Vibration Mode Shapes Research Articles

Proposal of an Improved Contactless Measurement Method Using an Advanced Light Emitting Diode Displacement Sensor for Vibration Stress of Piping Systems

Abstract This paper proposes an improved contactless measurement method for vibration stress of piping systems, by which the measurement time is shorter and the measuring works are more simple. The proposed method includes two processes, in which the bending mode shape of piping vibration is identified by a transmission-type light emitting diode (LED) displacement sensor and the vibration stress is calculated based on beam theory using the approximated curve of the bending mode shape. The proposed method uses one advanced LED displacement sensor to measure the vibration stress in contrast to multiple conventional LED sensors, which must be used in the previous method developed by the authors. Therefore, the measuring system could be reduced in size and a light-weight and portable measurement instrument was developed. The measurement accuracy and reliability of the new method were validated by the vibration experiment using a mockup piping system.

Unexpected soil amplification effect on seismic performance of highway bridges during the Aegean earthquake of October 2020, Mw 6.6

On October 30, 2020, an earthquake about 70 km away from the city center of Izmir with a 4.3 million population has shaken the city tremendously and has resulted in destruction of many building type of structures due to an unexpected high soil-amplified vibrations very similar to the Mexico City earthquake in 1985. The bridges at the soil-amplified sites has performed in elastic range with no damage at all. In the city of Izmir, the 42 year old twin bridges located on the main transportation route, were tremendously shaken by the earthquake had observed to have no seismic induced damage. Surprisingly twin bridges suffering from the alkali silica reaction (ASR) over the years did not even pound to each other despite the small size of longitudinal gap between them. As it has been known, the past performance of Turkish designed bridges are typically succesfull with almost no damage as observed in the Van 2011 and Sivrice 2020 earthquake mainly due to allowing movements at their joints and to flexible type of framing. The focus of the paper is given to understand the successful performance of bridges and to investigate the non-pounded twin bridges of the Izmir city. In this scope, a bridge inspection has been performed and the twin bridges have been analyzed for the recorded ground motion. The results have indicated that the structures have been subjected to 0.3 g at their vibration modes and the twin bridges have a synchronized motion due to having the identical vibration mode shape with a period of 1.5 seconds

Vortex-induced vibration of a flexible rectangular cylinder: Experimental investigation and comparative study of theoretical models

The vortex-induced vibration (VIV) of slender flexible structures is usually estimated based on two-dimensional aerodynamic parameters determined by sectional model wind tunnel tests. Before converting such test results into a prototype flexible structure, many complicated factors, such as the spanwise correlation of aerodynamic force and vibration mode shape, should be considered. In this study, wind tunnel tests of a 5:1 rectangular taut strip model and its sectional model with the same geometry were first conducted to analyze the relationship between the VIV amplitude of the flexible structure and sectional model. The results showed that the taut strip model sustained an approximately 15% higher VIV amplitude than the sectional model, indicating that the VIV amplitude could be underestimated if the sectional model test results were directly used. Next, based on the identified spanwise correlation function of the 5:1 rectangular cylinder, several existing VIV response estimation approaches based on Scanlan's semi-empirical vortex-induced force (VIF) model were examined and compared. Finally, an improved approach was proposed by which the correlation coefficients based on the partial spanwise correlation theory could be approximated with more comprehensive consideration of the spanwise correlation, allowing for a more accurate evaluation of the VIV amplitude for flexible structures.

Nonlinear dynamic analysis of nonlocal composite laminated toroidal shell segments subjected to mechanical shock

The present study deals with the nonlinear forced vibration of composite shallow nano toroidal shell segments. The improved Donnell shell theory is used with the nonlinear geometric strains, i.e., von Karman, Stein, and McElman’s assumptions. A coupled nonlinear partial differential equations using nonlocal theory, including the influence of nano-scales, is derived. The boundary conditions are simply supported and clamped–clamped, and the nano toroid shell is subjected to impulse radial dynamic excitation. The linear solution is verified by comparing natural frequencies with the available results using Galerkin’s method. The approximated linear mode shapes of the free vibration, which satisfy the boundary conditions, are used as the assumed multi-mode displacement functions required to solve the nonlinear vibration. Using Hamilton’s principle, the nonlinear equations of motion in generalized coordinates are solved using the displacement field by applying Galerkin’s and Runge–Kutta’s methods. Using the presented model, the effects of layer stacking, toroidal shell segments curvature (convex and concave toroidal shells), length scale parameter, and material properties are studied in detail, and some conclusions are surveyed. It is observed that the nonlocal parameter demonstrates a considerable effect on the dynamics behavior characteristics response of the toroidal shell segments. Furthermore, different fiber orientation effects, displacement fields, time response, and geometry parameters, i.e., thickness, and length are studied. Results indicate that natural frequencies significantly depend on the toroid dimensions, boundary conditions, layer stacking, curvature change, and considered wave numbers.

The Influence of the Grid Density of Measurement Points on Damage Detection in an Isotropic Plate by the Use of Elastic Waves and Laser Scanning Doppler Vibrometry.

Damage detection in structural components, especially in mechanical engineering, is an important element of engineering practice. There are many methods of damage detection, in which changes in various parameters caused by the presence of damage are analysed. Recently, methods based on the analysis of changes in dynamic parameters of structures, that is, frequencies or mode shapes of natural vibrations, as well as changes in propagating elastic waves, have been developed at the highest rate. Diagnostic methods based on the elastic wave propagation phenomenon are becoming more and more popular, therefore it is worth focusing on the improvement of the efficiency of these methods. Hence, a question arises about whether it is possible to shorten the required measurement time without affecting the sensitivity of the diagnostic method used. This paper discusses the results of research carried out by the authors in this regard both numerically and experimentally. The numerical analysis has been carried out by the use of the Time-domain Spectral Finite Element Method (TD-SFEM), whereas the experimental part has been based on the measurement performed by 1-D Laser Doppler Scanning Vibrometery (LDSV).

Detection of Material Degradation of a Composite Cylinder Using Mode Shapes and Convolutional Neural Networks.

This paper presents a numerical study of the feasibility of using vibration mode shapes to identify material degradation in composite structures. The considered structure is a multilayer composite cylinder, while the material degradation zone is, for simplicity, considered a square section of the lateral surface of the cylinder. The material degradation zone size and location along the cylinder axis are identified using a deep learning approach (convolutional neural networks, CNNs, are applied) on the basis of previously identified vibration mode shapes. The different numbers and combinations of identified mode shapes used to assess the damaged zone size and location were analyzed in detail. The final selection of mode shapes considered in the identification procedure yielded high accuracy in the identification of the degradation zone.

Determination of free vibration properties of masonry arch bridges using the dynamic stiffness method

Masonry arch bridges constitute over half of the European bridge stock. The dynamic response of these bridges to traffic loads is influenced by their free vibration properties, i.e. their natural vibration frequencies and mode shapes. However, these properties have not been systematically examined to date. This paper utilises the dynamic stiffness method (DSM) to obtain an improved fundamental understanding of the free vibration properties of idealised single and multi-span arch bridges. In this approach, the piers and arches are modelled as an assembly of linear Timoshenko beam segments, the backing and infill are represented with axial struts and the underlying soil with concentrated spring elements. These idealisations enable rapid but accurate calculations of in-plane free vibration properties. To demonstrate this, the proposed procedure is applied to a simply-supported arch structure, a small-scale single-span arch bridge model and a multi-span arch bridge, where published experimental data is available. Then, free vibration analyses of representative single and multi-span arch bridges are performed. The results obtained from DSM analyses agree with detailed 3D finite element models and reveal the critical influence of backing elements and the interactions between adjacent spans on free vibration properties. Lastly, a range of arch bridge geometries are examined to reveal the shortcomings of existing provisions for modal frequency estimation in codes of guidance.

Experimental Dispersion identification using a fitted state-space model

An identification method that can estimate the dispersion relation of waveguides experimentally using an efficient and accurate procedure is presented. The method fits a linear state-space model before resorting to a kinematic wave model in the frequency region near the pre-identified natural frequencies. The eigenvectors, or mode-shapes, are computed at the sensor locations, and based on the reduced-Bloch mode expansion method, the propagation modes are fitted to match the identified vibration mode-shape at these frequencies. Classical methods to identify the dispersion relation from measured data can be computationally expensive and time-consuming, with limited accuracy in cases of multimode propagation. Results show that the fitted dynamic model expands the frequency range of obtained dispersion curves and enhances the speed of computation and accuracy. The method is derived and verified for both lumped and distributed systems, using numerical finite element simulation. Experimental verification is carried out on two acoustical waveguides. The first is a circular ring-shaped array of coupled Helmholtz resonators, modeled using a lumped parameter model. The second is an air-filled acoustic wave-tube that is modeled as a distributed acoustic-elastic coupled waveguide. The method's strengths and weaknesses are discussed from the experimentally obtained dispersion curves, and its main feature, the ability to fit the dispersion model of weak modes, is highlighted.

GBT-based dynamic analysis of thin-walled members with circular axis

This paper presents an extension of the linear Generalized Beam Theory (GBT) formulation for members with circular axis, previously developed by the authors Peres et al., (2016), Peres et al., (2018), to the linear dynamic case. The proposed formulation makes it possible to calculate natural frequencies (and associated vibration mode shapes) and time-history responses of members with circular axis and arbitrary flat-walled cross-sections, undergoing complex global–distortional–local deformation. The remarkable modal decomposition features of GBT, stemming from the fact that the kinematic description of the beam is based on a superposition of structurally meaningful cross-section deformation modes, render the finite element implementation of the proposed formulation extremely accurate and computationally efficient, as demonstrated in the illustrative numerical examples presented throughout the paper. Both standard (displacement-based) and mixed (strain–displacement-based) finite elements are implemented and it is concluded that the predictive capacity of the latter is superior, as it is insensitive to the various forms of membrane locking appearing in curved members, even though it does not involve additional DOFs (the strain DOFs are eliminated from the element stiffness matrix).

Numerical validation of an analytical seal flutter model

An analytical model to describe the flutter onset of straight-through labyrinth seals has been numerically validated using a frequency domain linearized Navier-Stokes solver. A comprehensive set of simulations has been conducted to assess the stability criterion of the analytical model originally derived by Corral and Vega (2018), “Conceptual Flutter Analysis of Labyrinth Seals Using Analytical Models - Part I: Theoretical Support,” ASME J. Turbomach., 140 (12), pp. 121006. The accuracy of the model has been assessed by using a simplified geometry consisting of a two-fin straight-through labyrinth seal with identical gaps. The effective gaps and the kinetic energy carried over are retained and their effects on stability are evaluated. It turns out that is important to inform the model with the correct values of both parameters to allow a proper comparison with the numerical simulations. Moreover, the non-isentropic perturbations included in the formulations are observed in the simulations at relatively low frequencies whose characteristic time is of the same order as the discharge time of the seal. This effect is responsible for the bending of the stability limit in the <inline-formula><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline" overflow="scroll"><mml:mn>0</mml:mn><mml:mi>t</mml:mi><mml:mi>h</mml:mi></mml:math></inline-formula> ND stability map obtained both in the model and the simulations. It turns out that the analytical model can predict accurately the stability of the seal in a wide range of pressure ratios, vibration mode-shapes, and frequencies provided that this is informed with the fluid dynamic gaps and the energy carried over to the downstream fin from a steady RANS simulation. The numerical calculations show for the first time that the model can be used to predict accurately not only the trends of the work-per-cycle of the seal but also quantitative results.

An energy track method for early-stage rub-impact fault investigation of rotor system

Rubbing between rotor and stator may cause structural damage and lead to serious accidents. Diagnosing early-stage faults and determining the severity of rub-impact is of great significance for reducing the probability of rub-impact fault. At present, the conventional approaches based on the structure vibration displacement, velocity, acceleration, stress, and mode shape information are used to analyze and even diagnose faults. However, due to the slight impact force and the unobvious dynamic response characteristics, it is difficult to diagnose the early-stage faults accurately by applying those approaches. In this paper, a novel vibrational energy track method is adopted to diagnose the early-stage rub-impact faults. Aiming at a rotor rub-impact model considering the bending and torsional coupled vibration, a numerical method is applied to investigate the dynamic responses when early-stage rub-impact faults occur. Meanwhile, the dynamic response characteristics of early-stage faults and no rub condition are compared. Then, the strain energy theory and the energy track method are introduced to judge the faults. Finally, the energy track method is verified by experiments. The numerical and experimental results show that introducing the energy theory, especially using the energy track method can diagnose the early-stage rub-impact faults, by which the severity of faults can be well manifested. The method proposed in this paper provides a general and effective approach for the diagnosis of early-stage rub-impact faults of rotor.

Swell-induced flexural vibrations of a thickening ice shelf over a shoaling seabed

A solution method is developed for a linear model of ice shelf flexural vibrations in response to ocean waves, in which the ice shelf thickness and seabed beneath the ice shelf vary over distance, and the ice shelf/sub–ice-shelf cavity are connected to the open ocean. The method combines a decomposition of the ice shelf displacement profile at a prescribed frequency of motion into mode shapes of free vibrations, a finite-element method for the cavity water motion and a non-local operator to connect to the open ocean. An investigation is conducted into the effects of ice shelf thickening, seabed shoaling and the grounding-line conditions on time-harmonic ice shelf vibrations, induced by regular incident waves in the swell regime. Furthermore, results are given for ice shelf vibrations in response to irregular incident waves by superposing time-harmonic responses, and ocean-to-ice-shelf transfer functions are derived. The findings add to evidence that ice shelves experience appreciable flexural vibrations in response to swell, and that ice shelf thickening and seabed shoaling can have a considerable influence on predictions of how ice shelves respond to ocean waves.

Solution to Vibrations of Double-Beam Systems under General Boundary Conditions

The problem of vibrations of double-beam systems is of technological interest and has been extensively studied in the literature. However, a general solution is yet to be developed for the general boundary conditions. In this paper, a general solution of the free and forced vibrations of double-beam systems under classical, nonclassical, and mixed boundary conditions is derived. The obtained mode-shape solutions are exact and explicit. The characteristic frequency equation of a double-beam system is of order-4 and order-8 for the classical boundary conditions and the nonclassical boundary conditions, respectively. The solution of forced vibrations is developed with the classical modal expansion technique. In addition to the theory, applications of the proposed method are illustrated by numerical examples. The obtained frequencies for special problems are compared with the literature, and the agreement between the proposed method and the literature is remarkably good. This shows that the proposed shape function method may be applied reliably to determine the transverse vibration mode shapes and natural frequencies of the double-beam system under the general boundary conditions. The solutions of free and forced vibrations of the double-beam systems presented in the paper could easily be programmed into a computer for vibration analyses.

Low-Frequency Harmonic Perturbations Drive Protein Conformational Changes.

Protein dynamics has been investigated since almost half a century, as it is believed to constitute the fundamental connection between structure and function. Elastic network models (ENMs) have been widely used to predict protein dynamics, flexibility and the biological mechanism, from which remarkable results have been found regarding the prediction of protein conformational changes. Starting from the knowledge of the reference structure only, these conformational changes have been usually predicted either by looking at the individual mode shapes of vibrations (i.e., by considering the free vibrations of the ENM) or by applying static perturbations to the protein network (i.e., by considering a linear response theory). In this paper, we put together the two previous approaches and evaluate the complete protein response under the application of dynamic perturbations. Harmonic forces with random directions are applied to the protein ENM, which are meant to simulate the single frequency-dependent components of the collisions of the surrounding particles, and the protein response is computed by solving the dynamic equations in the underdamped regime, where mass, viscous damping and elastic stiffness contributions are explicitly taken into account. The obtained motion is investigated both in the coordinate space and in the sub-space of principal components (PCs). The results show that the application of perturbations in the low-frequency range is able to drive the protein conformational change, leading to remarkably high values of direction similarity. Eventually, this suggests that protein conformational change might be triggered by external collisions and favored by the inherent low-frequency dynamics of the protein structure.

Damage severity for cracked simply supported beams

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 zero frequency 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 zero frequency deflection and inversely proportional to first mode frequency.

Vibration analysis of thin-walled functionally graded sandwich beams with non-uniform polygonal cross-sections

In the paper, an analysis of thin-walled functionally graded straight and curved beams for general non-uniform polygonal cross-sections has been introduced for vibration problems. Higher order beam theory that has fully taken into account major deformations, e.g. in-plane distortion, out-of-plane warping, is adopted by means of beam frame modal approach. In addition with the effects of geometric parameters, the study also deals with some consequences resulting from material anisotropy which in turn tackles better in evaluation behaviors of thin-walled beams that enables analytical modeling to have the potential for further extension. Subsequently, the analysis can be applied for general closed cross-section beams with arbitrary curvatures whereas materials properties are assumed to be graded across the wall thickness following a predefined shape function. A numerical verification for the overall analysis and examples have been carried out based on the available results created from benchmark problems and finite element software simulation found in literature. Then, natural frequencies and vibrational mode shapes have been obtained through a robust framework of finite element method. Finally, a parametric study and discussion towards vibration phenomena are fully investigated referring to gradual law, skin-core-skin effects, etc.

Vibration Characteristics of Island-Bridge Structure on Porous PDMS Substrates for Stretchable Electronics

Abstract Stretchable electronics employing island-bridge structure design can achieve controllable and reversible stretchability. The use of a porous substrate, which provides excellent breathability for wearable devices bonded to skin, not only satisfies this static superiority but also has a profound impact on the dynamic performance of the stretchable electronics. In this paper, the vibration characteristics of the island-bridge structure based on porous polydimethylsiloxane (p-PDMS) substrates are studied by utilizing an analytical model, which takes account of geometric nonlinearity due to mid-plane stretching, buckling configuration, elastic boundary conditions considering the p-PDMS substrates and the mass of the island. In numerical examples, the accuracy of the analytical model is first verified by finite element analysis (FEA). After that, we investigate the effects of some primary factors, i.e., the prestrain of the substrate, spring stiffnesses at the ends of the interconnect, porosity and thickness of the substrate, and the mass of the island, on the natural frequencies and vibration mode shapes of the island-bridge structure. Results show that the vibration characteristics of the island-bridge structure can be tuned conveniently by adjusting the porosity of the substrate and the mass of the island, which are expected to be helpful to mechanical design and optimization of stretchable electronics in complex noise environments.

3D structural vibration identification from dynamic point clouds

Video-based measurement has received increased attention for modal analysis and nondestructive evaluation, playing an important role in the development of the next-generation structural sensing technologies. As these techniques have evolved, more quantitative approaches based on computer vision techniques have emerged on full-field unsupervised structural identification, exploiting the benefits provided by the use of video cameras such as high spatial sensor density and low installation costs. More recent work has started to explore the use of laser point cloud data for 3D mapping of scenes and structures. Sensors such as LIDAR provide huge amounts of measurements at high spatial resolution from which it is possible to estimate accurate structural geometry for applications such as the generation of CAD models. Unfortunately to-date, the frame rate and depth resolution of LIDAR and other sensors capable of 3D geometry measurements has not been sufficient for measuring structural dynamics. In this paper, we introduce an approach for efficient and extremely high resolution 3D structural dynamic identification/modal analysis from point cloud data acquired using a commercial, low-cost, time-of-flight imager. Vibration mode shapes and modal coordinates are extracted from this data by creating virtual Lagrangian sensors based on the point clouds parameters. First, time-varying point cloud data are collected from a vibrating structure. Then, a mesh of virtual sensors is created based on the dynamic point cloud data for tracking the structure’s displacement over time. Next solutions to the blind source separation problem are employed to estimate high resolution 3D mode shapes, modal coordinates, and resonant frequencies. We demonstrate the potential of our proposed approach on laboratory tests and compare the results to the data collected from conventional laser displacement sensors. This technique represents an advance towards efficiently exploring the full advantages of using dynamic point cloud data for practical monitoring applications and has the potential to be extended for a wide range of 3D motion decomposition problems.

Influence of dynamic properties on scaffoldings safety

Scaffoldings are used for works at height and in places that are hard to reach, which makes such works dangerous to employees and accidents occur frequently. Loads generated by scaffolding users cannot be avoided. Moving workers excite low-frequency (1–2 Hz) vibrations and scaffoldings as slender structures are prone to such dynamic action. The method for determining the probability of vibrations excitation is presented here. The quantity representing this probability is called the predictor of occurrence of a dangerous situation due to vibrations induced by a walking employee. The predictor of resonance with ith natural frequency requires an analysis of the scaffolding dynamic behavior. The frequencies and the natural mode shapes of vibrations were determined. Numerical dynamic simulations of the worker's movement on the penultimate decks of two scaffoldings were carried out, as well. Predictor analysis was made for single frequencies and combinations of frequency pairs. The predictor values calculated for the first frequency or combinations with it are the highest ones, however the probability of resonance is not only affected by the first frequency. To improve safety, the natural frequencies should be increased. For longitudinal vibrations, this can be done by adding more bracing or reducing lengths of anchors. Increasing the number of anchors gives good results in both directions. During scaffolding design of both typical and atypical constructions, one must determine the natural frequencies and then, if the first natural frequency is less than 4.0 Hz, perform a dynamic scaffolding analysis.

Evaluation of vibration mode shape using a mechanoluminescent sensor

This study proposes a method to measure vibration mode shape on beam structures by using the elastic mechanoluminescent (EML) material SrAl2O4: Eu2+. The EML phenomenon has attracted interest due to its potential for application to strain distribution, crack visualization, and structural health monitoring. In this Letter, the relationship between the distribution of luminous intensity and the vibration mode shape of beam structures is explored and is found to be quadratic. A comparison between the results of experiments and simulations shows that the proposed method can measure a structure's vibration mode shape. The study also shows that experimental parameters, such as the exposure time of the camera and the amplitude of excitation, do not affect the validity of this method. The robustness and convenience of the proposed method make it a more efficient technique than prevalent electrical or optical vibrational methods of analysis.