Mode Shapes Research Articles

Model Parameter Calibration for Vibration Fatigue Analysis by Means of Bayesian Updating and Artificial Neural Network Based Surrogate Models

Abstract In this paper, a methodology for model parameter calibration for vibration fatigue analysis is proposed. It combines Bayesian updating of uncertain model parameters and artificial neural networks (ANNs). The calibrated parameters are used to increase the accuracy of fatigue lifetime calculations for components submitted to vibrational loads. The Bayesian updating uses eigenfrequencies, mode shapes, total mass, and the frequency response functions (FRFs). These quantities are predicted by ANN-based surrogate models to accelerate the Bayesian updating process. A novel strategy for the prediction of the magnitude and phase of FRFs with ANNs is proposed. The frequency is used as an additional input variable, and a schematic selection of significant points of the FRF curves is presented. A high prediction accuracy of the surrogate models could be achieved. The procedure includes the analysis of the relevant frequency range and a sensitivity analysis based on the Morris method to identify appropriate modes and the most-influential parameters. The proposed framework is applied to a current vehicle component subjected to vibrational loads. An experimental modal analysis is used for the calibration and consideration of real parameter uncertainty. First, the accuracy of the surrogate models and Bayesian updating is verified by a nominal reference simulation and then validated with experimental data. The measurable control parameter thickness and component mass are used to examine the calibration accuracy. Finally, a decrease in the dispersion of the vibration fatigue distribution is obtained with the calibrated parameters.

Impact of Installation Deviations on the Dynamic Characteristics of the Shaft System for 1 Gigawatt Hydro-Generator Unit

The shaft system, transferring the kinetic energy of water flow into electrical energy, is the most critical component in hydropower plants. Installation deviations of the shaft system for a giant hydro-generator unit can have significant impacts on its dynamic characteristics and overall performance. In this investigation, a three-dimensional geometry of the shaft system of an operating hydro-generator unit prototype with a rated power of 1 GW is established. Then, the calculation model of the shaft system is generated accordingly with tetrahedral and hexahedral elements. By applying different boundary conditions, the finite-element method is used to analyze the influences of installation deviations, including shaft radial misalignment and angular misalignment, on the dynamic characteristics of the shaft system. The calculation results reveal that the installation deviations change the natural frequencies, critical speeds, and mode shapes of the shaft system to a certain degree. The natural frequencies of the backward precession motion with installation deviations are reduced by 23% and 38% for the rated speed and the maximum runaway speed. Furthermore, for the forward precession motion, they increased by 30% and 48%, respectively. The critical speeds for the shaft system with radial and angular deviations are 3.2% and 3% larger than the critical speed of the shaft system without any mounting deviations. The radial and angular installation deviations below the maximum permissible values will not result in the structural performance degradation of the 1 GW hydro-generator shaft system. The conclusion drawn in this research can be used as a valuable reference for installing other rotating machinery.

Evaluation of Boundary Stiffnesses for Beams with Generally Restrained Edges: Theory and Vibration Experiments

An effective methodology is proposed to evaluate the boundary stiffnesses of beams with generally restrained edges. The boundary conditions at the ends of the beams are implemented by the penalty method. The unified Jacobi polynomials are introduced to represent the assumed mode shape functions of the beam. Hamilton’s principle with the assumed mode method is employed to derive the equation of motion of the beam with generally restrained boundaries. Subsequently, a novel approach is proposed to evaluate the boundary stiffnesses of the beam using the measured natural frequencies of the beam by vibration experiments. The accuracy of this method is verified by comparing the results with the previously published literature, the finite element method (FEM), and the vibration experiments. The effects of the types and values of the boundary spring stiffnesses on the vibration properties of the beam are also discussed. The calculation method in this research can provide an effective prediction technique for the boundary stiffnesses of engineering structures with generally restrained edges.

Vibration behaviours of composite conical–cylindrical shells with damping coating: Theory and experiment

This study presents the theoretical and experimental investigations on vibration behaviours of carbon fiber/ resin composite conical-cylindrical shells (or CC shells) with damping coating. First, an analytical model of such a combined shell with coating material considers the effects of base harmonic excitation load under arbitrary boundary conditions is proposed using the first-order shear deformation theory, virtual artificial spring technique, modal superposition approach, the Rayleigh-Ritz method, etc. Subsequently, the solutions of fundamental frequencies, mode shapes, and vibration displacements in the frequency domain are acquired by deriving the corresponding energy expressions and motion equations. Convergence analysis is utilized to determine the virtual spring stiffness value and the appropriate truncation numbers. Both literature and experimental results are conducted to give an adequate validation of the current model, in which the continuous 3D printing technology is adapted to fabricate two CC shell specimens with one of the outside surfaces being sprayed by coating material using an atomization deposition approach. Finally, the effects of the coating thickness ratio, Young's modulus ratio of the coating and the substrate shell, the semi-vertex angle, and the fiber laying angle on the free vibration and forced vibrations of the coated structure are discussed, with some practical recommendations being provided to improve the vibration suppression of such a coated structure.

Primary Resonances in Inclined Cantilever Beam with Tip-Mass: A Parameter-Splitting Multiple-Scales Approach

The transverse static load, resultant from self-weight, significantly impacts both the response and the nonlinear dynamic behavior of a flexible inclined cantilever with a large lumping tip mass under harmonic base motion. However, the transverse static load was not taken into account and the mode shape of a pure cantilever beam was directly adopted during Galerkin’s procedure in the formulations of a cantilever carrying a large lumping tip mass in many previous studies. In this paper, (1) the static load effect caused by the self-weight in an inclined cantilever is considered by using a coordinate transformation, (2) the efficacy of adopting a pure cantilever beam mode shape to analyze a cantilever beam carrying a large lumping mass is studied and (3) a recently proposed semi-analytical method named parameter-splitting multiple-scales method is extended to analyze the cantilever studied to examine its effectiveness. First, the extended Hamilton’s principle is utilized to formulate the equation of motion of the cantilever studied. After that, the mode shape of a pure cantilever and the exact mode shape of a cantilever carrying a lumping mass are separately adopted in Galerkin’s method to transform the partial differential equation into ordinary differential equations. The frequency-response curves of the discretized nonlinear differential equations obtained by the numerical continuation method and the parameter-splitting multiple-scales method are compared to address ① the transverse static load effect on the frequency-response and nonlinear dynamical behavior of the beam, ② the discrepancies between the resultant frequency-response curves obtained by using two different mode shapes and ③ the efficacy of the parameter-splitting multiple-scales method on solving strongly nonlinear practical problem.

Spatial And Temporal Modes On A Generic Tandem Configuration In Transonic Shock Buffet

Certain combinations of upstream flow conditions in transonic flight may incite a highly unsteady shock-wave/boundary-layer interaction, often referred to as transonic buffet. The coupling of periodic shock motion and intermittent separation is seen to convey a characteristic unsteadiness in the evolution of the turbulent wake. This article uses high-speed focusing Schlieren and PIV measurements at high spatial resolution to characterize the temporal and spectral modes governing the flow past a generic 2D tandem configuration. This interaction, consisting of an OAT15A supercritical airfoil as the main wing and a NACA 64A-110 profile representing the tailplane, is investigated at chord-based Reynolds numbers of 2 million and 1 million, respectively and compared against fully-established shock buffet conditions in an isolated airfoil scenario operated at identical inflow conditions. The strongest buffet is observed in both configurations for a Mach number of 0.72 and an angle of attack of 5 deg, occurring at frequencies of 112 Hz and 103 Hz. Despite an equally periodic nature of the buffet unsteadiness, both shock amplitude and frequency are seen to decrease slightly in the tandem interaction. Global spectral analysis performed on the entire field of view identifies the buffet instability as the governing mode in both flow scenarios, imposing its unsteadiness even far downstream of its origin, allowing the extraction. of relevant sensor locations for a subsequent localized assessment of the involved time scales. Except for the buffet mode, complementing DMD analyses reveal a strong influence of a vortex-shedding mode throughout the entire buffet cycle. Mode shapes and associated frequencies furthermore suggest an intermittent low-frequency/large wavelength and high-frequency/low wavelength behavior, involving leading contributions between 2000 Hz and 4000 Hz.

Integrated Optical Diagnostics Of Cyclopentane Sprays To Characterize Transcritical Phase Transitions

This research investigates spray phase transitions under subcritical and transcritical conditions relevant to high-pressure combustion systems. Using advanced optical diagnostic techniques, including shadowgraphy (SH), Mie scattering (MS), and infrared radiation (IR) measurements, the study focuses on cyclopentane sprays in a high-pressure, high-temperature chamber filled with gaseous nitrogen. Three cases are studied to understand spray behavior under varying injection conditions. SH and MS analyses reveal significant differences across the cases. As the chamber temperature increases, the liquid core length shortens, indicating enhanced vaporization. This is particularly evident in the MS images, where step-like transitions in the axial distribution highlight abrupt phase changes from liquid to gas in subcritical conditions. Such transitions are absent under supercritical conditions, suggesting a smoother phase transition process. IR measurements provide additional insights into the spray dynamics, especially under transcritical conditions. The IR images display complex downstream (at x/D > 125) signal patterns with bimodal distributions. The application of inverse Abel deconvolution highlights negative intensities near the spray axis downstream, indicative of regions near the pseudo-critical temperature. This phenomenon, characterized by blocked background radiation due to critical opalescence, offers indirect evidence of the pseudo-critical temperature and the associated phase transition dynamics. Dynamic Mode Decomposition (DMD) analysis identifies dominant frequencies around 4.5 kHz, corresponding to injection pressure fluctuations. This correlation suggests that injection pressure variations significantly influence spray fluctuations. Differences in mode shapes among the cases are also observed; lateral fluctuations near the injector nozzle exit are present in Cases 1 and 2 but not in Case 3. Axial fluctuations beyond a certain length are consistent in all cases. The lateral fluctuations are related to the chamber pressure and the potential large-scale vortex formation or cavitation in the nozzle. The combined use of SH, MS, and IR methodologies provides a comprehensive understanding of transcritical spray dynamics, emphasizing the effectiveness of combining multiple optical diagnostic techniques. These findings will improve numerical simulations and contribute to the development of more efficient high-pressure combustion systems. The research underscores the complexity of phase transitions in sprays and the critical role of precise diagnostics in the advancement of combustion technology.

Efficient Vibration Measurement and Modal Shape Visualization Based on Dynamic Deviations of Structural Edge Profiles.

As a non-contact method, vision-based measurement for vibration extraction and modal parameter identification has attracted much attention. In most cases, artificial textures are crucial elements for visual tracking, and this feature limits the application of vision-based vibration measurement on textureless targets. As a computation technique for visualizing subtle variations in videos, the video magnification technique can analyze modal responses and visualize modal shapes, but the efficiency is low, and the processing results contain clipping artifacts. This paper proposes a novel method for the application of a modal test. In contrast to the deviation magnification that exaggerates subtle geometric deviations from only a single image, the proposed method extracts vibration signals with sub-pixel accuracy on edge positions by changing the perspective of deviations from space to timeline. Then, modal shapes are visualized by decoupling all spatial vibrations following the vibration theory of continuous linear systems. Without relying on artificial textures and motion magnification, the proposed method achieves high operating efficiency and avoids clipping artifacts. Finally, the effectiveness and practical value of the proposed method are validated by two laboratory experiments on a cantilever beam and an arch dam model.

An enhanced stochastic subspace identification incorporating time variables and its application to large-span sea-crossing suspension bridges

Structural health monitoring and assessment are essential to large-span sea-crossing bridges, and modal identification is the most effective method for achieving them, with the stochastic subspace identification (SSI) method dominating. However, when analyzed in open traffic scenarios with SSI for a large-span sea-crossing bridge, the contour of the vibration mode is distorted. Related affecting factors such as the measuring conditions, user-defined parameters, and traffic are explored, and traffic is verified as the essential component. To address this issue, an enhanced SSI algorithm incorporating time variables is created to automatically identify and rectify vibration mode shapes. Feature heterotopic signals are first matched with a common reference signal to perform modal identification. Advanced particle swarm optimization is applied to estimate critical parameters. The SSI algorithm with the density-based spatial clustering of applications with noise (DBSCAN) algorithm is then utilized to extract the modal characteristics. Both simulated misalignment and real records are applied to verify the suggested approach's performance. Comparative analysis with state-of-the-art techniques, such as the error norm, cross correlation, phase transform-β, mean phase deviation, and genetic algorithm, shows that the recommended approach is dependable and successful at identifying the large-span sea-crossing bridge's vibration mode shapes and computationally more accurate and efficient.

Data-driven sensor fault diagnosis for vibration-based structural health monitoring under ambient excitation

In vibration-based structural health monitoring (SHM), the early detection of sensor faults is key to preventing false alarms and misleading conclusions on the condition of monitored structures. Since sensor networks are exposed to hostile environments, they are prone to unexpected errors that might influence the quality of measured data. This paper proposes a novel method for detecting and isolating faulty sensors from vibration response data by establishing an overdetermined system between the measured signals and the actual motion. The method assumes a rigid body motion of the monitored system, describable by a limited number of degrees of freedom (DOFs), to define the overdetermined relation between the sensor outputs and the system’s DOFs. The concept is later extended to systems not governed by rigid body motions by considering their vibration mode shapes. The robustness of the proposed methodology is demonstrated using vibration response data from an experimental monitoring campaign.

A fast and efficient pseudo-direct identification method for orthotropic elastic properties of multi-layered composite plates

New types of laminated composites are continuously being developed to meet various requirements involved in engineering applications. Before mass production of these new composite materials can take place, it is necessary to identify their elastic properties. Although current identification methods based on vibration tests and optimization algorithms are highly accurate, their high computation cost is a drawback. To address this issue, this paper proposes a new method to identify orthotropic elastic properties of multi-layered composite plates. The proposed method is based on the Modal Stability Procedure, using the experimental vibration frequencies and corresponding numerical mode shapes of structures. It converges after only two or three iterations without requiring an optimization algorithm. Therefore, the computational cost of the proposed method is significantly lower than that of optimization-based methods. The effectiveness and robustness of this method are demonstrated through three examples with different material, stacking sequences, and geometries including curved plate.

Integrated hydrodynamic-structural analysis of flexible floating structures

This paper presents a comprehensive approach for the linearized frequency-domain analysis of hydrodynamic loading and structural responses on a deformable body. The structural and hydrodynamic analyses are integrated by employing the mode superposition method. Commencing with an eigenvalue analysis on the structural model, the shapes of selected modes, mass, and structural stiffness matrices are extracted as input to the subsequent hydrodynamic analysis, where the global response is solved in the modal space. The resulting hydrodynamic and inertial loads are then transferred to the same finite element model for stress assessment. To exemplify the proposed methodology, a bottom-fixed flexible cylindrical monopile and a long box-shaped barge model are investigated. For the barge model, the sectional loads are obtained from the integral of stresses in the cuts along the structure model, which are found to be consistent with those from the hydrodynamic analysis, demonstrating the consistency of the entire workflow. In particular, this paper introduces a straightforward formulation for evaluating the generalized restoring matrices, eliminating the need for spatial derivatives of the mode shape function and thereby significantly reducing numerical uncertainties in using the flexible modes derived from finite element model for hydrodynamic analysis.

Exact solution for mode shapes of cracked nonuniform axially functionally graded beams

This paper established the exact formulas for mode shapes of cracked nonuniform axially functionally graded (AFG) beams. The formula of the mode shape of cracked nonuniform AFG beam is derived in the form of a power presented as a recurrent relation. The power series solution of cracked beam is the same with the intact beam where the effect of cracks contributes to only the first four coefficients of the power series. The derivation of the formula of mode shapes is presented in details. Numerical simulations show that these effects on the natural frequency and mode shape are quite significant. The natural frequency and mode shape are more sensitive to cracks located at positions with low elastic modulus, small cross section and high mass density. For the case of tapered AFG cantilever beam, it is important to consider not only cracks located close to the fixed end but also cracks located close to the free end, especially when the free end has lower elastic modulus and higher mass density.

Research on Structural Damage Identification of Shear Buildings Based on Modal Curvature Change

In the past few decades, various techniques for structural damage identification based on dynamic properties have been widely studied, among which modal curvature is the research hotspot, due to its high sensitivity to local damage. To realize story-level structural damage identification of shear buildings, this paper first derives the patterns of variation in modal curvature of the damaged structure. Then, a new damage index sensitive to local damage called the Modal Curvature Change modified with Softmax (MCCSM for short), is proposed to improve the accuracy of structural damage identification. Afterward, the validity of the damage index is verified by a shaking table test of a scaled RC frame structure model and its corresponding numerical model. The research results show that the damage location indicated by MCCSM corresponds well with the damage phenomenon in the shaking table test and numerical simulation; compared with using the theoretical mode shape, the MCCSM index can give comparable damage identification results using the mode shape obtained through operational modal analysis from vibration time-history data, which proves the feasibility in engineering practices; through numerical simulation, it is proved that the MCCSM index is applicable in a wide range of damage cases with different story position, number of damaged story, and damage extent.

Identification of excitation frequencies of a vibrating beam using Dynamic Mode Decomposition

Identifying modal parameters for the different vibrational conditions for any structure or machine is very important for its structural health monitoring. Researchers have proposed different methodologies to determine that structure’s excitation or natural frequency. In the initial design phase, researchers have to rely on building the FE model and then analyze its natural and excitation frequencies. However, for the complex structure, it is difficult to handle a huge dataset of the vibrational information of the mesh for multi-frequency excitations and it becomes computationally expensive to analyze the system’s performance. This work presents the data-driven technique called Dynamic Mode Decomposition (DMD) which addresses this problem by discovering the system’s dynamics using system’s high-dimensional spatiotemporal data. The present work studies the FE vibration of fixed-fixed beams with single and multi-frequency harmonic forcing functions with and without noise for real-time scenarios. Here, DMD is applied to the sequential spatiotemporal data of displacement of the vibrating beams, and dominant frequencies and corresponding mode shapes are identified to predict the actual time dynamics of the system under different conditions in simulation and experimental conditions. This study also involves the failure of the DMD to capture the periodic oscillation (standing wave) in the beam and proposes the application of the Hankel matrix (time-delay coordinate) to capture the actual dynamical parameters of the system. The resulting predicted frequencies match the actual excitation frequencies, and the accuracy is improved with a number of stackings of the given time series data through the Hankel matrix at different rank truncations.

Investigating the modal behaviors of a deep beam with a transverse open crack

This paper investigates the modal behaviors of a deep beam with a transverse open crack. Unlike cracked shallow beams with aspect ratio (L/h) > 10, cracked deep beams with aspect ratio < 5 simultaneously exhibit near the crack both axial-bending coupling mode caused by mode I crack and shear (or sliding) mode caused by mode II crack, referred to as axial-bending-shear modal coupling. Therefore, for accurate modal behaviors prediction, appropriate compatibility equations accounting for both modes I and II crack should be employed for deriving a frequency equation. Because the derived frequency equation is nonlinear with respect to modal frequency, the modal frequencies are calculated through the Newton–Raphson method. The corresponding mode shapes are determined by substituting the calculated modal frequencies into the spectral solutions for either side of the crack. The validity of the proposed method was demonstrated through comparison to the finite element analysis and experimental results.

Bridging the Gap: commodifying infrastructure spatial dynamics with crowdsourced smartphone data

Structural information deficits about aging bridges have led to several avoidable catastrophes in recent years. Data-driven methods for bridge vibration monitoring enable frequent, accurate structural assessments; however, the high costs of widespread deployments of these systems make important condition information a luxury for bridge owners. Smartphone-based monitoring is inexpensive and has produced structural information, i.e., modal frequencies, in crowdsensing applications. Even so, current methods cannot extract spatial vibration characteristics with uncontrolled datasets that are needed for damage identification. Here we present an extensive real-world study with crowdsourced smartphone-vehicle trips within motor vehicles in which we estimate absolute value mode shapes and simulate damage detection capabilities. Our method analyzes over 800 trips across four road bridges with main spans ranging from 30 to 1300 m in length, representing about one-quarter of bridges in the United States. We demonstrate a bridge health monitoring platform compatible with ride-sourcing data streams that check conditions daily. The result has the potential to commodify data-driven structural assessments globally.

Integrated analytical, experimental, and numerical study of vibrational response in clamped circular plates with pressurised air cavities

This paper presents a comprehensive investigation, comprised of analytical, experimental and numerical approaches, into the interaction between the sound field produced within a pressurised air-filled back cavity and the vibration of a thin clamped circular plate subjected to its pressure. Previous investigations into the interaction of sound fields with vibrating boundaries have primarily been restricted to non-pressurised cavities. We extend the application of the weak modal coupling method to derive a solution for this unexplored physical problem by utilising classic plate theory, Fourier-Bessel series formulation and the uncoupled solutions for a pressurised cylindrical sound field and a thin clamped circular plate with uniform radial tension. We describe the vibration response of the coupled system in terms of the uncoupled cavity and plate modes. Using a set of geometric and physical properties for the system, the resonance frequencies and response functions are calculated and the coupling between the sound field and the plate is investigated as a function of cavity pressure, providing novel fundamental insights into the physical system. We construct an experimental rig to embody the conditions of the analytical investigation for the purpose of validation. Finite element analysis (FEA), employing modal analysis and transient dynamic analysis, is used to further validate and extend the analytical insights while more accurately mirroring the experimental conditions. The response functions, mode shapes, resonance frequencies and their dependence on the cavity pressure were experimentally measured and numerically simulated, with direct comparisons made with the analytical model. A high degree of accuracy for the analytical model is validated along with its ability to describe the underlying physical phenomena. The validated analytical model is then leveraged to perform fundamental explorations into the modal contributions as a function of cavity pressure and a parametric sensitivity analysis on the effects of plate radii and plate thickness on the coupled system resonance frequencies.

Multiscale Modeling of Elastic Waves in Carbon-Nanotube-Based Composite Membranes

A multiscale model is developed for vertically aligned carbon nanotube (CNT)-based membranes that are made for water purification or gas separation. As a consequence of driving fluids through the membranes, they carry stress waves along the fiber direction. Hence, a continuum mixture theory is established for a representative volume element to characterize guided waves propagating in a periodically CNT-reinforced matrix material. The obtained coupled governing equations for the CNT-based composite are found to retain the integrity of the wave propagation phenomenon in each constituent, while allowing them to coexist under analytically derived multiscale interaction parameters. The influence of the mesoscale characteristics on the continuum behavior of the composite is demonstrated by dispersion curves of harmonic wave propagation. Analytically established continuum mixture theory for the CNT-based composite is strengthened by numerical simulations conducted in COMSOL for visualizing mode shapes and wave propagation patterns.

A prediction method for railway-induced vibrations in building structures using modal properties in limited modes

This paper proposes a prediction method to estimate vibrations in building structures induced by to-be-constructed railways. The method derives modal equations of motion for building modeling by using the natural frequencies in limited modes, the corresponding damping ratios and mode shapes at selected locations. The participation vector for input ground motions is approximated by the corresponding mode shapes. Those modal properties can be obtained from ambient vibration testing, and hence the modeling does not require design documents to construct the mass, damping and stiffness matrices. The railway-induced motions at the column bases can be estimated priorly, and the dynamic responses at the selected locations are subsequently predicted by mode superposition. The fundamental characteristics of the proposed prediction method are studied by numerical simulations. The simulation result shows high participation-vector approximation accuracy when using few lower mode shapes at few locations; Higher modes exhibit more complicated mode shape amplitude distribution, and thus mode shapes at more locations are required for the sufficient approximation accuracy; Compared with seismic response calculation in the horizontal directions where few lower modes are sufficient, higher modes are required for satisfactory prediction accuracy. The method provides an alternative approach for predicting railway-induced vibrations and is helpful when design documents of the objective building are unavailable. The findings from the simulation provide guidance to the measurement-point design and the objective modes to identify in the implementation of the proposed method.