1 Introduction

Civil engineering infrastructure, including bridges, buildings, and dams, are inspected and maintained by federal, state, and private owners. Inspections are necessary to ensure continued functionality and safety as many structures approach the end of their design lives. The American Society of Civil Engineers (ASCE) reports that as of 2016, 56,007 (or 9.1%) of the 614,387 U.S. bridges are structurally deficient and that, on average, there were 188 million trips across structurally deficient bridges each day in 2016 [1]. ASCE estimates that the backlog of the nation’s bridge rehabilitation needs totals to $123 billion [1]. The structural integrity of other highway/traffic structures is also important to prevent catastrophic failures, which could result in injury or death of motorists.

Traditionally, structural monitoring has comprised visual inspections by certified inspectors and engineers. Although visual inspection is the most common monitoring technique in practice, the Structural Health Monitoring (SHM) field has developed as structural engineering researchers seek methods of determining structural “health” based on the quantitative structural response data. In general, SHM efforts involve the collection of response data (e.g., acceleration and strain) from electronic sensors physically attached to a structure to determine the structural response’s departure from baseline conditions [2,3,4]. Many methods focus on estimating the dynamic characteristics of the structure, such as natural frequency or mode shapes [5,6,7]; for example, material or geometric deterioration (e.g., cracking, corrosion) or wear of supports (e.g., bridge unseating) may cause significant changes in the bridge’s frequency of vibration. Numerous efforts have been made to develop automated monitoring systems for structural and mechanical systems [2,3,4, 8,9,10,11]. SHM methods have varying levels of success in field conditions and are frequently structure specific in application; as a result, no single, universal SHM technique has distinguished itself in the structural engineering and inspection community.

Recently, researchers have focused on the use of vehicles as mobile bridge inspection instruments [12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40]. Prior work focusing on the cases of a simply-supported single-span bridge [13, 20, 27, 30, 33, 34] and a two-span continuous bridge [37] has shown that the bridge frequency is identifiable in the vibrations recorded on the vehicle. The ability to identify the bridge’s fundamental frequency from the vibration of the vehicle has been shown to be dependent on parameters, including bridge support conditions, geometry, and material; vehicle suspension stiffness and damping; and vehicle velocity [22, 28, 30, 37]. This prior work has established the feasibility of an indirect monitoring system utilizing the vibration of passing vehicles.

Smartphones contain an accelerometer than can be used to record vehicle accelerations [41]. Leveraging the near-ubiquity of smartphones drastically increases the size of a potential data-gathering fleet of vehicles. Smartphones have been shown to be suitable for the use in indirect bridge monitoring in both single-sensor and crowdsourcing frameworks and cost a factor of 100 less than alternative specialized equipment [42,43,44,45].

Studies have shown that for both single-span and multi-span bridges, there exist frequency-shifted bridge frequencies within the data recorded from a passing vehicle that may cause difficulty in extracting the fundamental bridge frequency [34, 37]. Prior work by the authors presented an estimation technique that leverages the symmetry of these shifted bridge frequencies to obtain an accurate estimation of the bridge’s fundamental frequency [37]. Although this has been shown to be effective on a per-vehicle basis, it relies on some a priori knowledge of the expected bridge frequency. In practice; however, a bridge’s frequency may not be known prior to the start of monitoring. In this case, there is not enough information to consistently accurately estimate the bridge’s fundamental frequency.

Bridge natural frequencies are sensitive to changes in structural condition, particularly changes in flexural rigidity or support conditions [46]. Researchers have focused on the use of both the first natural frequency [47,48,49] as well as higher modal frequencies [50,51,52] for structural health monitoring and damage detection. This paper focuses on the identification of the first modal frequency of the bridge’s vibration, subsequently referred to as the bridge frequency.

This paper presents four postprocessing strategies for estimating bridge frequencies from smartphone acceleration data streams without any a priori information about the stiffness or mass of the vehicle or the bridge: single-signal estimations using FFT (SVE) and MUSIC (SVEM) algorithms and multi-signal, crowdsourcing estimations using FFT (CVE) and MUSIC (CVEM) algorithms. Estimations made from analytical finite element simulations and laboratory experiments illustrate the feasibility of bridge frequency estimation using multi-vehicle (i.e., crowdsourced) smartphone accelerations. The results from analytical simulations indicate that crowdsourcing methods identify the correct bridge frequency in all simulations of analytical signals; in all cases, estimations using smartphone acceleration are improved using a multi-vehicle approach. The effect of user interaction with the smartphone on performance is studied by including superimposed acceleration signals of user events on 25–100% of analytical results; user event signatures include a dropped smartphone and talking on a smartphone. Increasing the percentage of signals with superimposed user event acceleration in the pool of evaluated accelerations generally reduces performance. Crowdsourced Vehicle Estimation using MUSIC (CVEM) and a simple maximum peak estimation correctly identifies the bridge frequency with a percent error of less than 4% in all cases. Experimental validation is performed by passing an adjustable frequency vehicle over a small-scale bridge. As expected from analytical results, individual experimental acceleration time histories from single vehicles do not reliably identify the bridge frequency; both CVE and CVEM methods identify the bridge frequency with < 8.3% error.

2 Materials and methods

2.1 Vehicle–bridge interaction

Acceleration signals recorded from a smartphone within a vehicle as it traverses and excites a bridge will contain both the acceleration response of the vehicle as well as the acceleration response of the bridge. Figure 1 shows a simple vehicle–bridge system comprising a sprung mass traversing a single, simply-supported span at constant velocity. In this system, \(q_{{\text{v}}}\) is the vertical deflection of the vehicle mass, \(m_{{\text{v}}}\) is the mass of the vehicle, \(k_{{\text{v}}}\) is the stiffness of the vehicle’s suspension, \(v\) is the horizontal velocity of the vehicle over the bridge, \(u\) is the vertical deflection of the bridge which may be calculated at any point along the span, \(\overline{m}\) is the mass per unit length of the bridge, \(E\) is Young’s modulus for the bridge, \(I\) is the second moment of area for the bridge, and \(L\) is the span length. The primary frequencies that will manifest in an individual vehicle’s response are:

  • \(2{\upbeta }_{n} v\), also known as the vehicle driving frequency;

  • \(\omega_{{\text{v}}}\), the natural frequency of the vehicle;

  • \(\omega_{n} \pm {\upbeta }_{n} v\), two frequency-shifted versions of the nth bridge modal frequency.

Fig. 1
figure 1

Vehicle–bridge system consisting of a sprung mass traversing a single, simply-supported span

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Here, \(\beta_{n}\) is the spatial frequency of the bridge’s nth mode shape, which is a function of \(L\) as well as the bridge’s support conditions, and \(v\) is the velocity of the vehicle over the bridge. Full derivations for the vehicle–bridge system for a single-span simply-supported bridge and for a 2-span continuous bridge are given in [34, 37], respectively. For a simply-supported span,

$$ \beta_{n} = \frac{n\pi }{L}, $$
(1)

as presented in [34]. For this case, the relationship between \(\beta_{n}\) and \(\omega_{n}\) is:

$$ \omega_{n} = \beta_{n}^{2} \sqrt {\frac{EI}{{\overline{m}}}} . $$
(2)

Notably, although the bridge’s natural frequency, \(\omega_{n}\), does show up as a distinct frequency in the solution for the vehicle’s response, in practice it frequently does not show up as a distinct peak if a discrete Fourier transform (DFT) for the signal is calculated. The two shifted bridge frequencies, \(\omega_{n} \pm B_{n} v\), do manifest in the DFT. As shown in [37], in cases where \(\omega_{n}\) is desired, but does not appear as a distinct peak in the DFT, the symmetry of the shifted bridge frequencies may be leveraged to make an estimation:

$$ \frac{{\left( {\omega_{n} - {\upbeta }_{n} v} \right) + \left( {\omega_{n} + \beta_{n} v} \right)}}{2} = \frac{{2\omega_{n} }}{2} = \omega_{n} . $$
(3)

This estimation; however, requires the observer to have some prior knowledge of the expected bridge frequency; otherwise, the peaks for the shifted bridge frequencies will be indistinguishable from those for \(2\beta_{n} v\) and \(\omega_{{\text{v}}}\). Four postprocessing bridge frequency extraction strategies were investigated.

2.2 Bridge frequency estimation strategies

The general procedure of estimating the bridge frequency presented in this paper comprises recording vehicle acceleration responses as vehicles traverse a bridge and calculating frequency spectrums from these acceleration responses using either the DFT algorithm or the MUSIC algorithm. This is applied to signals from single vehicles to obtain bridge frequency estimations on a single-vehicle basis as well as to groups of vehicles to obtain bridge frequency estimations on a vehicle group basis using crowdsourcing. The general bridge frequency estimation procedure is shown in Fig. 2.

Fig. 2
figure 2

General bridge frequency estimation flow chart

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2.2.1 Single-vehicle DFT estimation (SVE)

After the vertical acceleration of a vehicle has been recorded as it traverses a bridge, the DFT of the vehicle’s acceleration response is calculated. Without prior knowledge of the expected bridge frequency, \(\omega_{{n,{\exp}}}\), there is no method by which to make an informed decision about which peaks represent the shifted bridge frequencies, \(\omega_{n} \pm \beta_{n} v\). Thus, a reasonable conclusion is to simply identify the peak with the highest magnitude in the DFT and search for additional peaks within \(\pm 2\beta_{n} v\) of that peak. Once any peaks within this range have been identified, the frequency values of the peaks may be averaged together to obtain an estimation of \(\omega_{n}\). Peaks with frequencies lower than 2 Hz are not considered as the frequency \(2\beta_{n} v\) falls within this range for most cases considered and 2 Hz is lower than can be reasonably expected for a typical bridge. Peaks with frequencies higher than 20 Hz are also not considered as they fall outside the range of frequencies that can be reasonably expected for a typical bridge.

2.2.2 Crowdsourced vehicle DFT estimation (CVE)

If multiple acceleration signatures recorded from different vehicles crossing a single bridge at different speeds are captured, a group estimation for \(\omega_{n}\) can be calculated which may lead to a more accurate and robust estimation. With a diverse vehicle set, the only commonalities between all of the individual DFTs for each vehicle are the bridge-related frequencies. To leverage this, all of the individual vehicle DFTs can be averaged together to obtain an average DFT for the whole vehicle set. From this average DFT, the peak with the highest magnitude is identified. Additional peaks within the range of anticipated frequency shifts of the maximum peak are also identified. All of the selected peaks are averaged together to obtain a single estimation of \(\omega_{n}\) for the entire vehicle set.

2.2.3 Single-vehicle estimation using MUSIC (SVEM)

Multiple signal classification (MUSIC) is an eigenspace method used for frequency estimation [53, 54]. The algorithm assumes that the input signal is a combination of a known number of complex exponentials. MUSIC may outperform DFT-based methods if noise or effects from outside interaction with the smartphone are present by leveraging the known number of components to ignore unwanted frequency components. Recently, MUSIC has been applied within the field of structural health monitoring [55,56,57].

After the vertical acceleration of a vehicle has been recorded as it traverses a bridge, the pseudospectrum of the vehicle’s acceleration response is calculated using the MUSIC algorithm. From the pseudospectrum, the peak with the highest magnitude is identified. Additional peaks within the range of anticipated frequency shifts of the maximum peak are also identified. All of the selected peaks are averaged together to obtain an estimation of \(\omega_{n}\) for the vehicle.

2.2.4 Crowdsourced vehicle estimation using MUSIC (CVEM)

In the SVEM method, the MUSIC algorithm is used to calculate the pseudospectrum for a single input signal; however, the MUSIC algorithm may be used to calculate the pseudospectrum of a group of signals. In this method, a group of vehicle acceleration time histories are gathered into a matrix. The MUSIC algorithm is used to calculate a single pseudospectrum for this matrix. From this pseudospectrum, the peak with the highest magnitude as well as any peaks within the anticipated frequency shift of this peak are identified. All of the selected peaks are averaged together to obtain a single estimation of \(\omega_{n}\) for the vehicle group.

2.3 Numerical simulation

2.3.1 Finite element model

To evaluate the four bridge frequency estimation strategies presented in Sect. 2.2, a numerical study using finite element analysis software was conducted. The vehicle–bridge system shown in Fig. 1 was modeled using the OpenSees finite element software framework using 0.5 m long displacement beam-column elements for the beam and 2D beam contact elements to simulate the contact interaction between the vehicle and the bridge [58]. The vehicle comprised seven nodes: two 2 degree of freedom (DOF) “wheel” nodes used to enforce contact with the bridge, two 2DOF nodes directly above the two wheel nodes, two 3DOF nodes at the same locations as the two top 2DOF nodes that are constrained to have the same vertical and lateral displacement as the 2DOF nodes, and a 3DOF node halfway between the two previous 3DOF nodes on which the vehicle mass and mass moment of inertia were applied. The 3DOF nodes were connected using rigid displacement beam-column elements. The node with the mass applied was the point from which vehicle vertical accelerations were recorded. A two-node link element was used to model the suspension of the vehicle connecting the “wheel” and “body” nodes with a material stiffness equal to \(k_{{\text{v}}}\). The vehicle starts with the leftmost (rear) wheel node 0.001 m to the right of the left bridge support and stops with the rightmost (front) wheel reaches the right bridge support. The left bridge support was modeled as a pin support, while the right support was modeled as a roller support. The TRBDF2 integrator was used to run the analysis in OpenSees with a time step of 0.001 s [58].

The vehicle parameters \(m_{{\text{v}}}\), \(\omega_{{\text{v}}}\), and \(v\) were varied to create different vehicle–bridge combinations. The vehicle stiffness, \(k_{{\text{v}}}\), can be calculated based on the selected \(m_{{\text{v}}}\) and \(\omega_{{\text{v}}}\) for a particular vehicle:

$$ k_{{\text{v}}} = \omega_{{\text{v}}}^{2} m_{{\text{v}}} , $$
(4)

where \(\omega_{{\text{v}}}\) is measured in radians per second. The mass of the vehicle is uniformly distributed along the vehicle’s length; thus, the mass moment of inertia for each vehicle was calculated as:

$$ I_{{\text{m}}} = \frac{1}{12}m_{{\text{v}}} L_{{\text{v}}}^{2} , $$
(5)

where \(L_{{\text{v}}}\) is the wheel-to-wheel length of the vehicle. Vehicles were created using all combinations of \(m_{{\text{v}}}\), \(\omega_{{\text{v}}}\), and \(v\) within the considered ranges. Table 1 gives the ranges of values considered for the vehicle parameters.

Table 1 Vehicle parameter value ranges
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Two different bridges were modeled for simulations with the goal of having one bridge with an \(\omega_{1}\) within the range of considered \(\omega_{{\text{v}}}\) and one bridge with an \(\omega_{1}\) outside of the range of considered \(\omega_{{\text{v}}}\). The bridge parameters for both bridges are given in Table 2.

Table 2 Bridge parameters
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2.3.2 Addition of user events

The vehicle acceleration signals output by OpenSees do not contain any interaction with the smartphone that might be encountered in a real-world application of the strategies presented in this paper. Accelerations due to user events were added to the OpenSees output to test the robustness of the four bridge frequency estimation strategies to events that may be encountered in real-world deployment. If smartphones are used as the sensor for data collection within the vehicle, the data is prone to human interaction with the smartphone as well as movement of the smartphone within the vehicle. iPhone 7 s were used to gather acceleration data for two different user event cases:

  1. 1.

    The phone being dropped onto a car seat (drop user event);

  2. 2.

    The smartphone user talking on the phone with the phone to their ear (talk user event).

After the results were obtained without the addition of these user events to the OpenSees simulation results, 25%, 50%, 75%, and 100% of the vehicle accelerations output by OpenSees were randomly selected to have the user event accelerations applied. If a signal was selected for user event application, the acceleration signature of one of the two different event types, i.e., Drop or Talk, was superimposed onto the vehicle acceleration signal output by OpenSees. In this way, the final signal contains responses from the bridge, the vehicle, and the interaction with or movement of the phone within the vehicle. The Appendix provides details on the superposition of signals from user events onto vehicle signals as a first-order approximation to structured noise caused by smartphone user interaction.

2.4 Effect of the quantity of vehicles for crowdsourced estimation

Since the crowdsourcing-based bridge frequency estimation strategies (CVE and CVEM) rely on the calculation of an average DFT (CVE) or group pseudospectrum (CVEM) for a vehicle group to aid in filtering out non-bridge-related frequencies, an investigation into the effect of vehicle group size on performance for these strategies was performed. 10 random groups each of 10, 50, 200, and 500 vehicles per group were tested using the CVE and CVEM estimation strategies on the dataset for Bridge 2 with no user events added.

2.5 Experimental setup

To evaluate the bridge frequency estimation strategies on data that better simulates reality, a scale model laboratory vehicle–bridge system was designed and constructed. The vehicle–bridge system is shown in Fig. 3.

Fig. 3
figure 3

Experimental laboratory vehicle–bridge system

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The bridge was constructed using a steel plate supported on both ends with vertical supports. The parameters for the steel plate are given in Table 3.

Table 3 Experimental laboratory bridge parameters
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The vehicle was constructed using a prebuilt chassis modified with flat steel plate supported by a four-point suspension. Different masses were added to the top plate of the vehicle to create different vehicle–bridge systems. An Arduino UNO was used to control the speed of the vehicle and drive it across the bridge at three different velocities. Each vehicle–bridge–velocity combination was repeated five times. The vehicle parameters used are given in Table 4.

Table 4 Vehicle parameter value ranges
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An iPhone 7 was mounted on top of the vehicle to record the vertical acceleration of the mass as the vehicle traversed the bridge, as shown in Fig. 4. A sampling frequency of 100 Hz was used for all experimental measurements.

Fig. 4
figure 4

Experimental vehicle with iPhone

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3 Results and discussion

3.1 Analytical study

Figures 5 and 6 show example DFT and MUSIC frequency spectrum outputs, respectively, for Bridge 1; Figs. 7 and 8 give examples for Bridge 2. Each figure shows both 5 randomly chosen frequency spectra for the single-vehicle estimation case (SVE or SVEM) and the group frequency spectrum for the crowdsourced estimation case (CVE or CVEM). The bridge frequency, \(\omega_{1}\) , as well as the range of possible values for the vehicle frequency, \(\omega_{{\text{v}}}\), are annotated on the figures. Each analytical model was run using a time step of 0.001 s (equivalent to a sampling frequency of 1000 Hz), while the iPhones used for the laboratory experimentation have a sampling frequency of 100 Hz. For consistency between analytical and experimental results, each analytical vehicle acceleration signal was downsampled to a sampling frequency of 100 Hz before frequency spectrums were calculated.

Fig. 5
figure 5

Analytical DFTs for Bridge 1, a 5 random individual vehicle DFTs, b average group DFT

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Fig. 6
figure 6

Analytical pseudospectrums from MUSIC for Bridge 1, a 5 random individual vehicle spectrums, b group spectrum

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Fig. 7
figure 7

Analytical DFTs for Bridge 2, a 5 random individual vehicle DFTs, b average group DFT

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Fig. 8
figure 8

Analytical pseudospectrums from MUSIC for Bridge 2, a 5 random individual vehicle spectrums, b group spectrum

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User events were added to different percentages of the vehicle acceleration signals output by OpenSees. From the full dataset containing these signals, 10 groups of 500 vehicles were randomly selected. Tables 5, 6, 7, 8 give the average classification accuracies (CA) and estimation percent errors (PE) for these 10 groups for all four strategies on Bridge 1 and Bridge 2 for the two types of added user events discussed in Sect. 2.3.2.

Table 5 Classification accuracy and estimation percent error for Bridge 1—drop event
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Table 6 Classification accuracy and estimation percent error for Bridge 1—talk event
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Table 7 Classification accuracy and estimation percent error for Bridge 2—drop event
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Table 8 Classification accuracy and estimation percent error for Bridge 2—talk event
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Classification accuracy represents how often a technique correctly identifies the bridge frequency peak within the DFT or pseudospectrum; an example of an incorrect classification would be a case where the peak corresponding to \(\omega_{{\text{v}}}\) has the highest magnitude within the DFT or pseudospectrum and is; thus, selected as the bridge peak used to estimate \(\omega_{n}\), resulting in large error.

The estimation percent errors represent the proximity of the estimate of \(\omega_{n}\) to the known value of \(\omega_{n}\); for the SVE and SVEM strategies, estimation percent errors are calculated using only the signals within a given vehicle group whose bridge peaks were correctly classified. For the CVE and CVEM strategies, estimation percent errors are calculated regardless of whether or not the bridge peak is correctly identified.

For Bridge 1, \(\omega_{1}\) falls within the range of \(\omega_{{\text{v}}}\) considered for the vehicles. On the dataset with no user events, all four techniques were able to accurately identify the bridge peak and, when the bridge peak was correctly identified, accurately estimate \(\omega_{1}\) within 4% error as shown in Tables 5 and 6. While the two strategies that make estimates on a per-vehicle basis, SVE and SVEM, were able to frequently identify the correct peak as the bridge-related peak, there were scenarios where an incorrect peak was selected. The crowdsourced strategies, CVE and CVEM, were both able to identify the correct peak as the bridge-related peak in all scenarios with a corresponding estimation error within 1.5%. These crowdsourced strategies are more robust against misclassifications due to them placing less importance on any single-vehicle estimation. Performance for the SVE and SVEM strategies steadily decreased as increasing amounts of user events were added, however the CVE and CVEM strategies proved to be robust against the addition of this event type.

For Bridge 2, \(\omega_{1}\) falls outside of the range of \(\omega_{v}\) considered for the vehicles. On the dataset with no user events, the CVE and CVEM techniques were once again able to identify the bridge peak in all scenarios and, when the bridge peak was correctly identified, accurately estimate \(\omega_{1}\) with a maximum error of 2.70% as shown in Tables 7 and 8. The performances of the SVE, CVE, and SVEM strategies declined with increasing amounts of drop events being added to the vehicle signatures; however, the CVEM strategy was able to identify the bridge response peak even with the addition of a Drop event to every vehicle signal. These same trends held for talk events, with the CVEM strategy proving robust against user interaction with the smartphone and the other strategies declining in performance as increasing amounts of user events were added.

3.2 Quantity of vehicles study for CVE and CVEM estimations

Table 9 shows estimation errors for CVE and CVEM estimations using the dataset for Bridge 2 without the addition of user events with 10, 50, 200, and 500 vehicles per group. Table 10 shows the same data, but with the addition of talk user events to 50% of the signals in the dataset. All the results shown are averages across 10 randomly-sampled vehicle groups.

The results presented in Table 9 show that, without the addition of accelerations due to user events, the CVE estimation strategy is capable of making an accurate bridge frequency estimation with as few as 10 vehicles in a group, while the CVEM strategy requires more vehicles to be effective. With the addition of talk user events to 50% of the signals in the dataset, CVE is no longer able to make an accurate estimation of the bridge frequency; CVEM is able to make an accurate estimation, but requires somewhere between 50 and 200 vehicles in a group to maximize accuracy.

Table 9 CVE and CVEM classification accuracy and estimation percent error for different vehicle group sizes for Bridge 2—no user event added
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Table 10 CVE and CVEM classification accuracy and estimation percent error for different vehicle group sizes for Bridge 2—talk user event added to 50% of signals
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3.3 Experimental study

The CVE and CVEM bridge frequency estimation strategies were used to compute estimates for \(\omega_{1}\) for the small-scale bridge discussed in Sect. 2.5. No user events were added to the measurements taken using this system. Table 11 gives the results from these estimations. In this experimental setup, \(\omega_{1}\) falls outside of the range of \(\omega_{v}\) used. Figures 9 and 10 show example frequency spectra from the laboratory validation calculated using the DFT and MUSIC, respectively. Both the CVE and CVEM strategies were able to estimate \(\omega_{1}\) within 8.30% error, as shown in Table 10. The results from these experiments show both that the bridge frequency estimation strategies presented in this paper are robust against the noise inherent in a real-world system.

Table 11 Vehicle parameter value ranges
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Fig. 9
figure 9

Experimental DFTs, a 5 random individual vehicle DFTs, b average group DFT

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Fig. 10
figure 10

Experimental pseudospectrums from MUSIC, a 5 random individual vehicle spectrums, b group spectrum

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4 Conclusions

This paper has proposed four different postprocessing strategies to estimate a bridge’s fundamental frequency, \(\omega_{1}\), from acceleration data recorded from a traversing vehicle when the mass and stiffness of the vehicle and the bridge (and, thus, the expected bridge frequency) are not known a priori. These strategies include single-vehicle estimation using vehicle acceleration DFTs (SVE), crowdsourced vehicle estimation using an average vehicle acceleration DFT for a group of vehicles (CVE), single-vehicle estimation using the MUSIC algorithm (SVEM), and a crowdsourced multi-vehicle estimation using the MUSIC algorithm (CVEM). Finite element simulations of various vehicles traversing bridges were performed and analyzed using these bridge frequency estimation strategies. Both a flexible bridge with \(\omega_{1}\) near the vehicle frequencies and a stiffer bridge with \(\omega_{1}\) outside of the considered vehicle frequencies were used for analysis and comparison between the different strategies. Additionally, an experimental validation was performed on a scale-model laboratory bridge.

The analytical results in this paper show that with multiple vehicles with diverse characteristics and no added user events, \(\omega_{1}\) can be estimated with no a priori knowledge of the mass and stiffness of the bridge or the vehicle using the simple method of identifying the peak in the DFT or pseudospectrum with the largest magnitude. If accelerations due to user interaction with the iPhone or some other phenomenon are included within the vehicle signatures, the MUSIC algorithm can be used in the CVEM strategy to make an accurate estimation of \(\omega_{1}\).

For scenarios where \(\omega_{1}\) and \(\omega_{{\text{v}}}\) are in close proximity, it is difficult to isolate the two frequencies on a single-vehicle basis. However, utilizing the crowdsourcing methodologies presented with a group of vehicles with diverse frequencies, the frequency content due to \(\omega_{{\text{v}}}\) becomes spread out; thus, making it easier to isolate the desired bridge frequency.

This paper establishes a crowdsourcing framework comparable to the state of the art without a priori information about vehicle or bridge mass, stiffness, and frequency and without the use of attached sensors. Future work includes investigating the effect of additional complexity, such as surface roughness or three-dimensional effects on the performance of the presented methodologies. Additionally, the use of the presented methodologies to estimate high-order modal frequencies for the use in structural health monitoring applications will be investigated.