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Data-driven damage detection methods are widely researched and implemented due to the availability of advanced sensing and cloud technologies, where machine learning models are used to process the various data collected from the sensors for damage diagnosis. Supervised methods have shown to be accurate in identifying damages given that they are trained with sufficient historical damage samples. As damage events are rare, supervised methods often face the cold-start problem, that is, insufficient labelled damage samples to initiate model training. Unsupervised methods do not require training and have shown to be practical up to detecting damage presence, but prior knowledge or labelled historical damage samples are still required as references to further locate and quantify the damage. To solve the cold-start issue, a modal-based method, which relates the modal parameters with damages, is proposed to bridge the gap between the unsupervised and supervised methods. Therefore, this work proposes an integration strategy of a modal-based method into a hybrid machine learning-based method for damage detection and localisation of beam-like structures. A two-stage scheme with unsupervised, supervised, and modal-based methods is proposed. The first stage detects damage presence using PCA-FRF as the damage-sensitive feature in an unsupervised manner, and the second stage locates the damages using the first mode shape. Mode shape assessment-based damage localisation is performed at the second stage when no trained model is available due to zero or limited damage samples for training to solve the cold-start and manual labelling issue. Supervised model training starts once there are at least two classes of damage, and the trained model is only used for damage localisation when it surpasses the 90% validation accuracy benchmark. Results showed accuracies of 100%, 100%, and 95.85% for the first stage unsupervised detection, second stage mode shape assessment-based damage localisation and the supervised model-based damage localisation, respectively. [ABSTRACT FROM AUTHOR]

This study aims to make investigation for the free vibration problem of simply supported axial-FGB by applying the Rayleigh method. The model of (power law) is adopted to describe the change in physical mechanical and properties through the axial direction. The computer program is built in this work. The accuracy of Rayleigh method is checked by comparison of the results of Rayleigh method with the results available in literatures and a very good accuracy was found. The combined effects of power-law index, modulus-ratio and density-ratio on the fundamental frequency and mode shapes of axial-FGBs are investigated. The results explain that the non-dimensional frequency parameter at any modulus-ratio is approximately constant when the power-law index increases. When the power-law index is smaller than (1), the effect of modulus-ratio is greater than the effect of density-ratio. While the effect of modulus-ratio is smaller than the effect of density-ratio when the power-law index is greater than (1). Also, when density-ratio equals (1), the dimensionless deflection (or amplitude) increases with reducing of power-law index (β). When the modulus-ratio is smaller than (1), the dimensionless deflection increases with increasing the power-law index (β). From results, the Rayleigh method is able to calculate the natural frequency and mode shape of simply supported axial-FGM beam. [ABSTRACT FROM AUTHOR]

A prevalent issue in rotors is mass unbalance, which leads to synchronous lateral vibrations. In this paper, the critical whirling speeds and global responses of an unbalanced rotor-bearing system with viscoelastic boundary condition are computed using the relationships between the vectors of solution coefficients of differential equations governing the transverse displacements. The mathematical formulation is based on the matrix form of the aforementioned relationships and boundary conditions in the resonance case. The Timoshenko beam model, including the effects of gyroscopic moments is employed. It was found that the presented approach in comparison with the transfer matrix method reduces the matrix product when shaft segments have the same geometric and mechanical properties. A good agreement is found between the results of this study and the findings of studies available in the literature. Furthermore, the eigenvectors for free vibrations of the system were calculated by combining Natanson's technique with the presented approach while considering the derived rotor's characteristic equation after removing the forced term in the equation that describes the forced vibrations. Each obtained forward whirling mode shape describes perfectly the global response in the vicinity of the corresponding forward critical speed. The presented approach can, therefore, be applied to obtain the unbalance responses, whirling speeds, and associated mode shapes of rotor-bearing systems with complex boundary conditions. [ABSTRACT FROM AUTHOR]

This study examined the vibration behavior of a beam with linear spring attachments using finite element analysis. It aims to determine the natural frequency with both consistent/coupled mass and lumped mass matrices. The natural frequencies and corresponding mode shapes were correctly determined which formed the basis of any further noise vibration and severity calculations and impact or crash analysis. In order to obtain eigenfrequencies subject to the attached spring, the characteristic equation was obtained by eigenfunctions expansion whose roots were extracted using the root-finding technique. The finite element method by coupled and lumped mass matrices was then used to determine complete mode shapes against various eigenfrequencies. The mode shapes were then analyzed subject to supports with varying stiffness thereby comparing the analytical and numerical results in case of consistent and lumped masses matrices so as to demonstrate how the present analysis could prove more valuable in mathematical and engineering contexts. Utilizing a consistent mass matrix significantly enhanced accuracy compared to a lumped mass matrix, thereby validating the preference for the former, even with a limited number of beam elements. The results indicated that substantial deflection occurred at the beam's endpoints, supporting the dynamic behavior of the spring-beam system. [ABSTRACT FROM AUTHOR]

This study examined the vibration behavior of a beam with linear spring attachments using finite element analysis. It aims to determine the natural frequency with both consistent/coupled mass and lumped mass matrices. The natural frequencies and corresponding mode shapes were correctly determined which formed the basis of any further noise vibration and severity calculations and impact or crash analysis. In order to obtain eigenfrequencies subject to the attached spring, the characteristic equation was obtained by eigenfunctions expansion whose roots were extracted using the root-finding technique. The finite element method by coupled and lumped mass matrices was then used to determine complete mode shapes against various eigenfrequencies. The mode shapes were then analyzed subject to supports with varying stiffness thereby comparing the analytical and numerical results in case of consistent and lumped masses matrices so as to demonstrate how the present analysis could prove more valuable in mathematical and engineering contexts. Utilizing a consistent mass matrix significantly enhanced accuracy compared to a lumped mass matrix, thereby validating the preference for the former, even with a limited number of beam elements. The results indicated that substantial deflection occurred at the beam's endpoints, supporting the dynamic behavior of the spring-beam system.

The first goal of this research was to document the process of using the MODAL analysis system of the ANSYS 2024R1 student edition to create a finite element model of the modes and frequencies of vibration of one basketball rim and backboard design. This finite element model included the use of steel for the rim and its mount, a tempered glass backboard, and an aluminum frame behind the backboard. After a mesh was created, fixed support boundary conditions were applied to the four corners of the aluminum frame, followed by the theoretical modal analysis. The second goal was to validate this model by comparing the finite element calculated mode shapes and frequencies to the empirical modal analysis previously measured at the United States Military Academy at West Point, New York. Five mode shapes and frequencies agreed rather well between the theoretical finite element analysis and previously published empirical modal analysis, specifically where the rim was vibrating in the vertical direction, which was the direction that the accelerometer was aligned for the empirical modal analysis. These five modes were addressed from a finite element model validation standpoint by a 99.5% confidence in a 98.09% cross-correlation with empirical modal analysis data, and from a verification standpoint by employing a refined-mesh. However, three theoretical mode shapes missed by the empirical modal analysis were found where the vibration of the rim was confined to the horizontal plane, which was orthogonal to the orientation of our accelerometer.

Functionally graded materials are innovative composites of hybrid ceramics and metals that exhibit excellent mechanical performance in harsh temperature environments and under various external loads. In this study, the free vibrations of Timoshenko circular arches, made of functionally graded materials in the axial direction, are investigated. The material properties of Young's modulus and mass density of the arch vary according to a symmetric quadratic function along the arch axis. Differential equations governing the free vibration of the arch including the rotatory inertia and shear deformation, called the Timoshenko arch, are derived. A novel numerical solution method is developed to calculate the natural frequencies and mode shapes of the arch. Parametric studies of the modular ratio, shear correction factor, shear modulus ratio, and slenderness ratio on the natural frequencies are conducted, and the results are reported in the tables and figures. [ABSTRACT FROM AUTHOR]

The effect of structure on the vibration response was explored for four piano soundboards with different but commonly adopted structures. The vibration response was obtained using the free-vibration method, and the values of the dynamic modulus of elasticity and dynamic shear modulus obtained using the free-vibration frequency method (EF and GF) were compared with the dynamic modulus of elasticity obtained using the Euler beam method (EE) and dynamic shear modulus obtained using the free-plate torsional vibration method (GT), respectively. It was found that the soundboards with different structures had different vibration modes and that excitation at different locations highlighted different vibration modes. For all the soundboards analyzed, the EE and GT were higher than EF and GF by 2.2% and 24.3%, respectively. However, the trends of the results of these methods were the same. The four piano soundboards with different structures possessed varying dynamic moduli of elasticity and dynamic shear moduli. These rules are consistent with the grain directions of the soundboards and the anisotropy of the wood (the direction of the units of the soundboards). The results show that the vibration mode of the piano soundboard is complex. The dynamic elastic modulus of the soundboard can be calculated using the Euler beam method. The results provide a reference for studies on the vibration response, material selection, production technology, and testing of piano soundboards. [ABSTRACT FROM AUTHOR]

Structures are subjected to a variety of environmental and loading conditions over time, and minor damage to structural elements may occur. By timely identifying damage and repairing damaged locations, it is possible to prevent the spread and development of damage to other elements and, as a result, the overall destruction of the structure. This article discusses the identification and determination of the location of damage in steel plates based on the use of the primary and secondary shapes of vibration modes and the analytical method of two-dimensional wavelet analysis. Modelling and frequency analysis of the plate were performed in ABAQUS software, and the primary and secondary mode shapes were extracted. To determine the location of the damage, a damage detection index (DDI) was proposed based on the angle between the primary and secondary mode shape vectors and the diagonal detail coefficients obtained from the wavelet analysis of the primary and secondary mode shapes. The results showed that by using this index, damage can be identified by identifying peaks resulting from irregularities and disturbances. Also, the DDI value of the damage was dependent on the severity of damage occurring in a damaged situation, and the height of the disorder peaks increased with increased damage only at that damage position.

The integration of Operational Modal Analysis (OMA) and Finite Element Model (FEM) techniques has proven to be a valuable approach for evaluating and maintaining the health of such structures. OMA extracts dynamic characteristics from a bridge's responses to ambient vibrations, while FEM employs computational models to simulate and compare these dynamic behaviours. This comprehensive methodology ensures an accurate representation of the bridge's behaviour and its correlation with real-world conditions. However, a significant challenge arises in assessing the Ultra High Performance Fiber Reinforced Concrete (UHPFRC) pedestrian bridge in Klang, Selangor, launched in October 2022, due to limited available information about OMA and FEM on UHPFRC. This research aims to develop FEM and conduct ambient vibration tests to obtain modal parameters. Moreover, the study investigates OMA techniques in the context of weak excitation sources, filling a critical knowledge gap. By comparing experimental results with FEM, this research provides insights into the feasibility and reliability of OMA under challenging environmental conditions. The principal results reveal significant percentage differences in natural frequencies between FEM and OMA, with the most notable disparities occurring in the first mode. Nevertheless, the mode shapes extracted from OMA closely resemble those from FEM. In conclusion, this research enhances the understanding of UHPFRC pedestrian bridge behaviour and modal analysis techniques. [ABSTRACT FROM AUTHOR]

In this study, the modified first-order shear deformable theory (FSDT) is applied to analyze the resonant characteristics of piezoelectric (PZT) moderately thick plate vibrators with different thicknesses. A simple and straightforward simplification method between mechanical and electrical functions is also proposed based on the modified FSDT. The natural frequencies and associated mode shapes of piezoelectric plates with completely free boundary are analyzed. The displacement fields of the flexural modes are presented based on the superposition method. This analytical method provides a strong-form novel solution for the flexural-dominated motion's out-of-plane and in-plane coupling displacement fields. The solutions of displacement functions, which are obtained from the superposition method, show excellent convergence for numerical calculation. To verify the validity of the theoretical solution, the resonant frequencies and the corresponding mode shapes are compared with that obtained by the finite element method (FEM) and experimental measurements using the amplitude-fluctuation electronic speckle pattern interferometer (AF-ESPI) technique. According to the electromechanical formula and the derived theoretical solution, four types of electrode are designed to excite the moderately thick PZT specimen for examining the application feasibility of the proposed method. It is shown in this study that the theoretical analysis, numerical calculation, and experimental measurement are fully compared, and excellent agreements of the piezoelectric moderately thick plate are also obtained. [ABSTRACT FROM AUTHOR]

It is crucial to perform routine bridge maintenance in order to evaluate the structure's current state. As a result, it is possible to guarantee that the bridge structure can offer services that are both comfortable and secure. The bridge structure being able to reach the service life as planned is another goal that can be accomplished. Visual inspection or the use of some currently popular sensors can be used to monitor the condition of the bridge. The dynamic properties of a structure including modal frequency and mode shape will be used to determine the structure's present and potential future conditions. Using a velocitymeter, vibration data collection is conducted as the first step. The next step is analyzing data to determine natural frequency. The fundamental frequency of the Tugu Suharto bridge structure in Semarang was determined to be 3.995 Hz. Future bridge structure condition monitoring can be done using frequency data and finite element model. The condition of bridge infrastructure in the future for one city is an important thing that must be considered. Some bridges are classified as structurally deficient, and many bridges are nearing the end of their design lives. The next generation of Semarang highway bridges is currently being designed and built, but existing bridges still need to be maintained through proper inspection and load rating. In order to incorporate structural modeling, instrumentation, and nondestructive testing into the design, construction, and management of bridges, this study proposes an objective load rating protocol. Using information gathered from structural health monitoring (SHM), a baseline structural model is developed and verified. The load rating factors of the bridge are then determined using the structural model under both real-condition and simulated damaged conditions. [ABSTRACT FROM AUTHOR]

This study investigates the feasibility of an alternative structural health monitoring (SHM) technique for an existing, post-tensioned, prestressed concrete bridge in Niigata, Japan. Currently, a static SHM system is in place to detect the progress of damages within the bridge. However, the existing system cannot properly monitor the structural health of the bridge including the periodic vibration, which is one of the damage-sensitive features of interest. Therefore, to effectively detect the real-time performances of the bridge, a three-dimensional (3D) Finite Element (FE) model was developed as a reference and verified using on-site loaddeflection test results. After validating the reference FE model, different damage scenarios, such as degradation of concrete, corrosion & rupture of steel tendons and missing tendons, were incorporated in the FE models. Based on non-linear structural and Eigenvalue analyses, natural frequencies and mode shapes of the bridge remain constant even after careful consideration of all types of damages in the FE model. However, vertical displacements are observed to increase for the damage scenarios. Although the effect of tendon rupture and corrosion showed negligible influences on the vertical displacement, the deterioration of the concrete largely influenced the vertical displacement. Additionally, crack widths were found to vary with damage types. Brifely, this study recommends some effective indicators to monitor the structural conditions of the bridge using FE analysis (FEA). [ABSTRACT FROM AUTHOR]

Analytical equations that describe the dynamic behavior of beams with a soft clamped end are very little treated in the literature. The paper aims to solve this problem by introducing a stiffness in the hinged end of the beam, respectively by comparing the bending moment in the clamped end with the slope in the hinge of the same end of the beam. The other end of the beam is permanently hinged. The characteristic equation for determining the eigenvalues and the modal function is deduced. The results show the first four vibration modes for seven stiffness values and the eigenvalues for eleven cases of soft clamped end. [ABSTRACT FROM AUTHOR]

Structures are subjected to a variety of environmental and loading conditions over time, and minor damage to structural elements may occur. By timely identifying damage and repairing damaged locations, it is possible to prevent the spread and development of damage to other elements and, as a result, the overall destruction of the structure. This article discusses the identification and determination of the location of damage in steel plates based on the use of the primary and secondary shapes of vibration modes and the analytical method of two-dimensional wavelet analysis. Modelling and frequency analysis of the plate were performed in ABAQUS software, and the primary and secondary mode shapes were extracted. To determine the location of the damage, a damage detection index (DDI) was proposed based on the angle between the primary and secondary mode shape vectors and the diagonal detail coefficients obtained from the wavelet analysis of the primary and secondary mode shapes. The results showed that by using this index, damage can be identified by identifying peaks resulting from irregularities and disturbances. Also, the DDI value of the damage was dependent on the severity of damage occurring in a damaged situation, and the height of the disorder peaks increased with increased damage only at that damage position. [ABSTRACT FROM AUTHOR]

This research extends a method for direct estimation of Poisson's ratio from mode shapes of plates with hinged-hinged boundary conditions to include combinations of clamped and free boundary conditions. Using finite element simulations, we observe the effects of boundary conditions on the estimate of Poisson's ratio using the combination of Time-Average Scanning Digital Holography and Cornu's method. First, we introduce the abstraction of the span-to-width ratio to an effective antinode region. This abstraction enables us to extend the direct estimation technique to plates with various boundary conditions. Second, for long and thin enough plates, the effects of boundary conditions become negligible and Cornu's method is sufficient for direct estimation of the true value of Poisson's ratio from an out-of-plane bending mode shape. We note that for many situations, the necessary plate dimensions could be a limitation. Therefore, as a third concept, we discuss the behavior of the direct estimate of Poisson's ratio with respect to each type of boundary condition. Ultimately, this research initiates a path forward for a common processing technique for making a direct estimate of Poisson's ratio across different types of opposing boundary conditions for thin plates.

Vibration-based damage detection is a range of methods that utilizes the dynamic response of a structure to evaluate its condition and detect damage. It is an important approach for structural health monitoring and has drawn much attention from researchers. While multiple reviews have been published focusing on different aspects of this field, there has not been a study specifically examining the recent development across the range of methods, including natural frequency, mode shape, modal curvature, modal strain energy, and modal flexibility-based damage detection methods. This paper aims to fill this gap by reviewing the recent application of these methods in civil structures, including beams, plates, trusses, frames, and composite structural members. The merits and limitations of each method are discussed, and research opportunities are presented. This broader review also provides an opportunity for critical comparison across this range of methods. While predominantly reviewing experiment-based studies, this review also considers some numerical studies that may motivate further research.

In this study the primary objective is to design prediction model for the free vibration analysis of thin circular cylindrical steel silos having various aspect ratios in empty and varying filled conditions for different types of closures. A finite element method (FEM) is used to carry out the free vibration analysis of steel silos. It is found that the effect of different aspect ratios slender, intermediate slender and squat steel silos is very significant for dynamic response of silo. The silos are considered having open, flat and cone type of closures at its top end. The clamped-free (cantilever) boundary condition is taken for this approach as actual silos are fixed at flat-base. The structural mass of thin cylindrical shell steel silo is made constant for a particular height and diameter. The eigenvalues of the thin cylindrical shell steel silos are extracted by using block Lanczos method. The free vibrations of thin cylindrical shell steel silo with different aspect ratios, radius to thickness ratio are studied. From the present studies it is seen that as aspect ratio increases the fundamental frequency is reduced in empty silo. It is more in the case of squat silo. It can be seen that the fundamental frequency is less in the case of flat closure in all the aspect ratios of the silo. The frequency values are more in the case of cone closure is observed. Also as the mode number increases the modal frequency value increases. Further, as the filling level is increased the modal frequency also increases. Finally, regression approach is adopted for predicting the mode frequency of empty and filled silos for wide range of aspect ratios.

Usually, only one surface or one side of 3D structures is measured in Scanning Laser Doppler Vibrometer (SLDV) tests due to test setup and instrumentation limitations. However, in this work, we demonstrate an approach to overcome these limitations while also using an SLDV that provides high spatial resolution measurement. In the case of a wind turbine blade, only one surface is typically measured. In most scenarios, only focusing on one surface of the blade is sufficient to characterize the dynamics of the global blade modes. However, as we show measurement of both surfaces offers valuable insights to explore both the global blade modes and the relative motions of the two surfaces of the blade. This work proposes and applies a new method of experimental modal testing on both surfaces of the wind turbine blade using a high spatial resolution 3D SLDV. The two surfaces of the wind turbine blade are scanned by the 3D SLDV respectively under the same global test coordinate system defined by several alignment objects. Then the two surfaces are stitched together to build the blade mode shapes of both surfaces. In the current testing, a total of over 1,500 points (4,500 response degrees of freedom for the 3D measurement) are scanned from both surfaces of the blade in a non-contact fashion to obtain the flap-wise, edge-wise, and torsional mode shapes of the blade. The orthogonality of the measured mode shapes, either one surface or both surfaces, is validated by the correlation tool, Modal Assurance Criterion (MAC). The effectiveness of the two-surface measurement is demonstrated by comparing the drive point Frequency Response Function and by correlating the mode shapes from the two surfaces. With the high spatial resolution 3D SLDV measurement, local panel modes of the wind turbine blade are observed from the mode shapes. Especially the panel breathing mode can only be revealed due to the two-surface measurement. This work also provides a useful reference for the wind turbine blade designers and researchers for design, structural analysis, and reliability study on the wind turbine blade. The two-surface measurement technique proposed in this work is demonstrated on a wind turbine blade, and it will also be applicable for other types of two-surface shell-type structures. [ABSTRACT FROM AUTHOR]

Wind turbine design and operation will benefit from a better understanding of blade dynamics. Usually, only one surface or one side of a 3D structure is measured in Scanning Laser Doppler Vibrometer (SLDV) tests due to test setup and instrumentation limitations. However, in this work, we demonstrate an approach to overcome these limitations while also using an SLDV that provides high spatial resolution measurement. In the case of a wind turbine blade, only one surface is typically measured; however, it is beneficial to investigate both blade global modes and the relative motions of the two blade surfaces by using both numerical models and experimental tests. This work creatively develops both experimental and numerical approaches to investigate the dynamics and the relative motions of both surfaces of the wind turbine blade from global and local perspectives. On the experimental side, experimental modal testing is conducted on both surfaces of the wind turbine blade with a high spatial resolution 3D SLDV. The two surfaces of the wind turbine blade are measured and stitched together to build the blade experimental mode shapes of both surfaces. A total of over 1500 points are scanned from both surfaces of the blade in a non-contact fashion to obtain not only the global bending modes (flap-wise and edge-wise), the global torsional modes, but also the localized panel mode shapes. On the numerical side, a finite element model of the blade is developed to obtain the numerical mode shapes. The experimental mode shapes on either one surface or both surfaces of the blade are used to validate the blade finite element model. The mode shape correspondence between the model and the test is also identified. With the availability of both experimental and numerical mode shapes, localized panel modes of the wind turbine blade are observed and characterized, and as a result the numerical model is validated. This work provides useful case studies for the design and structural analysis of wind turbine blades based on both the experimental observations and the validation of numerical models typical of those used for blade design and blade structural analysis. [ABSTRACT FROM AUTHOR]

Structural rigidity is a crucial factor that determines machining accuracy for computer-numerical-controlled (CNC) machines. Therefore, how to design a highly rigid CNC machine tool has been the focus of attention. In response to the rapid changes in various machine tools attributable to market needs, it is necessary to find an efficient way to examine and optimally design their structures. In this study, we propose an optimization methodology based on the finite element method (FEM) and sensor-based measurement to efficiently investigate and obtain an optimal structure with high rigidity of the selected target CNC movable-cross-beam double-column machining center (MDMC). The proposed methodology is mainly composed of the prototype design of a target machine, theoretical investigations via FEM, static as well as dynamic stiffness analysis, experimental measurements based on sensors, investigations on crucial parameters that mostly affect the whole structural strength, and the design of an optimum structure via synthesis and comparison. We found that a reduction as large as 1000 mm in the Z travel of a spindle head causes decreases as large as 69.37% in minimum static stiffness and 37.93% in minimum dynamic stiffness. It is suggested that, for optimally designing our target MDMC, the Z-travel length of the spindle head should be reduced to half the original size. This proposed methodology is a rapid, effective, and economical way to optimally design or modify the structure of a MDMC. It can also be used as an optimization guide of the structural design for other types of CNC machine tool. [ABSTRACT FROM AUTHOR]

Abstract Bolt connection is one of the main fixing methods of cylindrical shell structures. A typical bolted connection model is considered as a tuned system. However, in the actual working conditions, due to the manufacturing error, installation error and uneven materials of bolts, there are always random errors between different bolts. To investigate the influence of non-uniform parameters of bolt joint, including the stiffness and the distribution position, on frequency complexity characteristics of cylindrical shell through a statistical method is the main aim of this paper. The bolted joints considered here were simplified as a series of springs with random features. The vibration equation of the bolted joined cylindrical shell was derived based on Sanders’ thin shell theory. The Monte Carlo simulation and statistical theory were applied to the statistical analysis of mode characteristics of the system. First, the frequency and mode shape of the tuned system were investigated and compared with FEM. Then, the effect of the random distribution and the random constraint stiffness of the bolts on the frequency and mode shape were studied. And the statistical analysis on the natural frequencies was evaluated for different mistuned levels. And some special cases were presented to help understand the effect of random mistuning. This research introduces random theory into the modeling of bolted joints and proposes a reference result to interpret the complexity of the modal characteristics of cylindrical shells with non-uniform parameters of bolt joints.

The effect of structure on the vibration response was explored for four piano soundboards with different but commonly adopted structures. The vibration response was obtained using the free-vibration method, and the values of the dynamic modulus of elasticity and dynamic shear modulus obtained using the free-vibration frequency method (EF and GF) were compared with the dynamic modulus of elasticity obtained using the Euler beam method (EE) and dynamic shear modulus obtained using the free-plate torsional vibration method (GT), respectively. It was found that the soundboards with different structures had different vibration modes and that excitation at different locations highlighted different vibration modes. For all the soundboards analyzed, the EE and GT were higher than EF and GF by 2.2% and 24.3%, respectively. However, the trends of the results of these methods were the same. The four piano soundboards with different structures possessed varying dynamic moduli of elasticity and dynamic shear moduli. These rules are consistent with the grain directions of the soundboards and the anisotropy of the wood (the direction of the units of the soundboards). The results show that the vibration mode of the piano soundboard is complex. The dynamic elastic modulus of the soundboard can be calculated using the Euler beam method. The results provide a reference for studies on the vibration response, material selection, production technology, and testing of piano soundboards.

This paper uses the vibration-based damage detection method for structural health monitoring of a Railway Bridge. Hence, the bridge had been exactly modelled in finite element analysis software. Due to the elastomeric bridge bearing, direct calculating of the natural frequencies and extracting the mode shapes of the bridge is not suitable and effective. Therefore, a time-dependent transient analysis of the train movement on the bridge is done and then the vibrations of all truss cells during the train crossing period were extracted. Afterwards, the relevant data is transformed, which represents the natural frequencies of the bridge vibration and their amplitude at each point. In the following, the mode shapes of vibration and the distribution of vibration energy are calculated. Applying any artificial damages in the model, occurring changes in the natural frequencies, mode shapes, and the vibration energies of the modes, are examined. Consequently, detecting and locating damages in the structure has been done with acceptable accuracy. Finally, for validation of the results, an accelerometer is installed on the bridge truss in the middle part of the bridge to extract the acceleration of bridge vibration at the train crossing time interval. In the end, the results are compared and presented. [ABSTRACT FROM AUTHOR]

Context: Through experimental observations and reports, various challenges have been identified in carbon nanotubes (CNT), including Stone Wales (SW) flaws and position flaws. Among these imperfections, point vacancies are the most prevalent in the CNT lattice. However, there is currently no established method for detecting these issues, and the influence of these flaws on the vibrational properties of three-walled carbon nanotubes (TWCNTs) remains uncertain. This research paper introduces a novel approach that utilizes vibrational analysis to detect flaws in TWCNTs. By conducting the first investigation into the impact of point vacancies on the vibrational modal frequencies of TWCNTs, our study bridges these knowledge gaps. Methods: This study examines the impact of defect quantity on various types of TWCNTs and investigates the vibrational properties of TWCNTs with point vacancies using a molecular structural mechanics technique. A total of 432 TWCNT models were simulated using molecular structural mechanics (MSM), and their modes were identified through finite element (FE) analysis. The fundamental vibration's natural frequency in TWCNTs with defects was then determined. The findings indicate that the depth of the mode shape is influenced by the TWCNTs' diameter, the extent of point vacancy defects, and the boundary condition. It was observed that as the number of vacancy defects increases from 0 to 4%, the natural frequency decreases. The study also establishes the order of TWCNTs with the highest natural vibrational frequency at 0%-point vacancy and 10 - 21 g m a given attached mass, which follows the sequence of chiral, armchair, and zigzag TWCNTs. [ABSTRACT FROM AUTHOR]

• A symplectic approach is proposed for free vibration of stepped-thickness piezoelectric cylindrical shells. • Arbitrary number of steps in axial thickness can be solved by the proposed approach. • Exact frequency equations and vibration mode shapes are obtained without any trial functions. • Effects of stepped thickness on vibration characteristics are investigated and revealed. • Numerical results can be served as benchmarks for new numerical or approximate methods. An analytical study on free vibration of stepped-thickness piezoelectric cylindrical shells is performed under the framework of symplectic mechanics. In the proposed symplectic approach, the step-wise thickness in the axial direction is considered by regarding the overall shell as a combination of multiple segments with different uniform thicknesses. The Hamiltonian governing equations for free vibration of each segment are established by defining a new total unknown vector and the Reissner's thin shell theory. In this manner, analytical solutions can be represented by a series of symplectic eigenfunctions with undetermined coefficients. By using boundary conditions at the ends and interfaces, the coefficients of symplectic series are easily obtained by solving a set of algebraic equations. Consequently, the natural frequencies and analytical mode shapes are obtained simultaneously. The methodology is rational and rigorous, and the solution procedure is systematic with a step-by-step definition. Numerical results are compared with those reported in the open literature and good agreements are observed. A comprehensive parametric study of stepped thickness on vibration characteristics of piezoelectric cylindrical shells are carried out also. The results demonstrate that the local thickness thinning will decrease natural frequencies and lead to local deformation modes. Some of them can be served as benchmarks for other numerical or approximate methods. [Display omitted] [ABSTRACT FROM AUTHOR]

A large-span steel–concrete composite beam with precast hollow core slabs (CBHCSs) is a relatively new floor structure that can be applied to various long-span structures. However, human-induced vibrations may present serviceability issues in such structures. To alleviate vibrations, both the walking forces excited by humans and the associated floor responses must be elucidated. In this study, 150 load–time histories of walking, excited by 25 test participants, are obtained using a force measuring plate. The dynamic loading factors and phase angles in the Fourier series functions for one-step walking are determined. Subsequently, walking tests are performed on seven CBHCS specimens to capture the essential dynamic properties of mode shapes, natural frequencies, damping ratios, and acceleration time histories. The CBHCS floor system generally exhibits a high frequency (> 10 Hz) and low damping (damping ratio

The present work aims to determine the natural frequencies for a structure with Warren truss by changing the modulus of elasticity to two of the elements of the structure are keeping the same length of the elements. A numerical study is carried out by means of a 3D FE model of a Warren truss, simulating the dynamic interaction of beams with trusses.

Bolt connection is one of the main fixing methods of cylindrical shell structures. A typical bolted connection model is considered as a tuned system. However, in the actual working conditions, due to the manufacturing error, installation error and uneven materials of bolts, there are always random errors between different bolts. To investigate the influence of non-uniform parameters of bolt joint, including the stiffness and the distribution position, on frequency complexity characteristics of cylindrical shell through a statistical method is the main aim of this paper. The bolted joints considered here were simplified as a series of springs with random features. The vibration equation of the bolted joined cylindrical shell was derived based on Sanders' thin shell theory. The Monte Carlo simulation and statistical theory were applied to the statistical analysis of mode characteristics of the system. First, the frequency and mode shape of the tuned system were investigated and compared with FEM. Then, the effect of the random distribution and the random constraint stiffness of the bolts on the frequency and mode shape were studied. And the statistical analysis on the natural frequencies was evaluated for different mistuned levels. And some special cases were presented to help understand the effect of random mistuning. This research introduces random theory into the modeling of bolted joints and proposes a reference result to interpret the complexity of the modal characteristics of cylindrical shells with non-uniform parameters of bolt joints. [ABSTRACT FROM AUTHOR]

Finite element modal analysis can be performed for a truss that can be treated either as a truss or as a beam. In this paper we have considered a truss hinged at both ends of the Warren type. Where eigenfrequencies and modal shapes are presented for the girder beam where the elements are considered as truss and the beam where for one element of the structure the elasticity model has been reduced by 25%. The dynamic behavior of the structure has been analyzed for the first 5 modes of vibrations. The results indicate that if one element of the structure becomes more elastic the eigenfrequencies of the girder beam increase and the percentage deviation is not directly correlated with the modified element and the modes of vibrations. [ABSTRACT FROM AUTHOR]

The applications of plate like structures in different fields of engineering are increasing. In this paper, a new damage detection method investigated based on Gaussian process regression model (GPR). GPR is an efficient learning machines which has been used in different fields of engineering. To identify damage, mode shaped and natural frequencies of damaged structures used to train GPR. Finite element modelling of numerical examples and Gaussian process regression (GPR) model are carried out within the MATLAB environment. To show the effectiveness of presented approach, a two-fixed supported plate and a cantilever plate was studied. In other work, a comparative study has been done using a cantilever plates. The natural frequencies were contaminated with noise in above mentioned numerical examples. Results reveal that the proposed method works well using the only first mode data which may be noisy. In other word, GPR can be trained using limited sample numbers for training.

In this study, the dynamic structural and mechanical properties of the polymer composite with natural fiber reinforcement have been investigated experimentally and numerically. First of all, three different types, one-, three- and five-layered jute-epoxy and hemp-epoxy laminated composite, have been manufactured using the hand lay-up method. The tensile strength, bending strength and shear strength of the fabricated composite material have been investigated experimentally using Universal Testing Machine. The density is measured using a density measurement kit using Archimedes principle. Further, frequency and damping factors are also investigated using a vibration test setup. Later on, the numerical model is developed in finite element-based software ANSYS. Different natural frequency modes and corresponding damping factors have been investigated numerically. The mechanical and structural characteristics obtained experimentally and numerically are analyzed and discussed in detail. From the investigation, it is found that the hemp-epoxy composite has a greater strength and stiffness compared to the jute-epoxy composite, irrespective of the number of fiber layers used. Further, it is also found that with the increase in fiber layers or fiber percentage in the composite, the strength and stiffness of the fabricated composite increase significantly.

In this study, the dynamic structural and mechanical properties of the polymer composite with natural fiber reinforcement have been investigated experimentally and numerically. First of all, three different types, one-, three- and five-layered jute-epoxy and hemp-epoxy laminated composite, have been manufactured using the hand lay-up method. The tensile strength, bending strength and shear strength of the fabricated composite material have been investigated experimentally using Universal Testing Machine. The density is measured using a density measurement kit using Archimedes principle. Further, frequency and damping factors are also investigated using a vibration test setup. Later on, the numerical model is developed in finite element-based software ANSYS. Different natural frequency modes and corresponding damping factors have been investigated numerically. The mechanical and structural characteristics obtained experimentally and numerically are analyzed and discussed in detail. From the investigation, it is found that the hemp-epoxy composite has a greater strength and stiffness compared to the jute-epoxy composite, irrespective of the number of fiber layers used. Further, it is also found that with the increase in fiber layers or fiber percentage in the composite, the strength and stiffness of the fabricated composite increase significantly. [ABSTRACT FROM AUTHOR]

The first part of the paper presents a study regarding the dynamics behavior of thin circular plate clamped all around. An analytical investigation is performed and the results in terms of mode shapes are used to highlight the plate's dynamics. The modal shapes are obtained using Bessel functions and their graphic representation is presented in 3D by using MS Excel software. [ABSTRACT FROM AUTHOR]

Estimation and monitoring of cable tension is of great significance in the structural assessment of cable-supported bridges. For short cables, the traditional cable tension identification method via frequency measurement has large errors due to the influence of complex boundaries, which affect the accuracy of estimation. A new cable tension estimation method based on mode shape identification with a multiple sensor arrangement on the cable can take the influence of boundary conditions into account and its accuracy has been verified. However, it requires more sensors compared to the traditional frequency-based method, which will significantly increase the cost of long-term monitoring in practice. Therefore, a novel approach for cable tension monitoring considering both cost and accuracy is further proposed in this study. The approach adopts multiple sensors to measure the influence of boundary conditions. Then, only a single sensor is required for long-term monitoring of the cable. In this paper, an analytical model of the cable is firstly established. The influence of boundary conditions is calculated, which ensures the accuracy of mode shape identification. Furthermore, a field experiment is carried out to verify the effectiveness of the new approach. The results have demonstrated the effectiveness and accurateness of the proposed method in long-term short cable tension monitoring. [ABSTRACT FROM AUTHOR]

This paper presents a two-step method for damage localization in beams by combining natural frequencies and mode shapes. The general locations of the damage are first identified from an indicator developed using relative natural frequency change (RNFC) curves and the values of RNFCs. A curvature-mode-shape-based method is then utilized to determine the specific location of the damage in the second step. The proposed two-step method is verified by detecting damage in a simulated simply-supported beam. The identified damage location agrees well with the actual damage location. A strategy for fast and accurate damage localization based on general localization using natural frequencies and specific localization using mode shapes is the main novelty of the paper. [ABSTRACT FROM AUTHOR]

We have developed a general method, dubbed the split beam method, to solve Euler–Bernoulli equations for cantilever beams under multiple loading conditions. This kind of problem is, in general, a difficult inhomogeneous eigenvalue problem. The new idea is to split the original beam into two (or more) effective beams, each of which corresponds to one specific load and bears its own Young's modulus. The mode shape of the original beam can be obtained by linearly superposing those of the effective beams. We apply the split beam method to simulating mechanical responses of an atomic force microscope probe in the "dynamical" operation mode, under which there are a stabilizing force at the positioner and a point-contact force at the tip. Compared with traditional analytical or numerical methods, the split beam method uses only a few number of basis functions from each effective beam, so a very fast convergence rate is observed in solving both the resonance frequencies and the mode shapes at the same time. Moreover, by examining the superposition coefficients, the split beam method provides a physical insight into the relative contribution of an individual load on the beam. [ABSTRACT FROM AUTHOR]

In daily operations, there are lots of blade disk structures operating in water, e.g., hydraulic turbines and pumps, whose dynamic behavior may be severely affected by the added mass effect. Their mode shape change problem from vacuum to water due to the added mass effect has been studied in this paper. The added mass effect of simplified blade disk structures submerged in still water has been studied numerically first in this paper. Acoustic fluid–structure interaction technology has been used to simulate the fluid and structure coupling effect. Through comparing the strain energy percentage of the disk part in water with those in air under different disk material densities, it showed that different parts of the submerged blade disk structures can suffer from different average added mass factors, which makes the modal shapes of some modes with strong blade disk interactions change a lot from vacuum to water, thus produces a large influence on the optimal design of submerged blade disk structures. Based on the conclusions of the simplified model, the situation of a Francis turbine runner model was investigated, and the results fit the conclusions on the simplified model very well.

Present work concentrates on modal parameters of laminated composite plates composed of natural fibers reinforced with unsaturated polyester resin. The selected jute and ramie fibers underwent alkaline treatment to improve mechanical properties. Upon compression molding, the laminated composite plates such as ramie/polyester (R plate), jute/polyester (J plate) and combination of ramie and jute (RJ plate) are fabricated. The physical, mechanical and elastic properties are calculated as per ASTM standards. Using obtained elastic properties, the free undamped vibration analysis of composite plates are computed using finite element software ANSYS 18.1 for trapezoidal plates, with respect to a/b = 1 and 2, c/b = 0.25 to 1 and a/h = 50. The natural frequency and mode shape of the plates are obtained for various boundary conditions such as CFFF, CFCF, CCCC, SFSF and SSSS (C-Clamped, F-Free and S-Simply supported). The dimensionless frequency and mode shapes of R, J and RJ plates are compared with literatures which are in good agreement.

In this paper, layerwise finite element analysis for the free vibration behavior of two-dimensional functionally graded sandwich plates with different boundary conditions is presented. The plates consist of three layers; a functionally graded layer embedded between ceramic and metal isotropic layers. The layerwise approach is based on the third order shear deformation theory for the middle layer, while the first order shear deformation theory is used to model both the upper and lower isotropic layers. Quadrilateral 8-noded element with 13-degrees of freedom per node is used for this purpose. The present results show very good agreements with the published analytical results of plates consist of a single functionally graded layer. Furthermore, for sandwich plates good agreements were obtained when the present results are compared with similar problems solved by other methods in literature. Parametric studies were investigated for various plate parameters including applied boundary conditions, volume fraction exponents and plate side to thickness ratio.

Trusses are everywhere; they are used in bridges, antenna towers, cranes, even in parts of the International Space Station. And for good reason, they allow us to create strong structures while using materials in very efficient and cost-effective way. Trusses it is es-sentially a rigid structure made up of a collection of straight mem-bers. The type of truss depends on how the horizontal and diagonal beams are arranged.

Continuous beams simply supported with several intermediate sup-ports are very common in engineering achievements everywhere. The paper shows the evolution of the dimensionless wave number in 3D format, respectively of the eigenfrequencies for a continuous beam with three openings when the intermediate supports take any position inside the beam. The frequency equation for calculating the dimensionless wave number is presented and the modal func-tion is given with an example for the case where the eigenfrequency has the maximum value at fist vibration mode.

A theoretical framework is proposed to identify fundamental frequency, mode shape, and damping of a bridge simultaneously by a passing vehicle-trailers system with only accelerometers. The framework is the first attempt to use indirect method to identify all fundamental modal properties of a bridge simultaneously without shaker, laser sensors and pre-known information. The vehicle performs as both "tractor" and "excitor" and the trailers perform as "message carriers". The residual response defined as the response difference of one trailer to another one was usually adopted in the previous studies to remove the adverse effect of road roughness; however, it lacks theoretical support. Instead, the residual response of one trailer–bridge contact point to another one is used in this study, and theoretical analysis proves that it can effectively eliminate the adverse effect of road roughness. Different from previous studies, the residual response of contact point in this study is obtained in the frequency domain rather than the time domain. Frequency and damping can be evaluated by applying single degree-of-freedom (SDOF) model-based curve fitting technique to several discrete values from the Fourier transform of the residual response. Mode shapes can be reconstructed by dividing the Fourier transform of one residual response at the natural frequency by another one without time lag. The proposed framework is validated by independent finite element (FE) analysis and it shows great performance. Parametric study including vehicle speed, measurement noise, calibration error, road surface roughness and boundary condition has also been conducted to check its feasibility in various scenarios. [ABSTRACT FROM AUTHOR]

The distribution of seismic loading to each storey of a building depends on a number of factors, mainly on the periods, mode shapes and modal participation. The estimation of these dynamic characteristics is essential in analyzing the seismic response of multi-storey buildings. Based on a rigorous dynamic analytical model of the building, this research provides a novel database of vibration modes of multi-storey buildings with reinforced concrete (RC) shear walls. The Stodola–Vianello iterative method was used to determine these. The first modes of vibration, cumulating a modal mass participation ratio (MMPR) greater than 90%, are summarized and presented in a table form. The results are equally valid for uniform buildings with masonry infill walls. The research has highlighted that the eigenvectors are independent of the values of the mass and the mechanical characteristics of the structure (the modulus of elasticity, the moment of inertia and height) but also on the variation of these characteristics over the height of the building. Consequently, for shear wall buildings with identical storeys, the eigenvectors, the MMPRs and the modal participation factors are accurately determined for this type of structure. The significance of the research consists of providing a table that engineers and researchers can easily use to determine the dynamic characteristics of the building, thus avoiding repetition of the calculations relating to this type of buildings. The study has also highlighted the importance of selecting the appropriate dynamic behavior of the structure for seismic design. [ABSTRACT FROM AUTHOR]

Due to the limitations of the manufacturing processes, defect like crack is produced in the structures. Crack gives the catastrophic failure of the structures or beams; hence, the detection and classification of the cracks are important issues in the scientific research. For a long time, the free vibration-based crack detection methods have been used for the possible crack detection in the structures. On the other hand, the effect of arbitrary and random defect geometry, i.e. V-shaped, U-shaped open edge cracks on the applicability of these methods has been overlooked. In order to investigate this issue, three crack parameters, i.e. crack geometry, crack depth and crack locations, were considered on the cantilever beam to study its effect on the mode shapes of cracked beams. The stiffness of cracked cantilever beams was evaluated by the deflection method using ANSYS. Theoretical method, Numerical method and Experimental method have been used to compute the mode shapes of cracked beams. Through free vibration study, it is found that spring steel structures are somewhat sensitive to the change in crack geometries as far as the mode shapes are concerned. So, it is obvious that the mode shape based free vibration method can satisfactorily predict the crack depth and location in spring steel structures irrespective of the crack geometries. [ABSTRACT FROM AUTHOR]

Dynamic deformation measurement is an important approach to monitor structure stability. Due to its rapid imaging capabilities, ground-based multi-input–multi-output (GB-MIMO) radar has shown great application potential in bridge structure monitoring. This article proposes a dynamic deformation estimation method based on radar image series. First, an improved clutter suppression method is used to overcome the estimation error in complex working status. Second, a two-step pixel extraction method is adopted to ensure both the quantity and quality of sample pixels. Finally, frequency- and time-domain environmental stimulating methods are jointly considered for stable mode shape estimation. This article systematically measured the dynamic deformations of bridge structures, including two suspension bridges and a cable-stayed bridge. The time-series deformation, deflection, vibration frequency, and amplitude on the bridge structure are obtained by image-domain measurement. For the first time, the mode shapes of bridges are obtained through GB-MIMO radar. The correctness of structure parameter measurement is verified with the finite element model (FEM). Experimental results prove that with the proposed method, stable results could be acquired against weak and complex stimulation conditions. [ABSTRACT FROM AUTHOR]

This paper investigated the static and dynamic behaviors of isotropic cracked simply supported beam using finite element analysis (FEA), ANSYS software. Modal and harmonic vibration analysis of intact and damaged beam were performed in order to extract mode shapes of bending vibration, natural frequencies and obtain frequency response diagram. Static finite element analysis of undamaged and damaged simply supported beam was carried out to determine static deflection, then stiffness of intact and cracked beam was computed using conventional formula. Crack damage severity of damaged beam was calculated and it is noticed that as crack position is increased from left hand support of beam up to central point and crack depth is increased, then crack damage severity increases. The effect of mode shape pattern is investigated and it is found that the amount of decreasing of natural frequency is proportional to the normalized mode shape at position of crack. The exhibited correlation between results for damaged beam revealed that crack damage severity is proportional to static deflection and inversely proportional to first mode frequency.