Flexural Mode Shapes Research Articles

Higher-order shear deformation theory for accurate prediction of vibration behavior of thick piezoelectric disks and design of efficient surface electrodes

A modified higher-order shear deformation plate theory (MHSDT) is proposed to predict the resonance frequencies and related mode shapes of piezoelectric disks of various thicknesses. The theory explicitly accounts for the effects of the electrical potential on the displacement of the disk. The equations of motion for the disk are derived using Hamilton's principle in conjunction with a variational approach. It is shown analytically that the distribution of the electrical field on the surface of the disk is quantitatively similar to the distribution of the sum of the total stresses within the disk. Thus, the deformation of the disk can be approximated directly from the stress distribution on the disk, thereby greatly simplifying the solution process. The validity of the proposed MHSDT is confirmed by comparing the solutions for the natural frequencies, displacement fields, and electric field distributions of the disk for various radius-to-thickness ratios and boundary conditions with those obtained using Reddy’s third-order plate theory and COMSOL finite element simulations, respectively. The results show that the solutions obtained from the MHSDT method are consistently closer to the exact solutions than those determined using Reddy’s third-order method. Finally, the linear relationship between the electric field and the flexural displacement of the disk is leveraged to design surface electrodes capable of exciting specific flexural mode shapes of the disk under a free boundary condition. The efficiency of the designed electrodes is experimentally demonstrated using an amplitude-fluctuation electronic speckle pattern interferometer (AF-ESPI).

Progressive deterioration of reinforced concrete structures: effect on dynamic properties of flexural members

PurposeThe main purpose of this study is to examine the damage behavior of flexural members under different loading conditions. The finite element model is proposed for the prediction of modal parameters, damage assessment and damage detection of flexural members. Moreover, the analysis of flexural members has been done for the sensor arrangement to accurately predict the damage parameters without the laborious work of experimentation in the laboratory.Design/methodology/approachBeam-like structures are structures that are subjected to flexural loadings that are involved in almost every type of civil engineering construction like buildings, bridges, etc. Experimental Modal Analysis (EMA) is a popular technique to detect damages in structures without requiring tough and complex methods. Experimental work conducted in this study concludes that a structure experiences high changes in modal properties once when cracking occurs and then at the stage where cracks start at the critical neutral axis. Moreover, among the various modal parameters of the flexural members, natural frequency and mode shapes are the viable parameters for the damage detection.FindingsFor torsional mode, drop in natural frequency is high for higher damages as compared to low levels. This is because of the opening and closing of cracks in modal testing. When damage occurs in the structure, there is a reduction in the magnitude of the FRF plot. The measure of this drop can also lead to damage assessment in addition to damage detection. The natural frequency of the system is the most reliable modal parameter in detecting damages. However, for damage localization, the next step after damage assessment, mode shapes can be more helpful as compared to all other parameters.Originality/valueEffect on Dynamic Properties of Flexural Members during the Progressive Deterioration of Reinforced Concrete Structures is studied.

The Dynamic Behaviour of Symmetrical Laminated Nano-composite Containing Equal Numbers of Glass and Carbon Fibre Layers

Fibre-reinforced polymer composite has many uses in structural components that required high strength, stiffness, and damping capacity. Cross and quasi-laminated epoxy composites with and without nano Al2O3 were used in this investigation to determine flexural modulus, natural frequency, damping ratio, and mode shapes by using analytical, experimental, and numerical (ANSYS) methods. It was demonstrated that adding 2 % nano Al2O3 improved the flexural modulus and the damping ratio while decreased the natural frequency. Cross number 2 and quasi number 2 had the highest natural frequency for cross and quasi laminate groups which are equal to 23.5 Hz and 20.25 Hz experimentally, respectively. On the other hand, the higher damping ratio was achieved for cross number 1 with nano Al2O3 and quasi number 2 with nano Al2O3 for both cross and quasi laminates, which are equal to 0.707 % and 0.693 %, respectively. The flexural modulus and damping ratio are inversely related to each other. However, the novelty in this article is that by adding two glass plies at the outer surface of quasi group laminate the flexural modulus, natural frequency, and damping ratio are increased simultaneously, as in the configurations quasi number 2 and quasi number 2 with nano Al2O3 in comparison with quasi number 1 and quasi number 1 with nano Al2O3.

Identificação dinâmica de uma passarela estaiada utilizando um sistema de aquisição de dados de baixo custo

Abstract This study describes the dynamic identification of a cable-stayed footbridge using an alternative data acquisition system and output-only modal identification methods. Data were collected during dynamic tests performed using an acquisition system based on the Arduino platform, consisting of low-cost devices with on-board Micro-Electro-Mechanical System (MEMS) accelerometers. The Peak-Picking (PP) method was used for the stay cables. The Frequency Domain Decomposition (FDD) and Stochastic Subspace Identification - Unweighted Principal Components (SSI-UPC) methods were used for the complete structure. The proposed data acquisition system efficiently recorded the time series required for Operational Modal Analysis and the acceleration acquisition process provided stable results. At least four mode shapes were identified in all tests. A minimum of four high energy peaks in the 0 - 24 Hz frequency range of the spectrum were obtained by the acquisition system in all of the cable tests and selected by the method. In the complete structure tests, the low-cost data acquisition system and the identification methods provided the first four flexural mode shapes, within the 0 - 9 Hz frequency range. Results for the frequency domain method showed a maximum difference of 2.37% in the first experimental frequency when compared to a 3D finite element numerical model, while in the other frequencies the difference was between 1 and 2%. For the time domain method, the maximum difference was 1.74% in the fourth frequency, with differences of between 0.1 and 0.7% in the other frequencies. The mode shapes were evaluated using the Modal Assurance Criterion (MAC) index, and the results varied between 0.8513 and 0.9990.

Imaging vibrations of electromechanical few layer graphene resonators with a moving vacuum enclosure

Imaging the vibrations of nanomechanical resonators means measuring their flexural mode shapes from the dependence of their frequency response on in-plane position. Applied to two-dimensional resonators, this technique provides a wealth of information on the mechanical properties of atomically-thin membranes. We present a simple and robust system to image the vibrations of few layer graphene (FLG) resonators at room temperature and in vacuum with an in-plane displacement precision of ≈0.20 μm. It consists of a sturdy vacuum enclosure mounted on a three-axis micropositioning stage and designed for free space optical measurements of vibrations. The system is equipped with ultraflexible radio frequency transmission lines to electrically actuate resonators. With it we characterize the lowest frequency mode of FLG resonators by measuring its frequency response as a function of position on the membrane and by extracting its effective mass. We use the background noise of the undriven vibrational spectrum to calibrate in-plane displacement. Finally, we measure the first three vibration modes of a resonator whose membrane is partially folded and find that folds locally suppress vibrations.

Nonlinear vibration, stability, and bifurcation analysis of axially moving and spinning cylindrical shells

The nonlinear vibration characteristics of the rotating axially moving circular cylindrical shells in subharmonic regions are investigated in the present paper. The motion equations are carried out based on the Hamilton principle in cylindrical coordinates utilizing Donnell’s nonlinear shell theory. By introducing the suitable airy stress function, three equilibrium equations in the cylindrical coordinates are simplified into two nonlinear coupled nonhomogeneous PDEs, including a compatibility equation and the transverse motion equation. The compatibility equation solution is obtained employing the seven degrees of freedom for the flexural mode shape of the system. By implementation of the Galerkin method, the motion equation would be projected into seven nonlinear coupled nonhomogeneous ODEs. This set of equations is solved using a direct normal form method validated by the numerical method and available data. The effect of angular velocity and axial speed is investigated employing frequency and force response curves, bifurcation diagrams, time history, and the system’s phase portraits. Always axially moving and rotation speed of the system intensifies the nonlinear behavior of the frequency responses.

Study of flexural vibration of variable cross-section box-girder bridges with corrugated steel webs

In this study, the flexural vibration behavior of the variable cross-section box-girder bridge with corrugated steel webs is investigated using the energy variational method in conjunction with the extended spline finite point method. Detailed equations are provided for analyzing the dynamic characteristics of simply supported girders with flexural vibration. To predict the dynamic characteristics of continuous girders with flexural vibration, the method is further expanded to incorporate the spline subdomain method. To accurately analyze the flexural frequencies and mode shapes, the stress characteristics of the corrugated steel web and the shear lag effect of the box girder are also considered. The dynamic characteristics of girders with different variable cross sections are efficiently calculated and compared by the compiled calculation program. The predicted results are in good agreement with the experimental and finite element modeling results, which proves the validity and applicability of the proposed method. The results of a parameters analysis show that increases in the variable cross-section coefficient of the girder results in increases in the flexural frequency, and lead to a change in the shape of the first mode from ‘sinusoidal’ to ‘bell.’ In addition, increases in the width-to-span ratio of the girder rapidly decrease the natural frequency, which then gradually stabilizes.

Structural vibration measurement with multiple synchronous cameras

This paper presents a simulation and experimental study on the measurement with multiple, i.e. more than 2, synchronous cameras of the flexural deflection shape of a beam excited by a transverse force at the fundamental resonance frequency, which gives a good estimate of the first flexural natural mode shape. The first part of the paper presents the simplified optical model employed to simulate how the accuracy of the measurements varies with respect to: a) the distance of the cameras from the structure; b) the opening angle between pairs of cameras, which is analysed with respect to the azimuthal and elevation spherical coordinates, c) the resolution of cameras and d) the number of cameras. The second part of the paper provides experimental results taken on a beam rig and camera setup assembled using off-the-shelf devices. The study shows that the accuracy of the measurements significantly rises as the setup is equipped with an increasingly larger number of cameras, which should be arranged close to the vibrating structure with large opening angles between pairs of cameras characterized by large azimuthal and small elevation spherical coordinates. For instance, the average error of the measured flexural deflection shape falls from 15% to 3% when a setup with cameras arranged along a circumference oriented parallel the beam surface, with a small offset from the beam, passes from 2 cameras to 12 cameras.

Three-Dimensional free vibration analysis of composite FGM rectangular plates with in-plane heterogeneity: An EKM solution

A three-dimensional elasticity based approximate analytical solution is proposed for free vibration analysis of composite functionally graded Levy-type rectangular plates having an in-plane gradation of material properties. The density and elastic properties of the composite plates are considered to vary linearly along the x-direction. Modified Hamilton’s principle has been applied to derive the weak-form of governing equations in which all stresses and displacements consider as primary variables to ensure the exact point-wise satisfaction of all interlaminar continuity and edge support conditions. The extended Kantorovich method, in conjunction with the Fourier series and power series approach, has been used to obtain the approximate solution in analytical form. New benchmark numerical results are presented for a single-layered and multi-layered in-plane functionally graded rectangular plates. The influence of the in-plane heterogeneity, thickness ratio, and support conditions on the flexural frequencies and mode shapes of the plate are presented in detail. The present study shows that the free vibration response of the rectangular plate is affected significantly by the in-plane gradation of material properties, but the extent of influence largely depends on edge support conditions of the plate as well as on configuration. The present benchmark solution can be used to validate other 3D numerical approaches and two-dimensional (2D) plate theories.

Reliability of the Tensile Force Identification in Ancient Tie-Rods Using One Flexural Mode Shape

ABSTRACTThis article presents an investigation into the reliability of an experimental procedure for the axial load identification of tie-rods with unknown boundary conditions, that uses one vibration frequency and five amplitudes of the corresponding mode shape. The method adopted does not require the knowledge of the effective length of the tie-rod under examination, but only the flexural rigidity and the mass per unit length. In particular, the influence of measurement errors as well as of inaccurate estimates of geometric and elastic properties on the accuracy of the axial force identification is investigated. The effect of the added mass of the sensors is also analyzed. The proposed algorithm is verified by means of many numerical tests on ancient tie-rods having large scatter values of geometric and elastic properties. Good estimates of the axial forces are obtained. Moreover, it is shown that the reliability of the tensile force identification, using one flexural mode shape, relies on the magnitude of the measurement errors rather than on accurate guess of Young modulus.

A Comparison of the Vibrational Responses of Four Different Uniform and Tapered Composite Beams

A combined experimental and finite element approach was used to compare the vibrational response of the composite beams with different configurations of internal taper and uniform cross sections. Composite beams were made using the hand lay-up process. Glass fibre was used as reinforcement in the form of a bidirectional fabric and a general-purpose epoxy resin as the matrix for the composite material. Experimental dynamic tests were carried out on FFT analyzer using specimens with different taper configurations. From the results, the effect of different taper configurations on the flexural natural frequencies and mode shapes was investigated. Finite element analysis of the same problem was carried out on software package ANSYS and the results were validated using the experimental results. From the results, a thorough comparison was made between the natural frequencies, loss factors, displacements and mode shapes of the specimens with different taper configurations and uniform cross sections considered. The aerospace implications are provided.

Experimental Investigation on Flexural Response of Hybrid Steel Plate Girder

Flexural tests were conducted on five full scale I-shaped steel beams, three hybrid and two homogeneous, with two aims: 1) validate the inelastic flexural behavior of hybrid girder that is manufactured using lower grade steel 2) simplify finite element (FE) analysis procedure, for further parametric studies, by neglecting residual stresses when symmetrical welding sequence is adopted. Effect of material properties and member slenderness on flexural strength, ductility, and failure mode shapes is discussed. It is also found that failure mode shape and expansion of plastic field are mainly influenced by slenderness of member. Load-deflection and moment rotation response in tests are closely matching with non-linear finite element study. Close match is also observed in failure mode shapes and plastic field expansion. Considerable reduction is observed in residual stresses in symmetrically welded sections as compared to sections welded without proper sequence. This validates non-linear FE analysis procedure by neglecting residual stresses. Test results are compared with prediction equations for flexural strength, rotation capacity available in literature. Test results are also compared with provisions in EN 1993-1-1 and IS 800. Moment gradient (3-P) and uniform moment (4-P) loading have been considered to ensure that results are equally applicable in both cases.

Flexural Vibration of Prestressed Concrete Bridges with Corrugated Steel Webs

Prestressed concrete bridges with corrugated steel webs have emerged as a new form of steel-concrete composite bridges with remarkable advantages compared with the traditional ones. However, the assumption that plane sections remain plane may no longer be valid for such bridges due to the different behavior of the constituents. The sandwich beam theory is extended to predict the flexural vibration behavior of this type of bridges considering the presence of diaphragms, external prestressing tendons and interaction between the web shear deformation and flange local bending. To this end, a [Formula: see text] beam finite element is formulated. The proposed theory and finite element model are verified both numerically and experimentally. A comparison between the analyses based on the sandwich beam model and on the classical Euler–Bernoulli and Timoshenko models reveals the following findings. First of all, the extended sandwich beam model is applicable to the flexural vibration analysis of the bridges considered. By letting [Formula: see text] denote the square root of the ratio of equivalent shear rigidity to the flange local flexural rigidity, and L the span length, the combined parameter [Formula: see text] appears to be more suitable for considering the diaphragm effect and the interaction between the shear deformation and flange local bending. The diaphragms have significant effect on the flexural natural frequencies and mode shapes only when the [Formula: see text] value of the bridge falls below a certain limit. For a bridge with an [Formula: see text] value over a certain limit, the flexural natural frequencies and mode shapes obtained from the sandwich beam model and the classical Euler–Bernoulli and Timoshenko models tend to be the same. In such cases, either of the classical beam theories may be used.

Free flexural vibration of geometrically imperfect functionally graded microbeams

Free flexural vibration characteristics of functionally graded (FG) microbeams with geometric imperfection are explored numerically, taking into account the size effect phenomenon based on modified couple stress theory. This theory employs only one material length scale parameter to interpret the size-dependent mechanical behavior of microstructures. The mechanical and physical properties of FG microbeam are assumed to vary smoothly and continuously through the thickness direction according to a power-law distribution. Hamilton's principle in conjunction with Euler-Bernoulli beam theory is used to establish the coupled longitudinal-transverse equations of motion and associated boundary conditions. A weighted-residual method is utilized to evaluate the size-dependent free flexural vibration behavior of FG microbeams with clamped-clamped, clamped-pinned, and pinned-pinned boundary conditions. The influences of different dimensionless parameters i.e., maximum imperfection amplitude-to-length ratio, length-to-thickness ratio, flexural rigidity ratio, and power-law index on the flexural frequencies and mode shapes of FG microbeams are investigated. The mode veering phenomenon is also explored. Finally, the role of longitudinal displacement in the free flexural vibration of the geometrically imperfect FG microbeams is examined.

Dynamic analysis of maritime gasbag-type floating bridge subjected to moving loads

This paper studied the dynamic response of a new gasbag-type floating bridge under the effect of a moving load. The arbitrary Lagrangian-Eulerian (ALE) method was used to simulate the movement of seawater and air, and the penalty-based method was used to study the coupling between gasbags and fluid. A three-dimensional finite element model of the floating bridge was established, and the numerical model was verified by comparing with the experimental results. In order to prevent resonance, the natural frequencies and flexural mode shapes were analyzed. Based on the initial state analysis, the dynamic responses of the floating bridge subjected to different moving loads were investigated. Vertical displacements and radial deformations of gasbags under different loads were compared, and principal stress distributions of gasbags were researched while driving. The hinge forces between adjacent modules were calculated to ensure the connection strength. Besides, the floating bridge under wave impacting was analyzed. Those results can provide references for the analysis and design of this new floating bridge.

Free vibration analysis of functionally graded beams using an exact plane elasticity approach

Exact natural frequencies of functionally graded beams are determined using plane elasticity theory. The analysis yields infinitely many frequencies. For verification purposes, a comparison with the existing beam theory results is performed and a close agreement is observed for slender members. The elasticity solutions are general in the sense that they are valid for slender members as well as short and thick structural elements. Both flexural and axial free vibration mode shapes are presented for top and bottom surfaces and the effect of the beam thickness is discussed. The exact results presented herein can be used as benchmarks for future research of free vibration behavior of short and thick functionally graded material beams.

A novel differential frequency micro-gyroscope

We present a frequency-domain method to measure angular speeds using electrostatic micro-electro-mechanical system actuators. Towards this end, we study a single-axis gyroscope made of a micro-cantilever and a proof-mass coupled to two fixed electrodes. The gyroscope possesses two orthogonal axes of symmetry and identical flexural mode shapes along these axes. We develop the equations of motion describing the coupled bending modes in the presence of electrostatic and Coriolis forces. Furthermore, we derive a consistent closed-form higher-order expression for the natural frequencies of the coupled flexural modes. The closed-form expression is verified by comparing its results to those obtained from numerical integration of the equations of motion. We find that rotations around the beam axis couple each pair of identical bending modes to produce a pair of global modes. They also split their common natural frequency into a pair of closely spaced natural frequencies. We propose the use of the difference between this pair of frequencies, which is linearly proportional to the speed of rotation around the beam axis, as a detector for the angular speed.

Estimate of the axial force in slender beams with unknown boundary conditions using one flexural mode shape

This paper presents an experimental procedure for the axial load identification of slender prismatic beams with unknown boundary conditions by making use of one vibration frequency and of five amplitudes of the corresponding mode shape. In fact, this method does not require the knowledge of the effective length of the beam under examination, but only the flexural rigidity and mass per unit length. The proposed algorithm was verified by means of many numerical and experimental tests on tie-rods having different boundary conditions. Excellent estimates of the axial forces were obtained. Finally, the influence of the location of the instrumented sections on the estimation of the axial load was analyzed using a simply supported beam model.

Importance of flexural mode shapes in dynamic analysis of high-speed trains traveling on bridges

The effects of railway car-body flexibility on the dynamic analysis of high-speed trains traveling on bridges are studied. The flexible car-body is modeled as a uniform beam supported by the primary and secondary suspensions. A parametric sensitivity study is carried out to examine the effects of different parameters, namely the track irregularity, rail joint, traveling speed and the wheel flat, on the dynamic responses of the car-body and bridge. The rail surface roughness is regenerated by its power spectral density. Different types of rail joint geometries and wheel imperfections are mathematically modeled and included in the numerical simulation. It is found that the flexural mode shapes of the body structure can remarkably affect the calculated ride comfort index especially in the low frequency range.

A New Solution Method for Vibration Analysis of Circular Cylindrical Thin Shells with a Circumferential Surface Crack

In this paper, a new solution method is proposed for determining the natural frequency of a given mode for a finite-length circular cylindrical thin shell with a circumferential part-through crack. The governing equation of the cracked cylindrical shell is derived by integrating the line-spring model with the classical thin shell theory. The proposed method calculates the natural frequency from an initial trial to satisfy both the governing equations and appropriate boundary conditions through an optimization process. The initial trial is proposed to satisfy the governing equations by using the beam modal function to determine the modal wavenumbers and mode shapes of cylindrical shells in the axial direction, assuming the flexural mode shapes of cylindrical shells in the axial direction to be of the same form as that of a flexural vibration beam with the same boundary conditions. Four representative sets of boundary conditions are considered: simply supported (SS-SS), clamped-clamped (C-C), clamped-simply supported (C-SS), and clamped-free (C-F). Compared with the finite element (FE) method, the proposed solution method is verified to provide an accurate and efficient way to calculate the dynamic characteristics of both intact and cracked cylindrical shells.