Mode Shapes Of Beams Research Articles

Experiential learning with online vibro-acoustic demonstrators

After 25 years of delivering lecture-based learning at a university with mixed success I now supplement my short courses with online experiential learning demonstrators for visualizing and interacting with vibro-acoustic systems. The demonstrators use simple javascript coding and calls to the plotly free libraries. No special software is required (but stronger computers obviously perform faster). Learners can quickly adjust parameters like stiffnesses, masses, and damping using sliders and immediately visualize changes to vibration response and sound fields. Students may exercise demonstrators for one- and two-dimensional acoustic waves; vibrations and mode shapes of simple oscillators, beams, and plates; and sound radiated by and transmitted through panels. Some of the demonstrators are freely available at hambricacoustics.com (others are only available to my short course students).

Free vibration analysis of Euler-Bernoulli beams modeled by spatial-fractional differential equation

In this article, a more accurate dynamic model of Euler-Bernoulli beam is proposed, based on the spatial-fractional derivatives. Firstly, the governing differential equation is derived. After that, this equation is solved by two different methods: the first method uses only the left fractional derivative, while the second method tends to use both the left and right fractional derivatives, simultaneously. In the second method, the location of the "switching point" which connects the left and right derivatives is treated as a variable parameter. The effect of this parameter is investigated by analyzing several beams. As numerical results, the natural frequencies and mode shapes of beams with different boundary conditions are presented.

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.

A comparative analysis of the vibrational behavior of various beam models with different foundation designs

This article discusses the modal behavior of elastically constrained beams under various types of foundations and provides insights into the effects of different factors on the eigenfrequencies of beams. Numerical and analytical techniques, specifically the Galerkin finite element method (GFM) and the separation of variables, are utilized to determine the eigenfrequencies and mode shapes of beams. Modal analysis of Timoshenko, shear, Rayleigh, and Euler-Bernoulli beams that are elastically constrained and resting on Winkler, Pasternak, and Hetényi foundations, considering non-classical boundary conditions, is included in the study. The effects of factors such as flexural rigidity, transverse modulus, and Winkler foundation constant on natural frequencies of different beam models are investigated. The proposed method efficiently converges to the exact solution without shear locking in the stiffness element. The results demonstrate that the natural frequencies of the beam rise because of the shear layer, flexural rigidity, and foundation constant. Furthermore, the Hetényi elastic foundation affects the natural frequency of the beam, depending on the relative values of beam stiffness and foundation stiffness. Additionally, incorporating both shear deformation and rotary inertia has a greater impact on the eigenfrequencies of Euler-Bernoulli beams compared to incorporating only one of these effects. The findings of this work provide valuable insights into the behavior of beams under different foundation conditions and have potential applications in the design and optimization of structures incorporating beams, thereby enhancing the understanding of beam analysis.

Exact closed-form dynamic stiffness matrix of damaged frames comprising Timoshenko–Ehrenfest beams

The present work proposes the extension of an exact distributional model for cracked beams to include the local influence on axial deformation on shear deformable beams. Axial, shear and flexural concentrated flexibilities are taken into account to model the damaged sections. Shear deformability is accounted for by adopting the Timoshenko–Ehrenfest model and exact closed-form solution of the mode shapes for multiple cracked beams are derived. On this ground, the final achievement of the work consists in the original formulation of the Dynamic Stiffness Matrix (DSM) with multiple cracked sections in exact closed form. The closed-form expression of the DSM allows the assemblage of the global DSM of a structure to conduct the free vibration analysis of damaged framed structures as well as for the study of frames with semi-rigid joints. This approach has the advantage to avoid any numerical procedure for the construction of the DSM as well as the introduction of further degrees of freedom (dofs) in correspondence of the damaged sections, keeping the size of the problem the same as in the intact configuration. Natural frequencies of the frames are determined by means of the Wittrick–Williams algorithm (W–W), then the mode shapes and modal forces are subsequently obtained in closed form. The proposed approach is validated with results available in literature for two portal frames characterized by the presence of cracks and semi-rigid joints, respectively. Finally, the potentiality of the approach, also in view of practical applications, is demonstrated analysing a multi-storey frame with semi-rigid connections.

Deep learning enhanced metamodel design based on reduced mode shapes for delamination identification in composite structures

Structures requiring robust maintenance and accessibility pose challenges for structural integrity monitoring. This monitoring, crucial in various engineering applications, enables corrective actions to ensure machinery and equipment safety. Laminated composite materials, prevalent in aeronautical structures, often exhibit imperceptible delaminations or cracks, demanding the use of artificial intelligence for damage characterization. This study employs a numerical-experimental methodology, integrating artificial neural networks, to identify delaminations. Delamination, a structural damage form, was intentionally induced in epoxy-fiberglass composite structures. Modal tests captured characteristics in damaged and undamaged structures, revealing artificial intelligence effectiveness in pinpointing damage location and size. Damage significantly altered beam mode shapes, generating complex modes. Induced delaminations led to a measurable 15% reduction in natural frequencies, and damping ratios increased by approximately 10% in damaged structures. Artificial neural networks achieved an impressive 92% accuracy in predicting delamination location and size. Quantitative analysis of mode shape alterations validated the proposed methodology’s efficacy. This study provides valuable quantitative insights into damage detection in laminated composite structures, highlighting AI's potential for enhanced structural health monitoring. The numerical–experimental approach, supported by quantitative results, emphasizes the importance of advanced techniques for precise and efficient damage characterization in critical engineering applications.

Vibration and sound radiation of acoustic black hole beams on Pasternak foundation by the Riccati transfer matrix method

This paper aims to study the vibration and sound radiation of the acoustic black hole (ABH) beams on Pasternak foundations. We focus on the influences of the foundations on the ABH effect because foundations are essential and common parts of mechanical systems. To this end, we derive the exact transfer matrix of the ABH beam on the viscoelastic Pasternak foundation based on the Timoshenko beam theory. Using the presented method, we explore the influences of Pasternak foundations on the natural frequencies, modal damping factors, and mode shapes of ABH beams. In addition, the sound radiation behavior of ABH beams on Pasternak foundations is also investigated. Several case studies are presented to demonstrate the effects of nonuniform foundations on wavenumber-domain properties and the vibroacoustic response of ABH beams. The results suggest that viscoelastic Pasternak foundations can greatly influence vibration characteristics and increase the modal damping factors due to the shear effect of the Pasternak foundation, which enriches the physical understanding of the ABH structures. Finally, the experimental results are used to validate the presented model.

Vibroacoustics Analysis of Plates with Bioinspired Surface Modifications

This study presents a simplified semi-analytical model to analyze the dynamic behavior of a plate with shark skin biomimetic surface modifications. The base plate is considered to be thin isotropic homogeneous and modeled using a Kirchhoff–Love plate theory, whereas the dermal denticles are modeled as a distributed array of point mass and replicated using a Dirac delta function. A semi-analytical formulation, which takes into account closed-form beam mode shapes, with the Galerkin technique to minimize errors is utilized to study the free and forced vibration response of modified plates. Harmonic analysis is carried out to understand the dynamic behavior of the modified plate both quantitatively as well as qualitatively. A parametric study varying the array spacing and the boundary conditions is carried out to understand the effect of surface modification on power radiation characteristics. It appears that adding dermal denticles reduces the peak velocity amplitude and peak radiated power at all resonance frequencies. A separate study by varying the base plate material reveals that a higher reduction in the dynamic response and the radiated power can be achieved by increasing the mass ratio. Hence, surface modification using such a shark skin layer may prove beneficial to reduce acoustic radiation and dynamic response and to optimize the design.

Natural Frequency Maximization of Beam with an Internal Sliding Joint of Translational Restraint

The natural frequency variation of a uniform Euler–Bernoulli beam with an internal sliding joint of the translational restraint is studied intensively such that the optimal design of the connecting point is pursued for maximization of a natural frequency of the system. First, the compatibility conditions at the linkage are presented in detail, and the dynamic characteristic equation of the beam system is accordingly derived. Next, the influences of both the translational restraint stiffness and the position of the sliding joint upon the natural frequency and mode shape are described comprehensively. Then, the optimal position and the corresponding minimum translational stiffness of the internal joint are explored exclusively for maximizing a natural frequency of the beam structure. In this context, the essential requirements for the optimization of the internal sliding joint are presented, respectively, in two different ways to reveal their mechanical implications. It turns out that the optimal joint location can be determined readily from the corresponding mode shape of the equivalent continuous beam. Afterward, numerical results are provided with two classical beams of a single internal sliding joint to demonstrate the variations of their dynamic characteristics as well as to offer significant physical insight into the vibratory behaviors of the beam system with maximization of a natural frequency of interest.

Analytical solution and free vibration analysis for beams with step changes in stiffness

PurposeThis paper aims to investigate analytical solutions of natural frequencies and mode shapes of Euler-Bernoulli beams with step changes in the stiffness.Design/methodology/approachIn this work, analytical solutions for a beam with a single discontinuity was performed. Subsequently, based on an effective matrix formulation, the closed-form expressions of the single discontinuity beam could be conveniently extended to stepped beams with multiple stiffness discontinuities.FindingsThe results of the study show that the natural frequency of the beam can be adjusted by the local stiffness variation, and step location plays a significant role in free vibration responses.Originality/valueThe effects of the stiffness of the segment and step location on the natural frequencies of the stepped beams under different boundary conditions were examined using the proposed analytical scheme. This study provides insights into the design of variable-stiffness beam structures with the capability to adjust natural frequencies.

Damage detection of wooden beams based on the modal strain energy change and evidence theory

A damage detection and localization method for wooden beams was proposed based on the modal strain energy (MSE) change and evidence fusion. The fused damage indicator was deduced using the first three mode shapes of the wooden beams before and after damage. The finite element modal analysis of the free-supported undamaged and damaged wooden beams with different damage severities at one or two locations was performed. The first three mode shapes were extracted from the corresponding modal analysis, with which the damage indicator based on the MSE change and the fused damage indicator of each damage case were computed. The simulation results show that the fused damage indicator accurately detected and located the damage with different severities at one or two locations. Finally, the modal test was completed using the same damage cases as the finite element simulation. The frequency functions of the whole beam were first obtained, with which the first three experimental mode shapes were then acquired, and the damage indicator based on the MSE change and the fused indicator were further computed. The test results verified the validity and reliability of the proposed damage indicator.

Dynamic response of beams excited by moving oscillators: Approximate analytical solutions for general boundary conditions

In this paper, the dynamic response of an Euler-Bernoulli beam with general boundary conditions (BCs) and subject to a moving oscillator is examined. Notably, novel approximate closed-form expressions are determined for the vertical responses of both the beam and the moving oscillator, specifically considering the effect of damping in these systems, commonly omitted in standard approaches in the literature. In this regard, a modal superposition procedure is adopted and combined with an appropriate expansion-based approach of the dynamic response of the system, which naturally arises considering the oscillator-beam mass ratio to be reasonably small. Further, general boundary conditions are treated exploiting the use of a suitable set of orthogonal polynomial functions as beam mode shapes. In this manner, novel direct expressions for the response of the system are derived, in which the mode shapes coefficients explicitly appear. This leads to a straightforward application of the proposed solution, irrespective of the chosen BCs. Several numerical examples are presented to assess the reliability and accuracy of the proposed approach, considering different cases of beam BCs, and moving oscillator’s parameters. Results are validated by comparison with the data of finite element analyses, and numerical solutions of the complete system of governing equations.

Accurate determination of the elastic moduli of optimized cantilever beams by efficient time-averaged ESPI system

Elastic moduli, including Young’s modulus, shear modulus, bulk modulus, etc, are key parameters that are used to characterize the ability of a solid material to resist various types of deformation. The moduli can be extracted from the natural frequencies of a cantilever beam. In this paper, the relationships between moduli and natural frequencies, for the first time, are quantified by the finite element method. The optimized three-dimensional proportion of the cantilever beam is selected to be implemented simple error compensation. Experimentally, to precisely obtain the natural frequencies of the cantilever beam, an efficient time-averaged electronic speckle pattern interferometry(ESPI) system has been developed. The efficiency and precision are reflected in the following aspects: firstly, according to the slender character of the cantilever beam, a large shear optical path arrangement is designed to facilitate isolation from environmental interference; secondly, a resonance search method, based on the moiré effect is employed to recognize the natural frequencies accurately and efficiently; thirdly, a novel dynamic phase-shifting method is proposed based on the arrangement of the large shear optical path for clearer visualization of the mode shape of the cantilever beam. The proposed methods are verified by three kinds of common materials. The results suggest that Young’s modulus and shear modulus derived from natural frequencies are higher than the known value, and the error compensation can significantly reduce the calculation error. Furthermore, the experiments carried out on the woven carbon fiber reinforced plastic laminates illustrate the potential of the proposed methods in the evaluation of elastic moduli of composites. Given that the exciter attached to the specimen surfaces can be replaced with some special counterparts, the proposed ESPI system has considerable potential to test the objects loaded in some extreme environments, e.g. at high temperatures or underwater, where contact detection methods are difficult to be implemented.

The role of graphene oxide in the strength and vibration characteristics of standard and high-grade cement concrete

This study presents the results of an experimental study of the influence of graphene oxide (GO) on the static and dynamic mechanical characteristics of cement concrete. Four different dosages of GO, 0.05%, 0.10%, 0.15% and 0.20% by weight of cement and two different grades of concrete, standard concrete (M30) and high strength concrete (M60) were studied. The static properties such as compressive strength and chord modulus of elasticity were determined. The impact hammer technique was used to obtain dynamic characteristics such as fundamental natural frequency, damping ratios, and mode shapes of concrete beams under free-free condition. In addition, fundamental resonant frequencies under different modes of vibration were determined. The dynamic elasticity modulus, dynamic shear modulus and dynamic poisons ratio of all concrete mixes were also determined. Experimental results revealed that the compressive strength of the concrete with the incorporation of GO was remarkably enhanced. The highest increase in compressive strength was found at GO dosage of 0.15%, and the improvement at 7 and 28 days was 58.2% and 47.7% for standard concrete and 51.2% and 24.3% for high strength concrete respectively compared to control concrete. The impact hammer test revealed that the maximum increase in fundamental natural frequency was 8.0% and 6.6%, and the maximum decreased damping ratio was 29.1% and 26.3% for standard concrete and high strength concrete respectively when compared to the control concrete. Ultrasonic pulse velocity results indicated that the homogeneity of the concrete with GO addition was significantly improved. SEM and EDX microstructure investigations found that the addition of GO to concrete promoted the development of improved hydration phases leads to the dense microstructure. The findings of this study demonstrated that GO nanomaterial had a potential future as a reinforcing material in cement concrete for improved microstructure, strength, and free vibration characteristics.

New Formulations for Prediction of Buckling Loads in Steel Plate Girders Through Linear and Nonlinear Stability Analysis

Buckling analysis of steel structures can be performed using either linear (LBA) or nonlinear buckling analysis (NLBA). Each method is only effective in particular conditions and improper use might lead to inaccurate predictions. Despite the extensive use of these two methods, there is a lack of research on the accuracy of each method and their limitations regarding the range of application. In this paper, these two methods are evaluated and compared for steel plate girders with a wide range of geometry and the limitations of each method are presented. The numerical modeling of steel plate girders is developed using Abaqus software. The numerical models are verified with the existing experimental results in the literature. An extensive parametric study is conducted using 123 models to investigate the effect of web height, web thickness and beam length on the results. The buckling loads predicted from the numerical models using different analysis methods are compared with those determined based on the AISC 360-16 and EN 1993-1-5 formulations and recommendations are made. Results indicate that while the nonlinear buckling analysis provides accurate predictions of buckling load and mode shape of steel beams, the linear buckling analysis is accurate only for the case of elastic buckling mode without tension field action. A new formulation is proposed to modify the LBA results in order to obtain the advantages of both analysis methods. In addition, based on the NLBA results, a much simple yet accurate equation is proposed to predict the buckling load using plate geometry and material properties instead of the complicated code equations. The accuracy of the proposed equation in comparison with the code equations is also demonstrated for 500 plate girders selected with random geometry.

Dynamic Analysis of a Curved Beam With Tuning of Elastic Modulus and Mass Density in Circumferential Direction

Abstract Deformation and stress fields in a curved beam can be tailored by changing its mechanical properties such as the elastic modulus/mass density, which is typically done using functionally graded materials (FGM). Such functional gradation can be done, for instance, by using particles or fiber-reinforced materials with different volume fractions along the beam length. This article presents in-plane vibrations of functionally graded (FG) cantilevered curved beams. Both semi-analytical and finite element modeling are employed to find natural frequencies and mode shapes of such beams. The natural frequencies obtained from the analytical solution and finite element analysis are in close agreement with an error of 6.2% when the variance of material properties gradation is relatively small. In the analytical approach, the direct method is employed to derive the governing linear differential equations of motion. The natural frequencies and mode shapes are obtained using the Galerkin and the finite element methods. First, three natural frequencies and corresponding mode shapes are analyzed for different elastic modulus/mass density distribution functions. Furthermore, the natural frequencies of FG curved beams with a crack are also investigated. Our results indicate that larger cracks near the clamped side of the beam significantly decrease the first natural frequency. In the second and third vibration modes, cracks located in the area with a maximum moment result in the lowest natural frequency values. However, the second and third natural frequencies of the cracked curved beam are not affected by the presence of a crack, if the crack is located at the nodal points of the curved beam.

Analysis of free vibration of tapered cracked double nanobeams using Maclaurin series

In the study, the Maclaurin series technique is presented to analyse the vibration of cracked tapered double nanobeams. The equation of motion is derived from the Euler–Bernoulli beam theory based on the Hamiltonian principle and Eringen’s nonlocal theory of elasticity. The double-nanobeam system consists of two parallel nanobeams attached by a Winkler elastic layer. Both beams are identical and their widths vary along the x-axis. A single crack is considered at the upper beam of the system. The mechanical behaviour of cracked cross-sections is simulated by the local stiffness model. According to the model, the cracked double-beam system is divided into two intact segments. A numerical investigation is carried out to scrutinize the effects of nonlocal parameters, crack severity, taper ratio, and spring constant on the vibration of the double nanobeam. The results reveal that the effects of crack depth, crack location, nonlocal parameters, taper ratio, and spring constant influence the natural frequency and dynamic response of the system significantly. This study highlights that a crack at the upper beam influences the mode shape of the upper beam as well as the intact lower beam. Numerical results have been examined with the previously published works and found a good agreement with them.

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.

Vision-based vibration mode shape extraction for identifying cracks in a cantilever beam

Abstract Taking significant merits of high spatial resolution of image sensing, the vibration mode shape of a cantilever beam (CB) is obtained from the free vibration process for local crack identification and structural health monitoring. This vision-based vibration measurement uses a cost-effective rolling shutter camera and a computationally efficient singular value decomposition (SVD). It shows that vibration modes of a continuous structure such as a CB are mainly determined by spatial resolution and less rely on temporal resolution. Moreover, images recorded by the camera maintain the critical information about mode profiles though distortions are introduced into the image. Decomposing the images by SVD allows the vibration modes to be extracted with good degree of accuracy. Both simulation study and experimental verification shows that this novel approach allows different sizes of cracks and locations to be identified.

Improving performance of piezoelectric energy harvesterunder electrostatic actuation using cavity

This study aims to investigate the effect of cavity on electric energy harvesting from cantilever beam vibrations under electrostatic actuation. Electrostatic actuation is created by a layer of radioisotope materials that is placed on the opposite side of the beam emitting electrons. When the beam is charged, the electrostatic force is generated between the beam and the opposite plate and pulls the beam towards itself. After the beam strikes the radioisotope, it is electrically discharged and then released. The piezoelectric layer converts the released microbeam vibration into electricity. The equations of motion coupled with the electrical effects of the piezoelectric layer are extracted using Hamilton's principle and Gauss's law. The equations are discretized by Galerkin method. The exact mode shape of the cantilever beam with the piezoelectric layer is employed as the comparison function. By identifying the relations governing the system, the output voltage and consequently the amount of harvested electrical energy are obtained using various parameters such as thickness and position of the cavity and system electrical resistance. The results indicates that creating cavity has a significant effect on the energy harvesting.