Footbridge 2002 1

"Lively" Footbridges – a Real Challenge

Hugo BACHMANN
Prof. em. Dr. Dr. h.c.
Swiss Federal Institute of Technology ETH
Zurich, Switzerland

Summary
If footbridges are designed for static loads only they may be susceptible to vertical as well
as horizontal vibrations. Hence a dynamic design is often necessary. Based on an
understanding and formulation of the dynamic actions of one person walking, running or
jumping, respectively, special attention has to be given to the actions of several or many
persons including the "lock-in" effect, which can lead to the synchronisation of a
substantial percentage of the persons. Measures have to be taken, such as frequency
tuning of the structure, calculation of a forced vibration and limitation of amplitudes, etc.
In many cases the installation of one or several tuned vibration absorbers is an effective
alternative.

Keywords: footbridge; vibrations; dynamic actions due to pedestrians; walking; running;

jumping; lock-in effect; acceptance criteria; frequency tuning; tuned mass
dampers.

1. Even more vibration problems

With current design practice for footbridges vibrations are a growing problem – it seems
that footbridges are becoming increasingly prone to vibrations. There are several reasons
for this. Construction materials are becoming even more sophisticated, so that – under
static loads – they can be more highly stressed. This leads to more slender structures, i.e.
to smaller cross sectional dimensions or greater spans compared to those in older
structures. As a consequence, stiffness and mass decrease, whereby in nearly all cases
the reduction of the stiffness dominates, leading to smaller natural frequencies with a
greater danger of resonance. But also less mass has to be excited, which means that the
energy introduced by dynamic actions leads to higher vibration amplitudes. In addition, it
seems that people are becoming more sensitive to vibrations and therefore are quicker to
complain; perhaps this is partly a reaction to increasing environmental influences.
The increasing vibration problems encountered today show that footbridges should no
longer be designed for static loads only – even if the provisions in many codes give this
impression. Also, it may not be effective to design bridges initially just for static loads and
then to consider a "dynamic improvement" at a later stage. This can be a time-consuming
business leading to a substantially greater planning expenditure. It would be much better
and less expensive to consider the dynamic actions and the vibration behaviour of the
structure in a very early stage of planning.
Vibrations of footbridges affect above all the main girder carrying the deck. The vibrations
Footbridge 2002 2

may occur both vertically and horizontally, the latter either transversally or longitudinally
with respect to the direction of movement of people along the bridge axis.
Dynamic actions on beam-type footbridges stem mainly from walking and running
persons. Actions of cyclists are not important in comparison to the actions of persons on
foot. However, the so called vandal loading by rhythmical jumping on the spot and
perhaps also by rhythmical horizontal body movements of a single person or a group of
persons may also produce substantial dynamic actions.
In most cases the vibrations of footbridges lead to serviceability problems, i.e. the comfort
of the pedestrians is reduced or in the extreme case a bridge may no longer be used and
has to be closed. In rather seldom cases safety problems due to overstressing and/or
fatigue may also occur.
The following considerations concentrate on vertical and horizontal vibrations of the main
girder of footbridges under the actions of walking or running ("jogging") or jumping
people. The considerations are also valid for similar structures such as high span
stairways, ship gangways, etc. Primarily an understanding of the phenomena occurring
and how to overcome the relevant effects by engineering judgement and appropriate
measures are addressed. Hence, the following considerations have the character of an
overview of a fairly wide and demanding topic.

2. Dynamic actions induced by people

Substantial vertical and horizontal dynamic forces acting on footbridges result from
rhythmical body motions of persons. The following aspects are of importance:
− pacing frequencies of walking and running, or jumping frequency, respectively,
− time function of the vertical or horizontal dynamic action, respectively,
− number of persons involved
− "lock-in" effect

2.1 Actions due to one person

Typical pacing frequencies when a person is walking or running ("jogging") and
frequencies when jumping on the spot are given in Table 1 [1]. Rough mean values are 2
Hz for walking and 2.5 Hz for running and jumping. However, a great spread of possible
values must be considered.

Table 1 Pacing and jumping frequencies in Hz [1]`

total range slow normal fast
walking 1.4 – 2.4 1.4 – 1.7 1.7 – 2.2 2.2 – 2.4
running 1.9 – 3.3 1.9 – 2.2 2.2 – 2.7 2.7 – 3.3
jumping 1.3 – 3.4 1.3 – 1.9 1.9 – 3.0 3.0 – 3.4
Footbridge 2002 3

Of great importance is a proper understanding and the mathematical modelling of the

time function of the action force Fp(t) due to a single person. The vertical force can be
decomposed into a static part corresponding to the weight of the person and a dynamic
part as the sum of harmonic functions ("harmonics") with frequencies an integer multiple
of the pacing or the jumping frequency, respectively, which is the fundamental frequency
of the person’s action (so called Fourier decomposition):
Fp ( t ) = G + ∆G1 ⋅ sin(2 πf p t ) + ∆G 2 sin(4πf p t − φ 2 ) + ∆G 3 sin(6πf p t − φ3 ) (1)

G: Self weight of the person (in general = 800 N)

st nd rd
∆G1,2,3: Amplitude of 1 or 2 or 3 harmonic, respectively
fp: Pacing or jumping frequency, respectively
nd rd
φ2,3: Phase angle (timely phase shift) of the 2 or 3 harmonic, respectively, with
st
respect to the 1 harmonic
In the case of a horizontal action no static part appears.
In the formula for Fp(t) only the first three harmonics are considered. In rare cases,
th th
especially when jumping is considered, the 4 and even the 5 harmonic may also play a
certain role.

2.2 Vertical actions

Figure 1 shows the time function of the vertical force of a pedestrian walking with a
pacing frequency of 2 Hz. The total force produced by both feet and acting on the walking
surface, and the harmonic parts of it as well as the relevant amplitude spectrum (Fourier
spectrum) are shown. For the quantities ∆Gi und φi (i = number of the harmonic) the
values shown in the figure and recommended in [2] are used. However, the quantities
may considerably vary due to influences of the pacing frequency, the kind of the heel
impact and the roll off of the feet, and of the level of vibration of the deck [1].
During running and jumping the ground contact is interrupted. Figure 2 shows the time
function of the vertical force due to jumping on the spot of a person with 2 Hz and the
relevant amplitude spectrum. In this example the maximum force amounts six times that
th
of the self weight (!), and substantial dynamic forces up the 5 harmonic are produced.
nd th
Higher harmonics (2 ….. 5 ) of the force-time function of the action of persons may be
of great importance. Of course the most frequent case is that footbridges with a relatively
st
low fundamental frequency f0 are excited to resonance vibrations by the 1 harmonic of
the dynamic action with a pacing or jumping frequency fp, i.e. fp = f0. However, in some
cases bridges with a higher fundamental frequency f0 are excited by higher harmonics,
nd rd
i.e. 2fp = f0 or 3fp = f0. In such cases the 2 or the 3 harmonic produces so to speak an
nd rd
"impulse into every 2 or 3 wave trough of the structural vibration" (with phase shift).
Footbridge 2002 4

Fp/G

1.4
Fp(t)
1.2

1.0

0.8
Amplitude
∆G/G
spectrum
0.6 0.4

0.4 0.2

0.2 0
Total force 0 fp 2fp 3fp f
0
0 T1 t
∆G/G

0.6
Harmonic parts
1. harmonic (f = fp)
0.4 2. harmonic (f = 2 fp)
3. harmonic (f = 3 fp)

0.2 ∆G1 = 0.4 G ∆G2 = ∆G3 = 0.1 G

0
t
-0.2
ϕ2 = ϕ3 = π/2 Phase angle
-0.4
T3 = 1/3 fp

-0.6 T2 = 1/2 fp
Period T1 = 1/ fp

Fig. 1 Time function and relevant amplitude spectrum of the vertical force from walking of
a person at a pacing frequency of 2 Hz
Fourier force amplitude ∆G [kN]

1.5 1.38 kN

1.0

0.5

0
0 fp 2fp 3fp 4fp 5fp
Frequency [fp]

Fig. 2 Time function and relevant amplitude spectrum of the vertical force from jumping
on the spot of a person (G = 720N) at a frequency of 2 Hz [2]
Footbridge 2002 5

Example: A steel footbridge exhibited considerable vertical vibrations. During normal evening
pedestrian traffic frequencies and amplitudes were measured. The number of people crossing the
bridge varied between 30 and 55 per minute. It was observed that the bridge vibrated
nd
predominantly in its fundamental frequency of about 4 Hz. The bridge was excited by the 2
harmonic of the force-time function of the pedestrians with a pacing frequency of about 2 Hz.

2.3 Horizontal actions

Walking and running people also produce horizontal action forces on the deck. The forces
originate from the oscillation of the centre of mass with transverse displacement
amplitudes, in general, of about 1 to 2 cm, i.e. at right angles to the direction of
movement, or longitudinally relative to the spot in the case of a strictly constant moving
velocity. Note that the oscillation frequency, which is the fundamental frequency of action,
corresponds to half of the pacing frequency, and hence lies in the region of 0.7 to 1.7 Hz
for a pacing frequency of 1.4 to 3.5 Hz (Table 1). Keeping the designation of the
harmonics related to the pacing frequency ("1") the amplitudes of the harmonics are
st
∆G1/2, ∆G1, ∆G3/2, ∆G2 etc. ∆G1/2 in reality is the 1 harmonic of the time function of the
horizontal force; however, it is often called the "half harmonic".
Figure 3 shows amplitude spectra of measurements due to a person with a weight of 584
N walking at a pacing frequency of 2 Hz [3]. It can be seen that higher harmonics may be
of importance. In the direction "horizontal transverse" forces appear above all in the half
and the one and a half pacing frequency, whereas in the direction "horizontal longitudinal"
forces mainly appear in the one and the two fold but also in the half and the one and a
half fold pacing frequency. Depending on the person and the situation regarding vibration
of the deck, values of the deck amplitudes show a considerable scatter. In general, the
amplitudes increase with increasing vibrations of the deck. For example, for the direction
"horizontal transverse" maximum values ∆G1/2/G of up to 0.07 in the case of a stationary
deck and up to 0.14 in the case of a vibrating deck were measured [4].

horizontal transverse horizontal longitudinal

120 N

49 N

23 N 25 N 22 N
15 N 14 N
6N 7N 9N

∆G1/2 ∆G1 ∆G3/2 ∆G2 ∆G5/2 ∆G1/2 ∆G1 ∆G3/2 ∆G2 ∆G5/2
1 Hz 2 Hz 3 Hz 4 Hz 5 Hz 1 Hz 2 Hz 3 Hz 4 Hz 5 Hz

Fig. 3 Amplitude spectra of the horizontal forces of a person (G = 584 N) walking at a

pacing frequency of 2 Hz [3]
Footbridge 2002 6

Although the horizontal forces from walking and running are relatively small compared to
the vertical forces, they are sufficient to produce strong vibrations in the case of
horizontally soft and hence low frequency structures.
Example: A reinforced concrete footbridge at the Geneva airport has regular spans of about 15 m
(Figure 4). The columns of roughly 7 m height had a relatively small cross section of 65 cm x 30 cm
because they were only designed for static wind loads. At the end of a big public meeting, many
pedestrians streamed in one direction over the bridge. Strong horizontal vibrations occurred, mainly
in the transverse direction, but also in the longitudinal direction. The reaction of the persons
bordered on panic: some ran forwards to reach the end of the bridge, others turned around and
tried to get back. The motions of the bridge were compared to those of the ground in a strong
earthquake or of a large ship in high waves. Dynamic investigations indicated the fundamental
frequencies of the bridge shown in Figure 4. A look at the amplitude spectra of Figure 3 explains
the vibrations: The bridge was excited by the pedestrians in the horizontal transverse direction by
the half harmonic with the amplitude ∆G1/2 and in the longitudinal direction by the harmonic with
∆G1. The bridge had to be upgraded by stiffening the existing RC columns being carefully
roughened and enclosed by a new reinforcement cage and concrete cast in place. In addition, to
guarantee the fixing of the stiffened columns, the pile caps of the pile foundations hat to be
strengthened. By these measures the horizontal transverse and longitudinal natural frequencies
could be sufficiently increased, which corresponds to subsequent frequency tuning of the structure
(see section 5).

Fig. 4 RC footbridge vibrating horizontally both transversally and longitudinally

2.4 Actions due to more than one person

The foregoing sections deal with actions due to a single person. However, footbridges are
mostly excited simultaneously by several persons. The following kinds of action have to
be differentiated:
− random action
− synchronous action
In the case of a random action from walking or running, the pacing frequency and the self
weight of the relevant persons are distributed within a certain range according to a
Footbridge 2002 7

st
probability curve, whereas the phase angle ϕ1 of the 1 harmonic (related to an arbitrary
point in time) exhibits a completely random distribution. Therefore the actions of many
pedestrians will both enhance as well as partly compensate each other. Exact predictions
are difficult, because many different parameters and random effects play a role. However,
for practical purposes the approach given in [5] seems to be appropriate. A factor m is
defined for multiplying the vibration amplitude calculated for one person at the midspan of
a bridge:

m = λ ⋅ T0 (2)

λ: mean flow rate (person/s over the width of the deck) for a certain period of time
(maximum flow rate λmax ≅ 1.5 persons per s and m width)
T0: Time needed to cross a bridge of length L at speed vs, i.e. T0 = L/vs.
λ ⋅ T0 : Number of persons on the bridge simultaneously at the given mean flow rate
(corresponds to n introduced below).
Equation (2) already takes into account that an action uniformly distributed over a span
leads to only about 60 % of the displacement (deflection) caused by the same action
being concentrated in the midspan.
For example, for a 2 m wide and 26 m long bridge with a flow of 100 persons/minute (λ =
1.66 persons/s) and vs ≅ 1.5 m/s results a time T0 = 26/1.5 =17.3 s and a factor
m = 1.66 ⋅ 17.3 = 28.7 = 5.4 .
Equation (2) can be directly used in the case of footbridges with a fundamental frequency
f0 between 1.8 and 2.2 Hz. Lower and higher pacing frequencies are rather less frequent.
Therefore, in cases of fundamental frequencies f0 between 2.2 and 2.4 Hz and between
1.8 and 1.6 Hz the factor m may be reduced.

perfect synchronization
20 (m=n)
imperfect synchronization
Multiplication factor m

(dep. on harmonics, frequency

range, type of activity etc.)

10

no synchronization / random
(m = λTo = n)

1
1 5 10 20 30
Number of persons n

Fig. 5 Multiplication factor for the effects of more than one person
Footbridge 2002 8

A synchronous action by more than one person – for instance a group of pedestrians
walking in uniform step or synchronous jumping on the spot – in comparison to the action
st
of one person results in an increase which in the case of the 1 harmonic is nearly
proportional to the number n of the persons involved. In the case of higher harmonics,
due to the higher frequencies and shorter periods, respectively, relatively small
differences of time and other features (∆Gi/G, ϕi etc.) are more important and may lead to
a more or less imperfect synchronisation and subsequently to a reduction of the
multiplication factor (Figure 5). The amount of the reduction compared to a linear increase
with n being valid for a perfect synchronisation depends on the relevant circumstances
(frequency range, type of activity, etc.). However, at least for a small number of persons a
linear increase (as an upper bound) seems to be adequate. For example, for "vandal
loading" of footbridges 3 persons jumping on the spot with an ideal synchronisation may
be assumed.

2.5 Lock-in effect

Of substantial importance may be the so called lock-in effect. A walking or running person
adapts to and synchronizes his/her motions in frequency and phase (φ1) with a vibrating
deck if the displacement amplitude exceeds a certain threshold value. The threshold
value depends on the direction of vibration, the person's age, condition, etc. and for
vertical vibrations of ~ 2 Hz it lies in the range 10 to 20 mm [1]. For horizontal vibrations
with a frequency of ~ 1 Hz some persons begin to adapt their motions already when the
amplitude exceeds 2 to 3 mm.
If the individual threshold value of a person is exceeded, due to the synchronisation an
"impulse into every wave trough of the bridge vibration" occurs, which is a much more
adverse dynamic action than before. As a consequence, the vibration amplitude
increases, more persons are locked into synchronisation, etc. In certain cases a
synchronisation of more than 80% of the persons involved was observed.
Example: When the new Millennium Bridge over the Thames in the City of London with spans of 80
m – 140 m – 100 m was opened in June 2000, many pedestrians streamed over the bridge. Strong
horizontal transverse vibrations with frequencies of around 1 Hz occurred, reaching amplitudes in
the order of 70 mm (!). Videos have shown that an increasing percentage of the pedestrians
adapted their motions to and therefore synchronised with the bridge vibrations. However, many
pedestrians could no longer step forward and had to hold on to the railing. The bridge had to be
closed again. Within the framework of the studies for the dynamic upgrading of the bridge,
laboratory tests with a transverse vibrating deck were carried out [4]. The tests showed that for an
amplitude of 5 mm already 30 to 50 % and for 30 mm about 80 % of the pedestrians are locked into
synchronisation. For a transverse acceleration of 2 % g (see section 4) a synchronisation of about
30 % of the pedestrians may be assumed.

3. Dynamic properties and vibration behaviour of footbridges

Figure 6 shows the fundamental frequency of 67 beam-type footbridges as a function of
the (main) span [2]. Note that there is a large scatter and the given formula can hardly be
used for prediction purposes in a specific case.
Footbridge 2002 9

10
-0.73
f = 33.6· L

Fundamental frequency [Hz]

Steel
8 Concrete
Composite

0
0 10 20 30 40 50 60
Span [m]

Fig. 6 Fundamental frequency of 67 footbridges as a function of the span [2]

The damping of footbridges may be fairly low, especially in the case of steel or steel-
concrete composite bridges. Table 2 shows common values of the equivalent viscous
damping ratio ζ of beam-type footbridges measured when one pedestrian was walking at
the bridge’s fundamental frequency [2]. In the case of a higher vibration level and a
higher number of people dissipating energy by their bodies the damping ratio may
increase slightly. For cable-stayed bridges, suspension bridges, arch bridges, etc., the
damping ratio may be very different from that of beam-type bridges.

Table 2 Common values of damping ratio ζ for footbridges [2]

Construction type min. mean max.
Reinforced concrete 0.008 0.013 0.020
Prestressed concrete 0.005 0.010 0.017
Composite steel-concrete 0.003 0.006 --
Steel 0.002 0.004 --

Footbridges may show excessive vibrations due to rhythmical human body motions. In
general the stationary state achieved after the initial transient vibrations have died away,
which exhibits a longer duration in the case of a smaller damping ratio, has to be
considered. The displacement amplitudes may amount to a high multiple of that for the
static action of ∆G, for example 20 or even 100 times more. Hence, it is not feasible to
consider dynamic actions by increasing the static actions of persons. The dynamic effects
have to be treated independently.
Footbridge 2002 10

4. Acceptance criteria
Measured or calculated vibration amplitudes of footbridges may be compared with
acceptance criteria. Often acceptance criteria represent a practicable order of magnitude
rather than an exact "admissible" value. And in most cases acceptance criteria of
footbridges are related to physiological effects on people (mechanical, optical, acoustic
effects) representing serviceability problems rather than an overstressing of the structure
(deformation, strength, fatigue) representing safety problems.
The assessment of physiological acceptance criteria for vertical and horizontal footbridge
vibrations reveals difficult matters of judgement and discretion. Therefore, different codes
give different values. Relevant aspects may be:
− frequent or exceptional or very seldom occurrence
− experimentally approved or "only" calculated (uncertain) values for comparison
purposes
− desired level of comfort
− expected degree of acceptance by people (for example, there is a great difference
between a suspension bridge in a mountain valley and a stairway in a supermarket)
Basically, acceptance criteria are frequency-dependent. In general they are given in units
of acceleration rather than velocity. In the case of vertical vibrations, depending on the
2
circumstances, an acceleration of 0.5 to 1m/s , i.e. 5 to 10 % of gravity g, may be
accepted. People are much more sensitive to horizontal vibrations when walking or
running than to vertical vibrations. Therefore, an acceptance value of 1 to 2 % g is
recommended. In addition, displacement amplitudes of more than ~ 10 mm in the vertical
and ~ 2 mm in the horizontal direction cannot be tolerated if the lock-in effect and hence a
synchronisation of a significant percentage of the persons is to be avoided.

5. Frequency tuning
Measures against excessive vibrations of footbridges include:
− Frequency tuning
− Calculation of a forced vibration and limitation of amplitudes
− Introducing special measures
− Damping by vibration absorbers
With respect to the natural frequencies – especially the fundamental frequency – a
structure can be "tuned" in a similar way to a musical instrument. For a calculation of the
natural frequencies only stiffness and mass values are needed, and not the damping ratio
which is often difficult to predict.
Footbridge 2002 11

In the vertical direction, considering the most frequent pacing frequencies of pedestrians
(Table 1) and the relevant amplitude spectrum shown in Figure 1, natural frequencies f of
footbridges in the range
1.6 Hz < f < 2.4 Hz
nd
should be avoided. Moreover, due to the phenomenon of the 2 harmonic (see section
2.2) in the case of structures with a small damping ratio (mainly steel and composite
bridges) also natural frequencies in the range of
3.5 Hz < f < 4.5 Hz
should be avoided. Further, if footbridges are often crossed by running persons
("joggers"), natural frequencies in the range of 2.1 Hz to 2.9 Hz should be avoided too.
In the horizontal transverse direction, based on the amplitude spectrum shown in Figure 3
left, natural frequencies of around 1 Hz (i.e. 0.7 to 1.3 Hz) and in the case of very light
and lowly damped bridges perhaps also around 2 Hz or even 3 Hz should be avoided. In
any case a "high tuning" to f > 3.4 Hz is a safe solution.
In the horizontal longitudinal direction, based on the amplitude spectrum shown in Figure
3 right, similar considerations can be made and critical frequency ranges identified.
Although, in many cases, the friction of movable bearings is not overcome, attention must
be given to it, especially in the case of longitudinally soft frame bridges.
Figure 7 shows the critical frequency ranges of footbridges.
The frequency tuning of a footbridge is a relatively rough and "all-inclusive" measure
which, nevertheless, is successful in most cases.

always to avoid to avoid in relevant cases

Vertical Pedestrians
Runners Steel and
composite bridges

Horizontal transverse Ped.

Runners

Horizontal longitudinal Ped.

Runners

0 1 2 3 4 5 [Hz]

Fig. 7 Critical frequency ranges of footbridges being avoided by frequency tuning

Footbridge 2002 12

6. Calculation of a forced vibration and limitation of amplitudes

If initially a footbridge has been designed for static loads only, depending on the
circumstances frequency tuning may lead to substantial additional expenditure and costs.
In such cases more specific considerations may be required. Usually, the calculation of a
forced vibration is appropriate.
For a stationary resonant vibration (state after the initial transient vibrations have died
away) under the action of one person at midspan is:

d = d s / 2ζ ; v = 2πfd s / 2ζ ; a = 4 π 2 f 2 d s / 2ζ (3)

d, v, a: Amplitudes of displacement, velocity and acceleration, respectively, at midspan

ds: Static displacement due to ∆G at midspan
f: Natural frequency of the bridge and frequency of the relevant harmonic of
amplitude ∆G
ζ: Damping ratio of the bridge
Dynamic actions due to several people may be considered by multiplication by m or n
respectively (see section 2.4). Of course, more refined methods (e.g. [7]) and computer
programs may also be used; however, a check on the results using simple hand
calculations is always mandatory. In any case, a direct dependency on the damping ratio
ζ is given, which often makes the results questionable.
The resulting amplitudes can be compared with acceptance criteria (see section 4).
Consequently, measures for a limitation of the amplitudes by increasing the stiffness, etc.,
must be taken if necessary.
Experience has shown that in cases of footbridges with a vertical fundamental frequency
of around 2 Hz the calculation of a forced vibration with amplitude limitation leads more or
less to the same result as frequency tuning. More differentiated insights can be expected
in the case of footbridges with higher fundamental frequencies.

7. Special measures
As special measures, which usually increase the natural frequencies to avoid resonance
phenomena, the following may be considered:
− fixing the main bridge beam at one or at both abutments (only has to be effective for
dynamic actions)
− mounting of a stiffer railing ("serviceability structure")
− spanning of rods or cables (vertical, horizontal or inclined)
Increasing the damping by modifying the structure, connections, bearings, etc., may also
be considered, but often considerable practical problems arise. However, the installation
of one or more vibration absorbers may be an effective and relatively cheap alternative.
Footbridge 2002 13

8. Damping by vibration absorbers

A vibration absorber consists of a mass, a spring and a damper (or parallel springs and
dampers) attached to the primary system (bridge). Vibration absorbers are also called
"tuned mass spring dampers" because their natural frequency and damping ratio are
tuned to the relevant properties of the primary system resulting in an optimum frequency
and an optimum damping ratio of the absorber.
Vibration absorbers can be used within the framework of
− vibration upgrading of existing footbridges
− planning and dynamic design of new footbridges
A vibration absorber can reduce the vibrations of a certain natural frequency (often it is
the fundamental frequency) of the primary system. Hence the primary system can be
modelled as a single degree of freedom system with the relevant modal mass ms, spring
constant ks and damping ratio ζs. Attaching an absorber with mass mt, spring constant kt
and damping ratio ζt to the primary system leads to a two degree of freedom system
(Figure 8). The optimum frequency fopt and optimum damping ratio ζopt of the absorber
are:

fs 3( m t / ms )
f opt = ; ζ opt = (4)
1 + m t / ms 8(1 + m t / ms )3

cs ks
Primary
system ms

Fp (t)

ct kt xs
Vibration
absorber
mt

xt

Fig. 8 Dynamic model of the primary system with a vibration absorber [2]

Thereby the frequency of the primary system becomes fs = ks / ms /( 2π ) and of the

absorber f t = k t / m t /(2 π ) . mt/ms is the mass ratio which can be chosen in the range
0.01 to 0.05, depending on the damping ratio of the primary system and other
parameters. Design charts are given in [7] and further important hints can be found in [8].
Footbridge 2002 14

For the application of absorbers for the damping of vibrating footbridges the following
practical hints may be important:
− Vibration absorbers may be very efficient. However, the reduction of the vibration
amplitudes of the primary system is very susceptible to even small changes of the real
absorber frequency ft. Hence ft must correspond to the optimum frequency fopt as
closely as possible.
− On the other hand, differences between the real absorber damping ratio ζt and the
optimum damping ratio ζopt are less important.
− A vibration absorber is only efficient in a narrow frequency range and when exactly
tuned to a certain natural frequency of the primary system. An absorber does not work
efficiently if the primary system exhibits several narrowly spaced natural frequencies,
such as a bending and a torsional fundamental frequency.
− A vibration absorber is more efficient the greater the mass ratio mt/ms and the smaller
the damping ratio of the primary system ζs. This is mainly important in the case of steel
and composite footbridges which exhibit a smaller mass and a smaller damping ratio
than concrete footbridges and therefore are more prone to vibrations.
− The exact tuning of an absorber is best done on site by adding or subtracting small
parts of the absorber mass and less by varying spring properties.
− "De-tuning" of the two degrees of freedom system may occur above all by changes of
stiffness and/or mass of the primary system due for example replacement to paving or
providing the bridge with new railings, crack propagation or new cracks in RC beams,
etc. Therefore, vibration absorbers should always be placed and mounted in such a
way that they are well accessible for checking and re-tuning.
Example: A steel foot and cyclist bridge at Dietikon near Zürich with 4 spans including a main span
of 25 m exhibited excessive vertical vibrations even due to the crossing of just one pedestrian. The
fundamental frequency of 2.48 Hz lies outside the relevant critical frequency range (see section 5).
2
However, the damping ratio amounted to only 0.23 % for an acceleration of 1 m/s and 0.40 for 4
2 2
m/s , respectively. By one person jumping on the spot an acceleration of 9 m/s (!) was reached.
The dynamic upgrading was performed by the installation of two vibration absorbers in the middle
of the main span inside of the cross section of the steel girders. In this case the vibration absorbers
2
are very efficient; they reduce the acceleration to 0.5 m/s , i.e. by a factor of nearly 20.

In the future the vibration absorber technique will gain an even greater importance than
today. Among others, absorbers whose spring and damping properties can be readily
changed (e.g. electromagnetically) will become important; this offers the possibility of
damping several modes of vibration with different natural frequencies of the primary
system using only one absorber.
Footbridge 2002 15

9. Conclusions
Modern footbridges are often much more "lively" than older ones. Therefore, footbridges
can no longer be designed for static loads only. A dynamic design or at least a check on
the dynamic behaviour is essential. In many cases the dynamic design can be based on
simple rules, such as frequency tuning of the structure, calculation of a forced vibration
and limitation of amplitudes. In other cases, the introduction of special measures or the
installation of tuned vibration absorbers may be effective alternatives.

10. References
[1] BAUMANN K., BACHMANN H., "Durch Menschen verursachte dynamische Lasten
und deren Auswirkungen auf Balkentragwerke", Bericht Nr. 7501-3, Institut für
Baustatik und Konstruktion (IBK), ETH Zürich. Birkhäuser Verlag, Basel Boston
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