Experimental verification of DAS vibration response characteristics of equal strength beam under sinusoidal excitation.

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Learn more: PMC Disclaimer | PMC Copyright Notice Sci Rep . 2026 Apr 12;16:17009. doi: 10.1038/s41598-026-47097-2 Experimental verification of DAS vibration response characteristics of equal strength beam under sinusoidal excitation Chenyang Ma 1, ✉ , Jing Chai 1, 2 , Dingding Zhang 1, 2, ✉ , Zhi Li 1 , Lei Gao 1 Author information Article notes Copyright and License information 1 School of Energy and Mining Engineering, Xi’an University of Science and Technology, Xi’an, 710054 Shaanxi China 2 Key Laboratory of Western Mining and Disaster Prevention, Ministry of Education, Xi’an University of Science and Technology, Xi’an, 710054 Shaanxi China ✉ Corresponding author. Received 2026 Jan 16; Accepted 2026 Mar 30; Collection date 2026. © The Author(s) 2026 Open Access This article is licensed under a Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 International License, which permits any non-commercial use, sharing, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons licence, and indicate if you modified the licensed material. You do not have permission under this licence to share adapted material derived from this article or parts of it. The images or other third party material in this article are included in the article’s Creative Commons licence, unless indicated otherwise in a credit line to the material. If material is not included in the article’s Creative Commons licence and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder. To view a copy of this licence, visit http://creativecommons.org/licenses/by-nc-nd/4.0/ . PMC Copyright notice PMCID: PMC13230758  PMID: 41968139 Abstract Distributed Acoustic Sensing (DAS) technology holds great potential for structural dynamic monitoring; however, its quantitative capability for complex dynamic strain fields and the reliability of spatial resolution require systematic verification. This study employed a systematic “theoretical modeling–simulation analysis–experimental verification” approach, using an equal-strength beam under sinusoidal excitation to investigate the dynamic response characteristics of DAS. A complete dynamic strain transfer model from external excitation to DAS phase-change signals was established, defining the phase sensitivity coefficient as a key parameter for quantitative analysis. Finite element simulations revealed that while the beam’s strain is approximately uniform at low frequencies (< 5 Hz), it exhibits a gradient distribution—large near the fixed end and small near the free end—near the natural frequency, departing from the static “equal-strain” characteristic. Experimental results under 5 Hz excitation, utilizing synchronized measurements from accelerometers, FBG sensors, and DAS, demonstrated that DAS accurately captures the dominant vibration frequency. The measured phase sensitivity coefficient showed only a 7.3% error relative to theory, the response amplitude maintained high linearity with excitation and reference signals (R² ≥ 0.99), and the phase delay was negligible (0.02 rad). By incorporating the spatial averaging effect of the DAS gauge length, the theoretical strain ratio across channels is corrected, reducing the deviation from 23.4% to 1.7%, confirming DAS’s ability to quantitatively resolve spatial differences in dynamic strain distribution. Additional 50 Hz experiments validate the frequency applicability threshold and demonstrate DAS’s utility for event detection beyond the uniform-strain regime. Furthermore, by comparing DAS channel response ratios across spatial positions with theoretical strain ratios, the study confirmed DAS’s ability to quantitatively resolve spatial differences in dynamic strain distribution. This work systematically validates the quantitative accuracy and reliability of DAS as a distributed dynamic strain sensing tool, and the revealed dynamic-static strain distribution differences, together with the established validation methodology, provide solid theoretical and experimental support for refined DAS-based monitoring in engineering structure dynamic monitoring scenarios. Keywords: Distributed acoustic sensing (DAS), Equal-strength beam, Dynamic strain monitoring, Vibration response characteristics, Phase sensitivity coefficient Subject terms: Engineering, Materials science, Physics Introduction Structural Health Monitoring (SHM) is a core component of safe operation and maintenance in modern engineering, aiming to realize damage warning and lifespan assessment through real-time perception of changes in the physical state of structures 1 – 4 . Among various monitoring technologies, distributed optical fiber sensing technology has emerged as a research and application focus, benefiting from its advantages including immunity to electromagnetic interference, corrosion resistance, and the capability of long-distance continuous measurement 5 . Specifically, Distributed Acoustic Sensing (DAS), which relies on the phase-sensitive characteristics of coherent Rayleigh scattering, can acquire dynamic strain information along the fiber path with high spatial resolution. This provides a novel technical approach for capturing dynamic responses (e.g., vibration and impact) in structures 6 – 8 . Initially, research and applications of Distributed Acoustic Sensing (DAS) were primarily focused on fields such as safety monitoring of oil and gas pipelines, perimeter protection of critical zones, and seismic wave detection, with the core objective of determining the occurrence, type identification, and localization of vibration events 9 – 11 . With technological advancements, its application scenarios have expanded to structural health condition monitoring. For instance, Cheng Lun Kai et al. 12 systematically investigated the vibration responses of bridges and tunnels under traffic loads using DAS, verifying its feasibility in identifying vehicle loads and fundamental frequencies; Kinzo Kishida et al. 13 employed DAS-based dynamic monitoring to reveal the vibration response characteristics of bridges induced by train bogies; Julie Rodet et al. 14 utilized existing fiber-optic networks to simultaneously extract health indicators (e.g., vibration frequency, damping, and mode shapes) of multiple bridges, and further revealed periodic variations in the dynamic characteristics of bridges associated with temperature changes. Regarding the quantitative measurement of structural dynamic responses, James T. Xu et al. 15 conducted a shaking table test on an offshore wind turbine model, where they synchronously deployed DAS (Φ-OTDR) and OFDR to successfully capture the dynamic strain profile of the structure under simulated environmental loads. They also identified the local effects of bolt loosening at flange connections, thus verifying DAS’s capability to capture both local and global structural dynamic phenomena. Abdulfatah A. G. Abushagur et al. 16 found that the strain transfer efficiency of cables with different structures and protective layers differs significantly, which directly influences the amplitude of DAS signals. In addition, the robustness of monitoring results under complex environmental conditions is a common challenge faced by various technical means in structural health monitoring (SHM). For example, aiming at the problem of tunnel lining crack segmentation in complex environments, Duan et al. 17 proposed the DSYOLO model based on dynamic snake convolution, which demonstrates the importance of algorithm robustness in complex field environments. This study indicates a potential direction for future multi-sensor fusion: the combination of quantitative dynamic strain data provided by DAS and structural surface condition assessment from vision-based inspection can achieve a more comprehensive structural health diagnosis, which also puts forward higher requirements for the quantitative accuracy and environmental adaptability of DAS distributed monitoring technology. Despite igsnificant progress in the aforementioned research, the goal of establishing DAS as a reliable and universal quantitative sensing tool for dynamic strain fields still confronts core challenges 18 – 20 . Most existing studies are primarily focused on utilizing DAS signal-amplitude for event identification and frequency extraction. For the original physical quantity directly measured by DAS systems—the phase change rate—there remains a lack of systematic experimental calibration and theoretical verification regarding how to precisely establish a general quantitative mapping relationship with the actual dynamic strain rate on the structure surface. Furthermore, for typical mechanical models such as the equal-strength beam (which exhibits a uniform strain field under static conditions but shows strain distribution variations corresponding to mode shapes under dynamic conditions), experimental verification of whether DAS can quantitatively distinguish and reconstruct the spatial distribution characteristics of such dynamic strain gradients remains inadequate. This forms a key bottleneck in the current transition of DAS technology from “trend perception” to “accurate measurement.” The equal-strength beam, as a classical mechanical model, serves as an ideal platform for isolating and verifying the strain response of sensing systems, owing to its static “equal-stress” property 21 , 22 . This paper adopts it as the research subject, aiming to systematically investigate the dynamic response characteristics of DAS under sinusoidal excitation. A theoretical model correlating the DAS phase-change rate with the strain rate and acceleration of the equal-strength beam is established via theoretical derivation. ANSYS is employed to perform static and harmonic response simulations, thereby determining the test parameters. Finally, a comparative test system comprising DAS, FBG sensors, and accelerometers is established to examine the response linearity, phase characteristics, and distributed sensing capability of DAS under sinusoidal excitation. The core innovations of this study are reflected in the following three aspects. First, the study breaks through the limitation that most existing studies rely on DAS signal amplitude for qualitative analysis, and realizes the accurate quantitative mapping between the phase change rate (the original physical quantity directly measured by DAS) and the actual dynamic strain rate on the structural surface, with only 7.3% relative error between the measured phase sensitivity coefficient and the theoretical calculated value. Second, it reveals the essential difference between the static and dynamic strain distribution of the equal-strength beam, clarifies the low-frequency threshold of < 5 Hz for reliable quantitative measurement of DAS, and makes up for the defect that existing DAS calibration studies generally ignore the dynamic strain gradient characteristics of the calibration model. Third, it establishes a standardized verification system of “theoretical modeling–simulation analysis–experimental verification”, quantitatively corrects the 23.4% amplitude deviation caused by the spatial averaging effect of the DAS gauge length, and provides a reproducible methodological reference for subsequent DAS dynamic strain calibration research. The research results can provide theoretical support and experimental basis for dynamic strain field monitoring of engineering structures. Theoretical analysis of vibration monitoring for equal-strength beam based on DAS DAS sensing principle When an optical fiber is subjected to external physical parameters (e.g., vibration, stress, and temperature), its length and core refractive index undergo changes, which induce phase variations in the backscattered Rayleigh light. DAS technology realizes distributed vibration sensing by injecting coherent laser pulses into a long-span optical fiber and detecting phase changes in the backscattered light 23 , 24 . The measurement principle of DAS is illustrated in Fig. 1 . Fig. 1. Open in a new tab Schematic diagram of DAS measurement principle. The fiber phase change Δ ϕ is mainly composed of the fiber length change Δ and the refractive index change Δn caused by the photoelastic effect 25 , 26 : 1 Where, Δ ϕ is the phase of the backscattered light (rad); n is the refractive index of the optical fiber; λ is the wavelength of the laser pulse (m); L f is the gauge length of the DAS system (m); is the axial strain (µε). When the optical fiber is subjected to axial strain ε, it is accompanied by lateral contraction strains . When considering the coupling of lateral and longitudinal strains, the relationship between the refractive index change Δn induced by the photoelastic effect and the strain ε is expressed as: 2 Where: are the photoelastic coefficients; is the Poisson’s ratio of the fiber. Substituting Eq. (2) into Eq. (1) yields the relationship between the phase change monitored by DAS and the axial strain of the optical fiber as: 3 Where: is the fiber optic strain correction coefficient, which takes into account the influence of the elasto optical effect. The DAS system demodulates the phase signal at each sensing point and calculates its time derivative to obtain the phase change rate (i.e., the DAS phase change rate . By taking the time derivative on both sides of Eq. (3), the following expression is obtained: 4 Where 、 、 、 are all time-invariant constants, Eq. (4) demonstrates that, in theory, the DAS phase change rate exhibits a linear proportional relationship with the strain rate. The proportionality parameter is codetermined by the optical and mechanical parameters of the optical fiber. Theoretical analysis of equal-strength beam vibration The equal-strength beam is designed as a cantilever beam with linearly varying width along its length, so as to ensure uniform maximum normal stress at each cross-section. Let the beam length be , its thickness be h , and its fixed-end width be . Under static loading conditions, when a concentrated force F is applied at the free end, the theoretical strain of the equal-strength beam remains constant 27 : 5 When sinusoidal excitation is applied at the free end of the equal-strength beam (i.e., a harmonic force is applied at the free end), the beam undergoes forced vibration. According to Euler-Bernoulli beam theory, when the excitation frequency ω is close to the first-order natural frequency, the amplitude of higher-order modal responses is considerably smaller than that of the first-order mode. Thus, the displacement response is dominated by the first-order mode 28 , 29 : 6 Where: ψ ₁ (x) is the first-order mode shape function, is the modal response amplitude, is the phase. The standard form of the first-order mode shape for a cantilever beam is . The relationship between the axial strain on the surface of the equal-strength beam and its curvature is: 7 Substituting Eq. (6) into Eq. (7) yields the theoretical dynamic strain on the surface of the equal-strength beam under sinusoidal excitation: 8 The acceleration on the surface of the equal-strength beam corresponds to the second-order time derivative of the displacement: 9 Accordingly, the amplitude of the acceleration is given as: 10 Taking the time derivative of Eq. (8) yields the theoretical dynamic strain rate on the surface of the equal-strength beam: 11 Theoretical relationship between DAS phase change rate and vibration response of equal-strength beam Spatial averaging effect of the DAS gauge length The DAS system, based on coherent Rayleigh scattering, measures the phase change over a finite fiber segment rather than at a discrete point. For a given channel located at with gauge length , the output phase-change rate corresponds to the spatial average of the true strain rate over the interval . This process can be mathematically described as a convolution of the theoretical strain rate distribution with a rectangular window function: 12 Where: is the theoretical dynamic strain rate on the beam surface given by Eq. (11). Substituting Eq. (11) into Eq. (12) yields the amplitude of the DAS-measured equivalent strain rate: 13 Equations (12) and (13) quantify how the gauge length attenuates the measured amplitude relative to the point-wise theoretical value. A larger gauge length results in stronger spatial averaging and greater attenuation, especially in regions with high strain gradients. This spatial averaging effect is inherent to distributed sensing and must be accounted for when comparing DAS measurements with point-wise theoretical predictions or discrete sensor data. The spatially averaged strain rate obtained from Eq. (12) represents the effective mechanical input to the DAS system over the gauge length. This quantity is then converted to the DAS phase change rate through the opto-mechanical relationship established in Eq. (4). Due to the linear proportionality between phase change rate and strain rate, the ratio of phase change rates between two DAS channels is identical to the ratio of their corresponding spatially averaged strain rates. (2) Relationship with Vibration Response. To establish the relationship between the DAS phase change rate and the surface strain rate of the equal-strength beam, it is necessary to account for both the fiber-structure strain transfer effect and the frequency response characteristics of the DAS system. For this purpose, a complex phase sensitivity coefficient is introduced, where its magnitude represents the proportional relationship between the DAS output amplitude and the input strain rate amplitude, and its phase represents the phase delay introduced by the system. Consequently, the relationship between the measured DAS phase change rate and the theoretical dynamic strain rate on the beam surface is given by: 14 Substituting Eq. (11) into Eq. (14) and considering the time delay yields: 15 To eliminate the modal amplitude , which is difficult to measure directly, the acceleration signal can be used for normalization. From Eq. (10), . Substituting this into Eq. (15) gives the time-domain expression of the DAS phase change rate in terms of the acceleration amplitude: 16 Equation (16) completely describes the temporal evolution of the DAS phase change rate, incorporating both the amplitude response and the phase delay information of the system. Taking the amplitude of Eq. (16) yields the amplitude relationship convenient for experimental calibration: 17 By synchronously measuring the DAS phase change rate amplitude and the acceleration amplitude, combined with the known mode shape function and its second derivative , the phase sensitivity coefficient can be calibrated. This coefficient comprehensively reflects the strain transfer efficiency and the frequency response characteristics of the DAS system, and serves as the core parameter for quantitative analysis in this paper. Test design and theoretical verification based on finite element simulation Static simulation and theoretical model validation The ANSYS finite element analysis (FEA) software was employed to establish an equal-strength beam model for investigating its static characteristics. The distance from the fixed end to the load application point (denoted as l ) is 1200 mm; the width at the fixed end ( b 0 ) is 175 mm, and the thickness ( h ) is 5 mm. The beam is made of QA19-4 material, with a density of 7500 kg/m 3 , an elastic modulus of 116 GPa, and a Poisson’s ratio of 0.3. Following meshing, the model consists of 6685 elements and 20,668 nodes. Five equally spaced measurement points are arranged along the top surface of the beam, as illustrated in Fig. 2 . Fig. 2. Open in a new tab Model establishment and grid division. ( a ) Model size and measurement point location, ( b ) Grid division. A fixed constraint is imposed on the left end of the model, and static concentrated loads ranging from 10 N to 90 N are applied to the right end to conduct the simulation. The results indicate that under different static loads, the surface strain of the equal-strength beam presents an overall uniform distribution. The maximum strain variation arises at the intersection of the isosceles triangular and rectangular regions in the beam’s structure. The strain distribution in the left half of the beam surface remains largely consistent, while in the right half, it gradually decreases as it approaches the static load application point, as illustrated in Fig. 3 . Fig. 3. Open in a new tab The displacement variation diagram of the beam surface under static load. ( a )10 N, ( b )50 N, ( c )90 N. The average strain values on the beam surface were extracted and compared with the values calculated via the theoretical formula (5), as illustrated in Fig. 4 . The error remains minimal, with a maximum value of 0.0072 under a load of 90 N. As the load increases, minor plastic deformation arises locally in the beam, and slight elastic deformation emerges at the fixed-end constraint—which deviates from the ideal fixed boundary. To balance between measurement accuracy and signal strength while avoiding material nonlinear effects, the amplitude of the subsequent dynamic excitation force was selected as 50 N. Under this load, the stress is approximately 156 MPa (well below the yield strength of QA19-4, i.e., 350 MPa), and the corresponding error is only 0.0034 µε. Fig. 5. Open in a new tab Modal Analysis of Equal Strength Beam. ( a ) The first nine order natural frequency of equal strength beam, ( b ) The first order modal vertical displacement normalization results. Fig. 4. Open in a new tab Comparative Analysis of Beam Surface Strain Change and Theoretical Change Value under Static Load. Modal analysis and theoretical parameter extraction To obtain the first-order mode shape function and its second derivative for the equal-strength beam, first-order modal simulation analysis was conducted on the beam structure. The first nine natural frequencies are presented in Fig. 5 (a). Forty-nine measurement points were arranged along the centerline of the beam’s top surface at 25 mm intervals. At each point, the vibration displacement (perpendicular to the length direction of the beam) under the first-order mode was extracted. MATLAB was employed to normalize the displacement data. Based on the standard form of the first-order mode shape for an Euler-Bernoulli cantilever beam, a least-squares fitting procedure was carried out, as illustrated in Fig. 5 (b). The coefficient of determination R 2 for the fitting result was 0.99, which indicates high fitting accuracy. Considering the discrepancy between the actual boundary constraints of the beam and those of an ideal cantilever beam, the standard mode shape was modified to derive the first-order mode shape function of the equal-strength beam: 18 Where: 1.01 is the boundary constraint correction coefficient, obtained by fitting the deviation between measured displacement and the standard mode shape, reflecting the difference between actual and ideal constraints. Taking the second derivative of Eq. (18): 19 Harmonic response and vibration frequency threshold analysis of equal-strength beam To analyze the dynamic response characteristics and vibration frequency threshold of the equal-strength beam under different excitation frequencies, harmonic response analysis was conducted. A 50 N sinusoidal load was applied at the free end of the equal-strength beam, with the frequency range set to 0 to 100 Hz and a damping ratio of 0.02. The maximum displacement of the beam surface at different frequencies is presented in Fig. 6 . As the frequency increased, the maximum displacement exhibited a periodic “increase-decrease” trend: at 10 Hz, the displacement increased sharply and was much higher than that under static loading; at 30 Hz, the displacement decreased sharply. From 30 to 100 Hz, all displacements were smaller than the displacement under static loading. Additionally, displacement peaks appeared at 10 Hz, 39 Hz, and 93 Hz, which corresponded to the first, second, and third natural frequencies of the beam. Fig. 6. Open in a new tab Variation of maximum displacement of equal strength beam with frequency under harmonic response. The strain contour of the equal-strength beam under harmonic response is presented in Fig. 7 . As the frequency increased, the uniform distribution feature of the surface strain was disrupted, and the strain distribution became increasingly non-uniform at higher frequencies. Fig. 7. Open in a new tab Cloud Diagram of Surface Strain Change of Equal Strength Beam under Harmonic Response. The harmonic response results for displacement at each measurement point are presented in Fig. 8 . The frequencies corresponding to the maximum peaks at different positions on the beam surface are largely consistent, and are mainly concentrated around 10 Hz, 39 Hz, and 93 Hz. As the frequency increased, the variation in average displacement gradually decreased, which is consistent with the natural frequencies of the equal-strength beam. Under conditions far from the resonance frequencies, the average displacement amplitudes at each measurement point changed little. Fig. 8. Open in a new tab Harmonic response analysis of equal strength beam displacement change of each measuring point. The harmonic response results for strain at each measurement point are presented in Fig. 9 . Strain peaks also appeared at 10 Hz, 39 Hz, and 93 Hz; however, the peak strains (ranked from largest to smallest) corresponded to 39 Hz, 93 Hz, and 10 Hz. When the frequency was less than 5 Hz, the maximum and minimum relative deviations of strain values at each measurement point were 11.2% and − 12.8%, respectively, with average relative deviations ranging from − 5.44% to 8.49%. Therefore, the subsequent excitation test controlled the dynamic load frequency to be ≤ 5 Hz, so as to ensure the accuracy and interpretability of the test results. Fig. 9. Open in a new tab Harmonic response analysis of equal strength beam strain change. ( a ) Strain change, ( b ) Strain relative deviation. Analysis of DAS response characteristics under sinusoidal excitation of equal-strength beam Test system To investigate the response characteristics of the DAS system to sinusoidal excitation vibration signals of the equal-strength beam, comparative tests were conducted with accelerometer and FBG sensor data serving as references. The test system consisted of an excitation system and a monitoring system, as illustrated in Fig. 10 . Fig. 10. Open in a new tab Sinusoidal excitation test system layout diagram of equal strength beam. ( a ) Top view of sensor layout, ( b ) Front view of sensor layout, ( c ) Excitation test platform and sensor physical map. The excitation system mainly included the equal-strength beam, a personal computer (PC), a data acquisition unit, a power amplifier, and an exciter. The PC output a sinusoidal excitation signal, which was amplified by the power amplifier to drive the exciter, thereby applying a dynamic load at the free end of the beam. The monitoring system mainly included the DAS monitoring system, FBG strain monitoring system, and the accelerometer monitoring system. Accelerometers were placed equidistantly along the length of the beam at 200 mm intervals: A1 ~ A5 corresponded to measurement points A ~ E in the numerical simulation, while A6 was placed at the tip of the free end (with its direction opposite to that of A1 ~ A5) to capture acceleration responses at different positions of the beam. FBG sensors were attached at the positions corresponding to those of the accelerometers, with a bonding length of 20 mm, adhesive thickness of 0.1 mm, and bonding width of 2 mm. The distributed optical fiber was attached along the centerline of the beam’s top surface using the same bonding process as that for the FBG sensors to ensure identical strain transfer boundary conditions. The specific bonding process was as follows: first, the surface of the equal-strength beam was polished with 400-mesh sandpaper to remove the surface oxide layer, and then wiped with anhydrous ethanol to ensure the bonding surface was clean and free of impurities; second, the two-component epoxy resin adhesive was evenly applied to the polished area, the optical fiber was kept in a straight and slightly tensioned state during the bonding process to avoid initial bending strain; finally, the bonded optical fiber was cured at room temperature for 24 h to ensure the adhesive layer was completely cured, and the strain transfer between the optical fiber and the beam surface was stable. Based on the four-layer bonding structure strain transfer model, the strain transfer efficiency η of the FBG sensor is calculated as 27 : 20 Where: is the bonding length of the FBG, and is the shear lag coefficient related to the shear modulus of the adhesive layer and the elastic moduli of the fiber and substrate. The strain transfer calculation and bonding process adhere to the following core assumptions: (1) No relative slip occurs at the interfaces between the FBG fiber core, protective layer, adhesive layer, and beam substrate, ensuring coordinated deformation and uniform shear stress distribution; (2) The adhesive layer is treated as a homogeneous isotropic linear elastic material, with stable mechanical properties under the test temperature range; (3) The test is conducted in a constant temperature environment, and the temperature-induced strain of FBG is negligible compared to the vibration-induced strain, thus the cross-sensitivity of temperature and strain is ignored; (4) Only axial strain along the beam length direction is considered, and transverse strain, bending strain, and torsional strain are excluded due to the uniform low-frequency vibration characteristics. Substituting the bonding parameters and material parameters of this experiment, the theoretical strain transfer efficiency of the FBG sensor is calculated as , corresponding to a relative uncertainty of introduced by the adhesive bonding. The gauge length of the DAS system was set to m, with a channel spacing of 0.5 m. The laser wavelength was 1550 nm, and the phase demodulation accuracy was 0.01 rad. The sampling frequencies of the DAS demodulator, accelerometer data acquisition unit, and FBG demodulator were all set to 1 kHz. Through tapping tests at measurement points A–E, it was confirmed that points A–C (axial positions 200 mm, 400 mm, 600 mm) correspond to DAS Channel 13, whose central position is 400 mm and axial coverage is 150–650 mm. Points D–E (axial positions 800 mm, 1000 mm) correspond to DAS Channel 14, whose central position is 900 mm and axial coverage is 650–1150 mm. Results analysis Results analysis at 5 Hz excitation A sinusoidal excitation (with a frequency of 5 Hz and an amplitude of 500 mV) was applied at the free end of the equal-strength beam. All sensors were able to clearly and stably capture the vibration signal. As illustrated in Fig. 11 (a), the output signals of the accelerometers (A1–A6) were standard sinusoidal waveforms with a period of 0.2 s—this proves that the excitation system generates ideal linear forced vibration in the low-frequency range. Fig. 11. Open in a new tab The variation diagram under the action of dynamic load of 5 Hz sine wave. ( a ) Acceleration value, ( b ) FBG strain rate, ( c ) DAS phase change rate. Figure 11 (b) showed the dynamic strain signals monitored by the FBG sensors, which exhibited regular sinusoidal variations with a period that completely matched the excitation frequency. This preliminarily verified the accurate response of the FBG sensors to the surface strain of the equal-strength beam. Figure 11 (c) showed the phase-change rate signal demodulated by the DAS system. Although its time-domain waveform showed periodicity overall (at the same frequency as the excitation), obvious waveform distortion and high-frequency noise were observed. Fast Fourier Transform (FFT) was performed on 10 s of raw data from measurement point A, and the logarithmic transformation was applied to the results. These results are presented in Fig. 12 . Fig. 12. Open in a new tab 5 Hz, 500 mV sine excitation spectrum after FFT transform. ( a ) Acceleration value, ( b ) FBG strain rate, ( c ) DAS phase change rate. The spectra of the acceleration signal, FBG strain rate signal, and DAS phase-change rate signal all showed a single sharp peak at 5 Hz, with energy highly concentrated at this frequency. The calculated signal-to-noise ratios (SNR) at 5 Hz were 45 dB, 42 dB, and 38 dB, respectively. The amplitude of the nonlinear harmonic components was less than 5% of that of the fundamental frequency, indicating that the system was in a good linear operating state. All three monitoring methods were able to accurately recover the main frequency information of the external disturbance. To ensure the accuracy of subsequent quantitative analysis, a 4th-order Butterworth low-pass filter was applied to the raw signals based on the FFT results. The cut-off frequency of the filter was set to 100 Hz, and the zero-phase forward-backward filtering algorithm was used to avoid phase distortion of the original sinusoidal signal. The same filter parameters were used for the acceleration, FBG strain rate and DAS phase change rate signals to ensure the consistency of signal processing. The filtering effect is presented in Fig. 13 . The filtered signal remained undistorted, and high-frequency noise was effectively suppressed. The SNR of the filtered DAS phase-change rate signal increased to 41 dB. Fig. 13. Open in a new tab Comparison of filtering effect under 5 Hz, 500mV sinusoidal excitation. ( a ) Acceleration value, ( b ) FBG strain rate, ( c ) DAS phase change rate. Comprehensive time-domain and frequency-domain analysis showed that under 5 Hz excitation: the accelerometer signals had the highest regularity and fidelity; the FBG sensors accurately reflected the dynamic strain response on the beam surface; while the DAS system exhibited slight time-domain waveform distortion, its main frequency amplitude variation trend was completely consistent with that of the accelerometer and FBG signals. This consistency verifies the reliability of the accelerometer and FBG measurements, which can thus serve as a benchmark for validating the effectiveness of DAS monitoring. Comparative analysis at 50 Hz excitation Based on the 5 Hz sinusoidal excitation tests, the monitoring response characteristics of acceleration, FBG strain rate, and DAS phase change rate at measurement point A were investigated under a sinusoidal dynamic load with an amplitude of 1000 mV and a frequency of 50 Hz. The 50 Hz excitation test maintains consistent experimental conditions with the 5 Hz test to ensure comparability. Steady-state monitoring data within a 1-second interval were selected, and the time-domain waveforms of strain rate, acceleration, and phase change rate were plotted, as shown in Fig. 14 . Fig. 14. Open in a new tab The Variation Diagram under the Action of Dynamic Load of 50 Hz Sine Wave. ( a ) Acceleration value, ( b ) FBG strain rate, ( c ) DAS phase change rate. Unlike the clean sinusoidal waveforms observed at 5 Hz (Fig. 11 ), all three signals exhibit significant waveform distortion at 50 Hz, making it impossible to reconstruct the original vibration waveform from the time-domain data. The average peak values of strain rate, acceleration, and phase change rate are 26.11 µε/s, 15.64 m/s², and 13.25 rad/s, respectively, which no longer follow the linear relationship derived. On this basis, 10 s of raw monitoring data were selected, and frequency-domain analysis of the strain rate, acceleration, and phase change rate was performed using Fast Fourier Transform (FFT). The results are shown in Fig. 15 . Fig. 15. Open in a new tab 50 Hz, 1000 mV Sine Excitation Spectrum after FFT Transform. ( a ) Acceleration value, ( b ) FBG strain rate, ( c ) DAS phase change rate. As shown in Fig. 15 a, the acceleration signal monitored by the accelerometer exhibits characteristic peaks near 50 Hz, 100 Hz, 150 Hz, 200 Hz, 250 Hz, 350 Hz, and 450 Hz. This signal is also periodic with 50 Hz as the fundamental frequency, consistent with the experimental excitation conditions. Figure 15 (b) shows that the strain rate signal monitored by the FBG sensor exhibits characteristic peaks near 50 Hz, 100 Hz, 150 Hz, 200 Hz, and 250 Hz, indicating that this strain rate signal has 50 Hz as its fundamental frequency with multiple harmonic overtones, matching the experimental condition of sinusoidal vibration applied at the end. As shown in Fig. 15 (c), the phase change rate signal monitored by DAS exhibits significant characteristic peaks near 2.6 Hz and 50 Hz. Compared with the frequency-domain results from the FBG and accelerometer sensors, the DAS phase change rate signal demonstrates lower sensitivity in capturing high-frequency harmonic components of the vibration but exhibits superior sensitivity in capturing low-frequency components. Additionally, the presence of a clear peak at the 50 Hz fundamental frequency in the phase change rate signal indicates that DAS can effectively identify the primary frequency component of the excitation signal and can, to a certain extent, reconstruct the main characteristics of the vibration. Comparative analysis of high and low frequency operating conditions A comprehensive comparison of the monitoring response characteristics of the three sensor types under 5 Hz low-frequency and 50 Hz high-frequency sinusoidal excitation yields the following conclusions from both time-domain and frequency-domain perspectives: Time-domain characteristics: Under 5 Hz low-frequency excitation, the time-domain waveforms of strain rate, acceleration, and phase change rate monitored by the three sensor types all exhibit standard sinusoidal forms, accurately reproducing the waveform characteristics of the excitation load applied at the beam end. Each parameter shows a synchronized variation trend with changes in excitation amplitude. Under 50 Hz high-frequency excitation, the monitored waveforms of all three sensor types exhibit significant distortion, failing to reproduce the form of the excitation load. The synchronized variation trend among the parameters disappears, and the peak values no longer conform to the theoretically derived linear mapping relationship. Frequency-domain characteristics: Under both low-frequency and high-frequency excitation, all three sensor types can effectively identify the fundamental frequency component (5 Hz/50 Hz) of the excitation signal, consistent with the experimental excitation conditions. Among them, the FBG and accelerometer sensors are more sensitive in capturing high-frequency harmonics, while the DAS system exhibits a frequency-domain response characterized by “high sensitivity in capturing low frequencies, but relatively weak identification of high-frequency harmonics.” Model validation: The experimental phenomena observed under the 50 Hz high-frequency condition are fully consistent with the harmonic response simulation conclusions for the equal-strength beam. This validates that when the excitation frequency exceeds the 5 Hz threshold, the “equal-strain” characteristic of the equal-strength beam surface is disrupted, and the original theoretical derivation formulas based on the uniform strain assumption are no longer applicable. This is the fundamental reason for the disappearance of the linear relationships among the various parameters. Discussion To verify the reliability of the DAS monitoring system and establish a precise quantitative relationship between the DAS system’s monitoring signals and the surface strain of the equal-strength beam, variations in key parameters under sinusoidal dynamic loads with the same frequency but different amplitudes were analyzed based on the filtered signals. Phase sensitivity coefficient analysis To quantitatively derive the phase sensitivity coefficient, two approaches were employed respectively: an analysis method based on direct FBG measurements, and a comparison method based on accelerometer data and the simulation model. The actual strain rate on the surface of the equal-strength beam was calculated from the bonding parameters and strain transfer ratio of the FBG sensor 27 . Linear fitting was performed between this actual strain rate and the measured peak phase-change rate, as illustrated in Fig. 16 . According to Eq. (14), the slope of the fitted curve corresponds to the magnitude of the phase sensitivity coefficient at 5 Hz, which was determined to be 2.52714 rad/µε. Fig. 16. Open in a new tab Corresponding Relationship between Phase Transformation Rate and Actual Strain Rate under Different Amplitude Dynamic Loads. For the DAS system, the gauge length was set to L f =0.5 m, and the system wavelength was . The single-mode fiber parameters were , , , . Substituting these values into Eq. (4), the theoretical phase sensitivity coefficient is obtained: By substituting these values into Eq. (4), the theoretical phase sensitivity coefficient was calculated as 2.355 rad/µε. The error between the theoretically derived coefficient and the actual monitoring coefficient was only 7.3%, which indicates that the DAS phase-change rate effectively represents the strain rate of the equal-strength beam under sinusoidal excitation. Based on the measurement uncertainty propagation guidelines 30 , the 7.3% total error was quantitatively decomposed into the following uncertainty sources and contributions: 1) FBG strain transfer uncertainty , introduced by the shear lag effect of the adhesive bonding 27 , 31 ; 2) DAS phase demodulation uncertainty , introduced by the system’s 0.01 rad phase demodulation accuracy 32 ; Acceleration measurement uncertainty , given by the accelerometer’s calibration accuracy; Data acquisition synchronization uncertainty , introduced by synchronization errors among the three sensors sampling at 1 kHz 33 ; Theoretical model and actual boundary deviation , the core source of the total error, arising from differences between the actual beam boundary and the ideal cantilever model, as well as residual effects of the gauge length spatial averaging. The combined relative standard uncertainty of each component is: 21 The combined uncertainty shows good agreement with the total relative error of 7.3%, validating the rationality of the error decomposition. By comparing the filtered, time-synchronized DAS phase-change rate signal with the strain rate signal corrected from FBG measurements, the phase information at the 5 Hz fundamental frequency was extracted. The calculated phase difference is: 22 This phase delay corresponded to a time delay of 0.64 ms, which is shorter than the system sampling interval of 1 ms and does not introduce significant distortion to the 5 Hz sinusoidal signal. Sensor response linearity analysis Taking measurement point A as an example (with the excitation frequency fixed at 5 Hz and amplitudes set to 500 mV, 1000 mV, 1500 mV, 2000 mV, 2500 mV, and 3000 mV), the response characteristics of the acceleration signal, FBG strain rate signal, and DAS phase-change rate signal under different excitation amplitudes were analyzed. Figure 17 showed the changes in time-domain waveforms: under different excitation amplitudes, the time-domain signals of all three monitoring methods maintained well-defined sinusoidal shapes, and their amplitudes increased monotonically as the excitation amplitude rose. Fig. 17. Open in a new tab Waveform variation diagram under different amplitude sine wave vibration. ( a ) Acceleration value, ( b ) FBG strain rate, ( c ) DAS phase change rate. The average positive peak values of each signal were extracted, and the results are presented in Fig. 18 (a). The acceleration signal, FBG strain rate signal, and DAS phase-change rate signal exhibited a significant positive correlation with the excitation amplitude; the coefficients of determination (R 2 ) for linear fitting were 0.96 (SE ≈ 0.037 m/s 2 ), 0.99 (SE ≈ 0.773 µε/s), and 0.99 (SE ≈ 0.04 rad/s), respectively. Fig. 18. Open in a new tab Linear response of each sensor under 5 Hz sinusoidal dynamic loads with different amplitudes. ( a ) Peak variation of each sensor with excitation amplitude, ( b ) Correlation between acceleration, strain rate and phase change rate. With the positive peak value of the acceleration signal as the horizontal axis, the linear fitting results for the FBG strain rate signal and DAS phase change rate signal are presented in Fig. 18 (b), with R 2 values of 0.95(SE ≈ 1.373 µε/s) and 0.98 (SE  ≈ 0.785 rad/s) respectively. The bidirectional error bars in Fig 18 are plotted based on the expanded uncertainty at the 95% confidence interval (k = 2, the default coverage factor for engineering measurement 30 , which is derived from the 6.0% combined relative standard uncertainty (U = u c ×2 ≈ 12.0%). In contrast to the excellent linearity at 5 Hz, the linear correlation between all monitored parameters is completely invalid at 50 Hz sinusoidal excitation, which is attributed to the destruction of the uniform strain characteristic on the beam surface and the consequent inapplicability of the theoretical model. In terms of frequency response characteristics, the DAS system exhibits higher sensitivity to low-frequency signals but lower sensitivity to high-frequency components, reflecting its application advantages in structural low-frequency vibration monitoring. The high linearity demonstrated in Fig. 18 confirms the reliable quantitative relationship between DAS response and both excitation amplitude and acceleration signal. The slightly larger residual standard errors observed for FBG (SE ≈ 1.373 µε/s) and DAS (SE ≈ 0.785 rad/s) compared to acceleration (SE ≈ 0.037 m/s 2 ) arise from inherent differences in sensing mechanisms: DAS measures spatially averaged strain over the gauge length, while FBG provides localized point measurements. This deviation is consistent with the distributed nature of DAS and does not compromise the reliability of their linear mapping relationship. Verification of DAS distributed perception capability based on modal theory Finite element modal analysis results were compared with the measured data from DAS channels 13 and 14. From Eq.s (17) and (19), the theoretical strain amplitude ratio at any two points on the top surface of the equal-strength beam was completely consistent with the ratio of the second derivatives of the first-order mode shape (curvature, which is proportional to strain). The mathematical expression is: 23 Substituting the central coordinates of DAS channels 13 and 14 (400 mm and 900 mm) into Eq. (23), the discrete-point theoretical strain amplitude ratio was calculated. Combined with the measured peak phase-change rate amplitudes, the normalized comparison between theory and experiment is summarized in Table 1 . Table 1. Normalized Comparison of Theoretical Strain and DAS Measured Response. Channel combination Central position (mm) Theoretical strain amplitude ratio (Original Point-wise) DAS Measured phase change rate amplitude Ratio Relative deviation (%) 13/14 400/900 2.26 1.73 23.4 Open in a new tab As established in the theoretical section, the DAS output phase change rate is not the discrete-point strain rate at the channel center position, but rather the spatial average of the axial strain rate over the gauge length . Therefore, to achieve quantitative matching with experimental data, the original point-wise theoretical amplitude ratio must be corrected by accounting for the spatial averaging effect of the DAS gauge length. Spatial averaging correction based on gauge length Due to the finite gauge length m of the DAS system, the measured strain amplitude of a channel is the convolution result of the theoretical strain distribution over the channel coverage interval . According to the spatial averaging model in Eq. (13), the corrected theoretical strain amplitude for a DAS channel is the spatial average of the absolute value of the strain rate expression over the gauge interval. For Channel 13, whose axial coverage interval is 150 mm ~ 650 mm verified by tapping tests, substituting into the mode shape second derivative formula yields an average second derivative of m −2 over the interval, which is only 76.5% of the discrete-point value at mm. For Channel 14, with an axial coverage interval of 650 mm~1150 mm, the average second derivative is m⁻², only 1.8% lower than the discrete-point value at mm. This difference arises because Channel 13 is located near the fixed end of the beam, where the strain gradient is significantly larger than that near the free end (Channel 14), resulting in a more pronounced position-dependent influence of the spatial averaging effect. Quantitative comparison between corrected theory and measured data Based on the spatially averaged second derivatives, the corrected theoretical strain amplitude ratio for Channels 13 and 14 is recalculated as: 24 The DAS measured amplitude ratio is 1.73, resulting in a relative deviation of only 1.7% between the corrected theoretical value and the experimental value. This represents a significant reduction from the original 23.4% deviation based on point-wise theory. The high consistency after correction directly verifies that the spatial averaging effect of the DAS gauge length is the dominant factor causing the initial amplitude discrepancy. Error source quantification and contribution decomposition To further clarify the composition of the 23.4% total amplitude deviation, a quantitative error decomposition was performed. The total deviation was decomposed into four independent error sources, and the results are summarized in Table 2 : Spatial averaging effect of DAS gauge length : The core factor causing amplitude attenuation. After correction by Eq. (13), the deviation is reduced from 23.4% to 1.7%. Its contribution to the total deviation is calculated as ; Deviation between actual boundary constraints and ideal cantilever model: The actual fixed boundary of the beam differs slightly from the ideal fixed-free model. This effect has been partially corrected by the mode shape correction factor 1.01 in Eq. (18), with residual error contributing approximately 4% to the total deviation; DAS channel positioning deviation: The maximum coordinate deviation from tapping tests is ± 25 mm, far smaller than the system gauge length. Substituting this offset into the spatial averaging model yields an amplitude variation contribution of approximately 2% to the total deviation; Fiber-beam strain transfer loss + DAS phase demodulation noise: The FBG strain transfer efficiency reaches 99.7% 27 , with residual strain transfer loss of approximately 0.3%; the DAS system’s phase demodulation accuracy is 0.01 rad 32 , consistent with the experimental SNR of 38 dB. Together, these contribute approximately 2% to the total deviation. Table 2. Quantitative decomposition of 23.4% amplitude deviation. Error source Contribution (%) Quantitative description Spatial averaging effect of DAS gauge length 92 Core dominant factor; correction reduces deviation to 1.7% Deviation between actual boundary constraints and ideal cantilever model 4 Residual systematic error introduced by mode shape correction factor 1.01 DAS channel positioning deviation (± 25 mm) 2 Minor amplitude error from coordinate offset, negligible impact Fiber-beam strain transfer loss + DAS phase demodulation noise 2 FBG strain transfer efficiency > 99% 27 , DAS demodulation accuracy 0.01 rad 32 Open in a new tab Time-domain waveform consistency verification Figure 19 provides direct and quantitative verification of the spatial averaging model from two dimensions: time-domain waveform and amplitude comparison. Figure 19 (a) shows the time-domain comparison between the filtered DAS measured phase change rate of Channel 13 and the corrected theoretical signal under 5 Hz excitation. The Pearson determination coefficient between the two waveforms is , indicating an extremely high degree of temporal coincidence. The phase difference of 0.02 rad between the two curves is consistent with the system phase delay measured, validating the accuracy of the model in both amplitude and phase dimensions. Fig. 19. Open in a new tab Verification of the spatial averaging effect of DAS gauge length under 5 Hz sinusoidal excitation. ( a ) Time-domain waveform comparison of Channel 13 between filtered DAS measured signal and corrected theoretical signal; ( b ) Normalized amplitude comparison of original theoretical value, corrected theoretical value, and DAS measured value for Channels 13 and 14. Figure 19 (b) presents the normalized amplitude comparison for both channels. For Channel 13, the original point-wise theoretical amplitude (2.26) deviates significantly from the measured value (1.73), while the corrected theoretical amplitude (1.76) shows excellent agreement with the experimental data. For Channel 14, due to the small strain gradient, the spatial averaging effect is negligible, and all three sets of amplitudes are close to 1.00, clearly demonstrating the position-dependent characteristics of the spatial averaging effect. Combining the analysis results from Tables 1 and 2 , and Fig. 19 , after correction by the spatial averaging effect, the DAS measurement results show excellent agreement with the dynamic strain distribution law derived from modal theory, with the relative deviation between the corrected theoretical value and the measured value reduced to only 1.7%. This indicates that DAS is capable of not only qualitatively identifying structural vibration events but also quantitatively distinguishing the spatial distribution characteristics of dynamic strain fields, possessing the core capability to serve as a distributed monitoring tool for dynamic strains in engineering structures. Through the spatial averaging model, this study successfully corrected the 23.4% amplitude deviation, clarifying the applicability differences between point-wise theory and distributed measurements, and providing solid experimental support and theoretical basis for refined dynamic monitoring applications of DAS in engineering structures such as bridges, pipelines, and industrial equipment. Conclusion Through theoretical derivation, finite element simulation and sinusoidal excitation test of equal-strength beam, this paper systematically studies the response characteristics of DAS in dynamic strain monitoring. The main conclusions are as follows: A dynamic strain transfer model from external excitation to DAS phase response is constructed, and the phase sensitivity coefficient is proposed as the core parameter for quantitative analysis, which clarifies the linear mapping relationship between DAS phase change rate and the actual dynamic strain rate of the structure. The dynamic strain distribution law of the equal-strength beam is obtained: the surface strain is approximately uniform when the excitation frequency is less than 5 Hz, and presents a significant spatial gradient distribution near the natural frequency, which is fundamentally different from the static equal-strain characteristic. The 5 Hz frequency threshold for reliable quantitative measurement of DAS is determined. DAS has excellent quantitative monitoring performance in the low-frequency uniform strain range. Under 5 Hz excitation, the relative error of the phase sensitivity coefficient is 7.3%, the response linearity R² is up to 0.99, and the phase delay is less than 0.02 rad. After correcting the spatial averaging effect of the gauge length, the relative deviation between the measured and theoretical strain amplitude ratio is reduced from 23.4% to 1.7%, which verifies DAS’s ability to quantitatively distinguish the spatial distribution of dynamic strain. When the excitation frequency exceeds 5 Hz, the time-domain linear relationship between DAS signal and structural strain breaks down, but DAS can still reliably capture the fundamental frequency of excitation, which is suitable for vibration event identification and qualitative monitoring in high-frequency scenarios. The research results provide theoretical basis and experimental support for the engineering application of DAS in dynamic monitoring of engineering structures. In the follow-up, the quantitative method of this study can be extended to full-scale engineering structures, and the verification of DAS monitoring performance under complex loads and environments can be carried out to further improve its engineering applicability. Acknowledgements The work was supported by the financial supports from the National Natural Science Foundation of China (NSFC) (Grant number: 52474147), Shaanxi Qinchuangyuan “Scientists+Engineers” Team Building Project (Grant number: 2023KXJ-144) and the Xi’an Qinchuangyuan “Scientists+Engineers” Team Building Project (Grant number: 23KGDW0027-2022). List of symbols Δ Total phase change of backscattered Rayleigh light in the optical fiber (rad) n Refractive index of fiber core (dimensionless) λ Wavelength of laser pulse (m) L f Gauge length of the DAS system (m) Δn Change of the optical fiber core refractive index (dimensionless) ε Axial strain (με) µ Poisson’s ratio of fiber (dimensionless) P 11 , P 12 Photoelastic coefficients of fiber (dimensionless) ξ Strain correction factor of fiber (dimensionless) DAS phase change rate (rad/s) L Length of equal-strength beam (mm) h Thickness of the equal-strength beam (mm) b 0 Width at fixed end of the beam (mm) F Concentrated force (N) E Elastic modulus of beam material (GP a ) ε beam Strain on beam surface (με) F 0 Amplitude of harmonic excitation force (N) ω Angular frequency of excitation (rad/s) t Time variable (s) w (x,t) Transverse displacement of beam (mm) ψ 1 (x) First-order mode shape function (dimensionless) Q 1 First-order modal response amplitude (mm) θ 1 Phase of first-order modal response (rad) Second derivative of mode shape function (m/s 2 ) Acceleration on beam surface (m/s 2 ) Acceleration amplitude (m/s 2 ) Strain rate on beam surface (με/s) x 0 Central axial position of a DAS measurement channel (m) DAS-measured equivalent strain rate after spatial averaging over the gauge length (με/s) Complex phase sensitivity coefficient (rad/s)/(με/s) Phase delay of DAS system (rad) R theory Theoretical strain amplitude ratio (dimensionless) η FBG sensor strain transfer efficiency (dimensionless) β Shear lag coefficient m -1 u c Combined relative standard uncertainty (%) u η FBG strain transfer uncertainty (%) u DAS DAS phase demodulation uncertainty (%) u acc Acceleration measurement uncertainty (%) u sync Data acquisition synchronization uncertainty (%) u model Deviation between theoretical model and actual boundary (%) R corrected Corrected theoretical strain amplitude ratio (dimensionless) Author contributions Conceptualization: C.Y.Ma; Methodology: D.D.Zhang; Investigation: J.Chai, Z.Li; Data curation: C.Y.Ma; Writing—original draft preparation: C.Y.Ma; All authors have read and agreed to the published version of the manuscript. 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