Bespoke footbridge for studying pedestrian-structure interaction with vertical vibration

Bespoke Footbridge for Studying Pedestrian–Structure Interaction with Vertical Vibration Marta García-Diéguez 1 ; Beatriz Zapico-Blanco 2 ; Stana Živanovic 3 ; and José Luis Zapico-Valle 4 Abstract: Progress in quantifying and codifying pedestrian–structure interaction with a vertical structural vibration and the effects on both human beings and structures has been slow, primarily owing to a lack of experimental facilities that can simulate a wide range of vibration conditions. To accelerate the progress of pedestrian–structure interaction research, a new experimental facility (the UNIOVI footbridge) has been developed at the University of Oviedo, Gijón, Spain, and is presented in this paper. The UNIOVI footbridge is a unique laboratory structure, whose fundamental vertical vibration mode can be finely tuned in the frequency range between 1.6 and 9.3 Hz by altering its mass or stiffness. The clear separation of the first vibration mode from higher vibration modes and a low damping ratio make the structure ideal for interaction studies. The paper describes unique features of the facility and provides analytical expressions for modeling its dynamics. Time-domain procedures based on free decay response data are proposed to identify the dynamic parameters of both the structure and the human body in stationary postures. The use of the facility was successfully demonstrated by identifying body dynamics for six test subjects in three passive postures: one standing posture and two instantaneous postures extracted from the walking gait. The next task is to employ the facility in studying walking, for which the facility was primarily designed. DOI: 10.1061/JBENF2.BEENG-6562.This work is made available under the terms of the Creative Commons Attribution 4.0 International license, https://creativecommons.org/licenses/by/4.0/. Author keywords: Human–structure interaction; Laboratory footbridge; Vertical vibration. Introduction The use of lightweight materials and new construction techniques make modern structures prone to excessive vibration under dynamic loading. Consequently, vibration serviceability is often a governing limit state when designing footbridge and long-span floor structures subject to people walking. Estimation of the vertical vibration response of such structures has been based on applying the pedestrian’s dynamic load measured on rigid level ground to the dynamic model of the structure. This approach is no longer justifiable for structures where the presence of pedestrians might affect the dynamic properties (e.g., modal mass, stiffness, and damping ratio) of the vibrating system and where excessive vibration of the structure could alter the dynamic load generated by users. This pedestrian–structure interaction (PSI) could have a significant influence on the estimated vibration response and should therefore be included in the design models. Reliable modeling of the phenomenon requires the availability of high-quality experimental data. Hence, the development of experimental facilities that could provide such data is crucial for advancing PSI research and updating design guidelines. This paper reports the design and modeling of a bespoke laboratory footbridge intended for studying PSI. Initial tests were conducted to identify and model the dynamics of the unoccupied structure. In addition, further tests on the structure when occupied by a passive participant were conducted to verify a novel timedomain approach that is proposed here to identify human dynamics. The structure is unique in that both the stiffness and mass can be adjusted to give a first vertical mode of vibration in the frequency range from 1.6 to 9.3 Hz, which is the most relevant in the PSI research field. The adjustable frequency range and a low damping ratio in the first mode enable experimental studies to be conducted into how interactions between humans and structures affect different dynamic properties. At the same time, vibration modes beyond the interaction-prone frequency range are about 10 times more damped and thus do not interfere with the results. The next section includes an overview of related literature on experimental facilities, dynamics of the human body, and procedures of identification. Then follows a description of the components of the facility and its functioning. This section also includes finite-element (FE) and experimental modal analyses of the structure. A procedure for identifying the natural frequency and damping ratio of the first vibration mode of the structure is next proposed. The following section deals with the analysis of the structure when it is occupied by a test subject in three stationary postures. The section starts with a description of the experiments and continues with an identification of the dynamic properties of the human body. After this, the obtained parameters of the human model are critically evaluated against those from the literature. Conclusions are summarized in the final section. Overview of Literature This section provides a brief background on the development of the experimental facilities for PSI studies, modeling human body dynamics by utilizing a single-degree-of-freedom (SDOF) system, and numerical approaches to identifying the dynamic properties of the human body in various postures. 1 Dept. of Construction and Manufacturing Engineering, Univ. of Oviedo, 33203 Gijón, Spain (corresponding author). ORCID: https:// orcid.org/0000-0002-8960-5190. Email: [email protected] 2 Dept. of Building Structures and Geotechnical Engineering, Univ. of Seville, 41012 Seville, Spain. Email: [email protected] 3 School of Engineering, Univ. of Warwick, Coventry CV4 7AL, UK. Email: [email protected] 4 Dept. of Construction and Manufacturing Engineering, Univ. of Oviedo, 33203 Gijón, Spain. Email: [email protected] Note. This manuscript was submitted on July 10, 2023; approved on September 9, 2024; published online on October 25, 2024. Discussion period open until March 25, 2025; separate discussions must be submitted for individual papers. This paper is part of the Journal of Bridge Engineering, © ASCE, ISSN 1084-0702. © ASCE 04024102-1 J. Bridge Eng. J. Bridge Eng., 2025, 30(1): 04024102 This work is made available under the terms of the Creative Commons Attribution 4.0 International license. Experimental Facilities for PSI Studies Unexpected excessive sway of the Millennium Bridge in London on its opening day in 2000 motivated the development of experimental facilities to research the vibrations caused by pedestrian interaction with a structure. Early laboratory facilities were primarily developed to study PSIs that caused horizontal–lateral vibrations. These facilities were in the form of a suspended treadmill or laterally driven platform or treadmill instrumented with accelerometers or displacement gauges and direct or indirect means of capturing the force generated by a test subject’s walking (Dallard et al. 2001;Ricciardelli and Pizzimenti 2007;Carroll et al. 2013;Bocian et al. 2015). The development of experimental facilities for vertical PSI lagged behind, in part owing to priority being given to understanding the underlying causes of the Millennium Bridge problem, as well as the challenges of constructing a single laboratory facility that could enable the investigation of a range of conditions relevant for vertical vibrations resulting from PSI. Based on the authors’knowledge and experience, the ideal technical requirements (TRs) for such a facility for studying vertical PSI are that it should 1. allow for two-way interaction to develop, i.e., both pedestrian and walking surface (deck) are not restricted and are therefore free to respond to each other’s actions or vibrations; 2. have a sufficiently long vibrating deck that accommodates at least seven or eight consecutive steps, to enable insight into the adjustments that pedestrians make in response to perceived vibrations; 3. have an adjustable vibration frequency, ideally from 1.5 to 10 Hz (Mohammed and Pavic 2021), to enable the study of PSI for at least the first two pedestrian body modes; 4. have a low damping ratio, so as to vibrate at an amplitude perceptible by pedestrians, i.e., around 0.5 m/s 2 or more at the lower end of the frequency range; and 5. have a single pedestrian-to-structure modal mass ratio that is within the range 1%–10% observed on lightweight structures that are most prone to PSI (Wei et al. 2019). The development of a single facility to satisfy all these conditions is a complex task, as evidenced by existing facilities, all of which address some of the design requirements better than others. The first type of laboratory facility consists of a treadmill placed on a vibrating platform or a shaker. This setup enables shaking of the treadmill in a controlled manner, at target frequencies and vibration amplitudes. For example, Nessler et al. (2017) were able to impart vibrations having frequencies up to 3 Hz and vibration magnitudes up to 75 mm. Chadefaux et al. (2021) exposed pedestrians to vibration amplitudes of 2.5 mm and frequencies in the 2–12 Hz range. The VSimulator facility (UKCRIC 2024) can generate vibrations in the frequency range between 0.5 and 40 Hz and with amplitudes up to 2 m/s 2 . While these facilities, often accompanied with state-of-the-art hardware for monitoring human kinematics, offer a sophisticated insight into a pedestrian’s gait on a vibrating deck under a range of vibration conditions, they do not incorporate the two-way interaction (TR1), given that the movement of the platform is externally forced and unresponsive to the dynamic force generated by the pedestrian. The second type of experimental facility is the laboratory footbridge or walkway. These usually consist of simply supported elongated decks or beams that enable two-way interaction (TR1) to develop and provide a means of observing several successive steps (TR2). Such a facility used to exist at the University of Warwick, Coventry, UK, in the form of a steel–concrete composite bridge (Dang and Živanovic2016). It was employed for studying PSI at two different spans, resulting in vibration frequencies of 2.44 and 2.18 Hz, while the pedestrian-to-structure modal mass ratio was ≈1% in both cases. Similarly, a glass fiber–reinforced polymer (GFRP) laboratory footbridge at Monash University, Clayton, Victoria, Australia, was built to investigate pedestrian exposure to vibration frequencies in the vicinity of the fundamental vibration mode at 5.6 Hz, with the mass ratio being as high as 14% (Ahmadi et al. 2019). Conversely, the target vibration frequency for the concrete bridge at the Federal University of Rio de Janeiro, Brazil, was 3.1 Hz, with a mass ratio as low as 0.6% (Vega-Ruiz et al. 2022), while the steel beam utilized at Purdue University, West Lafayette, Indiana, vibrated at frequencies ≈2.1 Hz and had a mass ratio of ≈6% (Gómez et al. 2018). These and similar structures contributed to the development of understanding and numerical and empirical modeling of PSI, but lacked the adjustability of their dynamic properties required for a robust verification of the PSI models. The third class of facility suitable for PSI studies are actual asbuilt footbridges susceptible to PSI (e.g., Mulas et al. 2018;Tubino et al. 2016;Van Nimmen et al. 2021). These structures are naturally more restricted than laboratory structures, in terms of both the choice of their dynamic properties and the choice of equipment that can be utilized to monitor the structure and the pedestrian. Access to these structures or to data already collected on them is expected to be more valuable for validation of PSI models, rather than for developing them. In summary, the existing facilities or structures are often used for collecting kinematic and kinetic experimental data for an individual pedestrian’s interaction with a structure, and the pedestrian’s response to the perceived vibration, and are then used to calibrate dynamic models of the pedestrian. However, they do not offer scope for testing whether the developed models are applicable to vibration conditions unseen in the original tests (e.g., different vibration amplitudes and vibration frequencies). In addition, as the tests are usually performed for a limited population of pedestrians, the identified model parameters cannot easily be generalized and applied to other structures or facilities; this impairs the adoption of new models and updating of the design guidelines. One way forward would be to collate an open-access database that would store the existing data and experimental data collected in the future. While this might be the most promising approach, it requires standardization of data acquisition, reporting, and storage, which is unlikely to happen in the near future, owing to a lack of coordinated action between research groups. Another way forward is to develop more sophisticated facilities, such as the one presented in this paper, which can be used for both model development and verification. Dynamic Properties of the Human Body Studies of the relation between human body dynamics and structural vibrations started long before the need for modeling PSI materialized early in this century. Most proposals for modeling the dynamic interaction between a stationary (e.g., standing or sitting) person and a structure are based on adding a SDOF model of a human being, having modal mass m 2 , modal damping coefficient c 2 , and modal stiffness k 2 , to a SDOF model representing the relevant (usually the first) vibration mode of the structure, having modal mass m 1 , modal damping coefficient c 1 , and modal stiffness k 1 (Fig. 1). The modal parameters of human beings vary from one person to another, and they depend strongly on the posture adopted by the person. The SDOF model of a human being in the standing posture has been well researched and is widely accepted (Sachse et al. 2003;Shahabpoor et al. 2016). The mean value of the natural frequency of the standing human being is fairly consistent between © ASCE 04024102-2 J. Bridge Eng. J. Bridge Eng., 2025, 30(1): 04024102 This work is made available under the terms of the Creative Commons Attribution 4.0 International license. different studies. For the studies given in Table 1(Matsumoto and Griffin 1998;Zheng and Brownjohn 2001;Van Nimmen et al. 2015;He et al. 2022) and reporting data on Posture A, this ranges from 5.2 to 5.7 Hz. Conversely, the mean damping ratio varies more in the same studies, from 27% to 44%, demonstrating general difficulty in capturing a damping mechanism for such a highly damped system. Nevertheless, the uncertainty in the damping is not detrimental to using the model for structural engineering applications, as the vibration responses of the human–structure system are more sensitive to the frequency input. The success of SDOF modeling of the standing posture led to adoption of the same modeling approach for other passive postures, such as sitting and lying down. In addition, some passive postures that are not necessarily typically assumed by human beings over a prolonged time, but instead only represent time instants from other human postures, such as walking, have also been investigated. For example, Posture B in Table 1(Matsumoto and Griffin 1998;Van Nimmen et al. 2015) represents an instantaneous posture during single support, the so called midstance of the support leg phase of walking (or midswing of the other leg phase of walking) (Inman et al. 1989). In this posture, one straight leg is in contact with the support surface. In this case, the mean natural frequency still agrees well in the two studies (3.1 and 3.7 Hz). Both natural frequency and dampingratio,whenreported,werefoundtobelower thanforPosture A in the same studies. Another interesting case is Posture C (Table 1), which represents the toe-off phase of walking during the double support phase. Both legs are straight, with the leading foot fully supported and the trailing foot partially supported through toe contact with the deck (Inman et al. 1989). Van Nimmen et al. (2015) conducted a rare study into this posture, reporting a mean natural frequency of 3.3 Hz and a damping ratio of 26%. It is worth noting that the SDOF model is currently favored for modeling the dynamics of a pedestrian’s body, owing to the simplicity of its implementation and the relative familiarity of structural engineers with this type of model, as opposed to (numerically) more complex models of pedestrians from the inverted pendulum modeling family. An increasing number of researchers are proposing parameters for a pedestrian’s dynamics. However, the discrepancies between the results reported by different researchers are still significant. This state of the art suggests that further investigation is required before this model of a person in a walking posture can be adopted in the design. The experimental facility presented in this paper can play an important role in verifying the model. Procedures for Identifying Human Body Dynamics In the civil engineering field, the dynamic properties of the human body are most often identified using laboratory structures driven by electromechanical shakers (Zheng and Brownjohn 2001;Van Nimmen et al. 2015) or rhythmic human shakers (He et al. 2022). The experiments are performed in both the unoccupied structure and the structure occupied by one or several subjects in a passive posture. The modal parameters of the assumed lumped model of the structure itself and of the subjects are indirectly estimated by minimizing the discrepancies between the analytical model and the experimental data in the frequency domain. A procedure based on free decay experiments is proposed here to identify the modal properties of human bodies in the time domain. This approach has some practical advantages. The free decay response can be easily excited by hand in the presented structure. The resulting large variations of the response amplitude allow the potential amplitude-dependent modal parameters to be studied, while interaction with the excitation devices is avoided. Experimental Facility This section starts with a description of the experimental facility. This is followed by an insight into the modal properties identified using both numerical and experimental means. Description The facility consists of three parts: two access structures, the deck, and the adjustment system. The access structure at each end consists of a flight of stairs leading to the 2-m long landing platform [Fig. 2(a)]. The 33 mm thick plywood panel of the platform is supported by a stiff frame made of steel hollow sections (200 × 200 × 5 mm) and bolted to the laboratory floor. The access structures are dynamically independent of the main bridge structure. The footbridge deck is made of the same plywood paneling as is used for the stairs. The panels are bolted to a supporting steel structure, which consists of a grid of steel square hollow sections (200 × 200 × 5 mm), with 1.45 m spacing in the longitudinal direction and 0.65 m in the transverse direction. The supporting structure is composed of Part AB and Part BC [Fig. 2(a)]. The two parts are connected to each other by means of sliding bearings at the midspan (B). The deck structure resides on pin supports at its ends (A and C) [Fig. 2(a)]. For the safety of the test participants, a 1 m high handrail made of steel square hollow sections (50 × 50 × 2.5 mm) is installed on the access structures and along the footbridge. The elevation of the deck is 1 m, while the deck width and length are 1.5 m and 12 m, respectively. The adjustment system consists of a seesaw beam (DH) and a flexible beam (IK) [Fig. 2(c)]. Both beams are made of steel. The seesaw beam (square hollow section, 200 × 200 × 5 mm; length, 3.4 m) is supported at the middle by a pin bracket (E) bolted to Fig. 1. Dynamic model of human–structure system. Table 1. SDOF models of human beings for different postures Author (Number of test subjects) f 2 (Hz) ξ 2 (%) Posture Matsumoto and Griffin(1998) (12) 5.50 —A 3.75 —B Zheng and Brownjohn (2001) (30) 5.24 ±0.40 39 ±5A Van Nimmen et al. (2015) (6) 5.67 44 A 3.06 35 B 3.34 26 C He et al. (2022) (12) 5.47 ±0.31 27 ±6A Note: Mean ±standard deviation. © ASCE 04024102-3 J. Bridge Eng. J. Bridge Eng., 2025, 30(1): 04024102 This work is made available under the terms of the Creative Commons Attribution 4.0 International license. the laboratory floor. The seesaw beam is connected to the center of the deck through a hinge (F) and to the flexible beam via two steel rods (GJ) 12 mm in diameter [Fig. 2(b)]. The flexible beam (square hollow section, 100 × 100 × 5 mm; length 6 m) is simply supported by a pin support at one end (I) and a roller support at the other end (K) [Fig. 2(c)]. The pin supports A, C, and E are ball bearings, while hinge F consists of a roller bearing. The sliding bearing at the center of the deck, B, consists of polytetrafluoroethylene and stainless steel overlapping plates. The aim of this setup is to minimize inherent damping so that the structure is responsive to dynamic excitation by human beings. Fig. 3is a conceptual two-dimensional (2D) sketch of the structure. It includes the components of the structure idealized using line elements and ideal supports or connections. Note that the seesaw beam, perpendicular to Elements ABC and IJK in the actual structure, is sketched in the same plane to simplify the drawing, while still genuinely representing the dynamic behavior of the system. The mass of the structure can be adjusted by adding up to 24 individual steel plates, each having a mass of 19 kg. The plates are added at the ends of the seesaw beam (D and H). Precise positioning of these additional masses is achieved by a guide rod made of steel (diameter, 25 mm) and welded to the seesaw beam [Fig. 2(b)]. The stiffness of the structure is adjustable by varying the span of the flexible beam (L) from 2.6 to 5 m (Fig. 3). A close inspection of the structure and its dynamic behavior are shown in an online video recording (Universidad de Oviedo 2021). Modal Properties This section presents FE and measured modal properties of a configuration of the structure in which the span of the flexible beam (L) is set to 4.14 m, while the additional masses at the seesaw beam are varied from 0 to 24 plates (0–456 kg). Note that additional masses are always applied symmetrically, i.e., in any given configuration the mass added at Point D is equal to the mass added at Point H. FE Modeling A three-dimensional (3D) FE model of the facility was developed in Mecway v12 to simulate the dynamic behavior of the structure. Bernoulli beam elements were chosen to model the deck structure, the side handrails, and the adjustment system. The plywood panels were modeled using shell elements. Nominal values selected for the geometric characteristics of the cross sections were: Young modulus, 200 GPa for steel elements and 7 GPa for wooden elements; density, 7,850 kg/m 3 for steel elements and 750 kg/m 3 for wooden elements. Fig. 4shows the resulting shapes of the first four vibration modes, with no additional mass. In the first mode, both the deck and the seesaw beam behave as quasi-rigid bodies rotating around their hinges, while the flexible beam deforms by bending. As a result, this mode is sensitive to the use of additional masses or changes in stiffness of the flexible beam. Within the range 0– 456 kg of the additional masses and 2.6–5 m of the span of the flexible beam, the mode shapes of both the deck and the seesaw beam are almost invariant. Thus, the mode shape of the deck can be described accurately by the triangle shown in Fig. 5. Moreover, it is found that the modal stiffness of the first mode remains almost constant when the additional mass varies within the aforementioned ranges. This invariance of the modal stiffness allows the natural frequency of the first mode to be further formulated as an explicit function of the added mass for the chosen span of the flexible beam. The second and third modes are related to antisymmetric and symmetric bending of the deck, respectively, while the fourth mode corresponds to torsion of the deck (Fig. 4). Within the specified ranges of the additional mass, both the seesaw and the flexible beams remain almost stationary in the antisymmetric bending and torsion modes of vibration. This means that neither the additional (a) (b) (c) Fig. 2. (Color) Facility: (a) general view; (b) adjustment system; and (c) additional mass. Fig. 3. Conceptual two-dimensional representation of structure. © ASCE 04024102-4 J. Bridge Eng. J. Bridge Eng., 2025, 30(1): 04024102 This work is made available under the terms of the Creative Commons Attribution 4.0 International license. masses nor the span of the flexible beam have a significant influence on these modes. Conversely, the symmetric bending mode features a significant bending deformation of the flexible beam. Therefore, this mode is, similarly to the first mode, sensitive to the introduction of changes in mass or stiffness. Table 2lists the first four natural frequencies predicted by the FE model for the minimum and maximum values of the additional mass. As expected, Modes 2 and 4 are almost unaffected by the value of the additional mass. Modal Testing The natural frequencies and the damping ratios of the first four modes of vibration of the structure were experimentally determined, as described in this section. The results are only presented for one span of the flexible beam (L=4.14 m) with no additional mass. These results are compared with and critically evaluated against the frequencies predicted by the FE model. The structure was excited by applying vertical impacts with an ordinary (i.e., not instrumented) hammer at three locations, namely: at the end of the seesaw beam (D in Fig. 3), at an edge of the deck at the quarter-span (S in Fig. 6) and at the half-span of the bridge (T in Fig. 6). The corresponding vertical acceleration responses were measured at a single point on the deck in each case, as specified in Table 3and shown in Fig. 6(points P, Q, and R). A conventional uniaxial CSI accelerometer with a sensitivity of 100 mV/g and a mass of only 131 g was attached to the response point using a magnet base. The last column of Table 3indicates which modes were targeted in each of the three tests. The 30-s long segments of the response measured in each test were converted to the frequency domain using the fast Fourier transform (FFT). The measured frequencies were then extracted by peak-picking in the Fourier spectra. To estimate the damping ratios for each vibration mode, the free decay was band-pass filtered to isolate a single vibration mode. The peaks of the free decay vibration were identified and then used to estimate the logarithmic decrease, and therefore the damping ratio, for each mode. The obtained natural frequencies and damping ratios are given in Table 4. The natural frequencies are compared with the FE values in the (a) (b) (c) (d) Fig. 4. (Color) Natural mode shapes of structure from 3D FE model: (a) rigid deck, f 1 =4.9 Hz; (b) antisymmetric bending, f 2 =13.48 Hz; (c) symmetric bending, f 3 =15.93 Hz; and (d) torsion, f 4 =16.13 Hz. Fig. 5. Mode shape of deck for first mode of vibration. Table 2. Natural frequencies of FE model for L=4.14 m Natural frequency (Hz) Additional mass (kg) 0 456 f 1 4.90 2.23 f 2 13.48 13.47 f 3 15.93 13.63 f 4 16.13 16.13 Table 3. Setup of modal tests Test Excitation Measurement Mode 1 D P Rigid deck 2 S Q Bending 3 T R Torsion Fig. 6. Locations of impacts applied to deck (points S and T) and locations of vibration response measurement on deck (points Q, R, and P). Table 4. Natural frequencies and damping ratios of structure without additional mass Natural frequency (Hz) Mode of vibration Damping ratio (%) FE model Experimental Discrepancy (%) 1: Rigid deck 0.25 4.90 4.88 +0.4 2: Antisymmetric bending 2.4 13.48 13.66 −1.3 3: Symmetric bending 2.4 15.93 15.75 +1.1 4: Torsion 2.9 16.13 16.40 −1.6 © ASCE 04024102-5 J. Bridge Eng. J. Bridge Eng., 2025, 30(1): 04024102 This work is made available under the terms of the Creative Commons Attribution 4.0 International license. same table. The error in the FE results calculated with respect to the measured value is low for all modes, the maximum being a 1.6% underestimation of the natural frequency of the torsional mode. The agreement between the results of the FE model and the experimental results provides a high level of confidence that the FE model is an excellent representation of the dynamic behavior of the structure. The damping ratio of the first mode is extremely low, at 0.25%, while those of the other modes are approximately 10 times higher. This is because the deck moves as a rigid body in the first mode and, as a consequence, there is no significant friction between the plywood panels and the steel structure of the deck. Higher vibration modes involve deflection of the deck, which causes sliding and friction between the plywood panels and the steel structure, giving higher damping ratios as a result. It can be concluded that, in this configuration of the structure (span of the flexible beam, 4.14 m; additional mass varying from 0 to 456 kg), the first mode shape can be approximated by a rigid triangular displacement of the deck (Fig. 5), with a natural frequency between 4.90 and 2.23 Hz and a damping ratio of 0.25%. The higher vibration modes in all configurations have natural frequencies above 13 Hz, and damping ratios one order of magnitude higher. Fig. 7shows the acceleration response and Fourier transform for Test 2 of Table 3. A comprehensive analysis using the FE model revealed that the triangular modal shape hypothesis for the deck and the invariance of the stiffness of the first mode are acceptable for spans of the flexible beam greater than 2.6 m. In addition, the span of the flexible beam must be limited to 5 m, so as not to exceed its strength in the experiments.Consideringtheselimitationsandtherangeofvariation of the additional mass (0–456 kg), the natural frequency of the first mode of vibration of the structure can be adjusted from 1.6 to 9.3 Hz. Modeling First Vibration Mode of Unoccupied Structure A study of the differences between the properties of the structuredominated mode in the human–structure system and those of the structure itself is often the basis for quantifying the human–structure interaction. Therefore, accurate procedures of modal modeling and identification are needed to achieve reliable results. To this end, a preliminary study of the first mode of the structure is described next. From the results of this study, a procedure for identifying the modal properties (natural frequency and damping ratio) is then proposed. The problem is implicitly formulated in the time domain through the free decay acceleration. From this formulation, the values of the modal variables are determined based on acceleration measurements. Since the procedure supplies the modal variables corresponding to discrete values of the additional mass, it is referred to as the discrete approach. This approach is followed by an explicit formulation of both natural frequency and damping ratio as continuous functions of the additional mass and other spatial and modal parameters. This latter formulation is referred to as the continuous approach. Preliminary Study Free Decay Response Experiments The free decay response induced by manual excitation is chosen in this study for modal identification, as it offers practical advantages over the use of controlled, measured (shaker-induced or hammerinduced) excitation. The main advantages are the lower cost and faster speed of experiment execution. In addition, it is a simple way to avoid interactions between the structure and the excitation device attached to the structure, such as a shaker. Since only one accelerometer, having a mass of 131 g, is placed on the structure at any given time, its own interaction with the structure is negligible. Moreover, the free decay response presents vibration cycles from small to large amplitudes, allowing for potentially amplitudedependent modal properties to be determined. As mentioned earlier, it is the natural frequency and the damping ratio that are of primary interest in this work, and both can be easily determined from the free decay. The structure was excited manually at one of the ends of the seesaw beam (Point D, Fig. 3), with the aim of activating the resonance in the first vibration mode. The resulting vertical acceleration was measured at the midspan at the edge of the deck (Point R, Fig. 6). (a) (b) Fig. 7. (Color) Modal test results for Configuration 2: (a) time domain; and (b) frequency domain. © ASCE 04024102-6 J. Bridge Eng. J. Bridge Eng., 2025, 30(1): 04024102 This work is made available under the terms of the Creative Commons Attribution 4.0 International license. The excitation ceased when the acceleration reached ≈2 m/s 2 . This limit ensured that the acceleration at any point of the structure was smaller than the gravitational acceleration. Thus, neither the additional plates nor the flexible beam ever separated from the seesaw beam during an experiment, as required for correct functioning of the structure. The raw acceleration signal was filtered using a lowpass fourth-order Butterworth filter with a cut-off frequency of 10 Hz, to remove the high-frequency noise and isolate the first mode. The offset always present in the signals was not removed by filtering. Instead, it was addressed in the subsequent identification process, to avoid altering the signal. A preliminary experiment was carried out on the structure with no additional mass. During the experiment, the structure reached peak modal accelerations close to 2.5 m/s 2 (Fig. 8). The Fourier transform shown in Fig. 8(b) indicates that the response contains the contribution of the first mode only, as required for analysis of the free decay. Identification of Natural Frequency and Damping Ratio A procedure is implemented in this section to identify the natural frequency and damping ratio of the first mode of vibration of the structure. The identification is based on the premise that the free decay vibration of the structure is locally close to that of a pure linear dynamic model. The term locally implies an interval of only a few cycles of vibration. The procedure was successfully used in other similar cases (Zapico-Valle et al. 2013). It consists of the following steps. The position of the peaks in the preprocessed signal is identified. Then a signal segment is chosen from each peak. The segment length depends on the characteristics of the studied structure. Finally, each signal segment is fitted by a linear model. Assuming a constant offset and a linear dynamic behavior within each segment, the modal acceleration response can be formulated in the following discrete way: ˆ ai=Ae−ξ12πf1ticos 2πf1 (1 −ξ2 1) ti+ϕ  +Ω(1) in which f 1 =natural frequency; ξ 1 =damping ratio; A=amplitude of the modal acceleration response; ϕ=phase of the modal acceleration response; and Ω=offset of the modal acceleration response. The discrete variable t i denotes time: ti=iΔt;(i=0,1,2,...,N) (2) where Δt=sampling time interval. The parameters of Eq. (1) are identified by minimizing the normalized mean square error function: ε= i(ˆ ai−ai)2 ia2 i (3) where ˆ aiand a i represent the model-predicted [Eq. (1)] and measured modal accelerations, respectively. The minimization is carried out by means of an adaptive sampling algorithm developed previously (Zapico-Valle et al. 2010). The solution is sought by sampling the parameters to be updated within a bounded space in an iterative way. The characteristics of the sampling distributions are changed in each step of the iteration, depending on the results of the previous step. The process is stopped after 5,000 iterations. The bounds adopted for the parameters of the structure are given in Table 5. Amplitude Dependence of Modal Properties The identification procedure explained in the previous subsection was applied to determine the backbone curves corresponding to the natural frequency and damping ratio. Backbones represent the evolution of the modal parameters as a function of the acceleration amplitude, A[Eq. (1)]. It should be noted that these backbones do not represent the instantaneous values of the (a) (b) Acceleration Fig. 8. (Color) Low-pass-filtered free decay response: (a) time domain; and (b) frequency domain. Table 5. Bounds of parameters Parameter Upper bound Lower bound f 1 72 ξ 1 0.1 0.001 A10 0 ϕ0.5 −0.5 Ω0.2 −0.2 © ASCE 04024102-7 J. Bridge Eng. J. Bridge Eng., 2025, 30(1): 04024102 This work is made available under the terms of the Creative Commons Attribution 4.0 International license. modal parameters, but their equivalent linear values over each considered signal segment. Thus, the shorter the segments, the closer the equivalent linear values to the instantaneous ones. However, if the segment length is below an acceptable level, the noise that is inherently present in the signal renders the shape of the backbone irregular. After trial and error, a segment length of 5 s was found to be sufficient to provide reliable and smooth shapes of the backbones of the modal variables in all studied cases (Fig. 9). The backbone of the equivalent natural frequency decreases monotonically with an increase in the response amplitude. The maximum frequency corresponds to the minimum amplitude. Nevertheless, the variations of the frequency are extremely small (≈0.6%). Since the mass of the structure remains constant during the experiment, the results indicate that the stiffness of the structure is weakly nonlinear and features a light softening when the amplitude of the response increases. Closer inspection reveals a change in the trend of the frequency–acceleration amplitude curve for an amplitude of 0.6 m/s 2 . The structural softening is slightly lower for amplitudes below that bound value. For the equivalent damping ratio, the backbone exhibits a more complex shape (Fig. 9). There is a drastic change in the damping ratio at both sides of the aforementioned amplitude bound of 0.6 m/s 2 . The equivalent damping ratio varies around an average value of 0.21%, featuring a maximum of 0.245% and a minimum of 0.175%. Despite this variation in the damping ratio, the structure remains exceptionally lightly damped throughout. The shape of the damping backbone is very similar to that obtained for a 51.3-m single span suspension footbridge with fiber-reinforced composite deck (Wei et al. 2019) and is testament to the difficulties in predicting damping mechanisms engaged within a structure at different vibration amplitudes. Overall, the results indicate that the natural frequency of the structure is weakly nonlinear, while its modal damping is extremely low and strongly nonlinear. Discrete Approach Linear Approximation The intrinsic nonlinearities of the structure are circumvented in this section by identifying the natural frequency and damping ratio for a nominal value of the modal amplitude. The nominal value of the modal acceleration amplitude is set equal to 1 m/s 2 , for practical reasons. Namely, this amplitude of vibration of the deck is equal to the mean comfort limit defined by the Sétra guidelines (Sétra 2006) and is within the 0.5–2 m/s 2 range recommended by the BSI (2008). A procedure for identifying the natural frequency and damping ratio of the first mode of the structure is established as follows. 1. Excite the first mode, measure the free decay modal acceleration, and preprocess the signal, as explained previously. 2. From the preprocessed signal, select the peak closest to an acceleration of 1 m/s 2 . 3. Choose a signal segment of 5 s from the selected peak. 4. From the signal segment, identify the natural frequency and damping ratio, following the procedure explained previously. Experiments and Modal Identification Five configurations of additional mass attached at both ends of the seesaw beam were considered. They correspond to 0, 2, 6, 12, and 24 plates of 19 kg each (Configurations 1 to 5 in Tables 6and 7). Based on the modal data obtained from these configurations, the spatial parameter calibration method would be undetermined. This is so because a proportional variation of all unknown spatial parameters leads to the same natural frequency of the structure. To overcome this issue, two extra configurations of additional mass were accounted for by adding 12 and then 24 plates near the midspan of the deck (Configurations 6 and 7 in Table 6). Thus, the parameter calibration procedure becomes determined. Each configuration of the structure was tested twice, as explained previously. The corresponding natural frequencies and damping ratios, f 1 and ξ 1 , were obtained using the procedure explained (a) (b) Fig. 9. (Color) Backbones of structure with no additional mass (based on segment length of 5 s): (a) natural frequency; and (b) damping ratio. © ASCE 04024102-8 J. Bridge Eng. J. Bridge Eng., 2025, 30(1): 04024102 This work is made available under the terms of the Creative Commons Attribution 4.0 International license. previously. They are given in Tables 6and 7in the columns headed “Discrete”. Continuous Approach In this section, both the natural frequency and the damping ratio are formulated as explicit functions of the added mass, which also include some spatial and modal parameters. These parameters are calibrated on the basis of the results obtained using the discrete approach for the considered mass configurations. Modeling The first mode of the structure was modeled as a linear mass– spring–damper system. The additional mass was modeled as a rigid body coupled directly to the structure (Fig. 10). Thus, the modal mass of the model, m 1 , consists of the modal mass of the structure, m t , and the additional modal mass, m a : m1=mt+ma(4) The additional modal mass can be expressed as a function of the spatial values of the additional mass on the seesaw, M s , and on the deck, M d : ma=Msφ2 s+Mdφ2 d(5) in which φ s and φ d are the corresponding mode shape ordinates. Assuming that k 1 and m t are independent of the additional mass, the natural frequency of the structure can be expressed by the following explicit function: ˆ f1=1 2π k1 mt+Msφ2 s+Mdφ2 d (6) The damping ratio can be formulated via the following function of the modal parameters: ˆ ξ1=c1 2 k1m1 √(7) A proportional damping is proposed to model the damping coefficient (Ewins 2000). As the stiffness remains constant and the additional mass varies in the experiments, the following model is adopted: c1=α+βm1(8) Substituting this in Eq. (7) results in ˆ ξ1=α+βm1 2 k1m1 √(9) Calibration of Parameters The undamped model [Eq. (6)] contains four unknown parameters: k 1 ,m t ,φ s , and φ d . The parameter φ d was estimated from the position of the mass placed on the deck, assuming the linear shape established previously, φ d =0.975. The application of Eq. (6) to the 14 discrete cases available leads to a nonlinear and overdetermined system of equations. The calibration of the unknown parameters was posed as the minimization of the following error function: ε= 1 N N j=1 ˆ f1j−f1j f1j  2    (10) in which ˆ f1jand f 1j are, respectively, the model-predicted [Eq. (6)] and discrete natural frequencies (Table 6). Nis the total number of experiments: N=14. The minimization was carried out using the algorithm developed by Zapico-Valle et al. (2010) and outlined previously. Results were: m t =1, 617 kg; k 1 =1.504 MN/m; and φ s =3.66. The damping model [Eq. (9)] contains two unknown parameters: αand β. As the damping ratio is linear in these parameters, they were obtained using the least squares method from the damping ratios obtained for the discrete approach (Table 7) for Configurations 1 to 5. Outcomes were: α=120.8863 Ns/m and β= 0.0591 Ns/(m kg). Table 7. Damping ratio for different configurations Configuration Damping ratio (%) Discrete Continuous Discrepancy (%) 1 0.230 0.219 −4.635 0.229 0.219 −3.967 2 0.222 0.218 −1.757 0.222 0.218 −1.681 3 0.207 0.223 7.854 0.206 0.223 8.220 4 0.242 0.237 −2.119 0.241 0.237 −1.549 5 0.268 0.267 −0.132 0.269 0.267 −0.577 Table 6. Natural frequency for different configurations Configuration Additional mass (kg) Natural frequency (Hz) Seesaw, M s Deck, M d Discrete Continuous Discrepancy (%) 1 0 0 4.848 4.852 0.154 4.846 4.852 0.131 2 38 0 4.240 4.233 −0.156 4.240 4.233 −0.164 3 114 0 3.491 3.482 −0.251 3.491 3.482 −0.248 4 228 0 2.859 2.858 −0.031 2.859 2.858 −0.027 5 456 0 2.215 2.223 0.357 2.214 2.223 0.402 6 0 228 4.558 4.554 −0.111 4.558 4.554 −0.111 7 0 456 4.304 4.304 0.008 4.304 4.304 −0.003 Fig. 10. Model of first mode of structure with additional mass. © ASCE 04024102-9 J. Bridge Eng. J. Bridge Eng., 2025, 30(1): 04024102 This work is made available under the terms of the Creative Commons Attribution 4.0 International license.