Reduced-parameter damage detection in cantilever beams using magnetoelastically measured bending frequencies

1 Introduction

Localized damage such as cracks, voids, or material degradation can significantly influence the dynamic behavior of engineering structures by reducing stiffness and altering their vibrational response. Since these changes are directly reflected in the modal characteristics of a structure, vibration-based techniques have become widely adopted for structural damage identification in civil (Ali Zar et al., 2024a). Among the various dynamic parameters, natural frequencies are particularly attractive because they can be measured non-destructively while remaining highly sensitive to structural deterioration. The presence of damage generally produces measurable shifts in the modal frequencies, with the magnitude of these changes depending on the location, severity, and type of defect, as well as the vibration mode considered (Samourgkanidis and Kouzoudis, 2018). Consequently, considerable research effort has been devoted to developing analytical, numerical, and experimental methodologies capable of accurately correlating variations in modal properties with the existence of structural damage.

Various vibration-based methodologies have been proposed for structural damage detection using modal parameters such as mode shapes, modal curvature, damping, and natural frequencies (Charles and Worden, 2007; Chandrashekhar and Ganguli, 2009; Rucevskis et al., 2016; Cao et al., 2017; Sha et al., 2019). Several researchers have focused on mode-shape-based approaches due to their high sensitivity to localized stiffness reductions caused by cracks and other structural defects (Qiao et al., 2007; OBrien et al., 2016; Ghannadi et al., 2020; Modesti et al., 2025). In particular, methods based on modal curvature have been shown to enhance the visibility of local discontinuities in structural stiffness, allowing improved identification of damage regions compared to the direct use of displacement mode shapes (Abdel Wahab, 2001; Altunışık et al., 2019; Qiu et al., 2025). Other studies have employed wavelet transform techniques to decompose modal responses into different spatial frequency components, enabling the localization of singularities associated with crack positions (Rucka and Wilde, 2006; Sun et al., 2022; Nie et al., 2025). Similarly, modal strain energy methods have been extensively investigated because they provide information regarding the redistribution of deformation energy within damaged structures and can therefore be used to estimate both the location and severity of structural deterioration (Teng et al., 2019; Hassan Daneshvar et al., 2023; Zhang C et al., 2025). These approaches have demonstrated successful damage identification capabilities in beam-like, plate, and frame structures under both numerical and experimental conditions. However, despite their high sensitivity and accuracy, mode-shape-based methods generally require dense spatial measurements and a large number of sensing locations in order to reconstruct the modal response with sufficient precision. In practical structural health monitoring applications, this requirement increases the complexity of the measurement system, the instrumentation cost, and the sensitivity to measurement noise, thereby limiting the applicability of such techniques in large-scale or real-time monitoring scenarios. In contrast, natural-frequency-based methods provide a more compact and practically efficient alternative, since modal frequencies can be identified using relatively simple instrumentation and limited sensor placement (XU et al., 2007; He and Zhu, 2011; Pan et al., 2018; Pan et al., 2019; Kouzoudis et al., 2021; Saha and Yang, 2023; Demirtaş et al., 2026). As global dynamic quantities, natural frequencies inherently reflect the overall stiffness and mass distribution of a structure, which makes them well-suited for damage assessment without requiring spatially dense measurements. It is well established that the presence of cracks or localized stiffness reductions induces shifts in the natural frequencies of a structure, with these variations governed primarily by the crack location, crack severity, and vibration mode order (Liang et al., 1991; Nguyen and Tran, 2014; Grebla et al., 2023; Kindova-Petrova, 2025; Sun et al., 2025). In particular, different modes exhibit different levels of sensitivity to damage, reflecting the way strain energy is distributed along the structure. As a result, frequency-based approaches have been widely adopted within analytical, numerical, and experimental frameworks for damage identification in beam-like and general structural systems.

Beam-like structures have received particular attention in the context of vibration-based damage identification due to their widespread use in civil and mechanical engineering systems and their well-established analytical descriptions (Manoach et al., 2012; Altunışık et al., 2017; Samourgkanidis and Kouzoudis, 2021; Mansouri et al., 2025; De Maio et al., 2026). As fundamental structural elements, beams provide a convenient framework for investigating the effects of localized damage on dynamic response, while still capturing the essential physics of stiffness reduction and mode-dependent vibration behavior (Ciambella and Vestroni, 2015; Stoykov and Manoach, 2021). In particular, cracks in beam structures can be effectively modeled as localized reductions in bending stiffness, which alter the global dynamic properties and lead to measurable variations in natural frequencies and mode shapes (Sankara Babu Chinka et al., 2021; Kateel and Baddour, 2022). Because of their relatively simple geometry and boundary conditions, beam models have been extensively employed to establish theoretical relationships between crack characteristics and changes in modal parameters, serving as benchmark systems for both numerical simulations and experimental validation. Consequently, a large body of literature has focused on understanding how crack location and severity influence the vibration characteristics of beam-like structures under different support conditions and loading scenarios.

The effectiveness of vibration-based structural health monitoring techniques depends strongly on the capability of the sensing system to accurately capture the dynamic response of the structure. In experimental investigations, modal characteristics are typically identified from vibration measurements such as acceleration, velocity, or displacement under ambient or controlled excitation conditions. Among conventional sensing approaches, accelerometers are widely used due to their reliability, ease of implementation, and suitability for a broad range of frequency measurements (Ratcliffe et al., 2008; Bikash Chaudhury et al., 2014; Ali et al., 2019; Komariza et al., 2024; Kodrič et al., 2025). In addition to contact-based sensors, non-contact techniques such as Laser Doppler Vibrometers (Siringoringo and Fujino, 2006; Chen et al., 2016; Zhang X et al., 2025; Ali Zabihi et al., 2024b) and vision-based methods (Guo et al., 2019; Lai et al., 2022; Tan et al., 2023; Gua et al., 2025) are also employed when it is desirable to avoid mass loading or physical interference with the structure. Among emerging sensing technologies for vibration-based structural health monitoring, magnetoelastic materials have attracted increasing attention due to their inherent coupling between mechanical stress and magnetic response. Magnetoelasticity refers to the ability of certain ferromagnetic materials to alter their magnetic state when subjected to mechanical loading, while the inverse phenomenon, magnetostriction, describes the deformation of these materials under an applied magnetic field (Craig et al., 2002; Zhang et al., 2013; Xie et al., 2022; Singh and Perween, 2024). This bidirectional coupling enables the development of passive and contactless sensing systems, where mechanical vibrations or stress variations are indirectly transduced into measurable magnetic signals (Tapeinos et al., 2023; Samourgkanidis, 2026). In particular, metallic glass alloys such as Metglas-based ribbons have been investigated due to their high magnetoelastic coupling efficiency and favorable sensing characteristics (Samourgkanidis et al., 2018; Kouzoudis et al., 2020). By exploiting changes in magnetic permeability or resonance behavior under dynamic loading, these materials can be configured as compact vibration sensors capable of capturing structural response information (Tapeinos et al., 2025), including shifts in modal frequencies associated with damage-induced stiffness reduction. As a result, magnetoelastic sensors offer a promising alternative sensing approach within structural health monitoring frameworks, combining sensitivity to dynamic variations with the potential for wireless, low-power, lightweight, and cost-effective implementation using Metglas ribbons. Compared with conventional contact-based sensors such as accelerometers and strain gauges, they can reduce physical interference and simplify sensing configurations. Multiple Metglas ribbons can be attached at different locations of a structure and sequentially interrogated using a single coil, allowing the response at each location to be recorded without requiring individual wiring from the ribbons to the data acquisition system. This enables distributed sensing while reducing wiring complexity and instrumentation requirements. A key limitation, however, is their sensitivity to external magnetic fields, which can interfere with the magnetic response and affect reliable signal measurement.

The novelty of this work lies in demonstrating, both numerically and experimentally, a nearly material-invariant framework for crack identification in cantilever beams based on normalized modal frequency ratios combined with magnetoelastic vibration sensing. By introducing a finite element-fracture mechanics formulation and systematically analyzing the role of Young’s modulus, density, and Poisson’s ratio, the study shows that normalization of bending frequencies significantly reduces the dependence on stiffness and mass properties, leaving a weak but consistent sensitivity primarily to Poisson’s ratio. This establishes normalized modal frequencies as robust damage indicators that enable consistent comparison across different engineering materials. In addition, the use of Metglas-based magnetoelastic sensors enables accurate, non-contact extraction of modal frequencies, providing an experimentally viable route for real-time structural health monitoring. The combined numerical-experimental agreement further validates the proposed approach, positioning it as a lightweight and scalable methodology for material-agnostic damage detection in beam-like structures.

The novelty of the present work lies in extending to previous studies on crack identification in cantilever beams (Samourgkanidis and Kouzoudis, 2019), bending-frequency measurement (Samourgkanidis and Kouzoudis, 2021), and magnetoelastic sensing (Samourgkanidis and Kouzoudis, 2020) by introducing and systematically evaluating a normalized modal-frequency-ratio-based approach for damage detection across different materials. Unlike the earlier studies, which primarily investigated crack-induced changes in bending frequencies and the feasibility of magnetoelastic sensing for modal characterization (Samourgkanidis and Kouzoudis, 2018), the present study focuses on determining to what extent normalized modal frequency ratios can reduce the influence of material properties on crack-related frequency changes. To this end, a finite element-fracture mechanics formulation is employed to systematically examine the effects of Young’s modulus, density, and Poisson’s ratio on the normalized frequency response of cracked cantilever beams. The results indicate that normalization substantially reduces the dependence of the damage-sensitive frequency response on stiffness and mass properties, while a weaker but consistent dependence on Poisson’s ratio remains. The methodology is further assessed experimentally using multiple engineering materials and Metglas-based magnetoelastic sensors for contactless signal detection of the modal frequencies. The agreement between the numerical and experimental results provides validation of the proposed normalization approach and demonstrates its potential to improve the consistency of crack identification across different beam materials. Thus, rather than claiming complete material independence, the present work establishes a more systematic basis for reducing material-property effects in vibration-based crack detection and extends the applicability of magnetoelastic sensing for structural health monitoring of beam-like structures.

2 Beam formulation

The dynamic response of a cantilever beam, including the influence of localized damage, is modeled by combining the finite element method with concepts from fracture mechanics. The procedure is divided into three main steps, which are later integrated into a computational framework implemented in MATLAB. First, the mass and stiffness matrices of an intact beam element are derived analytically using the Euler–Bernoulli beam theory within a finite element context. Next, the stiffness degradation caused by a transverse crack is evaluated through fracture mechanics principles. Finally, all elements are assembled into global system matrices, boundary conditions are imposed, and the resulting eigenvalue problem is solved to obtain the bending natural frequencies (Samourgkanidis and Kouzoudis, 2019).

For the intact beam we consider a uniform beam element of length L e, defined by two nodes located at its ends (Figure 1). Each node is associated with two degrees of freedom: a transverse displacement and a rotation. The nodal displacement vector can be expressed as:where q 1 ( t ) and q 2 ( t ) represent the transverse displacement and rotation, respectively, at node 1, while q 3 ( t ) and q 4 ( t ) represent the transverse displacement and rotation at node 2.

q t = q 1 t , q 2 t , q 3 t , q 4 t T (1)

Using standard finite element procedures, the governing dynamic equation of the element is written as:

M e q ̈ t + K e q t = p t (2)

The vector q ( t ) denotes the nodal displacement field and collects the four time-dependent degrees of freedom q i ( t ). The vector p ( t ) represents the corresponding nodal force components, while [ K e ] and [ M e ] correspond to the elemental stiffness and mass matrices, respectively. This equation may be viewed as an alternative representation of Lagrange’s equation of motion, in which the kinetic and strain energy contributions of each beam element are explicitly embedded (Leonard, 1975). The element matrices for the undamaged beam take their classical Euler–Bernoulli form and can fully describe the vibration characteristics of the beam in the absence of defects:

M e = ρ A L e 420 156 22 L e 54 − 13 L e 22 L e 4 L e 2 13 L e − 3 L e 2 54 13 L e 156 − 22 L e − 13 L e − 3 L e 2 − 22 L e 4 L e 2 , K e = E I L e 3 12 6 L e − 12 6 L e 6 L e 4 L e 2 − 6 L e 2 L e 2 − 12 − 6 L e 12 − 6 L e 6 L e 2 L e 2 − 6 L e 4 L e 2 (3)

The introduction of a transverse crack within a structural element leads to a reduction in local stiffness, which in turn significantly modifies its dynamic response. Although both mass and stiffness properties are theoretically affected, the variation in the mass loss is neglected due to its minimal contribution relative to stiffness reduction. Therefore, only stiffness degradation is considered in the subsequent formulation. Figure 2 illustrates a beam element subjected to bending moments ( B 1 , B 2 ) and shear forces ( Q 1 , Q 2 ) in the presence of a transverse crack of depth α, located at mid-span of the element of length L e.

The static equilibrium of the cracked element can be expressed through force and moment balance. For shear forces:

∑ i = 1 2 Q i = 0 ⇔ Q 1 + Q 2 = 0 ⇔ Q 1 = − Q 2 (4)

while for bending moments:

∑ i = 1 2 B i = 0 ⇔ B 1 + B 2 + Q 2 L e = 0 ⇔ B 1 = − Q 2 L e − B 2 (5)

These relations can be written compactly in matrix form as:where C is the equilibrium transformation matrix of the cracked element.

Q 1 B 1 Q 2 B 2 = − 1 0 − L e − 1 1 0 0 1 Q 2 B 2 ⇔ Q 1 B 1 Q 2 B 2 = C Q 2 B 2 , C = − 1 0 − L e − 1 1 0 0 1 (6)

The mechanical response of the element is more conveniently described using the flexibility matrix D, whose components are defined through the strain energy W as:

D i j = ∂ 2 W ∂ P i ∂ P j , P i ∈ Q , B , i , j = 1,2 (7)

The stiffness matrix of the cracked element is then obtained through:

K e = C T D − 1 C (8)

and this energy-based formulation is valid for both intact and cracked configurations. Using (Equations 6–8) the stiffness matrix is obtained as:

K e = 1 D 11 D 22 − D 12 D 21 D 22 L e D 22 − D 12 − D 22 − D 12 L e D 22 − D 21 L e 2 D 22 − L e D 12 + D 21 + D 11 − L e D 22 + D 21 L e D 12 − D 11 − D 22 − L e D 22 + D 12 D 22 − D 12 D 21 L e D 21 − D 11 − D 21 D 11 (9)

For an undamaged beam subjected to a combined loading B T = Q L e + B, the strain energy is:

W 0 = 1 2 ∫ 0 L e B T 2 E I d L = 1 2 ∫ 0 L e Q L + B 2 E I d L = L e 2 E I Q L e 2 3 + B 2 + Q B L e (10)

When a crack is present, an additional energy term arises due to the discontinuity of the displacement field at the crack faces. This contribution is expressed in terms of stress intensity factors associated with the three fracture modes Figure 3: Mode I (opening), Mode II (sliding), and Mode III (tearing).

The additional strain energy is given by:

W c = ∫ S G I 2 + G I I 2 E ′ + G III 2 2 μ d S (11)

The plane strain Young’s modulus is defined as E ′ = E 1 − ν 2, while the shear modulus is given by μ = E 2 ( 1 + ν ), where E is Young’s modulus and ν denotes Poisson’s ratio. The quantities G I, G I I, and G III represent the crack-tip stress intensity factors corresponding to Modes I, II, and III, respectively. The variable S denotes the crack surface area. These stress intensity factors characterize the intensity of the stress field in the vicinity of the crack tip and can be interpreted as measures of the load transmission through the crack-tip region. In this work, the cracked beam element is defined by width b, height H, and crack depth α. Under combined shear and bending loading, typical of beam deformation, the stress intensity factors are expressed as follows:

G I = G I Q + G I B , G I I = G IIQ , G III = 0 (12)

These relations indicate that the stress intensity factors G consist of two separate contributions: one arising from shear forces Q and another from bending moments M. Since the G factors vary linearly with the applied stresses, the principle of superposition is valid, allowing the two effects to be combined additively. Specifically, Mode I includes contributions from both shear and bending, Mode II involves only the shear contribution, and Mode III does not involve either in driving crack development. Substituting Equation 12 into Equation 11, and noting that the crack cross-sectional area is given by S = b ⋅ α, yields:

W c = ∫ 0 α b E ′ G I Q + G I B 2 + G IIQ 2 d α (13)

The terms G I Q, G I B, and G IIQ in (Equation 13) depend on both the applied loads and the crack dimensions (Tada et al., 2000) and they can be expressed as:where

G I Q = τ I Q π α f I η , G I B = τ I B π α f I η , G IIQ = τ IIQ π α f I I η (14)

f I η = 2 π η tan π η 2 0.923 + 0.199 1 − sin π η 2 4 cos π η 2 , f I I η = 1.122 − 0.561 η + 0.085 η 2 + 0.18 η 3 1 − η (15)

and

τ I Q = 3 Q L e b H 2 , τ I B = 6 B b H 2 , τ IIQ = Q b H (16)

The stresses τ I , Q and τ I , B represent the normal stresses induced by shear and bending effects, respectively, which contribute to crack opening. The term τ I I , Q denotes the shear stress generated by shear forces that promotes crack sliding. The functions f I ( η ) and f I I ( η ) are geometry correction factors associated with each fracture mode, where η = α / H is the normalized crack depth. By substituting these expressions into (Equation 14), the stress intensity factors can be written in their explicit form as:

G I Q = 3 Q L e b H 2 π η H 2 π η tan π η 2 0.923 + 0.199 1 − sin π η 2 4 cos π η 2 G I B = 6 B b H 2 π η H 2 π η tan π η 2 0.923 + 0.199 1 − sin π η 2 4 cos π η 2 G IIQ = Q b H π η H 1.122 − 0.561 η + 0.085 η 2 + 0.18 η 3 1 − η (17)

Using (Equations 10, 13, 17), the total strain energy of the cracked beam element, W = W 0 + W c, can be expressed as:

W = L e 2 E I Q L e 2 3 + B 2 + Q B L e + π 1 − ν 2 E b 3 Q L e + 6 B H 2 ∫ 0 η η f I 2 η d η + Q 2 ∫ 0 η η f I I 2 η d η (18)

and the flexibility coefficients from Equation 7 the can be calculated as:

D 11 = L e 3 3 E I + 2 π 1 − ν 2 E b 9 L 2 H 2 ∫ 0 η η f I 2 η d η + ∫ 0 η η f I I 2 η d η D 12 = C 21 = L e 2 2 E I + 36 π L e 1 − ν 2 E b H 2 ∫ 0 η η f I 2 η d η D 22 = L e E I + 72 π 1 − ν 2 E b H 2 ∫ 0 η η f I 2 η d η (19)

3 MATLAB modeling

The theoretical formulation described in Section 2 was implemented in MATLAB to simulate the dynamic response of the beam. The implementation follows the three main stages mentioned: (i) undamaged element modeling, (ii) crack-induced stiffness modification, and (iii) global assembly and eigenvalue analysis. For the undamaged beam, the global stiffness and mass matrices are first assembled and the eigenvalue problem is solved to extract the natural bending frequencies. These frequencies serve as a reference baseline for the intact structure. Subsequently, the same procedure is repeated for the damaged configurations, where the elemental stiffness is modified to account for the presence of a crack at different locations and depths. The normalized bending frequencies are then evaluated as the ratio between the damaged and undamaged natural frequencies, providing a consistent measure of the influence of damage on the dynamic behavior of the beam.

Figure 4 presents the schematic flowchart of the MATLAB implementation used for the dynamic analysis of the beam. The procedure begins with the definition of the beam parameters, followed by the computation of the elemental mass and stiffness matrices for the undamaged structure. The flexibility and transformation matrices are then evaluated, and the global system of the intact beam is assembled to obtain the reference natural bending frequencies.

Subsequently, storage matrices are initialized for recording the results. A parametric analysis is performed through two nested loops: the outer loop iterates over different crack depths, while the inner loop considers all possible crack locations along the beam. For each configuration, the cracked element stiffness is computed and introduced into the global stiffness matrix, boundary conditions are applied, and the eigenvalue problem is solved to extract the natural bending frequencies. These frequencies are then normalized with respect to the undamaged case and stored. Finally, the procedure outputs the normalized bending frequencies for all crack scenarios.

Table 1 presents the beam and material parameters used in the MATLAB-based dynamic analysis. The dynamic analysis of the beam is defined using a set of geometric, material, and numerical parameters. The model consists of a discretized finite element mesh, leading to the global system degrees of freedom based on beam formulation. The total beam length defines the element size through uniform discretization. The material properties include Young’s modulus, density, and Poisson’s ratio, which govern the elastic and inertial behavior of the structure. The cross-sectional geometry is characterized by its width and height, from which the cross-sectional area and second moment of inertia are computed. Damage is introduced through a normalized crack depth parameter, which controls the severity of the simulated crack in the structure. Finally, a selected number of natural bending frequencies is extracted and used to evaluate the dynamic response of both the intact and cracked beam configurations. The geometrical characteristics were selected based on the experimental specimens used for the evaluation of this method. As can be seen in the table, the material properties of Young’s modulus, density, and Poisson’s ratio are not fixed single values but are instead defined over a range of values. This approach was adopted because the numerical simulations were performed for different combinations of crack depth and crack location, which were randomly selected, while varying the material properties. Specifically, Young’s modulus was varied from 65 to 210 GPa with a step size of 10 GPa, the density from 2000 to 9000 kg/m 3 with a step size of 1000 kg/m 3, and the Poisson’s ratio from 0.2 to 0.5 with a step size of 0.05. These ranges were chosen to represent a wide spectrum of engineering materials.

TABLE 1. Summary of geometric, material, and numerical parameters used in the MATLAB-based finite element dynamic analysis of the beam.

4 Experimental

The experimental validation of the present work was carried out using the setup shown in Figure 5. A 12-ton CARVER hydraulic press (Carver Inc., Indiana, United States) was employed to rigidly clamp one end of the beam, thereby establishing a cantilever boundary condition.

Clamping was achieved using a pair of custom-machined curved plates made from aluminum alloy 7075. The opposite end of the CB remained free and was positioned within a set of coils, consisting of a detection coil and Helmholtz coils (Table 2). The detection coil was used to capture the vibration signal generated by the beam and was wirelessly connected to the sound card of a laptop for data acquisition. In parallel, the Helmholtz coils were utilized to generate a uniform bias DC magnetic field, enhancing the sensitivity and amplitude of the recorded signal (Samourgkanidis and Kouzoudis, 2020). The applied DC magnetic field was maintained at a constant amplitude of 0.8 Gauss throughout all frequency measurements by supplying the Helmholtz coil with a constant DC current of 370 mA. The detection mechanism was based on the magnetoelastic properties of two equal-length annealed (Samourgkanidis et al., 2021; Samourgkanidis et al., 2025) ribbons of Metglas 2826MB3 (Table 2), manufactured by Metglas Inc (Conway, SC, United States) (Metglas Inc, 2025), which were mechanically attached to the free-end region of the CB using double-sided tape, aligned parallel to its longitudinal axis, and separated by a 5 mm distance, serving as vibration-sensing elements (Figure 5c). A sharp metallic cone attached to a spring was repeatedly released from a height of 30 cm above the CB, producing an impact near its fixed end before retracting due to the restoring force of the spring. As a result, the CB underwent mechanical vibrations, and the resulting mechanical strain was transferred to the Metglas ribbons, producing a corresponding variation in their magnetization through the magnetoelastic coupling effect. This time-varying magnetization generated a fluctuating magnetic field in the surrounding space. The magnetic response was detected by a nearby coil without direct electrical contact with the Metglas ribbons. According to Faraday’s law of electromagnetic induction, the time-varying magnetic field induced a voltage in the detection coil. Thus, the bending vibrations of the CB were converted into an electrical signal through magnetoelastic effect and contactless magnetic readout, and the resulting signal was recorded and further analyzed. The presence of the externally applied bias magnetic field from the Helmholtz coils enhanced this effect by optimizing the sensitivity of the ribbons to strain, thereby improving the signal quality.

TABLE 2. Geometrical, mechanical, magnetic, and electrical properties of the CBs, magnetoelastic ribbons, detection coil, and Helmholtz coils.

Figure 6a presents the signal measured by the detection coil during mechanical excitation of the CB, recorded using MATLAB via the laptop sound card, together with its corresponding FFT spectrum (Figure 6b). The acquisition time was 10 s, with a sampling rate of 48 kHz, resulting in a frequency resolution of 0.1 Hz. The eight dominant peaks observed in the FFT spectrum correspond to the first eight natural bending frequencies of the undamaged 6063 aluminum-alloy CB. The identification of these peaks as bending modes was confirmed using ANSYS software program (ANSYS 16.0, Inc., Canonsburg, PA, United States). Dynamic simulations were performed for each beam material to calculate the corresponding analytical natural frequencies and verify their agreement with the experimentally measured frequencies (Samourgkanidis and Kouzoudis, 2018). These peaks are characteristic of each beam material investigated in this study and are closely related to its mechanical condition, as any induced damage can cause a shift in their frequencies. Furthermore, the peak frequencies differ among beams made of different materials because, as discussed in Section 2, the natural bending frequencies are strongly dependent on the physical properties of the beam.

An additional experimental approach employed to improve result accuracy involved measuring bending frequencies under varying clamping forces applied using a hydraulic press, rather than relying on a single measurement (Samourgkanidis and Kouzoudis, 2018). Although an ideal cantilever boundary condition assumes a perfectly rigid clamp, practical implementations inevitably exhibit some compliance due to finite fixture stiffness. To mitigate this effect, a hydraulic press was used to apply a controlled clamping force, and frequency saturation behavior was observed as the applied load increased. Figures 6c,d illustrate the variation of each natural bending frequency of the undamaged 6063 aluminium alloy cantilever beam with respect to the applied pressure at the fixed end. The load was incrementally varied from 1 to 10 kN in steps of 0.5 kN. To enhance the assessment of experimental repeatability and measurement reliability, each experimental configuration was tested five times under identical loading and boundary conditions, and the reported modal frequencies represent the average values obtained from these five independent measurements. The frequency resolution of each individual measurement was 0.1 Hz, based on the sampling rate and recording duration. Across the five repeated measurements, the variation in the measured modal frequencies was less than 1 Hz for all investigated bending modes. Nevertheless, repeatability tests revealed stable modal peak locations with negligible variability, confirming the robustness of the experimental procedure and the consistency of the extracted vibration characteristics (Samourgkanidis and Kouzoudis, 2020). As expected, the frequencies exhibit saturation at higher pressure levels. The saturation values were determined by fitting the experimental frequency-load data for each vibration mode using exponential saturation functions. In OriginPro 2024 (OriginLab Corporation, Northampton, MA, United States) the Rank Models option was used to evaluate and compare a set of candidate saturation functions (e.g., y = y 0+Aexp(-x/t), y = y 0+Aexp(-(x-x 0)/t)), and the model providing the best fit was selected based on the adjusted R-square, with candidate models yielding values higher than 0.98 considered acceptable, and the highest-ranked model was selected. The corresponding asymptotic value of the selected exponential saturation function was then taken as the saturation frequency for each mode. The asymptotic region of the frequency-load curve was then interpreted as representing an effective clamped boundary condition in the high-stiffness limit. Repeatability tests under identical loading conditions further confirmed consistent frequency stabilization, supporting the reliability of this boundary approximation.

Table 2 summarizes the physical and geometrical properties of the beam materials used in this study, namely aluminium alloy 6063, aluminium alloy 7075, and AISI 1018 mild steel. All beam specimens were manufactured with identical nominal dimensions and precisely machined using CNC (Computer Numerical Control) processes to ensure geometric consistency across samples. This approach minimizes the influence of geometric variability, allowing the observed differences in dynamic response to be attributed primarily to material properties. The damages were introduced at predetermined locations along the beam by precision machining. Each damage consisted of a transverse machined cut terminating in a sharp V-shaped notch, with a nominal cut width of 1 mm and a V-notch height of 0.5 mm (Figure 5c). The crack-like notch depth and location were controlled during machining according to the experimental damage conditions considered. Because the damages were introduced by machining and had a finite width, they are more precisely described as crack-like notches. The selection of the specific materials was based on their availability from the machinery shop used for specimen preparation. Therefore, the present experimental results demonstrate the consistency of the proposed normalization for the selected metallic materials, rather than providing exhaustive validation across materials with widely different dynamic properties. A broader assessment of the influence of material properties is provided through the numerical results presented in the following section.

5 Results and discussion

Figure 7 shows the dependence of the natural bending frequencies (NFs) on Young’s modulus E and mass density ( ρ ) for the 1 s t, 4 t h, and 8 t h modes using 3D surface plots and contour maps. The results correspond to a representative damaged configuration in which the damage location and severity are kept constant, allowing direct comparison with the normalized formulation introduced later. This choice enables a consistent assessment of material-dependent effects under a fixed damage state, since the intact configuration would, by definition, yield a unity response in the normalized formulation. For clarity, only selected modes are presented, although other modes follow the same qualitative trends with different magnitudes.

Across all modes, a consistent physical behavior is observed: the natural frequencies increase with increasing stiffness and decrease with increasing density. This produces a characteristic warped surface, with maximum values occurring in the high-E/low-ρ region. Overall, the NFs exhibit a strong and nonlinear dependence on both parameters. Increasing density within the range 2000–9000 kg/m 3 reduces the frequencies due to increased inertia, whereas increasing Young’s modulus from 65 to 210 GPa raises the frequencies due to enhanced stiffness (Rao and Fah Yap, 1995). These opposing trends highlight the fundamental interplay between inertia and rigidity in governing the beam’s dynamic response. For a more detailed interpretation, Figure 8 isolates the behavior of the 8 t h mode, presenting it as a function of Young’s modulus for fixed density values and, conversely, as a function of density for fixed stiffness values. This representation confirms the monotonic influence of each parameter and further clarifies their individual contributions to the overall dynamic response.

The influence of Poisson’s ratio on the NFs is presented in Figure 9, revealing a systematic yet relatively weak effect compared to the dominant roles of Young’s modulus and mass density. As Poisson’s ratio increases from 0.20 to 0.50, all vibration modes exhibit a consistent upward shift in frequency, indicating a slight increase in the structural resonance levels. Its influence in the present model arises primarily through the fracture-mechanics-based formulation of the cracked beam, where Poisson’s ratio enters the crack-induced flexibility through the plane-strain and the shear modulus. Consequently, variations in Poisson’s ratio slightly modify the crack-related compliance and the effective stiffness distribution of the beam, thereby influencing the natural frequencies of the damaged beam (Graff, 2012; Achenbach, 2012). This effect is relatively small and becomes less effective for higher-order modes. Quantitatively, this sensitivity is more pronounced in the lower-order modes and gradually diminishes with increasing modal order. The natural frequencies increase by 1.38% for the 1 s t mode, 1.21% for the 2 n d mode, 0.89% for the 3 r d mode, and 0.78% for the 4 t h mode. This decreasing trend continues for higher modes, with variations of 0.61% (5 t h), 0.58% (6 t h), 0.48% (7 t h), and 0.38% (8 t h). Overall, the results confirm that although its magnitude is small, the consistency of this trend suggests that Poisson’s ratio introduces a secondary geometric-elastic coupling effect that is preserved even after normalization.

When expressed in normalized form (NFRs), the frequency response becomes largely insensitive to variations in Young’s modulus and mass density across the investigated range. For each bending mode, the NFR was obtained by normalizing the natural frequency of the damaged beam by the corresponding natural frequency of the undamaged beam as:

N F R = N F damaged N F undamaged (20)

This effect is clearly demonstrated in Figure 10, where 3D color maps of the 1 s t, 4 t h, and 8 t h NFRs are shown as functions of Young’s modulus and density for a fixed crack location and depth. Unlike NFs, which exhibit strong and monotonic sensitivity to both material parameters, the corresponding NFR surfaces become nearly uniform across the entire parameter space. This indicates that variations in Young’s modulus and density have a negligible effect on the normalized response and that the normalization provides a stable relationship between the damage state and the dynamic response.

It should be noted that the reduced sensitivity of the normalized frequency ratios to Young’s modulus and density is an inherent consequence of the normalization, since these parameters influence the damaged and undamaged natural frequencies through a common material-dependent scaling. Therefore, the cancellation of these effects is theoretically expected and is not considered, by itself, the primary novelty of the present work. The contribution of this study is instead the systematic demonstration of the robustness and practical applicability of the normalized frequency ratios under different material properties and damage conditions, while also examining the remaining influence of Poisson’s ratio.

Figure 11 presents the variation of the NFRs as a function of Poisson’s ratio for the first eight bending modes. While the normalization procedure successfully eliminates the dominant influence of Young’s modulus and mass density, this figure reveals that Poisson’s ratio maintains a distinct, mode-dependent influence on the dynamic signature of the damaged beam. As observed, all NFR curves exhibit a consistent positive monotonic correlation with Poisson’s ratio, where a shift from 0.20 to 0.50 yields a steady rise in values. Although this upward trend is uniform in direction, its magnitude is strictly governed by the modal order; the sensitivity to material properties is inversely proportional to the mode number, with lower-order modes exhibiting steeper gradients. Quantitatively, the fundamental mode demonstrates the highest degree of variation, with the 1 st NFR increasing from 0.936 to 0.949, representing a variation of 1.390%. This sensitivity diminishes progressively across the spectrum: the 2 nd mode increases by 1.098%, followed by the 3 rd (0.894%), 4 th (0.732%), 5 th (0.605%), 6 th (0.506%), and 7 th (0.425%) modes, until reaching the 8 th NFR, which exhibits the highest level of stability with a marginal increase of only 0.361% (shifting from 0.971 to 0.975). By comparing the percentage changes in Figure 9 with those in Figure 11, it is evident that the NFRs as a function of Poisson’s ratio preserve the same variation observed in the NFs. This confirms that the normalization procedure maintains an identical sensitivity profile, with consistent percentage changes between the normalized and absolute representations across all modes. From a structural health monitoring perspective, this reduced dependence of the normalized response on material stiffness and density is particularly significant, as it enables the development of generalized damage indicators that require minimal recalibration across different structural materials. This characteristic offers clear practical advantages in large-scale monitoring applications, where material variability and uncertainty often complicate the interpretation of modal measurements. By reducing the sensitivity of the diagnostic parameters to global material scaling effects, the proposed normalization framework facilitates transferable damage assessment methodologies applicable across multiple beam systems without requiring material-specific baseline adjustments.

To further substantiate these numerical findings, an experimental investigation was carried out, as detailed in Section 4. The objective of this stage was to assess the capability of the proposed numerical framework to accurately predict the dynamic behavior of cracked cantilever beams with different material properties and damage configurations. In contrast to the purely computational analysis, the experimental study provided direct physical measurements of the beam vibration response under controlled conditions, enabling a rigorous verification of the numerical predictions. A cantilever beam setup was established using a hydraulic press to approximate an ideal fixed-end boundary condition, while the free end of the beam was instrumented with a magnetoelastic vibration sensing system. This arrangement enabled accurate measurement of the natural bending frequencies under realistic operating conditions and facilitated direct comparison between the experimentally measured frequencies and the corresponding numerical predictions. The close agreement observed between the experimental and numerical results across all investigated cases demonstrates that the proposed normalization procedure effectively suppresses the dominant influence of Young’s modulus and mass density on the modal response. These findings therefore support the concept of approximately material-invariant damage characterization. Table 3 summarizes the experimentally measured natural frequencies (NFs) for each beam material under both undamaged (UnD) and damaged conditions using the proposed magnetoelastic sensing approach.

TABLE 3. First eight experimental NFs for undamaged and damaged beam configurations across different materials. Damage states are defined as: D1 (CL = 80.5 mm, CD = 1 mm), D2 (CL = 203 mm, CD = 2 mm), and D3 (CL = 290.5 mm, CD = 3 mm).

Three damage configurations were investigated, denoted as D1, D2, and D3. In configuration D1, the crack location (CL) was positioned 80.5 mm from the fixed end with a crack depth (CD) of 1 mm. For D2, the crack was located 203 mm from the fixed end with a depth of 2 mm, while for D3, the crack was positioned 290.5 mm from the fixed end with a depth of 3 mm. Table 4 presents the numerically predicted natural frequencies obtained from the finite element model for the same beam materials and damage configurations. The simulations were performed using identical geometric dimensions, material properties, and crack parameters as those adopted in the experimental investigation, thereby enabling a direct and consistent comparison between measured and predicted modal responses.

TABLE 4. First eight numerically predicted NFs for undamaged and damaged beam configurations across different materials.

To further evaluate the accuracy of the proposed numerical framework, the percentage differences between the experimentally measured and numerically predicted natural frequencies are illustrated in Figure 12. Across all eight vibration modes and structural conditions, the numerical predictions remain in close agreement with the experimental measurements, with deviations consistently within approximately 1.5%. This level of agreement is maintained for all investigated materials, indicating that the model presented reliably captures the dynamic response of both intact and damaged beams. The small discrepancies observed can reasonably be attributed to unavoidable experimental uncertainties, including minor variations in boundary condition enforcement, sensor positioning and crack configuration. Overall, the consistently low error levels confirm the robustness of the numerical framework and further demonstrate the effectiveness of the normalization procedure in minimizing material-dependent effects during damage identification.

Table 5 provides a direct comparison between the experimentally measured and numerically predicted NFRs for the three beam materials under the investigated damage configurations. The averaged NFR values reported in the table represent the mean value of the eight bending modes considered for each material and damage configuration, calculated as:

N F R avg = 1 8 ∑ i = 1 8 N F R i (21)

TABLE 5. Experimental and numerical NFRs for the investigated material configurations and damage conditions. The reported average NFR values correspond to the mean value of the first eight bending modes.

For Al 6063, the deviations between experimental and numerical NFRs range from 0.02% to 0.43%. Similarly, for Al 7075, all deviations remain below 0.22%, with no consistent tendency toward over- or under-prediction. For AISI 1018, the differences are also minimal, varying between 0.02% and 0.14%. Across all materials and damage cases, the maximum discrepancy does not exceed approximately 0.5%, and no systematic dependence on either material type or damage severity is observed. Furthermore, the experimental and numerical NFRs exhibit identical overall trends, with only negligible differences in magnitude between corresponding cases. This strong agreement demonstrates that the proposed normalization procedure produces highly consistent results in both experimental and numerical analyses while maintaining stability across varying material properties and crack configurations. The absence of significant outliers or material-dependent deviation patterns further confirms the reliability and general applicability of the proposed damage characterization framework.

6 Conclusions

This study developed and validated a vibration-based damage identification framework for cracked cantilever beams, combining finite element modelling, fracture mechanics, normalization of modal frequencies, and experimental measurements using Metglas-based magnetoelastic sensors. The proposed approach was evaluated across multiple materials, crack locations, and crack severities, focusing on the first eight bending modes.

The results demonstrate that the absolute natural frequencies are strongly governed by material stiffness and inertia, with clear dependence on Young’s modulus and mass density. However, once the frequencies are normalized, this strong material dependency is largely eliminated, leading to normalized frequency ratios that remain nearly invariant across all investigated materials. This outcome confirms that the normalization procedure effectively isolates the influence of damage from the underlying material properties, enabling consistent comparison between different beam materials and configurations. It was further observed that Poisson’s ratio has a relatively minor but systematic influence on the normalized response, particularly in the lower-order vibration modes. Although this effect is weak compared to stiffness and density, its consistency suggests that it cannot be entirely neglected in high-precision applications.

Experimental validation using Al 6063, Al 7075, and AISI 1018 cantilever beams equipped with Metglas-based magnetoelastic sensors showed excellent agreement with the numerical model. The measured natural frequencies closely matched the predicted values, with deviations remaining below approximately 1.5%. More importantly, the normalized frequency ratios exhibited even stronger agreement, with discrepancies below 0.5% across all damage cases and materials. These results confirm both the accuracy of the presented model and the reliability of the experimental sensing approach.

A limitation of the present study is that the proposed reduced-parameter framework was developed and validated for a single structural geometry, namely, a rectangular cantilever beam with fixed-free boundary conditions. Although the approach avoids the need for complex finite element modeling, its extension to more complex structures, such as multi-span bridges, requires further investigation. In such structures, multiple spans and different boundary and connection conditions can significantly influence the modal characteristics and their sensitivity to damage. Therefore, scaling the proposed method to complex structures would require a systematic recalibration of the reduced parameters, starting with a comparison between the experimentally measured and predicted natural frequencies for the corresponding structural configuration and boundary conditions. The influence of span arrangement, support conditions, and other structural parameters would then need to be incorporated into the calibration framework. Experimental validation using progressively more complex structural configurations will be an important direction for future research to assess the general applicability and robustness of the proposed approach.

Overall, the study demonstrates that normalized modal frequencies serve as robust, repeatable, and approximately material-invariant indicators of structural damage within the range of typical engineering materials. Furthermore, the integration of magnetoelastic sensors enables accurate, contactless signal detection of modal characteristics, providing a practical and scalable approach for structural health monitoring of beam-like structures. Future work will extend this framework to more complex structural systems, including plates, frames, composite materials, and scenarios involving multiple damage sites. Further investigations will also consider nonlinear crack behavior, environmental variability, and the incorporation of machine-learning-based methods for interpreting modal signatures in real time. The combination of normalized dynamic indicators with contactless magnetoelastic sensing thus opens new possibilities for lightweight, wireless, and scalable monitoring systems suited to next-generation smart infrastructure applications.

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Footnotes

Edited by: Dipendra Gautam, University of Iceland, Iceland

Reviewed by: Prajwal Giri, University of Perugia, Italy

Ying He, Nanjing University of Posts and Telecommunications, China