Tuned Mass Damper optimization for human-induced footbridge vibrations Federica Tubino1, Giuseppe Piccardo1 1Department of Civil, Chemical and Environmental Engineering, University of Genoa, Italy E-mail: federica.tubino@unige.it, giuseppe.piccardo@unige.it

Keywords: Dynamic response, Human-induced vibration, Footbridges, Tuned Mass Damper. SUMMARY. The design of Tuned Mass Dampers (TMD) is becoming more and more usual in the current design of footbridges. This paper analyzes the optimal TMD design for mitigating pedestrian-induced vibrations in the most frequent loading scenarios, i.e. the single pedestrian modeled as a resonant moving harmonic load and the normal unrestricted pedestrian traffic, modeled probabilistically through an equivalent spectral model. Two optimization criteria are set, concerning the minimization of the structural response and of the TMD displacement. The obtained results are compared with classic criteria given by the literature.

1 INTRODUCTION

Modern footbridges can be very sensitive to walking-induced vibrations because of their increasing slenderness and flexibility. Pedestrian-induced vibrations can occur both in the lateral and in the vertical directions, and can cause discomfort to footbridges’ users. Significant vibrations in the lateral direction can take place in very crowded conditions, while perceptible vibrations in the vertical direction can be induced even by a single walking pedestrian or by a small number of pedestrians. Footbridges’ serviceability assessment is currently a subject of great interest both from the scientific and from the technical point of view, due to the need for guaranteeing suitable comfort to people crossing the structure [1].

While lateral synchronization problems seem to be possible for very flexible footbridges only, moreover placed in strategic areas, troubles in serviceability for vibrations in the vertical direction are common for usual footbridges, even not extremely flexible and not crossed by significant pedestrian flows. For all the previous reasons, the design of Tuned Mass Dampers (TMDs) in order to get the required pedestrian comfort is becoming more and more usual in the current design of footbridges.

Different criteria are available in the literature for the optimal choice of the TMD characteristics (e.g., [2]), mainly deriving from the assumption of a harmonic loading condition [3] or of a random loading characterized by a uniform spectrum (white noise, e.g.,[4]). More recently, Krenk [5] identified optimal damping by a combined analysis of the dynamic amplification of the motion of the structural mass as well as the relative motion of the damper mass. All these criteria may not be optimal in the case of pedestrian-induced loading.

Two most common loading scenarios are of interest for footbridges’ serviceability assessment: the standard single pedestrian, modeled as a resonant moving harmonic load, and the normal unrestricted pedestrian traffic, where each pedestrian in free to cross the footbridge with his own walking characteristics (weight, walking velocity, step frequency).

This paper intends to analyze the optimal TMD design for mitigating pedestrian-induced vibrations in the vertical direction caused by a single pedestrian and by normal unrestricted pedestrian traffic. Due to the particular characteristics of footbridges, where TMDs are generally installed within the deck, an important aspect is also the limitation of relative structural-TMD displacements; thus, optimization is analyzed both for the structural absolute acceleration and for the TMD relative displacement. In order to obtain general results without the need to analyze a specific structure, equations of motion are written in non-dimensional form, both in the time and in the frequency domain.

Concerning unrestricted pedestrian traffic, particular attention is devoted to the simulation of the pedestrian load in a realistic way [1]. In principle, analyses could be carried out through time-domain probabilistic simulations of pedestrian-induced forces, modeling the TMD-structure system as an equivalent two degree-of-freedom oscillator; however, frequency domain analyses may in general be more straightforward and less time consuming. This aspect can be tackled starting from the Equivalent Spectral Model recently proposed by the authors [6]. Such a loading model allows to deal with the problem in the frequency domain without the need to perform burdensome Monte Carlo simulations of pedestrian-induced forces.

The results of TMD optimization for the single pedestrian and unrestricted pedestrian traffic scenarios are discussed and compared with classic literature proposals.

2 PROBLEM FORMULATION

In this Section, a footbridge equipped with a TMD is considered. Equations of motion are written in the time and frequency domain, as functions of suitably-defined non-dimensional parameters. Then, the equations of motion are solved in the two common pedestrian-loading scenarios: single pedestrian and unrestricted pedestrian traffic.

2.1 Equation of motion

Let us consider a footbridge, modeled as a linear structural system with classic viscous damping, equipped with a TMD. Focusing attention only on one structural mode, and assuming that the TMD is connected to the structure in the location where the mode shape is maximum (and equal to one), the equation of motion of the system can be written as follows (Figure 1):

s s s d s d d s d s d d

d d d d d d d s

m q t c c q t c q t k k q t k q t F t

m q t c q t k q t m q t

(1)

where t is the time, qs and qd are, respectively, the structural principal coordinate and the TMD relative displacement with respect to the structure. Furthermore, ms, cs, ks represent, respectively, the structural modal mass, viscous damping and stiffness, md, cd, kd represent, respectively, the mass, viscous damping and stiffness of the TMD, F is the modal external force applied to the structure.

Through simple analytical manipulations, Eq. (1) can be re-written in the following non-dimensional form:

2

1 2

2

s d s s s

d d d d s

q t q t q t q t F t

q t q t q t q t

(2)

where the following non-dimensional quantities appear:

02

02 2s d d d s d

s d s st s ds s d d s s st sts s

c c m F q qFt t F q q q

m m m F q qm

(3)

being F0 a representative value of the force (e.g. its maximum value). In Eq. (3), s and d represent, respectively, the natural circular frequencies of the structure

and of the TMD, and are given by:

s ds d

s d

k k

m m (4)

Eq. (2) can be re-written in the frequency domain as follows:

s s d dQ H F Q H F (5)

where / s is the non-dimensional circular frequency, sH and dH are the non-dimensional complex frequency response functions of the damped structure (equipped with TMD) and of the damper, and are given by:

2 2

4 3 2 2 2 2

2

4 3 2 2 2 2

2i

2i 1 1 4 1 2i

2i 1 1 4 1 2i

ds

d s s d s d

d

d s s d s d

H

H

(6)

The non-dimensional complex frequency response function of the structure in absence of the TMD suH can be obtained from sH in Eq. (6), setting =0, =0, d=0.

2.2 Single pedestrian

The conventional loading condition that has been traditionally considered for the analysis of pedestrian-induced vibrations of footbridges is the single pedestrian, modeled as a resonant harmonic moving load ([7], [8], [9]). In such a case, the modal force is expressed as:

sinp p s j pF t G t c t (7)

where p, Gp, cp represent, respectively, the dynamic load factor (DLF), the weight and the step velocity of the pedestrian, j is the structural mode shape.

Assuming F0 = pGp, the non-dimensional counterpart of Eq. (7) is given by:

sin j cF t t t (8)

being /c p sc L the non-dimensional step velocity (L is the bridge span length). If the structure is not equipped with a TMD, the dynamic response to a single pedestrian is

surely maximum when the pedestrian crosses the footbridge in resonance. In case of a structure equipped with a TMD, the maximum dynamic response can be achieved by a quasi-resonant condition. For this reason, the single pedestrian loading condition should be analyzed considering a pedestrian step frequency that does not coincide perfectly with the footbridge natural frequency. Assuming as a particular case that the structural mode shape is sinusoidal (the first mode shape of a simply-supported beam), and a quasi-resonant condition, Eq. (8) becomes:

sin sin cF t t t (9)

being the non-dimensional pedestrian step circular frequency.

2.3 Unrestricted pedestrian traffic

Dealing with pedestrian traffic on footbridges, a realistic loading scenario is characterized by pedestrians arriving in a random way and able to move undisturbed, each of them with their own characteristics in terms of loading amplitude, frequency, velocity and phase. In such a case, focusing attention only on the first walking harmonic for each pedestrian, the modal force induced by Np pedestrians can be expressed as [6]:

1

G sinpN

i i i i i j i i i ii i

LF t t c t H t- -H t- -

c

(10)

where (•) and H(•) are the Dirac function and the Heaviside function, respectively; furthermore, Fi (=iGi), i, i, ci and i are, respectively, the force amplitude, the step circular frequency, the phase-angle, the walking speed and the arrival time of the i-th pedestrian, while i and Gi are the DLF and the weight of the i-th pedestrian. All these quantities have to be probabilistically modeled in order to consider the inter-subject variability in walking forces induced by different pedestrians. In particular, the step circular frequency , the pedestrian weight G, the DLF and the walking velocity c can be considered as random Gaussian variables [10].

Eq. (10) can be re-written in non-dimensional form as follows:

1

1sin

pN

i i i i i j ci i i ii ci

F t G t t H t H t

(11)

where the following non-dimensional parameters are introduced:

i i i ii i i ci i s i

m m s s

G c, G , , ,

G L

(12)

being m and Gm the mean value of the DLF and of the pedestrian weight, respectively. Considering the particular case in which the structural mode shape is sinusoidal, sinj x x , focusing attention only on the randomness of the step frequency and under the

assumption that cm is very small compared with i , the following expression for the power spectral density function (psdf) of the non-dimensional modal force is obtained [6]:

4

pF

NS p (13)

where p is the probability density function of the non-dimensional circular step frequency

, which is a Gaussian random variable with mean value m m s/ and coefficient of

variation V. The dynamic response to unrestricted pedestrian traffic may be in general evaluated through a

Monte Carlo simulation approach. In particular, Monte Carlo simulations of the forcing function, Eq. (11), may be performed and the equation of motion (2) of the structure-TMD system may be solved step-by-step. The equivalent spectral model proposed by the authors [6] allows to analyze the problem through a spectral approach. Based on the classic methods of linear random dynamics, the psdf of the structural response and of the TMD relative displacement may be evaluated through the following expressions:

2 2 24

s dss s dF F FQ QQ

S H S S H S S H S (14)

The psdf of the structural response in absence of the TMD can be obtained from the first two equations in Eq. (16), replacing sH with suH .

3 OPTIMIZATION CRITERIA

Traditionally, the tuning of frequency and damping has been based on the dynamic amplification of the structural mass. The minimum structural response is achieved, in case of harmonic load, by minimizing the dynamic amplification of the structural mass. The classic procedure is based on the analysis of an undamped structure (s=0) and it consists of two separate steps: tuning of the frequency of the damper and selection of the optimal level of the TMD damping ratio. The frequency tuning is based on the observation that there are two frequencies where the dynamic amplification is independent of the applied damping, and optimal tuning is determined by setting the dynamic amplification at these two frequencies equal [2]. The optimal damping is then determined as the arithmetic mean of the values that give maximum dynamic amplification at the two frequencies. This procedure gives rise to the following optimal values of the frequency tuning and of the TMD damping ratio (so-called classic Den Hartog criterion):

opt opt

1 3

1 8 1d

(15)

A different optimization criterion for the damping has been proposed by Krenk [5], selecting a frequency located centrally in the interval between the two frequencies and selecting the damping of the TMD such that the dynamic amplification is equal at these three frequencies. Following this procedure, the optimal damping is given by:

opt,K

1

2 1d

(16)

4 TMD OPTIMIZATION FOR PEDESTRIAN-INDUCED VIBRATION

Classic optimization criteria are based on the assumption of harmonic loading or white noise input. Pedestrian loading on footbridges does not correspond exactly to any of these two conditions. In particular, as outlined in Section 2, two standard loading conditions may be of interest during a footbridge lifetime: the standard single pedestrian loading, modeled as a moving harmonic load, and unrestricted pedestrian traffic. In this Section, these two loading conditions are considered.

4.1 Single quasi-resonant pedestrian

Eq. (9) shows that the modal force exerted by a moving pedestrian can be expressed as the product of two sine functions; using well-known trigonometric identities, it can be seen as the sum of two harmonics with circular non-dimensional frequency c . Since c has usually a small value (generally lower than 0.01), the modal force may be decomposed as the sum of two harmonics with a circular non-dimensional frequency very close to . For this reason, it is expected that traditional optimization criteria for harmonic load may be satisfactory also as regards single pedestrian loading.

Numerical analyses have been performed assuming s = 0.005, = 0.01, =0.99 (the optimum value given by Eq. (15)), c =0.001. Figure 1 plots the maximum structural acceleration (a) and the maximum TMD relative displacement (b) as functions of d and of .

Fig. 1(a) confirms the conclusions that can be obtained considering a fixed harmonic load: if the load is perfectly resonant ( =1), the optimum TMD damping d is the smallest; however, adopting a small value of d causes a high structural dynamic response if the loading is not perfectly resonant (the diagram presents two peaks for around 0.95 and 1.05). A higher TMD damping d may represent a compromise value that allows to minimize the dynamic response for a quasi-resonant loading condition ( in the considered range). Fig. 1(b) shows that the dynamic response of the TMD always tends to decrease on increasing its damping ratio; furthermore, for a small value of the TMD damping ratio also the dynamic response of the TMD tends to be very high if the action is not perfectly resonant.

The classic optimization criteria furnishes opt = 0.06, while the Krenk criterion furnishes opt = 0.07: it can be seen from the three diagrams in Fig. 1 that, for such values of the TMD damping ratio, the surfaces are quite flat: thus, both classic criteria are quite satisfactory in protecting the structure from excessive vibrations when it is crossed by quasi-resonant harmonic loads.

0.90.95

11.05

1.11.15

0

0.05

0.10

5

10

15

20

25

30

35

c = 0.001

d

ddq s

(a)0.9

0.951

1.051.1

1.15

0

0.05

0.10

50

100

150

200

250

300

350

c = 0.001

d

q d

(b) Figure 1: Maximum dynamic response to a quasi-resonant moving pedestrian ( c =0.001) ((a)

structural acceleration, (b) TMD relative displacement).

4.2 Unrestricted pedestrian traffic Starting from the psdf of the quantity of interest, Eq. (14), the variance can be estimated

through an integration along the frequency domain. The TMD efficiency can be evaluated as a function of the reduction of the structural response.

Let us introduce two efficiency factors for the structural response sq and the structural acceleration sq ( sq and sq ), as the ratio between the variance of the response quantity in absence of TMD ( 2

suq , 2suq ) and its value for the structure equipped with TMD ( 2

sq , 2sq ):

22

2 2su su

s s

s s

q qq q

q q

(17)

The TMD efficiency in the reduction of the structural displacement and acceleration is thus

proportional to the efficiency factors sq and sq . The variance of the response quantities depends on the non-dimensional parameters

characterizing the frequency response function of the structure and of the TMD (, , s, d) and on the non-dimensional parameters characterizing the force spectrum (Np, m and V). Thus, the optimal properties of the TMD are functions of the response quantity of interest and on the force characteristics. Numerical analyses have been performed assuming the structural damping ratio s = 0.005 and the mass ratio = 0.01, fixing the coefficient of variation of the step frequency V=0.09, varying the frequency ratio , the damping ratio of the TMD d and the non-dimensional mean step frequency m . The number of pedestrians is set Np=10; however, the TMD optimization is not influenced by such a choice: being Np simply a factor in the force psdf (Eq. (15)), it only provides an amplification of the structural response.

Figure 2 plots the variance of the structural acceleration (a) and of the TMD relative displacement (b) as functions of the frequency ratio and of the TMD damping ratio d, in case of perfect resonance between the mean step frequency and the structural natural frequency ( m =1).

From Figure 2(a) it can be deduced that the structural acceleration is minimized adopting inter-mediate values of the frequency ratio and of the TMD damping d. Figure 2(b) shows that the TMD relative displacement is minimized by high values of the TMD damping d. The effect of the frequency ratio is opposite concerning the structural response and the TMD relative displacement: the structural response is minimized if the frequency ratio is around 1: in such a case, the response of the TMD is maximized (it absorbs more energy). If the location of the TMD can be designed in such a way that its displacement does not need to be contained within certain limits, then the optimal condition is given by values of and d that minimize the structural response without checking the level of TMD displacement; if the TMD has to be located within the bridge deck and its relative motion has to be limited, than the optimum frequency tuning can be set by a joint consideration of the structural and TMD responses.

Figure 3 plots the efficiency factors for the structural displacement (a) and for the structural acceleration (b) as functions of the frequency ratio and of the TMD damping ratio d. Figure 3 obviously confirms the observations from Figure 2 concerning the optimum TMD parameters (the maximum efficiency is obtained for the TMD parameters giving rise to minimum structural response), but it allows us to quantify the TMD efficiency. In particular, the variance of the response is reduced to approximately 1/6 when the TMD parameters are optimized.

0.90.95

11.05

1.11.15

0

0.05

0.10

100

200

300

400

500

m = 1

d

dd

q s

2

(a)0.9

0.951

1.051.1

1.15

0

0.05

0.10

0.5

1

1.5

2

x 104

m = 1

d

qd2

(b) Figure 2: Variance of the response, m =1, V=0.09: (a) structural acceleration, (b) TMD relative

displacement.

0.90.95

11.05

1.11.15

0

0.05

0.11

2

3

4

5

6

7

8

m = 1

d

qs

(a)0.9

0.951

1.051.1

1.15

0

0.05

0.11

2

3

4

5

6

7

8

m = 1

d

ddq s

(b)

Figure 3: Efficiency factor, m =1, V=0.09: structural displacement (a) and acceleration (b). With the aim of considering footbridges that are sensitive to human-induced vibration even if

not in perfect resonance with the mean step frequency, analyses have been carried out for values of the non-dimensional mean step frequency ranging between 0.8 and 1.2.

Figure 4 provides an overview of the optimization results in terms of the TMD damping d (Fig. 4a) and of frequency ratio (Fig. 4b) as functions of the non-dimensional mean step frequency as regards the two different response quantities (black solid line structural response, dark grey solid line structural acceleration), compared with the two optimization criteria considered in this paper, given by Eqs. (15) (dashed black line) and (16) (dashed grey line) for the TMD damping ratio, and by the first Eq. (15) for the frequency ratio (dashed black line). The optimization results in terms of TMD relative displacement are not reported, since from Fig. 2(b) it can be deduced that the TMD relative displacement is minimized by increasing its damping ratio d and by adopting a non perfect frequency tuning (i.e. assuming as far as possible from the unitary value).

From Figure 4(a) it can be deduced that the optimal TMD damping ratio for the minimization of the structural response is a function of the non-dimensional mean step frequency and it is minimum (and around 4.5%) in case of mean step frequency almost coincident with the structural natural frequency. In any case, optimum damping values are lower than the ones given by the literature criteria. The classic criterion (dashed black line) and the optimum damping proposed by Krenk (dashed grey line) provide a value of the TMD damping ratio that does not minimize the

structural response but, since the TMD relative displacement is minimized by increasing its damping ratio d, they can represent a compromise value: Krenk criterion favors the minimization of the TMD relative displacement, while the classic criterion provides a better solution for the minimization of the structural response. Figure 4(b) shows that the optimum frequency ratio is proportional to the non-dimensional mean step frequency as regards the minimization of the structural response; furthermore, optimization for the structural acceleration requires a higher frequency tuning with respect to structural displacement (but the difference is not very significant). The classic literature criterion is close to the optimal value for pedestrian loading if the non-dimensional mean step frequency close to 1, while it gets farther as the non-dimensional mean step frequency moves away from the unit value.

0.8 0.9 1 1.1 1.2m

0.04

0.05

0.06

0.07

0.08

d opt

(a)

0.8 0.9 1 1.1 1.2m

0.94

0.96

0.98

1

1.02

1.04

1.06

op

t

(b)

Figure 4: Optimum TMD damping ratio (a) and frequency ratio (b), V=0.09.

In order to get a general overview of the optimization problem, Figure 5 plots the efficiency

factor for the structural acceleration and the variance of the TMD displacement for two different values of the non-dimensional mean step frequency, (a) m =1 and (b) m =0.9, as a function of the TMD damping ratio. The different lines in the Figures correspond to different values of the frequency ratio, the black line representing the optimum frequency ratio, the dark grey line a higher value of , the light grey line a lower value ((a): black line = 0.995, dark grey line = 1.015, light grey line = 0.97, (b): black line = 0.98, dark grey line = 1, light grey line = 0.955). Dashed vertical lines are plot corresponding to the optimal TMD damping estimated for pedestrian traffic condition (opt = 0.045), the classic value (Eq. (18), (opt = 0.06) and the value provided by Krenk (Eq. (19), (opt = 0.07). From both Figures it can be deduced that a little change in the frequency tuning causes a significant reduction of the TMD efficiency in the limitation of the structural acceleration (grey solid lines rapidly move down from the optimal value) and a small reduction of the relative TMD displacement (dashed lines are all very close to each other). On the contrary, a small increase of TMD damping with respect to the optimal value causes a decrease of the TMD efficiency concerning the reduction of the structural acceleration, but also a significant decrease of the TMD relative displacement. For this reason, the TMD optimization for the pedestrian-induced vibrations can be carried out by increasing slightly the TMD damping ratio if the TMD relative displacement has to be limited within prescribed values.

5 CONCLUSIONS AND PROSPECTS

In this paper, the optimal TMD design for mitigating pedestrian-induced vibrations in the vertical direction has been studied. Two loading scenarios have been considered: the single pedestrian modeled as a quasi-resonant moving harmonic load and normal unrestricted pedestrian traffic, modeled probabilistically through an equivalent spectral model.

Analyses have shown that all the classic optimization criteria provide good results for the single pedestrian loading scenario in terms of minimization of the structural response. As far as

concerns normal unrestricted pedestrian traffic, the TMD optimization for the minimization of the structural response is strongly influenced by the expected value of the non-dimensional mean step frequency both in terms of frequency ratio and in terms of TMD damping ratio. The TMD optimization for the pedestrian-induced vibrations can be carried out by increasing slightly the TMD damping ratio if the TMD relative displacement has to be limited within prescribed values.

The search for an analytical expression for the TMD optimum characteristics as functions of the loading spectral properties is under investigation. Furthermore, the present study may be extended to multiple TMD configurations, that are quite usual in footbridges.

0 0.02 0.04 0.06 0.08 0.1d

2

4

6

8

qs

0

7000

14000

21000

q d

..

(a)

opt class Krenk

0 0.02 0.04 0.06 0.08 0.1d

2

4

6

8

qs

0

7000

14000

21000

q d

..

(b) Figure 5: Efficiency factor (solid lines) and variance of the non-dimensional TMD relative

displacement (dashed lines): (a) m =1, (b) m =0.9 (V=0.09).

ACKNOWLEDGEMENTS

This work has been partially supported by the Italian Ministry of Education, Universities and Research (PRIN co-financed program “Dynamics, Stability and Control of Flexible Structures”) and by University of Genoa (Progetto di Ateneo 2012 “Dinamica e stabilità di strutture flessibili”).

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[8] ISO10137. Bases for design of structures- Serviceability of buildings and walkways against vibration. Geneva: International Organization for Standardization (2007).

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