Mode Shapes Of Structure Research Articles

Free and forced vibration analysis of a Z-shaped structure with multi-span elastic supports

In this paper, an in-plane Z-shaped structure with multi-span elastic supports is proposed to investigate the natural frequency and transmission response through employing in-plane governing equation and transfer matrix method. Based on the solutions of transverse vibration and torsional vibration, the total transfer matrix for the Z-shaped structure with multi-span elastic supports is derived. Furthermore, natural frequencies and mode shapes of the Z-shaped structure are calculated. Finite element simulation method (FEM) is conducted here to verify the theoretical results. In order to effectively evaluate the vibration reduction performance, transmission response of the Z-shaped structure under different boundary supports and multi-span elastic supports is analyzed. This work is significant for the vibration reduction of in-plane Z-shaped structure, especially the multi-span elastic support case in engineering applications.

Frequency-domain method for non-stationary stochastic vibrations of train-bridge coupled system with time-varying characteristics

When a train passes over a bridge, the vibrations of the vehicles and the bridge are the result of non-stationary stochastic processes due to the time-dependent characteristics of the coupled vehicle-bridge system. The aim of this study is to generalize the frequency domain method to investigate the non-stationary random vibration of the coupled system subjected to the excitation of track irregularities with consideration of time-dependent characteristics. To illustrate the method, a three-span simply supported bridge traversed by a single railway vehicle is adopted as an example. The time-dependent frequency response function (FRF) of the coupled system is theoretically derived through solving ordinary differential equations with variable complex coefficients, and the perturbation method is adopted to improve the calculation efficiency. By combining this with Priestley’s Evolutionary Spectra theory, the evolutionary power spectral density (PSD) of the non-stationary random response of the system is then derived. The transitions will occur when the wheels cross the joints between each bridge span and between the bridge and the adjacent roadway. By adopting mode shapes of the full structure, the change of states of the vehicle crossing multiple bridge spans and moving onto the roadway can be solved as a continuous process without separation. The proposed method is validated by comparisons with the Monte Carlo method, showing higher accuracy and efficiency when calculating the time-varying standard deviation of the response. It is found that the vibration of the vehicle is approximately stationary but with large variance due to the random track irregularities, while the bridge vibration follows a strongly non-stationary process with small randomness and is more related to the moving mass effect.

Modeling and Analysis of a MEMS Vibrating Ring Gyroscope Subject to Imperfections

We present a new mathematical model for a vibrating ring gyroscope (VRG) in the presence of imperfections, namely, structural defects and material anisotropy. As a novelty, we calculate the mode shapes of the internal suspension structure to enable a more accurate and modular analysis of the VRG’s mass and stiffness distributions. Solving the associated eigenvalue problem shows that imperfections result in the frequency split between the gyroscope’s operating mode shapes, rotating their orientation with respect to the nominal drive and sense axes. We then use perturbation analysis to solve the VRG’s equations of motion and analyze the quadrature error that arises from frequency/damping mismatch between the mode shapes. We use our model to detail the various effects of the etching-related undercuts, structural uncertainties, and Young’s modulus anisotropy–in the form of suitable space-dependent functions–on the mode shapes and the quadrature error for the first time. The results reveal that rings are robust against imperfection, while the straight beams used in the suspension system are most likely responsible for the frequency split and quadrature error. For example, 50 nm (0.5%) width variation in a beam that connects the VRG’s suspension to an anchored internal structure leads to 4700°/s quadrature error. To validate our modeling, using the experimental data from a fabricated 59 kHz VRG, we provide rigorous, comparative simulations against the finite element method (FEM). [2022-0035]

A novel methodology for modal parameter identification of arch dam based on multi-level information fusion

Operational modal analysis plays an important role in the structural health monitoring and safety diagnosis of arch dam. However, due to background noise, it is difficult to accurately extract the effective characteristics information of arch dam from the vibration responses whose amplitudes are too small under ambient vibration, and the deviation caused by traditional identification methods will directly affect the estimation accuracy for structural modal parameters. Therefore, a novel methodology for modal parameter identification of arch dam based on multi-level information fusion is proposed in this paper. The proposed method is based on multi-sensor data-level fusion to identify the structural natural frequency and damping ratio, which greatly preserves and extracts structural modal properties in the vibration responses. Meanwhile, structural mode shapes are identified based on dynamic feature-level fusion, which significantly improves the identification accuracy. The effectiveness and feasibility of the proposed method are verified by the modal results of digital signals and simulated signals in the 7-DOF system. Prototype engineering case shows that the closely spaced and high-frequency modes can be decomposed and identified by the proposed method, and this method has a higher identification accuracy, which can provide a new idea for modal parameter identification of arch dam.

Pearson Correlation and Discrete Wavelet Transform for Crack Identification in Steel Beams

Discrete wavelet transform is a useful means for crack identification of beam structures. However, its accuracy is severely dependent on the selecting mother wavelet and vanishing moments, which raises a significant challenge in practical structural crack identification. In this paper, a novel approach is introduced for structural health monitoring of beams to fix this challenge. The approach is based on the combination of statistical characteristics of vibrational mode shapes of the beam structures and their discrete wavelet transforms. First, this paper suggests using regression statistics between intact and damaged modes to monitor the health of beam structures. Then, it suggests extracting quasi-Pearson-based mode shape index of the beam structures to use them as an original signal in discrete wavelet transforms. Findings show that the proposed approach has several advantages compared with the conventional mode shape signal processing by the discrete wavelet transforms and significantly improves damage detection’s accuracy.

Optimal design of dome structures with recently developed algorithm: Rao series

The modal parameters such as natural frequencies and mode shapes of structures are vital parameters that must be within the certain limits to ensure the desired structural behavior. Optimization of any type of structures regarding natural frequencies as constraints is considered as highly nonlinear and reaching the global solutions is very difficult. Furthermore, optimum design of small, medium and large-scale domes is known as a complicated optimization problem. In this study, the size optimization of the small, medium and large-scale truss dome structures subjected to multiple dynamic frequency constraints by using newly developed meta-heuristic algorithms named Rao algorithms (Rao-1, Rao-2 and Rao-3) is presented. The cross-sectional areas for the truss bar members of dome structures and first five natural frequencies are considered as the design variables and multiple dynamic constraints, respectively. To perform the optimization process of dome structures, a visual computer program is created in Matlab–GUI (Graphical User Interface). This program has combined the Rao algorithms and the finite element analysis using Eigen values of the stiffness matrix. Three different sizes such as small, medium and large-scale dome are examined to show the performance of the Rao algorithms. The examples for the small, medium and large-scale domes contain 120 bar, 600 bar, 1180 bar and1410 bar dome structures. The obtained near-optimal results are compared with the available results found in the literature to indicate the performance of the Rao algorithms. At the end of the study, it is concluded that the Rao algorithms performed similar to other methods and even better in some cases for designing the dome structures. Furthermore, Rao-2 algorithm has achieved the best results in the least number of generations compared to the Rao-1 algorithm.

Higher order theories for the free vibration analysis of laminated anisotropic doubly-curved shells of arbitrary geometry with general boundary conditions

The present work investigates the modal response of doubly-curved shells characterized by unsymmetric lamination schemes embedding anisotropic materials and externally constrained with general boundary conditions along the edges. The Equivalent Single Layer (ESL) methodology has been adopted for the assessment of the fundamental governing equations, employing a generalized formulation for the setup of each component of the kinematic field variable, leading to a higher order through-the-thickness expression of the displacement field. A two-dimensional non uniform discrete grid has been sat up within the mapped physical domain. The fundamental equations have been derived from the Hamiltonian Principle, and the problem has been solved employing a weak formulation of the field variable based on a higher order Lagrange polynomials-based interpolation algorithm. General boundary conditions have been obtained starting from a series of translational springs for each principal direction of the shell, each of them distributed following a generalized analytical expression. After some validations with respect to refined three-dimensional models, a systematic analysis has been performed, where the mode frequencies and shapes of structures with different syngonies and curvatures have been investigated. It has been shown that the modal response of anisotropic doubly-curved shells can be significantly oriented if governing distribution parameters are correctly selected.

Output-only complete mode shape identification of bridges using a limited number of sensors

Mode shape is an important structural property that needs be identified in structural condition monitoring and analysis. The conventional modal identification methods usually can only provide sparse and low-spatial resolution mode shape measurements because of the limited number of sensors, which could seriously affect the effectiveness and accuracy of the mode shape-based structural damage detection methods. In this paper, a new output-only method is proposed for identifying the complete mode shapes for beam-like bridge structures under a moving load using a limited number of sensors based on proper orthogonal decomposition (POD). Firstly, the physical interpretation of the POD of the responses of a beam-like bridge under a moving load is derived. It is proved that each component of the POD consists of a mode shape of the bridge and a dynamic component with the frequency corresponding to this mode. The contribution ratio of the component in the POD gives an excellent estimation of the modal contribution ratio (CR) in terms of the vibration energy. The mode shape can be identified by filtering out the dynamic information in each component of the POD. The identified mode shapes have a high spatial resolution that they can be regarded as the complete mode shapes of the bridge. To demonstrate the proposed method, a beam bridge with different boundary conditions is numerically modelled and a simply supported beam bridge model is tested in the lab. Both the numerical and experimental results demonstrate that this method can successfully identify the complete mode shapes with high resolution and modal contribution ratios with a limited number of sensors. This method breaks through the challenge on the requirement of a large number of sensors in vibration tests for full mode shape identifications in traditional methods.

A Parametric Study of the Free Undamped Torsional Vibration of One-Dimensional Structural Elements using the Differential Quadrature Method (DQM)

Recently, structures prone to torsional vibration have grabbed the interest of the structural engineering field whose aim has become to design structures with increased torsional stiffness. To the author’s limited knowledge, studies researching this topic couldn’t provide a simplified method to obtain the natural torsional frequencies and the corresponding mode shapes of structures providing ease of use by structural engineers having limited mathematical backgrounds. The purpose of this study is to establish a simplified numerical model of the torsional vibration of rods with non-uniform geometry and material with classical boundary conditions using the differential quadrature method (DQM). To account for geometric and material non- uniformity, any space varying functions in the longitudinal direction of a one- dimensional structural element can be used in the model. The numerical results obtained are compared with previously obtained analytical results in cases of rods with uniform cross sections and linearly non-uniform ones. The comparison proved significant agreement with the analytical results. The illustrated use of the DQM showed efficiency and acceptable accuracy that made it legible to create an alternative to the time-consuming analytical solutions that require a unique solution technique for every type of structural variation where a strong mathematical background of the user is needed.

Influence of Fluid Viscosity and Compressibility on Nonlinearities in Generalized Aerodynamic Forces for T-Tail Flutter

The numerical assessment of T-tail flutter requires a nonlinear description of the structural deformations when the unsteady aerodynamic forces comprise terms from lifting surface roll motion. For linear flutter, a linear deformation description of the vertical tail plane (VTP) out-of-plane bending results in a spurious stiffening proportional to the steady lift forces, which is corrected by incorporating second-order deformation terms in the equations of motion. While the effect of these nonlinear deformation components on the stiffness of the VTP out-of-plane bending mode shape is known from the literature, their impact on the aerodynamic coupling terms involved in T-tail flutter has not been studied so far, especially regarding amplitude-dependent characteristics. This term affects numerical results targeting common flutter analysis, as well as the study of amplitude-dependent dynamic aeroelastic stability phenomena, e.g., Limit Cycle Oscillations (LCOs). As LCOs might occur below the linear flutter boundary, fundamental knowledge about the structural and aerodynamic nonlinearities occurring in the dynamical system is essential. This paper gives an insight into the aerodynamic nonlinearities for representative structural deformations usually encountered in T-tail flutter mechanisms using a CFD approach in the time domain. It further outlines the impact of geometrically nonlinear deformations on the aerodynamic nonlinearities. For this, the horizontal tail plane (HTP) is considered in isolated form to exclude aerodynamic interference effects from the studies and subjected to rigid body roll and yaw motion as an approximation to the structural mode shapes. The complexity of the aerodynamics is increased successively from subsonic inviscid flow to transonic viscous flow. At a subsonic Mach number, a distinct aerodynamic nonlinearity in stiffness and damping in the aerodynamic coupling term HTP roll on yaw is shown. Geometric nonlinearities result in an almost entire cancellation of the stiffness nonlinearity and an increase in damping nonlinearity. The viscous forces result in a stiffness offset with respect to the inviscid results, but do not alter the observed nonlinearities, as well as the impact of geometric nonlinearities. At a transonic Mach number, the aerodynamic stiffness nonlinearity is amplified further and the damping nonlinearity is reduced considerably. Here, the geometrically nonlinear motion description reduces the aerodynamic stiffness nonlinearity as well, but does not cancel it.

Structural damage identification employing hybrid intelligence using artificial neural networks and vibration-based methods

The basis of vibration-based damage detection method is that when there are changes in physical properties of a structure, there will also be changes in vibration characteristics such as natural frequencies, mode shapes and damping ratios. Artificial neural network has become one of the most powerful approaches capable of pattern recognition, classification, and nonlinear modelling employing computational intelligence techniques to tackle damage detection as a complex problem. In this paper, ensemble neural networks based damage identification techniques were developed and applied for damage localization and severity identification in I-beam structures using dynamic parameters. Experimental modal analysis and numerical simulations of I-beam with triple-point damage cases were carried out to generate the natural frequencies and mode shapes of structures. In the procedure of damage identification, five different neural networks corresponding to mode 1 to mode 5 were constructed, and then an approach based on a neural network ensemble was proposed to combine the outcomes of the individual neural networks into a single network. The ensemble neural network has the superiorities of all the individual networks from different vibrational modes. The ensemble network produced better damage identification outcomes than the individual networks and the results revealed the ability of the ensemble neural networks to identify damage.

Exact wave propagation analysis of lattice structures based on the dynamic stiffness method and the Wittrick–Williams algorithm

This paper proposes two significant developments of the Wittrick–Williams (W–W) algorithm for an exact wave propagation analysis of lattice structures based on analytical dynamic stiffness (DS) model for each unit cell of the structures. Based on Bloch’s theorem, the combination of both the DS and the W–W algorithm makes the wave propagation analysis exact and efficient in contrast to existing methods such as the finite element method (FEM). Any number or order of natural frequencies can be computed within any desired accuracy from a very small-size DS matrix; and the W–W algorithm ensures that no natural frequency of the structure is missed in the computation. The proposed method is then applied to analyze the band gap characteristics and mode shapes of hexagonal honeycomb lattice structures and the results are validated and contrasted against the FE results. The effects of different primitive unit cell configurations on band diagrams and iso-frequency contours are thoroughly investigated. It is demonstrated that the proposed method gives exact eigenvalues and eigenmodes with the advantage of at least two orders of magnitude in computational efficiency over other methods. This research provides a powerful, reliable analysis and design tool for the wave propagations of lattice structures.

Radar Interferometric Experimental Reconstruction of Three-Dimensional Displacement Vectors and Mode Shapes for Masonry Constructions

Abstract Radar interferometry is an innovative measurement technique capable of remotely measuring vibrations in terms of displacement history. It apparently allows overcoming some major drawbacks of the plodding and time-consuming conventional accelerometric experimental setup for vibration testing. Indeed, radar interferometry does not require any access to the structure; moreover, its use is simpler and faster. Therefore, this technique appears very appealing for ambient vibration testing on structures, and in particular on architectural heritage masonry constructions. However, radar interferometry is still affected by some relevant limitations, that currently outweigh the advantages. One of the most important limitations is that displacements are measured only along the line-of-sight; therefore, it is not possible to determine the whole three-dimensional displacement vector for moving structural points and, consequently, to reconstruct three-dimensional mode shapes of structures, whose knowledge is crucial for characterizing the dynamic behavior of structures and for detecting eventual damages. In this paper, a theoretical and experimental approach for reconstructing both three-dimensional displacement vectors and mode shapes of masonry constructions is proposed. This approach is based on the simultaneous use of two synchronized radar interferometers, and on the application of a theoretical model assuming a specific kinematical constraint, which is generally plausible in the case of masonry constructions. The proposed approach has been validated through in-situ experimental tests on a masonry bell tower.

Distributed surface compliance for airfoil tonal noise reduction at various loading conditions

A novel concept of utilizing distributed surface compliance to achieve airfoil tonal noise reduction at various loading conditions is proposed. The aeroacoustics of airfoil configuration subjected to different loading conditions at angles of attack (AoAs) from 3° to 7° are numerically studied using high-fidelity two-dimensional direct aeroacoustic simulation at Reynolds and Mach numbers of 5×104 and 0.4, respectively. Initially, airfoil configurations mounted with single elastic panel (SEP) at individual AoA are designed with the knowledge of respective rigid airfoil flow characteristics. Stemming from the analysis of noise reduction potential of SEP configurations using a reduced-order modeling approach, a distributed surface compliance (DSC) airfoil configuration utilizing three resonating panels is designed to attain airfoil tonal noise reduction over entire range of AoA. Comprehensive acoustic analyses establish that the DSC airfoil could provide a maximum noise reduction ranging from 3 to 7 dB without any sacrifice in airfoil aerodynamics. The extent of noise reduction with DSC airfoil is found dependent on the flow-induced modal responses of the panels. At lower AoA, the panel(s) resonate in their designed structural modes, which remarkably weaken the flow instabilities convecting over the airfoil suction surface and eventually airfoil noise radiation. At higher AoA, the panel responses deviate from their designed structural mode shapes but could still give less noise reduction. Therefore, the designed DSC airfoil shows a feasible concept for tonal noise reduction over a wide range of operational AoA, which substantiates its applicability for aerodynamic devices at low Reynolds numbers.

Finite Cross-Section Method for Mode Shape Recognition of Highly Coupled Beam-Type Structures

Abstract The mode shapes of beam-type structures, such as aircraft wings and wind turbine blades, involve a high degree of coupling between bending and torsional deformations. In the case of wind turbine blades, different types of deformation are typically easily recognized by visual observation. However, this visual approach is sometimes challenging for high-order mode shapes, which involve complex coupling of both bending and torsional deformations. This work proposes a novel mode shape recognition algorithm, called the finite cross-section method (FCSM), for application to highly coupled beam-type structures not only to identify the deformation components of complex beam mode shapes but also more importantly to quantify their respective relative contribution. In the application case study for the FCSM, the entire structure is discretized into multiple cross sections. The flap-wise, edge-wise, and torsional deformation components of the entire structure are determined at the cross-section level. The deformation components of the entire structure and their respective contribution are obtained from assembling all cross sections. To validate the mode shape recognition performance, FCSM is applied to and demonstrated on four test cases: (1) numerical mode shapes of a simple cantilever beam, (2) numerical mode shapes from a straight wind turbine blade, (3) numerical mode shapes of a swept wind turbine blade, and (4) experimental mode shapes from a high spatial resolution 3D scanning laser Doppler vibrometer (SLDV) modal test. Both numerical and experimental studies demonstrate that FCSM can successfully recognize the quantitative contribution of flap-wise, edge-wise, and torsional deformation.

Influence of wind turbine design parameters on linearized physics-based models in OpenFAST

Abstract. While most physics involved in wind energy are nonlinear, linearization of the underlying nonlinear wind system equations is often important for understanding the system response and exploiting well-established methods and tools for analyzing linear systems. Linearized models are important for eigenanalysis (to derive structural natural frequencies, damping ratios, and mode shapes), controls design (based on linear state-space models), etc. In controls co-design, wherein methods often rely on linearized time-domain models of the physics, the physical structure (often called the plant) and controller are designed and optimized concurrently, so it is important to understand how changes to the physical design affect the linearized system. This work summarizes efforts done to understand the impact of design parameter variations in the physical system (e.g., mass, stiffness, geometry, and aerodynamic and hydrodynamic coefficients) on the linearized system using OpenFAST.

Output-only modal analysis of the Humen Bridge from video measurement

Abstract Video-based vibration testing is a promising non-contact method for structural health monitoring. This article develops an innovative methodology for estimating the structural modal parameters from videos without recovering the responses of structures. Firstly, an image processing (i.e., Gaussian smoothing) are adopted to smooth frames and alleviate the noises. Secondly, several regions of interest (ROIs) contained the projection area of structure are selected, and edge detection is performed to that to reduce the size of vision data. Then under some assumptions, the reduced vision data can be subjected to the frequency domain decomposition (FDD) similar to the structural modal analysis to identify the modal parameters of structure. Moreover, the structural mode shapes are recovering from such decomposition. The feasibility and accuracy of the proposed method were validated by a field experiment on the Humen Bridge in Guangzhou, China.

Bridge Flexibility Identification through a Reference-Free Substructuring Integration Method Driven by Mode Fitting

Flexibility plays an important role in bridge load-carrying capacity assessment. The flexibility of bridges has been extracted from the measured data of numerical impact testing and in laboratory experiments. Recently, substructuring impact testing has been increasingly investigated and developed because it avoids deploying sensors throughout bridges while obtaining the same results as full-structure testing. However, most substructuring testing methods require overlapping transition areas between the divided substructures, thereby affecting the operability and efficiency of the test. This study proposes a novel substructuring flexibility integration method that uses limited instruments to sequentially test different substructures subdivided from the whole girder bridge structure without requiring the overlapping reference points of adjacent substructures. Using the basis that the structural mode shapes satisfy the orthogonality and mode node theorem, the parameters of each substructure can be efficiently integrated and the entire flexibility matrix can be identified from the sparse flexibility matrices of the substructures. Case studies using experimental data from a cantilever beam and a continuous girder bridge are considered to verify the availability and effectiveness of the proposed method.

Employing adaptive learning rate method for analysis and efficient approach for predicting low-velocity impact performance in composite curved open-type shell

Dynamic stability analysis of the sandwich open-type shell on the viscoelastic substrate is presented. The structure is made of one graphene nanoplatelets layer and one piezoelectric layer. To simulate the force pertained to contact between the current impactor and structure, a contact model called Hertz is provided. For obtaining the boundary conditions (BCs) in addition to equations of motion associated with the sandwich open-type shell, first-order shear deformation theory and Hamilton’s principle are utilized to achieve the equations of motion. Then, Galerkin, as well as Newmark approaches, are coupled to solve the equations of the open sandwich-type curved shell subjected to a low-velocity impact. Also, a model employing an adaptively tuned deep neural network (DNN) is developed on the basis of the data of the problem to extract the results. The formulation and solution procedure are verified utilizing other papers. The outcome step has been presented the effect of the impactor’s initial velocity in addition to density on an indentation, absorbed energy, contact force, and indenter speed have been analyzed in detail. Last but not the list, the influence of the impactor’s initial speed and radius on the mode shape of the current structure will be discussed in detail.

High-frequency voltage-driven vibrations in dielectric elastomer membranes

This article deals with modelling and experimental characterisation of the continuum dynamic response of dielectric elastomer actuators (DEAs) subject to high-frequency voltage excitation.DEAs are capable of large deformations in response to an electrostatic stimulus, and they show large operating bandwidths of up to a few kilohertz. Although DEA systems normally make use of simple deformation patterns and a well-defined main actuation mode, they show complex structural dynamics and mode shapes if subject to high frequency voltage inputs. Taking advantage of these complex structural dynamics potentially allows developing multi-function actuators, audio devices and vibration isolators capable to perform different tasks using different vibration regimes.We first present a multi-domain model for the structural dynamics of DEAs, accounting for the contribution of the air pressure loads generated the DEA membrane vibrations, which play a relevant role in the DEA dynamics.We then present an extensive experimental characterisation of DEA samples’ structural dynamics based on laser Doppler vibrometer measurements. In particular, we measure the complex structural mode shapes and the velocity spectra generated through a broadband excitation in correspondence of a wide set of different design and control parameters (namely, the DEA geometric layout, the mechanical-preload and the applied voltage bias).Based on the experimental results, we validate the proposed continuum model, and we demonstrate that the forced response of the DEA can be efficiently described using a lumped-parameter computationally efficient reduced reformulation.