Free-free Beam Research Articles

A viscoelastic harvester-absorber for energy generation and vibration control

Abstract A vibration energy harvester harnesses the oscillatory movement and
converts it into electrical energy. On the other hand, a (dynamic) vibration absorber
dissipates the mechanical energy of a host structure (primary system). The proposal for
a harvester-absorber device (HAD), comprinsing a piezoelectric element (conventional
harvester) bonded with a viscoelastic layer and a steel constrained layer, is based on the
dual purpose of generating energy and at the same time reducing the vibratory response
of the primary system. Introducing a viscoelastic dissipative layer in a conventional
harvester, also offers the advantage of simultaneously reducing wear on the piezoelectric
element, thereby extending its lifespan and preventing fatigue-related breakage. The
paper presents a model of a harvester-absorber built from a bimorph piezoelectric
beam, a layer of viscoelastic material (E-A-R C1002-01PSA) and a beam of steel, which
acts as a constrained layer. A lumped parameter model is proposed for the theoretical
description of the HAD which is successfully used to experimentally determine its
parameters. After that, and using the equivalent stiffness concept, the HAD was
coupled to a free-free beam with two masses at its ends to study its perfomance over a
real host structure. The inertance of the composite system and voltage/force frequency
response function of the HAD were obtained and compared with experimental results.
The accuracy of the results justifies and validate the model which has the advantage
of simplicity and can contribute to reducing wear and increasing the lifespan of brittle
piezoelectric structures.

Uniformly Loaded Logarithmic Beam Mode with Spatially Varying Flexural Rigidity

AbstractThis analysis explores natural leading modes represented by logarithmic functions, achieved by imposing four boundary constraints at the ends of an elastic inhomogeneous beam. The beam possessing constant material inertia, is assumed to be uniformly loaded, and is composed of material with variable stiffness. It is sought analytical expressions for beam deflections in terms of logarithmic functions. Our findings demonstrate that such formulae can be derived for a beam under axially uniform load and with spatially distributed flexural rigidity. Subsequently, the beam shapes and material properties for four specific scenarios are identified: free-free logarithmic beam, cantilevered logarithmic beam, simply-supported logarithmic beam, and simply-supported sliding logarithmic beam. Explicit logarithmic beam responses, governed by a limited number of shape parameters, are illustrated graphically using normalized deflections with respect to the maximum deflection. Highly deflected elastic logarithmic modes emerge as a consequence of high flexural rigidity influenced by the uniformly applied transverse load. These elucidated logarithmic beam modes offer potential practical applications in the structural design of functionally graded materials. They also serve as valuable testing platforms for numerical techniques employed in the analysis of more complex beam problems.

Forced vibration of an axially moving free-free beam with internal resonance

In this paper, the nonlinear governing equations of motion for an axially moving cylindrical shell with free ends which modeled as a free-free beam are established by the generalized Hamilton’s principle for the first time. The Galerkin method and multiple-scale method are adopted to solve the governing equations. Stability of the motion determined by the velocity and axial stiffness is investigated according to the linear equation firstly. Next, multiple-scale approach method is used to investigate the frequency response by adjusting the detuning parameters of the first two natural frequencies considering the internal resonance. Stability of periodic motion is studied through the trajectories of eigenvalues and phase of trajectories with adjusting the detuning parameter. Influences of its velocity and flexible force of the moving beam on the periodic motion and its stability are addressed through bifurcation diagrams and Lyapunov exponents. Correctness of the presented method for investigating dynamic behavior of the moving beam has been validated through Runge–Kutta numerical calculation. The goal of the research lies in the application of the method in the free-free moving beam and revealing the nonlinear dynamic behavior of the beam.

Comparative Analysis of Measurement Points for Structural Damage Identification in Free-Free Aluminium Beam using Response Surface Methodology and FRF Curvature

This research paper presents a comprehensive comparative analysis of measurement points for structural damage identification in free-free aluminium beams using Response Surface Methodology (RSM) and Frequency Response Function (FRF) curvature. The primary objective is to determine the optimal measurement points for establishing a relationship between the input parameters and response features, thus forming a surrogate model. The investigation focuses on analyzing the influence of the number of measurement points on RSM's capability to identify structural damage. Three distinct Design of Experiment (DOE) techniques are explored: central composite design (CCD), Box-Behnken design (BBD), and D-optimal design, each incorporating different numbers of measurement points (10 elements, 15 elements, and 20 elements). The study's results highlight BBD's strong capability in detecting damage with fewer measurement points, while CCD design accurately identifies damage in finer measurement points. The findings reveal a clear trend: as the number of measurement points increases, the model becomes more refined, yielding higher stiffness reduction factor (SRF) values. The results underscore the significance of selecting an appropriate number of measurement points to enhance RSM's efficiency in identifying structural damage, thereby optimizing damage identification processes. This study contributes valuable insights into the selection of measurement points for effective structural damage identification, offering practical implications for engineering applications.

Exploring the Limits of a Deflection-Based Method for the Estimation of Dynamic Stress in Beams

Estimation of the dynamic stress in structures, such as beams and plates, has previously been made using the relationship between stress and velocity spatial maxima based on farfield assumptions. This paper presents a method for the estimation of dynamic stress in a beam using Euler-Bernoulli beam theory, where deflection data from a grid of measurement points on the surface of the beam is used to estimate the dynamic bending stress in the structure. The limitations of the method are investigated via response data provided by a numerical model of a free-free beam. A non-dimensional wavenumber analysis is used to determine the number of points required for an accurate estimate of stress. Beams with a range of material and geometric parameters are modelled in order to explore the limits of the estimation method, and parameters representative of several real-world materials are used to assess the suitability of the method for practical applications.

Feature extraction and classification of multiple cracks from raw vibrational responses of composite beams using 1D-CNN network

Structures and components need to be constantly monitored to ensure good service life and avoid failures. Structural health monitoring is a wide-ranging concept that tries to ensure the safety of components and structures. Vibration responses of small and large structures contain information about these damages. In the present study, this information is extracted and used to classify the location and severity of cracks. For this purpose, an FE model of Glass Fibre Reinforced Plastic (GFRP) free-free beam is created for beams without any crack and beams with three crack locations and three crack severities. Finite Element (FE) model is validated using modal analysis and is used to generate vibration responses from simulated hammer strikes at different locations on the beam. The vibration data consists of responses of beams that have only one crack present at a time and when two cracks are present. A 1D-CNN network is trained on the generated vibrational responses to classify damage labels assigned for each crack location and crack severity. The network was tested using datasets containing only single crack instances and only dual crack instances. The trained networks were found to be 95% and 93% accurate in determining the damage class respectively. Furthermore, datasets of these two instances combined, containing responses when single cracks and dual cracks are present, were also investigated. These networks had an accuracy of 92%. Hence, in all these instances, the 1D-CNN network classifies the healthy and damaged conditions satisfactorily. There are two advantages of using such a technique. Firstly, this approach simplifies the two-step approach: one for feature extraction and another one for damage prediction into a single step. Secondly, the use of raw vibration data for damage prediction means that real-time damage detection is possible.

The Effect of the Traction Rod on the Vertical Vibration Behavior of the Railway Vehicle Carbody

Although research has shown that through the additional rigidity introduced in the secondary suspension, traction rods can affect the vertical dynamic performance of railway vehicles, this topic has been less studied by researchers in the field. In this paper, the effect of a traction rod on the vertical vibration behavior of a carbody of a railway vehicle is analyzed, using the results obtained through numerical simulations. Numerical simulation applications are developed based on a vehicle model, where the vehicle carbody is modeled using a free-free equivalent beam Euler–Bernoulli, and the bogie chassis and wheelsets are represented by rigid bodies linked together by Kelvin-Voigt systems that model the secondary suspension and the primary suspension. The novelty element of this paper is found in the model of the traction rod. This includes traction rod damping, which has been neglected in previous research. The stiffness and damping of the traction rod are represented by a longitudinal Kelvin–Voigt system integrated into a secondary suspension model. The effect of the traction rod on the vertical vibration behavior of the vehicle carbody is analyzed based on the power spectral density of the acceleration, the root mean square of acceleration, and the ride comfort index, for three cases for analysis: a ‘without traction rod’ case, a ‘with traction rod—with damping’ case, and a ‘with traction rod—without damping’ case. The conclusions of the paper highlight the influence of the stiffness and damping of the traction rod on the vibration level of the carbody, especially in its middle. Depending on the stiffness of the traction rod, significant increases in the ride comfort index are obtained, which at high velocities can exceed 300%. Damping of the traction rod reduces the ride comfort index by up to 10%.

An improved version of mode shape based indicator for structural damage identification

Damage detection is crucial for the operation and maintenance of an existing bridge or building. A vibration-based method is a popular approach that can be used to identify failures based on changes in the dynamic properties. However, this approach often requires information on both healthy and unhealthy states. In this current study, the structural failure determination is carried out through a damage index based solely on the unhealthy state. A combination of the gapped smooth and modal curvature methods is the core of calculating the proposed indicator. In particular, damage scenarios by means of assumed stiffness reduction, and cuts are investigated. Three categories of beam-like structures consisting of a simply supported beam, a cantilever beam, and a free-free beam are used to validate the applicability of the proposed approach. For comparison purposes, the traditional gapped smooth method is implemented. The location of damage can be identified using only displacement mode shapes. Then, damage quantification at the identified location is estimated using a stochastic optimization process. The promising findings indicate that the proposed approach can enhance the effectiveness of damage identification based on vibration data in the unhealthy state of the structures

On 3D exact free torsional-bending vibration and buckling of biaxially loaded isotropic and anisotropic Timoshenko beams with complex cross-section

This research reports on Three Dimensional (3D) exact free torsional-bending vibrations and buckling of Timoshenko beams possessing homogeneous and nonhomogeneous properties and complex cross-section. The proposed exact solution is derived from 3D elasticity equations. The Hamilton’s integral equation is initially employed to establish the 3D exact governing Partial Differential Equations (PDEs) of motion and buckling, taking into account the effect of biaxial load applied to Timoshenko beams possessing complex cross-section. The PDEs are then solved using integral transform and trigonometric series differentiation procedure. The current solution procedure does not require characteristic deformation function to be predetermined and has fast convergence. A root finding algorithm is proposed to screen and filter the discontinuity and breakdown points as well as imaginary roots of the determinant function associated with bending eigenfrequency and eigenbuckling matrices. Two algorithms are further developed to capture the 3D dynamic and buckling mode shapes of biaxially loaded Timoshenko beams without any restriction imposed on the complexity of cross-section and degree of material anisotropy. The accuracy and convergence of the proposed 3D exact solution are first investigated by results comparison with the experimental and theoretical results in the published literature for a particular case in which bending natural frequencies response and modes shapes of a free-free isotropic Timoshenko beam with rectangular cross-section are sought. Finally, the effect of material isotropy/anisotropy, cross-section complexity, and biaxial load on torsional-bending natural frequencies, critical buckling loads, and 3D vibration and buckling mode shapes are thoroughly explored using the proposed exact solution procedure and validated by Abaqus finite element package.

Numerically Stable form of Green’s Function for a Free-Free Uniform Timoshenko Beam

Beam models are widely applied in civil engineering, transport, and industry because the beams are basic structural elements. When dealing with the high-order modes of beam in the context of applying the modal analysis method, the numerical instability issue affects the numeric simulation accuracy in many boundary conditions. There are two solutions in literature to overcome this shortcoming, namely refinement of the asymptotic form for the high order modes and reshaping the terms within the equation of the modes to eliminate the source of the numerical instability. In this paper, the numerical instability issue is signalled when the standard form of Green’s function, which includes hyperbolic functions, is applied to a free-free Timoshenko length-long beam. A new way is proposed based on new set of eigenfunctions, including an exponential function, to construct a new form of Green’s function. To this end, it starts from a new general form of Green’s function and the characteristic equation is obtained; then, based on the boundary condition, the Green’s function associated to the differential operator of the free-free Timoshenko beam is distilled. The numerical stability of the new form of the Green’s function is verified in a numerical application and the results are compared with those obtained by using the standard form of the Green’s function.

Material Damping Prediction in Timber Beams Based on Timoshenko Free-Free Beam Model

Damping in vibrating structures is much less understood than their stiffness and inertial properties and in timber structures it is even less understood how damping is generated in motion characteristic of bending and shear separately. We analyse these contributions through application of the so-called method of complex moduli, arising in constant-hysteretic models of damping. We separate the influences on overall damping due to these independent material properties by considering a partial differential equation of undamped motion and substituting Young's and shear moduli with their complex counterparts. We keep the rotary inertia present in the model to assess its own damping contribution and design an experimental setup in which vibration of a free-free beam is simulated so that no reaction (i.e., no contact damping) should be present, which allows us to assess the contribution due to bending, shear and rotary inertia to material damping only. We show that, in the present model, the overall loss factor is indeed smaller than in the literature, while the damping due to shear effects grows in higher vibration modes and beams with smaller length-to-height ratio. Finally, we propose a damping prediction model built on simple linear regression in which the vibration modes are processed separately.

A 10-MHz FREE-FREE BEAM MICROMECHANICAL RESONATOR FABRICATION PROCESS USING SURFACE-MICROMACHINED TECHNOLOGY

The fabrication process for the designed MEMS resonator using surface-micromachined technology is presented in this paper. A 10-MHz Free-Free beam MEMS resonator is designed to vibrate in the second-mode shape, which is significant improvement compare to the fundamental mode. The design showed a Q value as high as 75,000, which is significant improvement compared to 8,400 VHF F-F beam MEMS resonator by K. Wang; and very low motional resistance (18kΩ). The surface-micromachined technology is used as the standard process for the design. The process is briefly described from the layout design to the experimental fabricated device.

Desain sensor massa resonator MEMS menggunakan struktur free-free beam

MEMS is an integrated electro-mechanical device which the technology has been applied to various applications, one of which is for the fabrication of microsensors, such accelerometers, microsensors for flow sensing, pressure sensing and mass sensing. In the MEMS implementation, there are several approaches and structures that can be used. This study designed a MEMS resonator mass sensor by the means of a mechanical resonator approach with free-free beam structure and actuated electrostatically. The design is expected to produce a MEMS based mass sensor with a high performance.

Structural vibration mode identification from high-speed camera footages using an adaptive spatial filtering approach

As a non-contact and full-field testing method, high-speed camera-based modal analysis has become a feasible and acknowledged approach. However, extracting small displacements from noisy images has experienced high level of difficulty, especially in high frequency range. This paper proposes a novel adaptive spatial filtering (beamforming) algorithm to extract the displacement signals using high-speed camera. In the proposed algorithm, one pixel is considered as a sensor measuring displacement and a set of pixels are therefore taken as the elements of sensor array. Then, an adaptive spatial filtering acting on this sensor array is proposed. The proposed approach mainly includes three steps. Firstly, a set of pixels are selected to compose a sensor array according to signal to distortion and noise (SINAD). Secondly, a node/antinode searching scheme is proposed based on sinusoid-based piecewise functions, which works as an adaptive filter to match mode shape and enhance modal displacement. Finally, the output of spatial filtering is adopted as the system response for the identification of modal parameters. To validate the performance of the proposed method, simulation and experiment studies are conducted based on measuring the vibration model properties of a free-free beam, which includes a comparison with LK optical flow method and conventional accelerometer-based method. The results show that the SNR of estimated displacement and computational efficiency is significantly improved without using additional sensors. The proposed method paves a way for broadening the applications of using high-speed camera for full-field vibration measurements.

An FRF-based perturbation approach for stochastic updating of mass, stiffness and damping matrices

Most of the current approaches and studies in stochastic FE model updating are based on experimental modal data considering updating of uncertain parameters related to stiffness and mass matrices. In many instances, it is desired that the stochastic FE model is able to predict the variability of dynamic response. This calls for identifying the variability of damping matrix also along with variabilities of mass and stiffness matrices. In the present paper, a new stochastic approach to FE model updating is developed based on variability of measured Frequency Response Functions (FRFs) using a perturbation-based framework. The objective of the method is to identify mean and coefficient of variation of uncertain parameters related to mass, stiffness and damping matrices. Three numerical studies related to, a discrete three degree of freedom spring-mass-damper system, a damped cantilever beam and an airplane model (DLR-AIRMOD structure) are presented to validate the method. The experimental FRFs are simulated using Perturbation and Monte-Carlo approaches. The effect of number of samples of test-structures, effect of modelling error and the effect of noise and incompleteness of the simulated data on the performance of the method are studied. Experimental validation of the method is carried out using different samples of geometrically similar free-free beams to identify mean and variability of coefficient of elasticity and damping.

A temperature compensated biaxial eFM accelerometer in Epi-seal process

This paper reports on the implementation of a vacuum-encapsulated bi-axial resonant accelerometer utilizing the electrostatic frequency modulation (eFM) technique. A novel flexure structure is designed to enable fully-decoupled in-plane displacement of the proof-mass to minimize cross-axis sensitivity. A differential readout scheme leveraging pairs of high quality factor (>12k) free-free beam resonators mitigates the first-order nonlinearities due to temperature effects and unwanted response to proof-mass spurious modes. The fabricated device measures a scale factor of 45.8 Hz/g with negligible cross-axis sensitivity (<2 %). Furthermore, the smooth characteristics of temperature coefficients of frequency (TCf) of the resonators enable an accurate temperature sensor using the common-mode frequency of the resonator pair. Applying a temperature compensation scheme utilizing the common-mode frequency output and pre-characterized TCfs, the accelerometer demonstrates superior performance measuring a VRW of 5.8 μg/√Hz and BI of 5.7 μg with a scale factor stability of 0.38 % and zero-g-output variation of 8.3 mg over temperature range from –20 °C to 80 °C.

A hybrid computational intelligence approach for structural damage detection using marine predator algorithm and feedforward neural networks

Finite element (FE) based structural health monitoring (SHM) algorithms seek to update structural damage indices through solving an optimisation problem in which the difference between the response of the real structure and a corresponding FE model to some excitation force is minimised. These techniques, therefore, exploit advanced optimisation algorithms to alleviate errors stemming from the lack of information or the use of highly noisy measured responses. This study proposes an effective approach for damage detection by using a recently developed novel swarm intelligence algorithm, i.e. the marine predator algorithm (MPA). In the proposed approach, optimal foraging strategy and marine memory are employed to improve the learning ability of feedforward neural networks. After training, the hybrid feedforward neural networks and marine predator algorithm, MPAFNN, produces the best combination of connection weights and biases. These weights and biases then are re-input to the networks for prediction. Firstly, the classification capability of the proposed algorithm is investigated in comparison with some well-known optimization algorithms such as particle swarm optimization (PSO), gravitational search algorithm (GSA), hybrid particle swarm optimization-gravitational search algorithm (PSOGSA), and grey wolf optimizer (GWO) via four classification benchmark problems. The superior and stable performance of MPAFNN proves its effectiveness. Then, the proposed method is applied for damage identification of three numerical models, i.e. a simply supported beam, a two-span continuous beam, and a laboratory free-free beam by using modal flexibility indices. The obtained results reveal the feasibility of the proposed approach in damage identification not only for different structures with single damage and multiple damage, but also considering noise effect.

Static and Dynamic Stiffness of Reinforced Concrete Beams Strengthened with Externally Bonded CFRP Strips.

This paper presents experimental investigations of reinforced concrete (RC) beams flexurally strengthened with carbon fiber reinforced polymer (CFRP) strips. Seven 3300 mm × 250 mm × 150 mm beams of the same design, with the tension reinforcement ratio of 1.01%, were tested. The beams differed in the way they were strengthened: one of the beams was the reference, two beams were passively strengthened as precracked (series B-I), two beams were passively strengthened as unprecracked (series B-II) and two beams were actively strengthened as unprecracked (series B-III). Moreover, the strengthening parameters differed between the particular series. The parameters were: CFRP strip cross-sectional areas (series B-I, B-II) or prestressing forces (series B-III). The beams were statically loaded, up to the assumed force value, in the three-point bending test and deflections at midspan were registered. After unloading the beams were suspended on flexible ropes (the free-free beam system) and their eigenfrequencies were measured using operational modal analysis (OMA). The static measurements (deflections) and the dynamic measurements (eigenfrequencies) were conducted for the adopted loading steps until failure. Static stiffnesses and dynamic stiffnesses were calculated on the basis of respectively the deflections and the eigenfrequencies. The qualitative and quantitative differences between the parameters are described.

Damage detection in structures using modal curvatures gapped smoothing method and deep learning

This paper deals with damage detection using a Gapped Smoothing Method (GSM) combined with deep learning. Convolutional Neural Network (CNN) is a model of deep learning. CNN has an input layer, an output layer, and a number of hidden layers that consist of convolutional layers. The input layer is a tensor with shape (number of images) × (image width) × (image height) × (image depth). An activation function is applied each time to this tensor passing through a hidden layer and the last layer is the fully connected layer. After the fully connected layer, the output layer, which is the final layer, is predicted by CNN. In this paper, a complete machine learning system is introduced. The training data was taken from a Finite Element (FE) model. The input images are the contour plots of curvature gapped smooth damage index. A free-free beam is used as a case study. In the first step, the FE model of the beam was used to generate data. The collected data were then divided into two parts, i.e. 70% for training and 30% for validation. In the second step, the proposed CNN was trained using training data and then validated using available data. Furthermore, a vibration experiment on steel damaged beam in free-free support condition was carried out in the laboratory to test the method. A total number of 15 accelerometers were set up to measure the mode shapes and calculate the curvature gapped smooth of the damaged beam. Two scenarios were introduced with different severities of the damage. The results showed that the trained CNN was successful in detecting the location as well as the severity of the damage in the experimental damaged beam.

Review of the measurement methods for elastic moduli and internal frictions of solids

Elastic moduli and internal frictions are fundamental properties of solid materials and their measurement accuracy and convenience are of great significance to industrial production and scientific research. This paper reviews the measurement methods of elastic moduli and internal frictions of solid materials in the past 100 years, which can be divided into four types: quasi-static method, low frequency method, resonance method and wave propagation method. Firstly, the measurement principle of each type method is introduced and evaluated. Then the resonance methods, including free-free beam method, impulse excitation technique, resonant ultrasound spectrum and piezoelectric ultrasonic composite oscillator technique (PUCOT) are presented and discussed in detail. After that, a new method called modified piezoelectric ultrasonic composite oscillator technique (M-PUCOT), proposed by the current authors, are introduced. This new method is based on the principle of electromechanical impedance, and can measure the Young’s modulus/shear modulus and the related internal frictions simultaneously, accurately and quickly. Finally, the significance and prospective of this new method are discussed.