2019/10/8

Vehicle-Bridge Interaction System
Introduction to IDE (1)Yamamoto Kyosuke, Asst. Prof.

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Road unevenness induces
vibrations on a travelling vehicle

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The simplest model of the travelling vehicle isa mass-spring model2019/10/8

Suspensions are usually modeled byspring and damping

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Equation of motion of the vehicle: (𝑚𝑎𝑚: Mass𝑧 𝑡
Vehicle Vibration

𝑚𝑧𝑐 𝑧𝑟damping force

𝑘 𝑧𝑟

restoring force

𝑘: stiffness
𝑐: damping

𝑟 𝑡Road Profile

Linear Differential Equation
System Parameters: 𝑚, 𝑐, 𝑘

Output: 𝑧 𝑡

Input: 𝑟 𝑡𝐹)
𝑑𝑑𝑡𝑑𝑑𝑡2019/10/8

Road Profile: 𝑟 𝑡
Displacement (m)Time (s)System Parameters:

Displacement (m)Mass: 𝑚Damping: 𝑐

Stiffness: 𝑘

10000 [kg]2000 [kg/s]

10000 [N/m]

Vehicle Vibration: 𝑧 𝑡Time (s)2019/10/8%-- set vehicle parametersfprintf('setting vehicle parameters...¥n')m = 10000; %-- mass (kg)c = 2000; %-- damping (kg/s)
k = 10000; %-- stifness (N/m)=(kg/s/s)
v = 10.0; %-- speed (m/s)
dt= 0.001; %-- time increment;
T = 60;%-- end time%-- calculate inputfprintf('calculating vehicle input...¥n')
R = load('profile-1.txt'); %-- load: R(x)x = 0:0.20:(length(R)-1)*0.20; %-- dx=0.20(m) for R(x)t = 0:dt:T; %-- time, dt=0.001(s)xx= v*t; %-- vehicle position: x(t)r = spline(x,R,xx); %-- spline interpolation: R(x) to r(t)=R(x(t))
dr= 1/dt*gradient(r); %-- numerical differential
r=r-r(1); %-- reference value setting: r(0)=0
dr=dr-dr(1); %-- reference value setting: dr(0)=0
f=c*dr+k*r; %-- right-hand-side of equation of motion of the vehicle%-- simulate output
fprintf('vehicle vibration simulation: newmark-beta method...¥n')G=1/2; %-- gammma coefficient for newmark-beta method
B=1/4; %-- beta coefficient for newmark-beta method[ t, z, dz, ddz ] = newmark_beta( m,c,k,f,1/1000,zeros(1,3), G, B );function [ t, x, dx, ddx ] = newmark_beta( m,c,k,f,dt,X, G, B )%% initialization%-- memory space
x=zeros(size(f));
dx=zeros(size(f));
ddx=zeros(size(f));%-- input initial valuesx(:,1)=X(:, 1);dx(:,1)=X(:, 2);ddx(:,1)=X(:, 3);%-- time vector
T=length(f);t=(0:T-1)*dt;%% solving
A = m + dt*G*c + dt^2*B*k; %-- global matrix
M = inv(A); %-- inverse matrix of A
for tt=2:T%-- solving preparation
b1 = -c * ( dx(:,tt-1) + (1-G)*dt*ddx(:, tt-1) );b2 = -k * ( x(:,tt-1) + dt*dx(:,tt-1) + (1/2-B)*dt^2*ddx(:,tt-1) );b = f(:,tt) + b1 + b2; %-- right hand side%-- calculation of acceleration at t=t+1ddx(:,tt) = M*b;%-- visualization
fprintf('outputting the simulation results...¥n')figure(1)subplot(2,1,1); plot(t,r,'-b')subplot(2,1,2); plot(t,z,'-r')%-- animation
figure(2);set(gcf, 'Unit', 'pixel', 'Position',[120, 280, 800, 300]);
writerObj = VideoWriter('ex_01.avi','Motion JPEG AVI'); %-- setting movie
classwriterObj.FrameRate = 20; %-- flame rate (slide/sec)open(writerObj); %-- initialization of movie object
for tt=1:100:length(t)plot(xx(1:tt)',[r(1:tt);z(1:tt)+0.01]');hold onplot(xx(tt),r(1,tt),'bo');plot(xx(tt),z(1,tt)+0.01','ro');hold off
xlim([0 max(xx)]);ylim([-0.004 0.020]);
frame = getframe(figure(2)); %-- get the figure as a frame of theanimation
writeVideo(writerObj,frame); %-- add the gotten frame to the movie object
end
close(writerObj); %-- end of animation%-- calculation of velocity and displacement responsesdx(:,tt) = dx(:,tt-1) + dt*(1-G)*ddx(:,tt-1) + dt*G*ddx(:,tt);x(:,tt) = x(:,tt-1) + dt*dx(:,tt-1) + dt^2*(1/2-B)*ddx(:,tt-1) + dt^2*B*ddx(:,tt);
endend2019/10/8

Equation of motion of the vehicle: (𝑚𝑎𝑠 𝑡

𝑚 𝑠

𝑐 𝑠

Sprungmass
Vibration

𝑚 𝑧

𝑐 𝑠𝑧

𝑘 𝑠𝑧

𝑐 : sprungdamping

𝑚 𝑠𝑐𝑠

𝑘 𝑠𝑐𝑧

𝑘 𝑧𝑧 𝑡𝑚 𝑧𝑐𝑧

𝑘𝑘 𝑧𝑚𝑠𝑧𝑚 : sprung-mass

𝑘 : sprungstiffness𝑚 : unsprungmass𝑘 : unsprungstiffness

Unsprungmass
Vibration𝑟 𝑡𝑚𝑧𝑐𝑐𝑘 𝑠𝑐𝑐𝑧
𝑘 𝑧𝑟𝑐𝑠𝑘 𝑠

𝑘 𝑟𝑠𝑧

𝑘𝑘𝐂𝒛

𝐊𝒛

𝑘𝑘𝑘

Road Profile𝐌𝒛𝐹)

𝒇𝑠𝑧

0𝑘 𝑟

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Matrix and Vector
System of Equations:

Using Matrix:

𝑎 𝑥

𝑎 𝑥

𝑏𝑎 𝑥

𝑎 𝑥

𝑏𝑎 𝑥

𝑎 𝑥

𝑎 𝑥

𝑏𝑎 𝑥

𝑎 𝑥

𝑎 𝑥

𝑏𝑎 𝑥

𝑎 𝑥

𝑎 𝑥

𝑏𝑨𝒙

𝒃𝑏𝑏𝑎𝑎𝑎𝑎𝑥𝑥

𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎

Solution:

𝒙𝑥𝑥𝑥

𝑨 𝒃
Inverse matrix

𝑏𝑏𝑏

2019/10/81) Download: http://www.kz.tsukuba.ac.jp/~yamamoto_k/material/matlab_for_ide.zip
2) Decompress the downloaded zip file3) Launch Matlab and set the decompressed directory as current directory4) Run the following script:>> sample_02

2019/10/8

Two quarter-cars can be easily
extended to a half-car model2019/10/8𝑠𝑑𝑚G𝑠 𝑡𝑃Definition:𝑠𝑟𝜃

𝑠𝑑𝑟

𝑚
𝑃

𝜃

𝑠𝑟𝜃

𝑠𝑟𝜃
𝑚𝑠 𝑡

𝑚𝑑 𝑚𝑠𝑑𝑑𝑚𝑠
𝜃𝑠𝑑𝑑 𝑚𝑠𝑑𝑑𝑚 𝑠𝑠𝑑

Equation of motion of mass point 𝑚𝑚𝑠
Law ofAction andReaction𝑓internal force
𝑓𝑃𝑃𝑟

𝑓𝑑𝑃𝑑𝑃

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𝑠𝑑𝑚G𝑠 𝑡𝑃𝑠𝑑𝑟𝑚𝑃𝜃

Summation of Equation of motion𝑠 𝑡

𝑚𝑠Equation of motionof translation: 𝑚𝑎 𝐹𝑚 𝑠𝑓𝑃𝑃2019/10/8

𝑠𝑑𝑚G𝑠 𝑡𝑃𝑠𝑑𝑟

𝑚
𝑃

Summation of Equation of motion × arm length

𝑚 𝑟𝑠

𝜃𝑠 𝑡

𝑟𝑓

Equation of motionof rotation: 𝐼𝜃 𝑇𝑚𝑟𝜃𝑑 𝑃

TorqueInertia
MomentAngularAcceleration𝑑 𝑃

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Equation of Motion of Translation

𝑚𝑑𝑑 𝑚𝑠𝑑𝑑𝑠 𝑡𝑘𝑧 𝑡𝑟 𝑡𝑐𝑚𝑐𝑠𝑧𝑐𝑠𝑧

𝜃𝑘𝑠𝑧𝑘𝑠𝑧𝑠 𝑡

Equation of Motion of Rotation

𝑑G𝑘𝑚𝑑 𝑚𝑠𝑑𝑑𝑐

𝐼𝑑𝑧 𝑡𝑟 𝑡

𝑑𝑚 𝑧

𝑚 𝑧

𝐼𝑠𝑑𝑐𝑐𝑠𝑠𝑑𝑧𝑧

𝑠𝑑 𝑐𝑠𝑧

𝑑 𝑐𝑠𝑧

𝑑 𝑘

𝑠𝑧

𝑑 𝑘𝑠𝑧

𝑘𝑠𝑧

𝑘𝑧

𝑟

𝑘𝑠𝑧

𝑘𝑧𝑟

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𝑑 𝑚𝑑𝑑𝐼𝑑𝑑𝑑 𝑚𝑑𝑑𝐼𝑑𝑑𝑐
𝑑 𝑐𝑐𝑠𝑠𝑧𝑧

𝑚𝑐
𝑑 𝑐𝑐𝑐
𝑑 𝑐𝑐𝑐
𝑑 𝑐𝑐𝑠𝑠𝑧𝑧

𝑘𝑑 𝑘𝑘𝑘𝑑 𝑘

𝑘𝑑 𝑘𝑘𝑘𝑘

𝑚𝐌𝒛 𝑡

𝐂𝒛 𝑡

𝐊𝒛 𝑡

𝒇 𝑡

Newmark-β method𝐀𝒛 𝑡𝒃 𝑡𝒛 𝑡

𝐀 𝒃 𝑡
Solving the system of equations

𝑘𝑑 𝑘𝑘𝑘𝑠𝑠𝑧𝑧𝑘 𝑟𝑘 𝑟

2019/10/8𝐌𝒛 𝑡

𝐂𝒛 𝑡

𝐊𝒛 𝑡𝒇 𝑡Consider the next time step

𝐌𝒛 𝑡

Δ𝑡Newmark-β method

𝐌Δ𝑡𝛾𝐂𝐂𝒛 𝑡

Δ𝑡

𝐊𝒛 𝑡𝒛 𝑡Δ𝑡𝒛 𝑡𝒛 𝑡Δ𝑡𝒛 𝑡

Δ𝑡 𝛽𝐊𝐀𝒛 𝑡𝒛 𝑡

Δ𝑡Δ𝑡𝒃

Δ𝑡𝒇 𝑡

Δ𝑡

Δ𝑡 1𝛾 𝒛 𝑡Δ𝑡𝛾𝒛 𝑡

Δ𝑡

Δ𝑡𝒛 𝑡

Δ𝑡

12𝛽 𝒛 𝑡Δ𝑡 𝛽𝒛 𝑡Δ𝑡𝒇𝐂 𝒛𝑡𝑡 Δ𝑡
Δ𝑡 1 𝛾 𝒛 𝑡
1𝐊𝐂𝒛 𝒛𝑡 𝑡 Δ𝑡𝒛 Δ𝑡𝑡 1Δ𝑡 𝛾 𝒛𝛽 𝑡𝒛 𝑡2𝐊𝒛 𝑡
Δ𝑡 1 𝛾 𝒛 𝑡

𝒇 𝑡

Δ𝑡Once 𝒛 𝑡 , 𝒛 𝑡 , 𝒛 𝑡 and 𝒇 𝑡 Δ𝑡 are obtained,
we can solve 𝒛 𝑡 Δ𝑡 , 𝒛 𝑡 Δ𝑡 and 𝒛 𝑡 Δ𝑡 .

2019/10/81) Download: http://www.kz.tsukuba.ac.jp/~yamamoto_k/material/matlab_for_ide.zip
2) Decompress the downloaded zip file3) Launch Matlab and set the decompressed directory as current directory4) Run the following script:>> sample_03

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Consider the effect ofBridge Vibrations

2019/10/8

Vehicle and Bridge can be modeled by
Half-car model(Rigid Body Spring Model)

andBeamFinite Element Methodusing 1D Finite Beam Element Model2019/10/8

Vehicle Inputs include Bridge vibrations
as well as Road unevenness
𝑠 𝑡𝑠 𝑡𝑧 𝑡𝑢 𝑡𝑧 𝑡𝑢 𝑡𝑢 𝑡𝑟 𝑡𝑦 𝑡

𝑢 𝑡𝑟 𝑡𝑦 𝑡

road unevenness

bridge vibration

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The Vehicle Vibrations affect on
the Contact Forces acting on the Bridge𝑠 𝑡𝑠 𝑡𝑃 𝑡𝑧 𝑡

𝑧 𝑡
𝑃 𝑡

𝑃 𝑡

𝑃 𝑡

𝑑 𝑚
𝑔 𝑠 𝑡
𝑑𝑑𝑑 𝑚
𝑔 𝑚 𝑔𝑑𝑑𝑑 𝑚
𝑔𝑑𝑑𝑑 𝑚𝑔𝑑𝑑weight𝑚𝑘𝑔𝑧𝑧 𝑡
𝑢𝑠 𝑡

𝑚𝑔𝑧 𝑡

𝑚 𝑔𝑘𝑧𝑢restoring forceof unsprung-stiffness

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Bridge Deflection Vibrations can be calculatedby Structure Mechanics
𝑥 𝑃 𝑡𝑥

𝑃 𝑡

𝑦 𝑥, 𝑡𝜌𝐴𝑦 𝑥, 𝑡

Mass perunit length

𝜕𝜕𝐸𝐼𝑦 𝑥, 𝑡𝜕𝑥
𝜕𝑥
FlexuralRigidity𝑥

𝛿 𝑥𝑥 𝑡 𝑃 𝑡

DeltaFunction

𝛿 𝑥𝑥 𝑡 𝑃 𝑡

𝛿 𝑥

0 , when 𝑥0

𝛿 𝑥∞ , when 𝑥0

𝛿 𝑥 d𝑥1

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The equation of motion of Vehicle System:𝑑 𝑚𝑧𝑑𝑑𝐼𝑧𝑑𝑑𝑑 𝑚𝑧𝑑𝑑𝐼𝑧
𝑑𝑑𝑡𝑡𝑡𝑘𝑡𝑧

𝑑𝑡𝑧𝑡𝑘𝑧𝑡𝑡𝑧𝑡𝑘𝑧𝑡𝑧𝑑𝑘𝑧

𝑚𝑧𝑡𝑘𝑧𝑡𝑧𝑡𝑐𝑧𝑡𝑧𝑡𝑘𝑧𝑡𝑢 𝑡𝑚𝑧𝑡𝑘𝑧𝑡𝑧𝑡𝑐𝑧𝑡𝑧𝑡𝑘𝑧𝑡𝑢 𝑡𝑑𝑧𝑑𝑚𝑡𝑐𝑧𝑧𝑡𝑧𝑡𝑧𝑑𝑡𝑐𝑐𝑧𝑢 𝑡𝑢 𝑡

𝑅 𝑥 𝑡
𝑅 𝑥 𝑡

𝑦 𝑥 𝑡 ,𝑡𝑦 𝑥 𝑡 ,𝑡𝑘𝑐 𝑘𝑚𝑚𝑧𝑢𝑘𝑐𝑦 𝑥, 𝑡𝑃 𝑡𝑢
𝐸𝐼 𝜌𝐴

The equation of motion of Bridge System:𝜌𝐴𝑦 𝑥, 𝑡

𝜕𝜕𝐸𝐼𝑦 𝑥, 𝑡𝜕𝑥𝜕𝑥𝛿 𝑥𝑥 𝑡 𝑃 𝑡𝛿 𝑥𝑥 𝑡 𝑃 𝑡𝑧𝑑 𝑚𝑑𝑑𝑑 𝑚𝑑𝑑𝑧
𝑘𝑡𝑧𝑡𝑡𝑑𝑐𝑧𝑡𝑧𝑡

Contact Forces:
𝑃 𝑡

Input Profile:𝑡𝑧𝑔𝑧𝑡𝑚𝑔𝑧𝑡𝑔𝑧𝑡𝑚𝑔𝑧𝑡

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FEM is a method for discretizinga Continuum Body𝑥𝑦 𝑡⋮𝒚 𝑡𝑦 𝑡⋮2019/10/8

The equation of motion of Bridge can be also
expressed by matrices and vectors𝑥𝑦 𝑡

MassMatrix

𝐌 𝒚 𝑡

𝐊 𝒚 𝑡

𝑳 𝑡 𝑷 𝑡StiffnessMatrix

Equivalent Nodal ForceDistribution Matrix

Contact Force

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𝐊𝐊

12𝐸𝐼 6𝑙
12𝑙
6𝑙

6𝑙
4𝑙
6𝑙
2𝑙

126𝑙
126𝑙6𝑙
2𝑙6𝑙4𝑙2019/10/8

𝐌 𝒛 𝑡

𝐂 𝒛 𝑡𝑘𝑧𝑘𝑢𝑦 𝑡

𝐌𝒛𝒚𝒓 𝑡𝑳 𝑡 𝐌 𝒈𝒛 𝑡

𝑚𝑐 𝑘𝑚𝑚𝑐

𝑳 𝑡 𝒚 𝑡𝑧𝑧𝑘𝑢𝑟 𝑡𝑟 𝑡

𝐌𝑳 𝑡 𝐌𝐊𝑑𝑑𝑧

𝐊 𝒛 𝑡𝑦 𝑡𝐌 𝒚 𝑡

𝐊 𝒚 𝑡

𝐂𝒛𝒚𝐂

𝐊𝐊 𝑳 𝑡𝐊𝒛
𝒚𝐊 𝒓 𝑡𝑳 𝑡 𝐌 𝒈2019/10/8

Equation of Motion of VBI system
is NOT a linear differential equation
𝐌𝑳 𝑡 𝐌

𝐌𝒛𝒚𝐂

𝐌 𝑡 𝒙 𝑡𝐂𝒛𝒚𝐂𝒙 𝑡Coefficients temporary changes

𝐊𝐊 𝑳 𝑡𝐊

𝐊 𝑡 𝒙 𝑡𝒛
𝒚𝒇 𝑡

𝐊 𝒓 𝑡
𝑳 𝑡 𝐌 𝒈2019/10/8

Newton-Raphson Method
for solving a non-linear problem

Convergence calculation

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Road ProfileVehicle
Sprung-mass:𝑚 (kg)
Front: 𝑖=1
Rear: 𝑖=2

9000

Unsprung-mass:𝑚 (kg)

Sprung-damping:𝑐 (kg/s)

Sprung-stiffness:𝑘 (N/m)Unsprungstiffness:𝑘 (N/m)Distance from G:
𝑑 (m)

500

2000

4500

60000

1.75

500

2000

4500

60000

1.75

Vehicle Vibration

Input Profile

Contact Force

Bridge Vibration
BridgeFlexural Rigidity:
𝐸𝐼 (GNm2)

Mass per unit Length:𝜌𝐴 (kg/m)

Span Length:
𝐿 (m)Element Number

156

3000

10020

Travel speed:
𝑣 (m)
10.0

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BridgeInput Profile

Road Profile

Sprung-mass vibration

Unsprung-mass vibration

1) Download: http://www.kz.tsukuba.ac.jp/~yamamoto_k/material/matlab_for_ide.zip
2) Decompress the downloaded zip file3) Launch Matlab and set the decompressed directory as current directory4) Run the following script:>> sample_04>> sample_04_application

2019/10/8

VBI simulation is applied toBridge Design and MaintenanceTraffic Load for Bridge Design
Vibration-based Structure Health Monitoring
Future Technology (Watching Logistics)

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Traffic Loadfor Bridge Design

Maximum Value of 𝑃 of 10t vehicle is about 50 (kN)
(100kN)(NOTICE: Only considering very smooth road profile)2019/10/8

Vibration-basedStructure Health Monitoring
Bridge Acceleration Vibrations
𝐿/4

𝐿/2

3𝐿/4

2019/10/8

Watching Logistics(On-going Identification of Vehicle-Bridge-Road)

Only Measuring
Vehicle Acceleration Vibrations and Position

Identifying
Vehicle Parameters, Bridge Parameters
and Road Profile2019/10/8