2.1. LSTM Architecture

The combined use of TensorFlow and Keras streamlined the construction of the LSTM network, leveraging their comprehensive features and optimizations. TensorFlow’s widespread adoption across different domains underscores its reliability and scalability in real-world applications [63].

The LSTM model aims to enhance the accuracy of a low-cost accelerometer system, known as LARA, by using data from commercial PCB sensors (considered a reference for higher precision). This refined model output is then used to determine the modal shapes of structures in a vibration analysis context. Specifically, data from six commercial sensors (PCB) and three LARA sensors were incorporated into this study. The placement and utilization of these sensors were customized for the data from this case study, with further details provided in subsequent sections.

The architecture of the network is specifically customized based on the data from the case study, further details of which are provided in the following sections.

2.2. Advancements in Intelligent Low-Cost Sensor Monitoring System

In this section, the application of low-cost sensors in SHM and their significance in the industry are introduced. The review begins with a discussion on the state-of-the-art use of various low-cost sensors for SHM, specifically focusing on bridges. Subsequently, LARA, a standout low-cost accelerometer, is presented alongside its recent enhancements, with an in-depth look at its hardware and software components.

Commercial accelerometers are highly regarded in academic and industry circles for their widespread availability, dependability, and proven performance metrics [64]. One of their major benefits is the thorough calibration and validation processes they undergo. Manufacturers meticulously test and calibrate these devices to guarantee precise and consistent data, giving researchers a solid foundation for their monitoring projects and analysis [65]. The detailed specifications and performance indicators available for commercial accelerometers are the key factors in their broad acceptance within the scientific community.

Accelerometers hold a vital role in the SHM of bridges and buildings [66]. Essential characteristics of commercially available accelerometers, including name, acceleration range, frequency range, spectral noise density, price (VAT excluded), axis, and references, are summarized in Table 1. It is noteworthy that certain accelerometers, such as PCB TLD356A17 and PCB 356B18 shown in Table 1, require additional data acquisition equipment, which averages a cost of €700 per channel. The rest of the sensors introduced in this table generally do not need extra data acquisition equipment under the circumstances indicated on their datasheets.

The analysis of the commercial solutions in Table 1 reveals a significant price variation, ranging from €700 to €1820. Accelerometers with higher prices (such as IOLITEi0-3x-ACC) usually provide a lower measurement range and higher resolution.

Civil engineering literature suggests that short-span bridges (with spans not exceeding 40 m [70]) primarily exhibit natural frequencies between 3.0 and 30.0 Hz [75]. Conversely, medium-to-long-span bridges display primary eigenfrequencies ranging from 0.1 to 8.0 Hz [76], underscoring the suitability of sensors with a more modest frequency range for SHM applications that demand lower acceleration ranges [77].

A prominent strategy in developing affordable accelerometers is leveraging MEMS sensors [78]. MEMS accelerometers present a cost-effective, low-power, and sensitive alternative with a sufficient sampling rate compared to their piezoelectric [79] counterparts [28]. Their ability to connect to economical microcontrollers, such as Arduino (known for its various communication ports, including the interintegrated circuit [I2C]) further underscores their utility [80].

The landscape of low-cost MEMS accelerometers is continually enriched by emerging prototypes. For instance, Pardeshi et al. [81] employed an ADXL335 accelerometer alongside an Arduino Mega2560 to assess vehicle tire pressure through wheel hub vibrations. Similarly, Patange et al. [81] devised a mechanism for identifying different failure modes in single-point cutting tools, utilizing machine learning and Arduino-based vibration detection equipment featuring an ADXL335 MEMS accelerometer.

Despite their versatility and benefits, MEMS accelerometers inherently exhibit higher noise density values compared to other accelerometer types [82]. Furthermore, unlike piezoelectric accelerometers that necessitate costly, precision external signal conditioning, MEMS accelerometers integrate this feature onboard, a factor that can introduce additional noise [70].

To mitigate the accuracy limitations typically associated with low-cost accelerometers, Komarizadehasl et al. [70] developed the LARA, distinguishing itself with the employment of five MPU9250 MEMS accelerometers to amplify performance. A comparative analysis in Table 1 highlights LARA’s significantly lower noise density (50.5 μg/√Hz) against alternatives such as the ADXL345 (300 μg/√Hz) [83], suggesting a notable advancement in reducing noise interference.

LARA signifies an innovative leap in cost-efficient sensor technology, incorporating open-source prototyping platforms, namely, Arduino and Raspberry Pi, alongside the economical MEMS accelerometer MPU9250. Though such accelerometers are rarely used in civil engineering due to precision concerns, LARA’s innovative approach (averaging readings from five accelerometers at a singular point through a multiplexer) marks a pioneering effort in literature to reduce overall noise density, balancing the signal-to-noise ratio [84] with the optimal number of accelerometers to maintain system sampling frequency.

Figure 1 details LARA’s sensor configurations, showcasing both the wiring setup with an Arduino Due microcontroller and its compact form. Through processing data from these devices, LARA corrects individual accelerometer discrepancies, achieving a precision akin to commercial-grade sensors. Its efficacy for dynamic monitoring of structures such as buildings and bridges has been affirmed.

To enhance usability, LARA’s framework has been reimagined to autonomously begin data collection upon power connection, ceasing after 2 min. Employing a Raspberry Pi, it establishes internet connectivity via Wi-Fi, LAN, or SIM card, accessing a Google Drive to download Python scripts for data acquisition. This setup ensures data management efficiency, with programmed tasks facilitating regular data transfer to Google Drive, thus preventing local storage overload. This innovative information flow, depicted in Figure 2(a), outlines LARA’s operation from initialization to data processing, showcasing its adaptability and precision-enhancement capabilities for SHM.

3.1. Footbridge Description

The experiments were conducted on the HUMVIB footbridge situated within the Campus Lichtwiese of TU Darmstadt in Germany. This bridge is a composite (steel–concrete) structure with a span length of 13.24 m. It is built of a pair of longitudinal steel beams with a cross-section of HEB 240. These beams are supported at both ends using articulation. The bridge deck consists of 13 prefabricated concrete slabs resting on these beams (Figure 3(a)). To prevent any shifting of the concrete against the steel, 5mm-thick dummy blocks were strategically placed as separators. It was observed that three such blocks per concrete unit offered greater stability than four. As illustrated in Figure 3(b), these dummy blocks are aligned in alternating orientations for adjacent concrete slabs. Moreover, the bridge’s lateral rigidity has been enhanced by incorporating a crossbeam at three junctures: the start, the middle, and the end of the footbridge. This crossbeam, a U160 × 65 × 7.5 profile, links the main beams via angle steel (L120 × 120 × 12) and bolted joints, as depicted in Figure 3(c). The overview of the chosen footbridge for the laboratory experiments is shown in Figure 3(d).

It should be noted that, although equipment appears to be visible under the bridge in Figure 3, all necessary precautions were taken to ensure that no equipment or objects interacted with the structure during the experiments.

3.2. Bridge Instrumentation

In this study, the bridge was instrumented with an array of sensors consisting of six PCB TLD356A17 accelerometers and six LARA accelerometers. The structure was divided into three equal segments: one at a quarte-span, the midspan, and three-quarters-span. Two nodes were selected within each segment, resulting in a total of six nodes for sensor placement, as depicted in Figure 4(a). Each LARA sensor was positioned adjacent to a PCB sensor, ensuring the longitudinal (X) axis of all sensors was uniformly aligned. The setup of the sensors on the bridge, featuring both a PCB TLD356A17 and a LARA accelerometer, is illustrated in Figure 4(b).

In addition, for clarification throughout this paper, the sensors are identified according to their node location on the bridge. For instance, LARA1 and PCB1 refer to the low-cost and commercial sensors situated at Node 1, respectively.

It is important to note the differences in the instrumentation process. While each LARA operates autonomously, the PCB accelerometers require a data acquisition system connected to a computer for data streaming and storage, as demonstrated in Figure 4(b). In contrast, the LARA accelerometers offer a more time-efficient solution, negating the need for additional cables, data acquisition equipment, or a computer for operational control. Instead, they execute preprogrammed instructions via a modifiable Python script, which simplifies the setup process, as shown in Figure 2.

3.3. Bridge Excitation Methodology

The bridge was subjected to dynamic excitation by a person walking across it. To mitigate the variability in pedestrian excitation, the walking speed was set to approximately 2 Hz using a pendulum as a guiding mechanism. The pendulum helped pedestrians maintain a steady pace, ensuring consistency in the excitation frequency. The chosen speed of 2 Hz aligns closely with the first eigenfrequency of the bridge, thereby enhancing the excitation of the bridge’s dynamic response. This controlled excitation not only improves data consistency but also facilitates the identification of additional mode shapes by resonating with the bridge’s natural frequencies. The action of walking was systematically repeated three times to ensure consistency. In each instance, the individual traversed the bridge from the left to the right abutment, maintaining the target frequency of 2 Hz. Subsequently, the bridge was allowed to freely oscillate for 2 min to observe its natural response. This procedure was precisely replicated at intervals of 0, 3, and 6 min.

The simplicity of the experimental setup was intentional, with conditions controlled to remain as consistent as possible. It is crucial to acknowledge, as illustrated in Figure 5, that despite the aim for uniformity in the walking frequency, there was an element of human variability. These variations, rather than solely posing as a limitation, served a dual purpose. They prevented the AI tool employed for processing LARA’s data from potential overfitting, thereby enhancing the robustness of the study’s findings.

After completing the experiment and extracting data from the data acquisition systems of both PCB and LARA sensors, a postprocessing step was initiated to align the Z-axis and Y-axis data. This alignment ensured a comprehensive analysis of the bridge’s response by standardizing the orientation across all axes and sensors.

Subsequently, the datasets underwent synchronization and were resampled to a uniform sampling frequency of 333 Hz. This frequency was selected based on the maximum sampling capability of the LARA sensors, necessitating the resampling of the PCB sensor data, which was originally collected at 600 Hz, to match.

Upon completing the alignment, synchronization, and standardization of the sampling frequency, data for the Z, X, and Y axes were thoroughly extracted. Given the simply supported nature of the footbridge’s structural design, the analysis primarily concentrated on the Z-axis data. This focus was deliberately chosen to test the training model’s proficiency in refining LARA sensor data to ILARA data. The goal was to ascertain the model’s impact on enhancing data reliability and accuracy for SHM applications. Figure 6 displays the time-domain output from the three PCB sensors used for the training of ILARA.

While standard practices often advocate for a fixed percentage split (e.g., 60-20-20), the availability of a large dataset in this study enabled the prioritization of comprehensive training, ensuring the model’s effectiveness in real-life conditions rather than merely in a laboratory setting. A broader range of training data was adopted to enhance the model’s robustness beyond controlled environments, rather than strictly adhering to regions most compatible with the test datasets. This decision was driven by the necessity of replicating real-world scenarios, where data quality varies due to sensor accuracy and imposed loads. By incorporating a diverse dataset, the LSTM model was trained to handle fluctuations and imperfections present in practical applications, such as monitoring an operational bridge using ILARA sensors. In addition, this approach minimized overfitting, improving the model’s ability to generalize when applied to new datasets.

3.4. Architecture of the AI Tool

This subsection illustrates the design and operational mechanics of the LSTM model tailored for analyzing SHM data derived from bridge sensors. The model is trained on the training dataset, and hyperparameters are optimized using the hold-out cross-validation dataset. Finally, the test dataset is used to evaluate the model’s ability to generalize to new data. This thorough process confirms the reliability of the model and ensures it avoids overfitting to the training data. The core of our deep learning strategy involved training the LSTM model to discern complex temporal relationships within the bridge’s dynamic response data, a crucial step towards enhancing the predictive accuracy of low-cost sensor data. It is essential to emphasize that the role of the commercial accelerometer in this study is to validate the proper calibration of LARA readings during the training of the proposed LSTM. The calibration process leverages the LSTM model to address the unique noise characteristics of low-cost accelerometers. By combining data from all available sensors, the model estimates the output of each LARA sensor (e.g., Sensor number 1) and compares this prediction with the sensor’s actual readings and historical data stored in its memory. As a RNN, the LSTM processes data in sequences, incorporating temporal dependencies to refine predictions. This allows the model to account for both spatial relationships among sensors and temporal patterns in the data, effectively identifying and reducing noise.

Unlike traditional calibration methods for commercial sensors, which often rely on linear transformations or static correction factors, the LSTM model dynamically adjusts based on the context provided by prior and concurrent measurements. While the LSTM model functions as a “black box,” limiting the interpretability of its internal mechanisms, its ability to extract meaningful patterns and reduce noise has been validated through experimental results. This trade-off between interpretability and performance is a common characteristic of advanced machine learning models and does not diminish the practical utility of the approach.

It should be noted that overfitting was mitigated by dividing the data into three distinct datasets: a training dataset, a hold-out cross-validation dataset, and a test dataset. The model was trained on the training dataset, and hyperparameters were optimized based on the performance on the hold-out cross-validation dataset. Finally, the model’s generalization ability was evaluated using the test dataset. This process ensured that the model did not overfit the training data, thereby maintaining reliability and robustness when applied to unseen data. By separating these datasets and following this rigorous validation strategy, the model’s performance and generalization capabilities were effectively validated.

The monitoring data are divided into two distinct sets: (1) the training set, comprising the first 140,000 records of the data, (2) the hold-out cross-validation dataset, between 140,000 and 1,500,000, and (3) the test dataset, representing the remaining data from 150,000 to 180,000 records. The hyperparameters are determined through an experimental analysis of the model performance using hold-out cross-validation datasets. A crucial preprocessing step involved standardizing the input features via the “StandardScaler,” ensuring a consistent scale across the diverse dataset. This preparation was vital for the LSTM network to effectively process and learn from the temporal sequences.

The selected architecture for the LSTM model included six input neurons, which encompassed LARA sensors at Nodes 1, 4, and 5, in conjunction with PCB sensors at Nodes 2, 3, and 6 as can be seen in Figure 3(a). This unique combination of input data, spanning across different sensor types and locations, is chosen to encapsulate a broad spectrum of the bridge’s dynamic behavior. The training dataset, aimed at guiding the model’s learning process, comprised the responses from PCB sensors 1, 4, and 5 in its output layer. This selection was instrumental in refining the model’s ability to accurately predict the bridge’s dynamic characteristics, underpinned by the high-quality measurements provided by these specific PCB sensors. For the test dataset, the output dataset is represented as ILARA outputs which are enhanced versions of LARA measurement for Nodes 1, 4, and 5. The model’s architecture is designed with a sequence of three LSTM layers, each consisting of 30 units and one feed-forward layer. Figure 7 shows the models’ architecture.

The ReLU activation function is used in LSTM layers. The model variables for the considered LSTM are listed in Table 2.

The resultant ILARA output, which indicates the model’s enhanced data acquisition capabilities, represents a significant improvement in the quality of insights derived from LARA sensors. This effort to leverage deep learning to enhance SHM data quality through LARA sensors, while strategically integrating PCB sensor data for model training, signifies a major advancement. It not only increases the feasibility of using fewer high-cost PCB sensors across extensive bridge networks but also moves us toward a more cost-effective and reliable approach to SHM.

Figure 8 presents a series of scatter plots that compare PCB accelerometer measurements (horizontal axis) with two types of LARA sensor outputs (vertical axis). Each row corresponds to a specific sensor node on the bridge: the left plot in each row displays raw LARA data (i.e., uncalibrated) versus PCB, while the right plot shows ILARA data (i.e., calibrated via the LSTM model) versus PCB. Every individual point on the plots represents a paired measurement from the PCB and LARA sensors. The red dashed line (y = x) indicates where the two measurements would be in perfect agreement. Comparing the left and right columns reveals that the calibrated ILARA data exhibit improved alignment with the PCB measurements, illustrating the effectiveness of the proposed LSTM–based calibration approach in reducing noise and enhancing measurement accuracy.

The analysis of the scatter plots in Figure 8 reveals a clear improvement in data accuracy achieved through the ILARA calibration process. In the left column, which compares PCB reference data with raw LARA measurements, there is noticeable scatter indicating inconsistencies and lower precision in the raw LARA data relative to the PCB sensor. In contrast, the right column shows that the ILARA data are more closely aligned with the ideal (y = x) line, signifying enhanced consistency and accuracy after calibration.

To quantify the signal recovery capabilities of the LSTM–based network, a detailed error analysis was conducted, comparing the raw LARA sensor data and the enhanced ILARA data against the reference PCB sensors. The mean squared error (MSE), root MSE (RMSE), and mean absolute error (MAE) were computed for each monitored node to provide a localized assessment of the corrections achieved by the LSTM–based model. The results presented in Table 3 illustrate a significant reduction in error across all nodes, demonstrating the effectiveness of the proposed AI–driven calibration approach.

The analysis of Table 3 shows that, compared to the uncorrected LARA data, the ILARA outputs exhibit a substantial decrease in MSE, RMSE, and MAE at each node. Overall, these results demonstrate that the current ILARA calibration approach is effective to a certain extent. However, the exact scope of these improvements can only be determined through a subsequent modal analysis, which will help identify which signal components are most improved and uncover any previously hidden mode shapes now discernible in the calibrated data.

It should be noted that the application of AI to low-cost sensors is a relatively novel approach, and this study represents an early attempt to validate its feasibility. Currently, the LSTM model must be trained specifically for each bridge to account for its unique characteristics and conditions. Although the results presented here are case-specific, they provide a foundation for broader applications and demonstrate the potential of this methodology in SHM.

Future research will focus on enhancing the scalability and versatility of this approach by optimizing the methodology for bridges with varying structural characteristics. One promising direction is the use of transfer learning, which would allow a model trained on one bridge to be adapted to another with minimal additional training. This advancement could significantly reduce the need for extensive retraining, improving both the practicality and generalizability of the proposed method.

The bridge selected for this study was chosen due to its specific structural conditions, which provided an ideal setting to validate the methodology under controlled circumstances. The complexity of the selected bridge also underscores the robustness of the proposed model in handling challenging structural dynamics. In addition, the algorithm’s reliance on temporal calibration of low-cost sensors using commercial ones makes it inherently adaptable to other systems. Such calibration can be seamlessly integrated into standard bridge inspection protocols. For newly constructed bridges, calibration aligns naturally with mandatory static and dynamic load tests often required by regulatory standards. For older bridges, it can be incorporated into scheduled dynamic load tests using temporary instrumentation, ensuring compatibility with existing maintenance practices and maximizing the utility of the proposed system.

4.1. Eigenfrequencies and Mode Shapes From PCB Sensors

This subsection presents the FDD [87] analysis, conducted using Z-axis data from PCB sensors.

Figure 9 illustrates the selected tips for the FDD analysis based on data from PCB sensors at Nodes 1, 4, and 5 (Figures 3(a)).

To gain deeper insights into the mode shapes of the bridge, additional modal analysis was performed using data from PCB sensors located across Nodes 1 through 6. The comprehensive dataset from all nodes facilitated the creation of Figure 10, which displays the bridge’s mode shapes for the first 16 modes.

4.2. FDD Analysis of LARA Sensors Data

In this subsection, attention is turned towards the modal value estimations derived from the LARA sensors, employing the FDD methodology consistent with the preceding analysis. Figure 11 displays selected tips for the FDD analysis based on data from LARA sensors at Nodes 1, 4, and 5.

The comparative analysis between Figures 9 and 11 highlights the differences in accuracy and resolution between LARA sensors and PCB sensors. It is evident that while PCB sensors detected more than 16 modes, LARA sensors identified only 5 tips. It is important to note that an equal amount of data from both LARA and PCB sensors was used for this analysis. In addition, the sampling frequencies of both systems have been synchronized. Considering these measures, the difference in resolution between the LARA sensors and the commercial PCB sensors is clearly demonstrated by analyzing Figures 9 and 11.

4.3. Insights From ILARA–Enhanced Data

In this subsection, an examination is conducted on the impact of LSTM processing on enhancing the data quality of ILARA sensors, specifically regarding the accuracy and reliability of modal value estimations. Observations on improvements or discrepancies when compared to LARA and PCB sensors’ data are highlighted, illustrating the benefits and limitations of the LSTM enhancement process.

Figure 12 displays selected tips from an FDD analysis on the ILARA–enhanced data.

The analysis of Figure 12 indicates that the accuracy of ILARA’s enhanced data has improved, although it raises concerns as the tips and the shape of the FDD closely resemble those in Figure 9. This similarity may be due to the LSTM model’s configuration, which includes six input neurons from LARA sensors at Nodes 1, 4, and 5, combined with PCB sensors at Nodes 2, 3, and 6. This strategic combination has led to better identification in the eigenfrequency analysis of the bridge. However, it is important to note that the chosen PCB nodes within the input neurons were not adjacent to the LARA nodes.

4.4. Comparative Evaluation of Modal Value Estimations

This subsection presents a comprehensive comparison of the modal values obtained from PCB, LARA, and ILARA datasets. The analysis begins by comparing the modal values of the different systems, calculating their differences relative to those of PCB sensors. The mode shapes of different systems have been also compared with each other in this subsection using the modal assurance criterion (MAC) [88] value. The MAC value is a statistical measure that quantifies the similarity between two mode shapes, ranging from 0 to 1. A value of 1 indicates identical mode shapes, while a value closer to 0 suggests a greater discrepancy [89].

Table 4 compares the modal estimations of PCB sensors with the data from LARA.

Analysis of Table 4 shows that LARA identified only 5 tips in the FDD analysis. However, the modal values measured by LARA sensors are well captured, as evidenced by MAC values near 1 when comparing the mode shapes of LARA (Nodes 1, 4, and 5) with those of PCB sensors (Nodes 1, 4, and 5), demonstrating LARA’s high capability. Nevertheless, due to LARA being a low-cost MEMS accelerometer with higher noise density compared to PCB sensors (Table 1), it was not able to identify more modes.

Table 5 compares the modal estimations of PCB sensors with the enhanced data from ILARA.

Analysis of Table 5 indicates that ILARA performed a better eigenfrequency analysis than LARA, marking an important improvement. It is also noted that ILARA identified 7 modes, six of which have a MAC value over 0.9. This high accuracy can be considered a significant step towards utilizing AI to enhance the modal analysis of low-cost sensors. However, the analysis also reveals that for higher mode numbers, the MAC values were insufficient. It is proposed that with more complex training, such as using a shaker to excite the bridge or conducting more experiments on the bridge under study, the training of ILARA could be enhanced, potentially resulting in a better system. This paper, for the first time in the literature, attempts to improve the modal analysis of low-cost sensors by combining the acquired data of low-cost and commercial sensors, which may pave the way for future improvements and advancements.

Figure 13 visualizes and compares the frequencies and MAC between PCB sensors and two sensor types: with data coming from LARA (Table 4) and ILARA (Table 5).

The bar chart on the top left of the figure compares the eigenfrequency values measured by PCB sensors with those computed for LARA. The results show minimal differences between PCB and LARA eigenfrequencies for most modes, confirming the consistency of measurements between the two systems. However, it is evident that eigenfrequencies for Modes 5–7 are unavailable, indicating gaps in LARA’s ability to detect certain higher-order frequencies. Despite this limitation, the available frequencies exhibit strong agreement.

The chart on the top right illustrates the MAC values for PCB and LARA mode shapes. MAC values close to 1.0 (100%) indicate excellent correlation, and in this Figure 13, all modes analyzed (Modes 1–3 and Mode 8) have MAC values above 0.85, reflecting strong alignment between PCB and LARA mode shapes.

The bar chart on the bottom left of the Figure 13 compares the eigenfrequency values measured by PCB sensors with those computed for ILARA. ILARA demonstrates the capability to identify up to 16 modes of eigenfrequencies, with a generally strong agreement between the two systems.

The chart on the bottom right illustrates the MAC values for PCB and ILARA mode shapes. The results are more variable compared to LARA. Modes 1–7 and Mode 10 exhibit MAC values above 0.9, indicating strong correlations. However, Modes 8, 9, 11, and 15 exhibit lower MAC values, suggesting less similarity between the mode shapes detected by ILARA and PCB. This variability highlights the challenge of detecting higher mode shapes accurately. While eigenfrequency detection remains relatively straightforward, mode shape detection for higher modes is more sensitive to noise and requires advanced noise cancellation and data correction techniques. The algorithm proposed in this paper aims to minimize data postprocessing and relies on a consistent OMA code applied to both systems, demonstrating reliable performance within these constraints.

In the context of bridge maintenance and OMA, the first mode or the first three mode shapes are typically prioritized, as they provide critical insights into the global behavior of the structure. Higher modes are often excluded from practical analyses due to their sensitivity to noise and limited relevance in standard maintenance protocols. The LARA system, by itself, can accurately identify up to the first four mode shapes. However, with the implementation of the proposed algorithm, the ILARA system extends this capability to reliably analyze up to the first seven mode shapes. This performance is more than sufficient for most OMA tests in civil engineering, ensuring that the system effectively meets the demands of SHM while maintaining simplicity and computational efficiency.