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Spectacular torsional-flexural autovibrations of a Volga bridge

The autovibrations of the bridge over Volga river in the city of Volgograd(Russia) were filmed (not by me) today on 05/21/2010. The amplitude reachedabout 1 m. The bridge opening was just six months ago...

Apparently, either the bridge frame drag was too high or torsional stiffness / damping too low.

 

Comments

Zhigang Suo's picture

Thank you for posting this link.  I've checked online, and found several very brief reports.  Here is one.  Not sure if this incident has a similar mechanism as that of the Tacoma Narrows Bridge.

After visual inspection (http://top.rbc.ru/incidents/21/05/2010/410041.shtml), the vice minister of the Transportation Dept. of Russia, Mr. Belozerov said

"The bridge is absolutely OK". Layer of paint deposited on the metal frame of the bridge forms a stiff coating. The official said, that cracking of the paint layer could give evidence of the deformations however, nothing of the kind can be observed. He also added, that the paving asphalt is also absolutely OK, all the structural parts are where they belong to. Preliminary, he said, the wind speed of 16 m/s could be responsible for the bridge vibrations.This happens periodically in Volgograd.

According to his words, the experts are now trying to determine the reasons for the onset of the observed vibrations. "According to the videorecording, the amplitude of vibrations amounted to about 40 cm. But this was allowed for in terms of the permissible metal elongation for this structure.

The official has also reassured, that the joinings (weld?) of the bridge will be inspected by ultrasound measurements prior to the bridge reopening.

 Width of the Volga river under the bridge is about 1 km. The actual span
of the bridge over the Volga river is about 2.9 km according to my
measurements using the Google Earth.

 http://en.wikipedia.org/wiki/Volgograd_Bridge

 http://ru.wikipedia.org/wiki/%D0%92%D0%BE%D0%BB%D0%B3%D0%BE%D0%B3%D1%80%...

ChangyongCao's picture

Good video for display.

I guess the material of the bridge is strong/good enough, or it has been broken or severely damaged at such a situation. The structure design may be wrong and should be adjusted.

Mike Ciavarella's picture

 

Londoners nicknamed the bridge the Wobbly Bridge after
participants in a special event to open the bridge (a charity walk on
behalf of Save the Children) felt an unexpected (and, for some,
uncomfortable) swaying motion on the first two days after the bridge
opened. The bridge was closed later that day, and after two days of
limited access the bridge was closed for almost two years while
modifications were made to eliminate the wobble entirely. It was
reopened in 2002.

 

The bridge's movements were caused by a 'positive feedback'
phenomenon, known as Synchronous Lateral Excitation. The natural sway motion of
people walking caused small sideways oscillations in the bridge, which in turn caused
people on the bridge to sway in step, increasing the amplitude
of the bridge oscillations and continually reinforcing the effect.[3]

The bridge opened on an exceptionally fine day, and it was included
on the route of a major charity walk. On the day of opening the bridge
was crossed by 90,000 people, with up to 2,000 on the bridge at any one
time.

Resonant
vibrational modes due to vertical loads (such as trains, traffic,
pedestrians) and wind loads are well understood in bridge design. In the
case of the Millennium Bridge, because the lateral motion caused the
pedestrians loading the bridge to directly participate with the bridge,
the vibrational modes had not been anticipated by the designers.

The lateral vibration problems of the Millennium Bridge are very
unusual, but not entirely unique.[4]
Any bridge with lateral frequency modes of less than 1.3 Hz, and
sufficiently low mass, could witness the same phenomenon with sufficient
pedestrian loading. The greater the number of people, the greater the
amplitude of the vibrations. Other bridges which have seen similar
problems are:

  • Birmingham NEC Link
    bridge, with a lateral frequency of 0.7 Hz
  • Groves Suspension Bridge, Chester, in 1977 during the Jubilee
    River
    Regatta
  • Auckland Harbour Road
    Bridge, with a lateral frequency of 0.67 Hz, during a 1975 demonstration[5]

After extensive analysis by the engineers [1], the problem was fixed by the retrofitting of 37
fluid-viscous dampers (energy dissipating) to control horizontal
movement and 52 tuned mass dampers (inertial) to control
vertical movement. This took from May 2001 to January 2002 and cost £5m.
After a period of testing, the bridge was successfully re-opened on 22
February 2002. The bridge has not been subject to significant vibration
since.

An artistic expression of the higher-frequency resonances within the
cables of the bridge were explored by Bill Fontana's 'Harmonic Bridge'
exhibition at the Tate Modern museum in the summer of 2006. This
utilised acoustic transducers placed at strategic locations on the
cabling of the Millennium Bridge and the signals from those transducers
were amplified and dynamically distributed throughout the Turbine Hall
of the Tate by a program Fontana entered into the sound diffusion engine
of the Richmond Sound Design AudioBox [2].

 

 

Michele Ciavarella, Politecnico di BARI - Italy, Rector's delegate.
http://poliba.academia.edu/micheleciavarella
Editor, Italian Science Debate, www.sciencedebate.it
Associate Editor, Ferrari Millechili Journal, http://imechanica.org/node/7878

Mike Ciavarella's picture

From the video, you clearly hear a lot of noise due to wind.

 

Now, the 1940 Tacoma Narrows Bridge opened to traffic on July 1, 1940,
and dramatically collapsed into Puget Sound on November 7 of the same year.
At the time of its construction (and its destruction) the bridge was
the third longest suspension bridge in the world in terms of main span
length, behind the Golden Gate Bridge and the George Washington Bridge.

Construction on the bridge began in September 1938. From the time the
deck was built, it began to move vertically in windy conditions, which
led to construction workers giving the bridge the nickname Galloping
Gertie
. The motion was observed even when the bridge opened to the
public. Several measures aimed at stopping the motion were ineffective,
and the bridge's main span finally collapsed under 40-mile-per-hour
(64 km/h) wind conditions the morning of November 7, 1940.

Following the collapse, the United States' involvement in World
War II
delayed plans to replace the bridge. The portions of the
bridge still standing after the collapse, including the towers and
cables, were dismantled and sold as scrap metal. Nearly 10 years after
the bridge collapsed, a new Tacoma Narrows Bridge
opened in the same location, using the original bridge's tower pedestals
and cable anchorages. The portion of the bridge that fell into the
water now serves as an artificial reef.

The bridge's collapse had a lasting effect on science and
engineering. In many physics textbooks the event is presented as an
example of elementary forced resonance with the wind providing an
external periodic frequency that matched the natural structural
frequency, even though its real cause of failure was aeroelastic flutter.[1]
Its failure also boosted research in the field of bridge
aerodynamics-aeroelastics, the study of which has influenced the designs
of all the world's great long-span bridges built since 1940.

 

Attempt to control
structural vibration

Since the structure experienced considerable vertical oscillations
while it was still under construction, several strategies were used to
reduce the motion of the bridge. They included[5]

  • attachment of tie-down cables to the plate girders, which were
    anchored to 50-ton concrete blocks on the shore. This measure proved
    ineffective, as the cables snapped shortly after installation.
  • addition of a pair of inclined cable stays that connected the main
    cables to the bridge deck at mid-span. These remained in place until the
    collapse, but were also ineffective at reducing the oscillations.
  • finally, the structure was equipped with hydraulic buffers installed
    between the towers and the floor system of the deck to damp
    longitudinal motion of the main span. The effectiveness of the hydraulic
    dampers was nullified, however, because it was found that the seals of
    the units were damaged when the bridge was sand-blasted before being
    painted.

The Washington Toll Bridge
Authority
hired Professor Frederick Burt Farquharson, an engineering
professor at the University of Washington, to make
wind-tunnel tests and recommend solutions in order to reduce the
oscillations of the bridge. Professor Farquharson and his students built
a 1:200-scale model of the bridge and a 1:20-scale model of a section
of the deck. The first studies concluded on November 2, 1940—five days
before the bridge collapse on November 7. He proposed two solutions:

  • To drill some holes in the lateral girders and along the deck so
    that the air flow could circulate through them (in this way reducing
    lift forces).
  • To give a more aerodynamic shape to the transverse section of the
    deck by adding fairings or deflector vanes along the deck, attached to
    the girder fascia.

The first option was not favored because of its irreversible nature.
The second option was the chosen one; but it was not carried out,
because the bridge collapsed five days after the studies were concluded.[4]

[edit] Collapse

The 1940 Tacoma Narrows Bridge collapsing,
in a frame from a 16mm Kodachrome motion picture film taken by Barney
Elliott

The wind-induced collapse occurred on November 7, 1940, at 11:00 am
(Pacific time), due to a physical phenomenon known as aeroelastic flutter.[1]

From the account of Leonard Coatsworth, a Tacoma News Tribune editor who
was the last person to drive on the bridge, he was driving with his dog
over the bridge when the bridge started to vibrate violently.
Coatsworth was forced to flee his car:


Just as I drove past the
towers, the bridge began to sway violently from side to side. Before I
realized it, the tilt became so violent that I lost control of the
car...I jammed on the brakes and got out, only to be thrown onto my face
against the curb...Around me I could hear concrete cracking...The car
itself began to slide from side to side of the roadway.

On hands and knees most of the time, I crawled 500 yards (460 m) or
more to the towers...My breath was coming in gasps; my knees were raw
and bleeding, my hands bruised and swollen from gripping the concrete
curb...Toward the last, I risked rising to my feet and running a few
yards at a time...Safely back at the toll plaza, I saw the bridge in its
final collapse and saw my car plunge into the Narrows.[6]

 

Film of collapse

Film of the 1940 Tacoma Narrows bridge
collapse

Tacoma Narrows Bridge destruction.ogg

Play<br />
			video

Footage of the Tacoma
Narrows bridge collapsing. (19.1 MB
video, 2:30)

Produced by
Barney Elliott

The collapse of the bridge was recorded on film by Barney Elliott,
owner of a local camera shop, which shows Leonard Coatsworth leaving the
bridge after exiting his car. In 1998, The Tacoma Narrows Bridge
Collapse
was selected for preservation in the United States National Film Registry by the Library of Congress as being culturally, historically,
or aesthetically significant. This footage is still shown to engineering,
architecture, and physics
students as a cautionary tale.[12]
Elliot's original films of the construction and collapse of the bridge
were shot on 16 mm Kodachrome film, but most copies in circulation
are in black and white because newsreels of the day copied the film onto
35 mm black-and-white stock.

 

Michele Ciavarella, Politecnico di BARI - Italy, Rector's delegate.
http://poliba.academia.edu/micheleciavarella
Editor, Italian Science Debate, www.sciencedebate.it
Associate Editor, Ferrari Millechili Journal, http://imechanica.org/node/7878

Mike Ciavarella's picture

It could even be that this bridge is so badly designed that it suffers both instabilities.

In fact, in the video you see very few people, yet they are walking.

In the original event leading to the closing in the news "Europe's longest bridge closed after it wobbles By
Channel 4 on 21 May 2010re,  I read:

 

A seven-kilometre-long bridge over the Volga River
has been closed after the structure started to wobble.

The bridge in Volgograd, southern Russia was built last year at
a cost of £55m, and opened in October 2009.

Several people were travelling across the bridge in cars when
it started moving, but there were no injuries reported.

It was closed last night, and remained shut this morning as
emergency services looked to see if there had been further structural
damage.

Engineers were examining whether strong river currents caused
by melting snow upstream had loosened one of the bridge's vertical
supports.

Its closure is a blow to drivers and inhabitants of Volgograd, a
large industrial city.

The nearest bridge is a hundred kilometres away and the only
alternative is a ferry.

 

So maybe it suffers both problems, the walking instability, and the aerolastic one at high speed of wind.

Michele Ciavarella, Politecnico di BARI - Italy, Rector's delegate.
http://poliba.academia.edu/micheleciavarella
Editor, Italian Science Debate, www.sciencedebate.it
Associate Editor, Ferrari Millechili Journal, http://imechanica.org/node/7878

Mike Ciavarella's picture

 

Commission of the Federal
Works Agency

A commission formed by the Federal Works Agency studied the collapse of the
bridge. It included Othmar Ammann and Theodore von Kármán. Without drawing any
definitive conclusions, the commission explored three possible failure
causes:

  • Aerodynamic instability by self-induced vibrations in the structure
  • Eddy formations that might be periodic in nature
  • Random effects of turbulence, that is the random fluctuations in
    velocity and direction of the wind.

 

Michele Ciavarella, Politecnico di BARI - Italy, Rector's delegate.
http://poliba.academia.edu/micheleciavarella
Editor, Italian Science Debate, www.sciencedebate.it
Associate Editor, Ferrari Millechili Journal, http://imechanica.org/node/7878

Mike Ciavarella's picture

Tacome collapsed under 40-mile-per-hour (64 km/h) wind, which is higher but not far from the 16m/s wind conditions you indicate.

Also, the official says the bridge is covered with stiff paint to control the cracks.  This is quite surprising.  If the excursions are 40 cm, surely there must be some deformation, even by visual inspection.

It is true, however, to distinguish a self-excited phenomenon, that we have to consider the energy input from the wind, but also to consider if this is resonating or not.   A self-excited phenomenon in theory doesn't need any significant amount of initial perturbation.  The initial perturbation grows unbouded exponentially in time. If you start small, you just take longer to reach the non-linear part where there is some "damping" and additional resistance.

So I am not sure now if your case is or not self-excited phenomenon or not.   I would need to check the video.  Was the oscillation constant in time, or it appeared to increase?  We need more quantitative descriptions.

However, to say something quantitative, we need a model of the bridge.

It is not a single span bridge, so we need at least the model of a single structure between two pillar.

If you have these data, imechanica (or perhaps just my students) can make a FEM model easily!

 

Michele Ciavarella, Politecnico di BARI - Italy, Rector's delegate.
http://poliba.academia.edu/micheleciavarella
Editor, Italian Science Debate, www.sciencedebate.it
Associate Editor, Ferrari Millechili Journal, http://imechanica.org/node/7878

Mike Ciavarella's picture

This may be off track, but the London Bridge phenomenon is attracting some interest, see this interesting paper from an italian young mathematician with whom I am discussing on a good online journal of the highlycited scientists in Italy.

 

Unfortunately in italian.

 

Matematica

Pedoni in risonanza

La dinamica delle folle ha stimolato in tempi recenti l'interesse dei
matematici applicati. Ciò è avvenuto soprattutto in conseguenza di
eventi di grande portata, spesso catastrofici, che hanno fatto capire
l'importanza di affiancare ai tradizionali metodi di indagine
scientifica, quali l'osservazione sperimentale, strumenti di simulazione
e predizione.

londo<br />
millenium bridge
London Millennium
Bridge

Emblematico è il caso del London Millennium Bridge,
un ponte pedonale sul Tamigi chiuso due giorni dopo la sua
inaugurazione nel 2000 a causa di forti oscillazioni latera­li che
compromettevano la sicurezza dei passanti. Appro­fondite (e costose)
analisi supplementari accertarono che le instabilità erano dovute a
fenomeni di risonanza inne­scati dagli stessi pedoni. Il ponte venne
definitivamente riaperto due anni più tardi, ma le sue vicissitudini
resero evidente la necessità di studiare l'accoppiamento folla-struttura
in fase di progettazione di infrastrutture pedona­li.

Ponte<br />
jamarat
Ponte Jamarat
durante un
pellegrinaggio

Un altro esempio significativo è il ponte delle
Jamarat
a Mina, una città dell'Arabia del Sud vicino a La Mecca,
annualmente meta del Hajj, il pellegrinaggio che i musulmani
fisicamente ed economicamente abili devono compiere almeno una volta
nella vita. L'elevato numero di pellegrini che affollano il ponte in
quelle occasioni ha provocato, negli anni Novanta, la morte di numerose
persone schiacciate dalla folla. I mo­delli matematici hanno aiutato a
studiare la dinamica del flusso di pellegrini, suggerendo contromisure
per miglio­rare la sicurezza dell'evento.

Anche senza chiamare in
causa eventi così importanti, si può fare quotidianamente esperienza di
luoghi affollati: centri commerciali, stazioni, aeroporti, stadi, nella
cui progettazione rientra sempre più spesso la simulazione virtuale.
Essa consente infatti di studiare ambienti che ri­spondano ai necessari
criteri di sicurezza per i loro fre­quentatori, senza costruirli
veramente se non in ultima battuta.

Costruire
un modello matematico

La costruzione di un modello per il moto di
folle richiede di descrivere con il linguaggio della matematica un
sistema fortemente non standard. Infatti i pedoni obbediscono solo in
minima parte alle leggi della meccanica newtoniana e per il resto sono
capaci di prendere decisioni e di auto-orga­nizzarsi.

Uno degli
approcci modellistici più in voga nella letteratura scientifica è quello
macroscopico, così chiamato perché adotta un punto di vista a
larga scala e guarda alla densità dei pedoni, assimi­lando
idealmente questi ultimi ad un mezzo continuo (ad esempio un fluido)
distribuito nello spa­zio. Il principio di base dei modelli macroscopici
è la conservazione della massa: durante il moto i pedoni si spostano da
un luogo all'altro, ma la loro quantità totale non varia almeno finché
essi non escono dalla zona di osservazione. Ciò significa che la
variazione nel tempo della quantità di pedoni in una qualsiasi
sotto-zona, diciamo S, è dovuta unicamente al flusso
di pedoni che ne attraversano il contorno:

variazione dei pedoni
contenuti in S = (flusso entrante in S) - (flusso
uscente da S).

I modelli considerano inoltre che la
dinamica del sistema è determinata essenzialmente da due fattori: da un
lato la volontà di ogni pedone di raggiungere una certa destinazione,
dall'altro le inte­razioni che possono far deviare i pedoni dalle loro
traiettorie preferenziali (infatti le persone gene­ralmente non amano
stare troppo a contatto e tendono ad evitare le zone di maggiore
affollamento). Risolvendo le equazioni dei modelli con un computer si
può visualizzare la densità dei pedoni a di­versi istanti di tempo e
trarre indicazioni utili sul raggiun­gimento di particolari livelli
critici di congestione in alcu­ne aree.

Simulazione<br />
di flusso pedonale attra¬verso passaggi stretti.
Simulazione
di flusso pedonale
attraverso passaggi stretti

Un modello
matematico per la simulazione dei flussi pedonali mette a disposizione
un "laboratorio virtuale" per studiare, in tempi relativamente brevi e a
costo quasi nullo, molte situazioni spesso non facilmente
sperimenta­bili in pratica. Infatti, non è semplice realizzare
esperi­menti per raccogliere dati sul comportamento dei pedoni, perché
le persone possono essere  influenzate dal fatto di sapersi osservate o
dal fatto di sapere che stanno effet­tuando una simulazione.  Inoltre, i
dati sperimentali non costituiscono da soli strumenti di indagine
sufficienti, poi­ché sono istantanee statiche di eventi particolari,
mentre i fenomeni più interessanti avvengono a livello dinamico e sono
diversi caso per caso (Si veda qui a fianco una simulazione
bidimensionale di un flusso di pedoni attraverso passaggi
stretti. I colori rappresentano i diversi valori assunti dalla densità
della folla nei punti del piano, in base alla scala mostrata sulla
destra
). I modelli matematici colgono invece il caso medio, ne
descrivono l'evo­luzione dinamica e permettono di riprodurlo un numero
illimitato di volte.

Andrea
Tosin

Matematica, Politecnico di Torino

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Il tuo voto: Nessuno Media: 4.3 (3 voti)

15
ottobre, 2009
da Andrea Tosin

Commenti

#1 Volevo segnalare un caso forse simile di un ponte sul
Volga

ritratto di Michele Ciavarella

25 maggio, 2010 -
10:31
da Michele Ciavarella

Su un blog di Harvard di meccanica, che io frequento molto, e che ha
ben 20
mila iscritti e forse 200 mila lettori, è stato recentemente segnalato
questo caso http://www.imechanica.org/node/8280 "The autovibrations of
the
bridge over Volga river in the city of Volgograd(Russia) were filmed
(not by
me) today on 05/21/2010. The amplitude reachedabout 1 m. The bridge
opening
was just six months ago... Apparently, either the bridge frame drag was
too
high or torsional stiffness / damping too low. " di cui si trova video
su
Youtube
http://www.youtube.com/watch?v=uWP5d2t2JVE&feature=player_embedded
Io
ho fatto dei miei commenti sulle possibili cause, e non si capisce bene
se il
ponte presenta fenomeni di autoeccitazione tipo il flutter aerodinamico
del
famoso ponte Tacoma, studiato da Von Karman e che ha generato decenni di
studi, oppure quello del Ponte del Millenium Bridge. Sarebbe utile un
confronto/dibattito sul tema, e invito i lettori di scienzainrete,
nonchè il
matematico Andrea Tosin, a ragionarci insieme. Si trovano sul sito anche
dei
commenti relativi alla valutazione di ufficiali russi sull'evento, che
hanno
smorzato il panico facendo riferimento a vernici fragili sul ponte, che
dimostrerebbero la non pericolosità delle oscillazioni, ma sollecito
maggiore approfondimento. After visual inspection
(http://top.rbc.ru/incidents/21/05/2010/410041.shtml), the vice minister
of
the Transportation Dept. of Russia, Mr. Belozerov said "The bridge is
absolutely OK". Layer of paint deposited on the metal frame of the
bridge
forms a stiff coating. The official said, that cracking of the paint
layer
could give evidence of the deformations however, nothing of the kind can
be
observed. He also added, that the paving asphalt is also absolutely OK,
all
the structural parts are where they belong to. Preliminary, he said, the
wind
speed of 16 m/s could be responsible for the bridge vibrations.This
happens
periodically in Volgograd. According to his words, the experts are now
trying
to determine the reasons for the onset of the observed vibrations.
"According
to the videorecording, the amplitude of vibrations amounted to about 40
cm.
But this was allowed for in terms of the permissible metal elongation
for
this structure. The official has also reassured, that the joinings
(weld?) of
the bridge will be inspected by ultrasound measurements prior to the
bridge
reopening. Width of the Volga river under the bridge is about 1 km. The
actual span of the bridge over the Volga river is about 2.9 km according
to
my measurements using the Google Earth.
http://en.wikipedia.org/wiki/Volgograd_Bridge Tacome collapsed under
40-mile-per-hour (64 km/h) wind, which is higher but not far from the
16m/s
wind conditions you indicate. Also, the official says the bridge is
covered
with stiff paint to control the cracks. This is quite surprising. If the
excursions are 40 cm, surely there must be some deformation, even by
visual
inspection. It is true, however, to distinguish a self-excited
phenomenon,
that we have to consider the energy input from the wind, but also to
consider
if this is resonating or not. A self-excited phenomenon in theory
doesn't
need any significant amount of initial perturbation. The initial
perturbation
grows unbouded exponentially in time. If you start small, you just take
longer to reach the non-linear part where there is some "damping" and
additional resistance. So I am not sure now if your case is or not
self-excited phenomenon or not. I would need to check the video. Was the
oscillation constant in time, or it appeared to increase? We need more
quantitative descriptions. However, to say something quantitative, we
need a
model of the bridge. It is not a single span bridge, so we need at least
the
model of a single structure between two pillar. If you have these data,
imechanica (or perhaps just my students) can make a FEM model easily! I
miei
contatti sono allegati e sono a disposizione per la discussione.

Prof. Michele Ciavarella
Politecnico di BARI, Delegato del Rettore al CNR
http://rettorevirtuoso.blogspot.com
Associate Editor, Ferrari MilleChili Journal http://imechanica.org/node/7878
Editor, Italian Science Debate, www.sciencedebate.it

 

Michele Ciavarella, Politecnico di BARI - Italy, Rector's delegate.
http://poliba.academia.edu/micheleciavarella
Editor, Italian Science Debate, www.sciencedebate.it
Associate Editor, Ferrari Millechili Journal, http://imechanica.org/node/7878

I attended a seminar talk by Prof. Yozo Fujino - advisor for the vibration control of the Millenium Bridge (http://www-e.civil.t.u-tokyo.ac.jp/lab/project1.html). He mentioned that the lateral sway of Millenium Bridge was due of the periodic motion of the crowd moving in an already swaying bridge (because of its architecture, the lateral stiffness of the Millenium bridge was very low and small lateral vibrations were norm). The periodic motion of the crowd was observed because the people moving on the swaying bridge start balancing them, thereby making their 'balancing' steps synchronous. It was similar to the beating retreat of an army.

He also mentioned one case of a cable stayed bridge in Tokyo (I forget the name of the bridge) which starts vibrating whenever there is rain. Prof. Yozo Fujino found out the reason of the vibration was the droplets of rain moving along the cables! (However, the hitting of the drops of rain on the cables with a certain frequency may be one of the reasons of its vibartion).

The similarity in the above cases is that, apparently, the excitation harmonic frequency (be it due to crowd or due to rain drops hitting the cables) was very less as compared to that of the fundamental frequency of the structures. And, because of nonlinearity, the structure seems to resonate even under such low excitations (I conducted simulations on closed cylindrical shells subjected to point harmonic excitations and found that for some cases, the cylindrical seems to resonate under an excitation that is 1/65th of the fundamental frequency of the cylinder).

 

Tacoma Narrows was another classic case of flutter due to the horizontal wind and it became the milestone for considering the adverse effects of horizontal wind on large slender structures.

 

Ibrahim

Research Scholar, Department of Applied Mechanics, IITDelhi

 

 

 

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