Leadless pacemakers require permanent power sources to support long-term operation and eliminate the need for replacement surgeries. This study introduces a dual-folded beam piezoelectric energy harvester (PEH) designed to harvest energy from heartbeat-induced vibrations for powering leadless pacemakers. To simplify the design and reduce computational complexity, a discrete modeling approach is developed under base excitation with the lowest acceleration amplitude of heart vibrations. The results show that incorporating a dual-folded beam configuration with both a link mass and a tip mass allow tuning of the harvester’s resonant frequency within the range of heart vibrations. Parametric analyses of the main system variables including link mass, tip mass, piezoelectric thickness, substrate thickness, and electrical load resistance indicate that optimal values can be determined for these parameters to maximize the harvested output power. The Genetic algorithm is used to optimize the value of system parameters to maximize the total average output power while satisfying dimensional constraints. The optimized model achieves a resonant frequency within the cardiac band (1–50 Hz), ensuring dimensional compatibility with the leadless pacemaker. Simulation results demonstrate that the optimized device increases the average output power from 4.83 μW to 257.64 μW, while simultaneously reducing the resonant frequency from 52.46 Hz to 20.21 Hz, indicating its potential as a compact power supply for leadless pacemakers.
This study investigates the effects of thickness reduction ratio (TRR), die angle, and friction coefficient on maximum stress, residual stress, and plastic deformation during AA1100 aluminum cup ironing using finite element analysis supported by experimental observations. A three-factor, three-level design generated 27 simulation combinations, and mesh verification identified a 1.00 mm element size as an efficient and stable configuration. The results show that maximum stress ranged from 348.99 to 363.00 N/mm², while residual stress ranged from 119.11 to 163.33 N/mm². ANOVA screening identified TRR as the dominant factor for maximum stress with a contribution of 84.21%, followed by die angle at 9.82%, whereas friction had only a minor effect. For residual stress, TRR and friction were the most influential parameters. Increasing TRR also increased the mean maximum equivalent plastic strain from 0.3980 at 20% TRR to 0.5915 at 30% TRR, while the mean residual stress decreased, indicating that active forming stress, accumulated plastic strain, and post-forming stress follow different response patterns. Experimental results supported the physical consistency of the model through forming-force and dimensional comparisons. Hardness increased from 44.3 to 52.0 BHN after ironing, while metallographic and SEM observations showed deformation-related morphological changes. These findings indicate that TRR primarily governs deformation severity and maximum stress, whereas friction is more relevant to the post-forming stress state. The results provide a practical basis for selecting ironing parameters according to the targeted mechanical response, although direct experimental measurement of residual stress is still required.
Electric vehicles raise higher durability demands for powertrain rubber-metal mount bushings owing to compact structural layout and complex random excitations. Traditional fatigue evaluation methods depending on nominal load and global stiffness fail to identify local fatigue cracks of rubber components with prominent hyperelasticity. This work proposes a road-spectrum-driven closed-loop framework for fatigue crack prediction and durability optimization of such bushings. Virtual road spectrum loads are acquired and preprocessed, and a Yeoh hyperelastic finite element model is built to analyze nonlinear stress-strain responses. Combining Miner linear damage rule, fatigue critical regions are predicted, and bench tests are adopted to validate simulation accuracy. Results reveal that severe stress-strain concentration occurs at rubber transition zones and rubber-metal interfaces, with an average fatigue safety factor of only 0.75 under 1.0 × 10⁶ cycles. Experimental crack initiation positions are highly consistent with simulation predictions. Targeted local geometric optimization enhances both stiffness and fatigue performance. The optimized experimental and simulated stiffness values reach 482.93 N/mm and 487.51 N/mm respectively, both surpassing the design requirement. The low-fatigue-safety-factor areas are greatly reduced. This framework offers an effective solution for fatigue localization and structural durability improvement of automotive rubber bushings.
E-glass fiber reinforced epoxy composites were fabricated by the hand lay-up technique with varying fiber layers (2, 3, and 4) under constant matrix content to investigate the influence of fiber weight fraction. Nano-silica fillers (2, 4, and 6 wt.%) were incorporated within each laminate to investigate their effects on mechanical properties, post-aging mechanical performance under neutral (pH 7) conditions, and moisture absorption behavior in acidic, neutral, and alkaline environments. Increasing the number of E-glass fiber layers significantly enhanced the tensile, flexural, and hardness properties of the composites due to the higher fiber reinforcement content, while the 4-layer E-glass/epoxy composite reinforced with 4 wt.% nano-silica demonstrated the best overall mechanical performance among all developed composite configurations. Post-aging in pH 7 medium for 55 days resulted in moderate reductions in mechanical properties across all composite variants. Finite element analysis (FEA) was employed to validate the experimental tensile and flexural results, demonstrating good agreement between numerical and experimental findings. Fracture surface morphology analyzed by scanning electron microscopy (SEM) showed fiber pull-out, matrix cracking, and filler dispersion, which correlated well with the observed mechanical performance, while energy-dispersive X-ray spectroscopy (EDS) verified the elemental composition of the composites. XRD analysis identified crystalline α- quartz nano silica within an amorphous epoxy/glass matrix in the composite, corroborated by FTIR functional group retention. Water absorption over 55 days followed Fickian diffusion, with acidic medium (pH 3) exhibiting the highest uptake, followed by alkaline (pH 8), then neutral (pH 7).
This work is concerned with the influences of surfactants on the poromechanical response of unsaturated porous materials. Surfactants modify the surface tension at the interface between the wetting and nonwetting fluids in the interconnected pores thereby affecting the mechanical and hydrological responses of unsaturated media. This work employs a Biot-type poroelasticity theory for unsaturated porous media with a surfactant-modified capillary modulus to investigate the coupled fluid transport and solid deformation in surfactant-laden unsaturated porous materials. Analytical expressions of pore pressures and settlement in a finite porous column filled by a wetting fluid and a nonwetting fluid are obtained using the Laplace transform method. Numerical results for a clay texture contaminated by perfluorooctane sulfonate (a major PFAS compound) show that the pore water pressure decreases with an increase in the aqueous PFAS concentration. On the other hand, the pore air pressure is increased by the presence of the PFAS. The surface displacement also increases with an increase in the PFAS concentration. Finally, the influences of the PFAS on the pore fluid pressures and deformation become less significant at high water saturations towards the saturated moisture content in the clay.
The transient bending response of Bernoulli–Euler nanobeams is investigated within the framework of two-phase strain-driven nonlocal integral elasticity. A central novelty of this study is the introduction of a physically motivated functional relationship between the nonlocal phase fraction and the nonlocal length-scale parameter, eliminating the conventional assumption that these quantities are independent. Three functional forms are proposed for this dependence: linear, exponential saturation, and power-law models, each satisfying the requirement that nonlocal contributions vanish in the local limit and grow monotonically with increasing internal length scale. The governing integro-differential equation of transverse motion is derived via Hamilton's principle and converted into an equivalent sixth-order partial differential equation supplemented by constitutive boundary conditions. The resulting initial-boundary value problem is solved analytically using the Laplace transform technique in the temporal domain. For the simply supported beam conFig.uration, closed-form time-domain solutions are obtained for impulsive, step, and harmonic loading cases through exact inverse transformation; for other boundary conditions, the full sixth-order system must be solved numerically in the Laplace domain. A systematic parametric study reveals that the scale-dependent phase fraction coupling amplifies the softening character of nonlocality relative to constant-fraction models, with significant consequences for natural frequencies, peak transient displacements, and resonance characteristics. The present analytical solutions serve as benchmarks for numerical implementations of two-phase nonlocal beam formulations.
This paper presents a discrete mechanical formulation for the static, buckling, and free-vibration analyses of a tapered two-dimensional functionally graded material portal frame. The structure is composed of two columns and a top beam, whose cross-sectional dimensions may vary along the member axis and whose material properties are graded both through the thickness and along the axial direction. The thickness-wise gradation is described by a power-law distribution, whereas the axial variation is modeled by an exponential law. The frame is discretized by a reduced number of generalized coordinates, leading to a total of 3N+1 degrees of freedom. The structural flexibility is modeled by discrete rotational springs, while the inertia is represented by lumped masses located at the discrete stations of the frame. This reduced-order representation is a major advantage of the proposed approach, since it preserves the main geometric and mechanical features of complex frame structures while keeping the formulation simple and computationally inexpensive. The model is therefore particularly attractive for large parametric investigations. The governing equations are derived from the total strain energy, kinetic energy, and geometric energy associated with axial compression in the columns. The linear stiffness matrix, mass matrix, and geometric stiffness matrix are obtained in explicit form and assembled into a unified discrete system. The formulation enables the prediction of static response, critical buckling loads, and natural frequencies within the same framework. Due to its simplicity, efficiency, and ability to capture the effects of geometric non-uniformity and bidirectional material gradation, the proposed discrete model provides an effective tool for the analysis of advanced tapered portal frames.
This study investigates the efficacy of alkali treatment on JUCO fabric-reinforced epoxy composites to improve their structural integrity and environmental resilience. JUCO fibers were treated with sodium hydroxide (NaOH) at varying concentrations (1-5 wt. %) for 48 h. To date, JUCO fiber has not been subjected to alkali treatment in different NaOH concentrations. Using a hand lay-up and cold press method, composites were fabricated using four layers of fabric and epoxy resin. The mechanical characteristics, hydrophilicity, and long-term aging were evaluated in three different mediums. Fick's law was applied to analyze water diffusion kinetics, while finite element modeling (FEM) was used to validate tensile and flexural behavior. The alkali treatment significantly improved the mechanical strength of the composites. This enhancement is attributed to superior fiber-matrix interfacial adhesion, which is supported by scanning electron microscopy (SEM) and energy-dispersive spectroscopy (EDS) analyses. In addition, X-ray diffraction (XRD) revealed that composites made with NaOH-treated fibers exhibited higher crystallinity compared to those with untreated fibers. The 3% NaOH treatment demonstrates a significant reduction in moisture uptake from 4.92% to 0.82% in distilled water and a corresponding decrease in diffusion rates. The findings confirm that the surface modification of JUCO-based composites by NaOH treatment presents a sustainable and effective strategy for developing moisture-resistant materials appropriate for structural applications. An ideal balance between improved performance and long-term durability is offered by the 3% NaOH treatment.
This study develops a predictive model to estimate the critical buckling load of pultruded composite columns using a finite element–group method of data handling (GMDH) neural network, considering the effect of fiber orientation. Four types of composite columns with different cross-sectional shapes are analyzed. A total number of 243 tests are designed using the Design of Experiments (DOE) method to generate an optimized dataset for neural network training. The predicted buckling loads serve as the starting point for damage analysis, where the peridynamic method is employed to investigate crack initiation, branching, and overall damage evolution in the columns. The proposed approach provides a predictive relationship between fiber orientation and critical buckling load based on the GMDH model. The novel aspect of the proposed model is that it can evaluate the effect of different fiber orientations on the buckling load without the need for additional numerical simulations or expensive experimental testing. The proposed approach also compares the traditional Hashin failure criterion with the peridynamic results, indicating that peridynamics captures damage patterns and crack propagation more accurately compared to experimental results. The results indicate that closed-section composite columns can increase the critical buckling load by approximately 40%. Furthermore, validation against experimental data shows an average error of approximately 9.8% in the predicted critical buckling loads, demonstrating the accuracy of the proposed GMDH neural network model. This integrated framework demonstrates how predictive modeling of buckling combined with peridynamic damage analysis provides an efficient and reliable framework for designing and assessing the structural performance of composite columns.
To ensure the safety and reliability of structures, accurate identification of both the magnitude and location of impact loads is critically important. In this paper, we propose a method for estimating these parameters using the Cuckoo Optimization Algorithm (COA). Assuming that the structure undergoes elastic deformation, the temporal variations of the rotational angles measured by wireless gyroscopic sensors mounted on the surface and the impact loads are related through a response matrix equation. The impact location was determined using the COA, whereas the magnitude of the impact load was computed using the Moore–Penrose generalized inverse. Consequently, the proposed approach enables efficient identification of the time history of impact loads and their locations without requiring a large number of sensors. To verify the effectiveness of the proposed method, it was applied to impact problems involving a simply supported beam and a plate. The results confirm that the method can accurately estimate both the magnitude and location of the applied impact load in these cases.
This study introduces the Analytical Artificial Neural Networks Method (AANNM), a groundbreaking framework that systematically converts the discrete, black-box outputs of neural network solvers into closed-form analytical solutions. The efficacy of AANNM is demonstrated by solving the differential equation governing the Kelvin-Voigt viscoelastic model. First, a high-fidelity numerical solution is obtained using a Physics-Informed Neural Network (PINN). The core innovation of AANNM is then deployed: the discrete PINN data is used to construct a system of algebraic equations, the solution of which yields the coefficients for a precise polynomial analytical expression. The derived AANNM solution is directly validated against the known exact analytical solution, demonstrating exceptional agreement and providing a more rigorous benchmark than comparisons with purely numerical methods. Crucially, while demonstrated with PINNs, the AANNM framework is solver-agnostic, designed to convert discrete solutions from any artificial neural network into analytical form. This inherent flexibility ensures the method's applicability to future ANN advancements, making it both timeless and adaptable. The proposed framework establishes AANNM as a transformative pipeline that bridges data-driven numerical models with rigorous analytical mathematics, significantly enhancing the interpretability, utility, and trustworthiness of machine learning in computational science.
In this study, a thin-walled high-strength S960 rectangular hollow section T-joint is cyclically tested at a stress ratio of R=0.1. The work focuses on the effect of weld profiling conducted as post-weld treatment by grinding of the weld seam. Numerical simulations using the real weld seam geometry are carried out and compared with strain gauge measurements. Thereby, it is shown that the stress states of both methods are in sound agreement and the crack initiation site can be assessed well by the numerical approach. The resulting nominal S/N curves reveal an increase of the fatigue strength by about 33% at two million load-cycles due to weld profiling, which represents a high potential for this post-weld treatment technique. Furthermore, the notch stress approach using the common procedure by modelling a reference radius of rref=1 mm at the weld toe as well as modelling the real weld toe geometry after weld profiling are applied. The results are compared to the recently published values in the Recommendations for Fatigue Design of Welded Joints and Components by the International Institute of Welding (IIW) and highlight that both different modelling methods lead to a sound notch stress fatigue assessment of the weld-profiled condition.
In the current study, we investigate the dynamic response of cracked rotating FG-beams with two variable boundary conditions. The crack is considered to be simulated by a massless torsional spring model. The beam motion equation is obtained based on Hamilton’s concept. In this study, a power-law exponent describes graded beam materials as they vary through the beam's thickness. The beam's natural frequencies are established by solving the vibration equations with the Galerkin method. The study investigates the effect of geometrical and material properties, rotating speed, distributed force, hub length, and crack parameters on these frequencies. The analysis shows that the power index decreases the dimensionless natural frequencies for all end conditions, with or without a crack. In the absence of cracks, the ratio of the reduction in frequency of the double-simply supported FG beam is 17.58%, and the ratio of the reduction in frequency of the clamped-free end conditions is 15.95%. The frequency decreases by 22.75% and 19.60% in S-S and C-F, respectively, with a crack.Also, the dimensionless vibration frequency decreases with increasing tangent follower force, by 14.97% in S-S and 10.65% in C-F. Also, the results exhibit that crack depth lowers the dimensionless vibration frequencies. Moreover, the analysis shows that the hub radius ratio raises the dimensionless vibration frequencies, irrespective of the presence of a crack, across all end conditions. The findings provide useful insight for the vibration analysis and design of rotating FG structures in practical applications.
Classical finite‐element model updating (FEMU) requires iterative updates of the structural model. Alternative approaches are often computationally expensive or oversimplify the actual physical behavior of the structures. This paper introduces a novel approach, an enriched spring-based Bayesian finite element model updating (ES-BFEMU) method, which reduces computational cost while preserving high fidelity. In the proposed model, each structural element is divided into two beam-like sub-elements and a rotational spring. The stiffness matrix is derived by enriching the strain energy of both the sub-elements and the spring. This formulation enhances the physical interpretability of local stiffness degradation by explicitly representing rotational flexibility in potential damage zones. The enriched stiffness matrix is updated to provide a more realistic finite element model by detecting stiffness reductions within a Bayesian FEMU framework that considers the uncertainties. The computational efficiency is improved by adopting an adaptive transitional Markov chain Monte Carlo (TMCMC) algorithm to obtain the posterior probability of parameters. The proposed model is applied to the Salar Bridge, a six-span structure instrumented with accelerometers, displacement transducers, and strain gauges. Structural damage is simulated by introducing stiffness reduction coefficients into the elastic modulus of selected elements, with scenarios defined at reduction levels of 2%, 5%, 10%, and 15%. The introduced damages were successfully detected with a deviation of <2%. The proposed ES-BFEMU was also compared with surrogate and reduced-order FEMU approaches, demonstrating improved computational efficiency and higher accuracy in damage identification. The proposed model serves as a bridge between the surrogate and reduced-order FEMU. It reduces the computational cost associated with the reduced-order FEMU by a factor of 3.57 and enhances the accuracy of the surrogate method by 33%. The stress-time results of ES-BFEMU show a prediction error of <6.15% when compared to experimental results. These results demonstrate that ES-BFEMU provides a computationally efficient, physically interpretable, and reliable framework for structural health monitoring and damage identification.
The designs and improvements in the damage tolerance field for engineering constructions subjected to variable amplitude loading are crucial research topic in the modern technology. In the present work, the combined impact for nickel-multiwalled carbon nanotubes coating with the variable amplitude loading on the fatigue crack growth performance has been investigated in the 7075-T6 aluminum specimens. Electroless plating technique was used to coat the adopted aluminum specimens. Six conditions of the variable amplitude loading were applied individually on the coated and un-coated aluminum specimens. Multifunctional fatigue equipment integrated with high magnification camera and a computer software was utilized to applying the fatigue loading conditions and measuring crack lengths. FESEM and EDXRF techniques were adopted to examine the topography, modification in composition of chemicals and quality of the nano-coated specimens' surfaces. Numerical modeling used an effective algorithm contains Fortran and ABAQUS software has been utilized in order to validate the measured results. The extracted results indicated that there is an enhancement of 62.2 % in the full width failure of the nano-coated specimens in comparison with the un-coated specimens. A single overload condition led to a significant retardation in crack growth and it rates (reach to 27,300 delayed cycles) as compared with constant amplitude and other loading. Utilizing the nano-coating over aluminum specimens resulted in decrease the intensity of the retardation of crack growth rates in comparison with that in un-coated specimens for all loading conditions. The behavior similarity of the crack propagation and its path are the main findings of the results' validations that extracted from numerical modeling and fatigue testing.
Civil as well as military facilities, vehicles and applications must increasingly meet higher safety standards to ensure the highest possible protection against extraordinary stresses such as ballistic and/or air-blast loading. The use of lightweight sandwich constructions is considered an efficient and promising measure for enhancing passive safety and maintaining structural integrity. In addition to increased bending stiffness compared to monolithic plates of the same weight, they also exhibit a more favorable behavior under dynamic loading scenarios. The influence of open and closed auxetic core geometries on the relevant mechanical parameters under impact loads of spherical rigid projectiles (rsph = 15 mm, msph = 10 g) with velocities of 200 ms−1 will be analyzed in the present study in order to be able to draw principal conclusions on the auxetic mechanisms and their effectiveness. A new performance indicator is suggested in this context. Numerical studies were performed using the commercial finite element code ABAQUS/Explicit. This included a validated material model for the aluminum alloy EN AW-7108 T6, which considers strain-rate dependent plastic material behavior and typical failure criteria. A comparison with a monolithic reference plate of the same mass and conventional non-auxetic core topologies allows a final efficiency assessment of the sandwich designs with a modified internal structure. The displacements of the rear face surfaces as well as the resulting stresses on supporting structures can be reduced by up to 90 percent and the plastically dissipated energy can be increased by up to 15 percent for some core variants.
The thermal behaviour of viscoelastic fluids, particularly those described by the Maxwell model capable of capturing stress-relaxation effects, exhibits several intriguing and practically relevant characteristics. Likewise, flow induced by a contracting surface presents distinctive and non-classical boundary layer features. Motivated by these aspects, the present theoretical study investigates heat transfer in Maxwell fluid flow over a porous contracting flat surface embedded in a porous medium, incorporating a temperature-dependent and spatially-dependent non-uniform heat source/sink. The governing partial differential equations are reduced to self-similar ordinary differential equations using appropriate similarity transformations, and the resulting nonlinear system is solved with high accuracy using the shooting method along with fourth-order Runge–Kutta (RK-4) technique and Secant method. For sufficiently strong suction, dual solutions of the transformed equations are obtained, and a stability assessment confirms that the upper branch solutions are stable, while the lower branch solutions are unstable. The comprehensive analysis uncovers several notable physical insights. The analysis shows that viscoelasticity of Maxwell fluids plays a key stabilizing role: increasing the Deborah number weakens vorticity production, lowers the suction required to maintain an attached boundary layer, and broadens the parameter range in which similarity solutions exist. Permeability of the porous medium also strongly influences boundary layer behaviour; higher resistance suppresses vorticity generated by sheet contraction, enhancing skin friction and heat transfer in the upper branch but diminishing both in the lower branch. Both temperature- and space-dependent heat sources reduce heat transfer rates, while their sink counterparts enhance cooling, though spatial heat generation/absorption produces a far more pronounced thermal response. Suction strengthens and compresses the boundary layer in the stable upper branch but weakens the flow in the unstable lower branch. Overall, the study clarifies how elasticity, permeability, heat generation mechanisms, and mass transfer collectively shape the dual solution structure of Maxwell fluid boundary layers, offering detailed insights relevant to thermal control in viscoelastic–porous systems.
Hybrid two or more fiber-reinforced composites are generally prepared to enhance different properties as compared to single-fiber reinforced composites. Sheep wool and sisal fibers are natural fibers that can be obtained from animal and plant sources respectively. After extracted and treated the fibers, the woven yarn fiber mat was prepared. The woven hybrid composite was fabricated with a 20% weight fraction of fiber by using hand layup fabrication techniques. Composites samples were prepared under five different weight percentage ratios of sheep wool to sisal fiber 0:20, 5:15, 10:10, 15:5, and 20:0. And each weight percentage of a sample was fabricated with two different angles (0°-90° and ±45°) of orientations. From the experimental test results, it was observed that the tensile, compressive, flexural strengths increase directly with increase sisal fiber weight percentage of composite samples in both 0-90° and ±45°angle of orientations. However, between the two angles of orientation, the tensile and flexural strengths of the hybrid sheep wool and sisal fiber epoxy composite samples were highest in 0-90°angle orientations composite samples. On other hand, the compressive and impact strengths were highest in ±45°angle orientation of the composite samples. Overall, the composite sample with a 15:5 sisal fiber-sheep wool ratio (SA4 and SB4) demonstrated the best mechanical performance. The maximum tensile and flexural strengths of 95.73 MPa and 358.80 MPa, respectively, were obtained for the 0°–90° oriented composite, whereas the highest compressive strength of 95.73 MPa and impact strength is 746.77 kJ/m² were observed in the ±45° oriented composite. The experimental test result shows that the hybrid sheep wool and sisal fiber epoxy composite are alternative materials for the interior part of automotive applications like interior roof and door panels.
Roller bearings are commonly designed by using non-Hertzian submodels to simulate the roller-raceway contacts. The paper aims to analyze how to accurately model the roller-raceway contacts of a large wind turbine slewing bearing with significant structural deformation. To this end, a global FE model of a rotor blade bearing inside a test rig is used. This global FE model uses simplified roller-raceway contact models, the results of which are then inserted into a more accurate analytical submodel based on Reusner. Various methods can be used to perform the global FE calculation and then to insert the result into the submodel. The paper analyzes which method is required in the global FE model and compares two submodels against each other. One of the submodels, which uses displacement as input, is from the literature, and the paper derives from it another one which uses load as input. The model using load as input is identified to be more suitable for the analyzed applications. Roller-raceway contacts in the global FE model do not appear to require much detail for accurate submodel simulations. The results of the paper can support engineers perform detailed rolling contact fatigue life calculations of roller bearings.
Accurate knowledge of dynamic load locations and time histories is a critical input for structural design but is often infeasible to measure directly. While numerous load identification methods exist, they predominantly address the localization and time-history reconstruction separately, relying on the prior assumption that one of the two is known. This paper introduces a novel and efficient integrated approach that combines Blind Source Separation (BSS) with Structural Modal Shape Matching (SMSM) to concurrently identify both the spatial location and temporal profile of dynamic loads. The proposed methodology is founded on the principle that modal loads and physical loads are mutually convertible. Initially, truncated modal loads are stably reconstructed in the modal space using a shape function method with Tikhonov regularization. These recovered modal loads are then interpreted as blind mixtures of the unknown physical load source signals, with the structural modal shape coefficients acting as the mixing matrix. BSS is subsequently employed to separate the equivalent load time histories and estimate the mixing matrix. Since the mixing coefficient vector is linearly related to the structural mode shape vector at the load application point, SMSM is implemented by quantifying the intersection angles between the estimated mixing vectors and candidate modal shape vectors to pinpoint the most probable load locations. Finally, the actual load time histories are accurately retrieved using the reconstructed modal loads and the identified modal shape matrix. The efficacy of the proposed method is rigorously demonstrated through two numerical examples involving a complex ropeway tower and a rectangular plate.
