edge, the time range is l l , a

v v

  

 

  .

Then equations (3.2.3) and (3.2.4) can be written as:

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1 1 1 1 1 1 1 1

( ) ( )

( ) ( ) 0

t t s s t s s t

t t c t t c

M y k y y a c y y a

k y y c y y

 

 

     

    

   

  (3.4.14)

2 2 2 1 2 2 2 2

2 2 2 2 2 2 2 2

( ) ( )

( ) ( ) 0

t t s s t s s t

t t c t t c

M y k y y a c y y a

k y y c y y

 

 

     

    

   

  (3.4.15) According to the compatibility conditions the bridge function (3.3.24) can be changed to

 

2 1 1 1 2 2 1 1 2 2

2

1 1 1 1 2 2 2 2

1 1 1 2 2 2

2

n n n n

s s

n t t n t t n n n n n n

n n

a a

m y J

a a

m y m y

W W

        

         

   

 



    

  

 

 

 

(3.4.16) We can get the system model of the vehicle-bridge as,

       

M t Y   C t Y   K t Y  Q t (3.4.17) Where,

  ,   ,  

M t C t K t

     

      are respectively n 4 orders mass matrix, damping matrix, stiffness matrix.

Q t is also    ⁿ ^ ⁴  orders vector matrix, Y is  ⁿ ^ ⁴  orders free vector.

Then we get:

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

1 2 1 2



T

s t t n

Y  y  y y     (3.4.18) This is the displacement vector of the system. The matrices are given by:

 

1

2

2 11 1 1 21 2 11 1 21 2

11 1 1 21 2 2

2 12 1 1 22 2 12 1 22 2

12 1 1 22 2 2

2 1 1 1 2 2 1 1 2 2

1 1 1 2 2 2

0 0 0 0 0 0

1 0 0

0 1 0

0 0 0

s

t

s t t

n n n n

s n t n t

m

J

m

a a

m J m m

M t a a

a a

m J m m

a a

m J m m

a a

           

 



 



       

1  

 

 

(3.4.19)

 

1 2 1 1 2 2 1 2

2 2

1 1 2 2 1 1 2 2 1 1 2 2

1 1 1 1 1 1 11 1 1 12 1 1 1 1

2 2 2 2 2 2 21 2 2 22 2 2 2 2

1 1

2 2

0 0 0

0 0

0 0 0 0 2 0 0

0 0 0 0 0 2 0

0 0 0 0 0

s s s s s s

s s s t t t t n

c c c a c a c c

c a c a c a c a c a c a

c c a c c c c c

C t

     

 

   

  

     

    





       

0 0 2  

_n _n

 

 

 

(3.4.20)