Vib Rati Ons in Bri Dge S: Submitted To: Sir DR Asif Israr

Submitted to:
Submitted By
Vib
rati
ons
in
Bri
dge
Sir Dr Asif Israr
s
Muhammad Shaheer Naqqash

Moseeab Zaman Mian
Contents
OBJECTIVE.................................................................................................................. 2
INTRODUCTION........................................................................................................... 2
MATHEMATICAL MODELLING....................................................................................... 3
VIBRATIONS REDUCTION............................................................................................ 4
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Vibration Analysis of A Bridge

OBJECTIVE
Vibrations exists in all type of structures and are responsible for their
collapse so in order to increase the stability of the structure these vibrations
must be minimized. The purpose of this project is the mathematical
modelling of a real time problem exhibiting mechanical vibrations. We will be
performing mathematical modeling of the vibrations induced in a bridge.
INTRODUCTION
A bridge is a structure built to span physical obstacles such as a body of
water, valley, or road, for the purpose of providing passage over the
obstacle. To the naked eye, bridges appear to be fixed in place, unmoved by
forces like wind and rain. But in fact, these large structures do experience
imperceptibly small vibrations that, depending on their frequency, may
indicate instability or structural damage. Structures like bridges can start to
oscillate i.e. vibrate at a regular rate for many reasons. Indeed,
environmental factors like traffic, a high footfall or powerful machinery can
trigger vibrations. These vibrations amount to decrease the life of the bridge
structure and, in some cases, increased vibrations can lead to a sudden
collapse of a bridge.
Bridges can collapse due to
Natural disasters (like earth quakes, floods, strong winds)

Human Interference (like soldiers marching together)
One of the most famous examples of a resonance disaster is the 1940

Tacoma Narrows Bridge collapse in Washington, USA. This came about not
simply as a result of mechanical resonance, but also aeroelastic flutter a
process that occurs when complex, varying oscillations are caused by
passing winds
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Broughton Suspension Bridge was an iron chain suspension bridge built in

1826 to span the River Irwell between Broughton and Pendleton, now
in Salford, Greater Manchester, England. On 12 April 1831, the bridge
collapsed, reportedly due to mechanical resonance induced by troops
marching in step. As a result of the incident, the British Army issued an
order that troops should "break step" when crossing a bridge.
MATHEMATICAL MODELLING
The equation of motion of bridge system is
m1 1 = kx1 + k2 (x2 x1) + c (x2 x1) F1 sint
m2 2 = kx1 + k2 (x2 x1) c (x2 x1)
By using sinusoidal steady state solution can be derived for this system of the form
X1 = A sin( t + 1)
X2 = A sin( t + 2)
||
cs 2
(2 ) +( s 2r 2 )2
cc
A
=
2
F1
cs
( ) (s 21+ s 2)2 +( r 2 s 2(s 21)(s 2r 2 ))2
k1
cc
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Where , 1 and 2 are as above, r =
2
1
, cc =2m2 1 and s =
Therefore the particular unit of measurement is
| |(
A
=
F1
1+
k1
1
( )
( ) )( ( ) )
1
VIBRATIONS REDUCTION
Bridge Bearings
Bearing assumes the functionality of a bridge by allowing translation and
rotation to occur while supporting the vertical loads. Vibrations makes it
necessary to have a device in between the substructure (base) and the superstructure
which allows for the required movement, while also giving stability and having the capacity
to bear the loads placed on the bridge.
Tuned mass dampers

The effects of resonance can be countered by installing tuned mass
dampers, or harmonic absorbers. Tuned mass dampers stabilize against
violent motion caused by harmonic vibration. A tuned damper reduces the
vibration of a system with a comparatively lightweight component so that
the worst-case vibrations are less intense. Roughly speaking, practical
systems are tuned to either move the main mode away from a troubling
excitation frequency, or to add damping to a resonance that is difficult or
4 | Page
expensive to damp directly. An example of the latter is a crankshaft torsional

damper. Mass dampers are frequently implemented with a frictional or
hydraulic component that turns mechanical kinetic energy into heat, like an
automotive shock absorber. The worlds largest tuned mass damper is a 660ton pendulum in the Taipei 101 tower in Taiwan.
Gussaphalt
Gussaphalt is a building material, and belongs to the group of asphalts. This
is a mixture of fine and coarse aggregates and bitumen, the casting during
installation and is spreadable. Therefore, when using mastic asphalt no
compression work is needed. Mastic asphalt is used for road surfacing,
flooring and waterproofing.
Break step
Troops should "break step" when crossing a bridge.
5 | Page

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Submitted to:

Muhammad Shaheer Naqqash

Vibration Analysis of A Bridge

Natural disasters (like earth quakes, floods, strong winds)

One of the most famous examples of a resonance disaster is the 1940

Broughton Suspension Bridge was an iron chain suspension bridge built in

Where , 1 and 2 are as above, r =

Therefore the particular unit of measurement is

Tuned mass dampers

expensive to damp directly. An example of the latter is a crankshaft torsional

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