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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]

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]

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]

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]

There are well-known expressions for natural frequencies and mode shapes of a Euler-Bernoulli beam which has classical boundary conditions, such as free, fixed, and pinned. There are also expressions for particular boundary conditions, such as attached springs and masses. Surprisingly, however, there is not a method to calculate the natural frequencies and mode shapes for a Euler–Bernoulli beam which has any combination of linear boundary conditions. This paper describes a new method to achieve this, by writing the boundary conditions in terms of dynamic stiffness of attached elements. The method is valid for any boundaries provided they are linear, including dissipative boundaries. Ways to overcome numerical issues that can occur when computing higher natural frequencies and mode shapes are also discussed. Some examples are given to illustrate the applicability of the proposed method. [ABSTRACT FROM AUTHOR]

High-frequency transverse vibration of stepped beams has attracted increasing attention in various industrial areas. For an n-step Timoshenko beam, the governing differential equations of transverse vibration have been well established in the literature on the basis of assembling classic Timoshenko beam equations for uniform beam segments. However, solving the governing differential equation has not been resolved well to date, manifested by a computational bottleneck: only the first k modes (k ≤ 12) are solvable for i-step (i ≥ 0) Timoshenko beams. This bottleneck diminishes the completeness of stepped Timoshenko beam theory. To address this problem, this study first reveals the root cause of the bottleneck in solving the governing differential equations for high-order modes, and then creates a sophisticated method, based on local coordinate systems, that can overcome the bottleneck to accomplish high-order mode shapes of an n-step Timoshenko beam. The proposed method uses a set of local coordinate systems in place of the conventional global coordinate system to characterize the transverse vibration of an n-step Timoshenko beam. With the method, the local coordinate systems can simplify the frequency equation for the vibration of an n-step Timoshenko beam, making it possible to obtain high-order modes of the beam. The accuracy, capacity, and efficiency of the method based on local coordinate systems in acquiring high-order modes are corroborated using the well-known exact dynamic stiffness method underpinned by the Wittrick-Williams algorithm as a reference. Removal of the bottlenecks in solving the governing differential equations for high-order modes contributes usefully to the completeness of stepped Timoshenko beam theory. [ABSTRACT FROM AUTHOR]

This study concerns damage detection in plate-type structures using fractal surface singularities with noncontact laser measurement of structural dynamic responses. The fractal dimension analysis aided by linear isomorphism is used to deal with mode shapes for a plate with damage. With this method, the linear isomorphism is utilized to remove the local extrema while preserving the damage information of a mode shape, giving a retrofitted mode shape. The retrofitted mode shape is processed by Katz’s fractal dimension analysis to produce a fractal dimension surface with its singular peak indicating the presence and location of the damage. In the method, a series of high-resolution mode shapes of the plate are used, which are acquired by using a noncontact measurement system consisting of a piezoceramic transducer as the actuator and a scanning laser vibrometer as the sensor. The capability of the method to locate and quantify damage is numerically demonstrated using a two-side-clamped aluminum plate with cracks of various depths; the efficacy of the method in identifying complex cracks is experimentally validated using a suspended aluminum plate bearing a cross-like crack. The numerical and experimental results show that the proposed method can accurately identify complex damage in plates, requiring neither benchmark models for the entire structure under investigation, or any prior knowledge of the material properties and the boundary conditions of the structure. [ABSTRACT FROM AUTHOR]

Stepped beams with elastic end supports have been extensively investigated due to their importance in structural engineering fields, including active structures, structural elements with integrated piezoelectric materials, shaft-disc system components, turbomachinery blades and many other structural configurations. Considering the importance of the use of discontinuous structures in the engineering, the authors propose a mathematic modeling which advantage is that the results are independent of the degree of mesh refinement. The analysis is based on the classical Euler-Bernoulli beam theory. In comparison with the published literature on the transverse vibration of single cross section change beams, there are relatively few works covering beam vibration when there is more than one change in the beam cross section. In the present study, the natural frequencies and the mode shapes of beams with variable geometry or material discontinuities are investigated. The mode shapes of a beam with multiple step changes in cross section are discussed theoretically and experimentally. Numerical results obtained by Euler-Bernoulli beam theory are compared with experimental results. [ABSTRACT FROM AUTHOR]

When the velocity of fluid flow in a cantilevered pipe is successively increased, the vibration characteristics of the system may vary remarkably. This paper is concerned with exploring the evolution of the actual mode shapes of the pipe with increasing flow velocity. Results show that the mode shapes of the cantilevered system may dramatically change due to the increment of flow velocity. At higher flow velocity, these mode shapes, indeed, differ much from those of the classical cantilevered beam. When a critical mass ratio at which the so-called ‘mode exchange’ phenomenon occurs was chosen, the corresponding two modes of the cantilevered pipe would have the same shape. In addition, the nonlinear responses of the system have also been linked to the lowest three mode shapes by comparing the calculated mode shapes with the limit-cycle motions obtained experimentally. [ABSTRACT FROM AUTHOR]

The purpose of the present paper is to apply the differential transformation method (DTM) to deal with the vibration problem of stepped beams with general end supports and elastically constrained ends. The method demonstrates several advantages, such as rapid convergence, high accuracy, and computational stability. Unlike some approximate approaches which require correct assumed admissible function, the differential transformation method gives all natural frequencies and their mode shapes without any frequency missing. By using the DTM algorithms fully provided in this paper with general mathematical software packets, natural frequencies and mode shapes of these beams can be obtained easily for every boundary condition. Aspects such as boundary conditions, spring constant values, stepped beam types, step ratio and step location, which have a significant impact on frequencies and mode shapes, are taken into investigation in this paper. [ABSTRACT FROM AUTHOR]

For a surface mounted fiber Bragg grating (FBG) strain sensor, this study used an improved bonding method that distributes cement only on the two ends of the grating and leaves the grating free from cement. According to a recent numerical study, for a conventional surface mounted FBG strain sensor its relationship between average strains (within its glued grating) and voltage signals is nonlinear and changeable depending on the variation patterns of strain gradients (within its glued grating) in a strain history. By contrast, the relationship for the improved bonding one is linear because of the constant zero strain gradient within the glue-free grating. To experimentally verify the linear relationship for the improved bonding fiber Bragg grating (IBFBG), this study set up a cantilever with both the IBFBG and a resistance strain gauge surface mounted in close proximity, and the vibration responses were induced by impactions of a free-falling steel ball on the cantilever's surface. Experimental results indicate that dynamic calibrations of an IBFBG by using a commonly used resistance strain gauge are possible because both sensors in close proximity have their changes of voltage signals almost proportional to each other. [ABSTRACT FROM PUBLISHER]

Modal models are developed for vibro-acoustic response computation of fluid-structure interaction problems on the basis of modal frequencies and modal damping ratios. The uncoupled equations are built using in vacuo mode shapes and fluid-loaded undamped real mode shapes and are solved for the coupled response of the fluid-loaded structures. Conventional modal analysis techniques are used to obtain the response of the coupled fluid-structure systems from a reduced-order modal model. The accuracy and efficiency of the technique is demonstrated with examples of computing the vibro-acoustic response of a baffled thin plate and a baffled thick stiffened plate. The comparisons in terms of mean-square normal velocity and sound power between the present models and the conventional coupling method show a good agreement. The present modal models for computing the vibro-acoustic response of the fluid-loaded structures have the advantages of constructing a model from modal parameters that could be obtained from either numerical computation or experimental measurement; a small dimension of using only a very small number of modes; and of being applicable to conventional modal analysis techniques. These features may make the present technique a powerful tool for dynamics and the control of fluid-loaded structures. [ABSTRACT FROM PUBLISHER]

An analytical model to study the coupled transverse and longitudinal vibrations of a single lap adhesive joint is described in this paper, which includes partial differential form of the motion equations. A joint consists of two identical adherents of mild steel that are lap jointed over a certain length by a viscoelastic material, epoxy resin (araldite). Adherents are modeled as Euler-Bernoulli free-free beam. Both transverse and axial deformation of adherents and longitudinal shear and transverse peel stresses at the adhesive joint interface are considered in deriving the equations of motion. The numerical solutions of the governing equations for free vibrations yield the system’s natural frequency and mode shapes. Experimentation was carried out on both monolithic and adhesively jointed beams to observe the effect of the joint; natural frequencies of the system were measured experimentally and compared with those obtained theoretically. The fundamental frequency of a free-free jointed beam was more sensitive to joint overlap ratio. However natural frequency depended on the accelerometer location in the system, which was attributed to its mass contribution to the overall system mass. Theoretical frequency response function is generated for a beam with and without accelerometer mass to show the mass loading effect of the transducer (accelerometer). [ABSTRACT FROM PUBLISHER]