Free-free Beam Research Articles

A covariance based framework for the propagation of uncertainty through inverse problems with an application to force identification

Inverse problems are widely encountered in fields as diverse as physics, geophysics, engineering and finance. In the present paper, a covariance based framework for the estimation of their uncertainty is presented and applied to the problem of inverse force identification. A key step in its application involves the propagation of frequency response function (FRF) uncertainty through a matrix inversion, for example, between mobility and impedance. To this end a linearised inverse propagation relation is derived. This relation may be considered a generalisation of work presented in the particle physics literature, where we consider both complex valued and non-square matrices through a bivariate description of their uncertainty. Results are illustrated, first, through a numerical simulation where force and moment pairs are applied to a free-free beam model. An experimental study then illustrates the in-situ determination of blocked forces and their subsequent use in the prediction of an operational response. The uncertainties predicted by the proposed framework are in agreement with those acquired through Monte-Carlo (MC) methods for small input variance but are obtained at much lower computational cost, and with improved insight. In the process illustrating the propagation framework, matrix condition number, often taken as an indicator of uncertainty, is shown to relate poorly to a more rigorous uncertainty estimate, leaving open the question as to whether condition number is an appropriate indicator of uncertainty.

Free vibration analysis of a free-free Timoshenko beam carrying multiple concentrated elements with effect of rigid-body motions considered

A modified mode-superposition method (MMSM) was presented in this paper to solve the free vibration problem of the “unconstrained” beams that the conventional mode-superposition method (CMSM) cannot easily tackle. For convenience, a beam without any attachments is called the “bare” beam, and the beam carrying any concentrated elements (CEs) is called the “loaded” beam, in this paper. Furthermore, the mode-superposition method (MSM) with only the “elastic” modes of the bare beam considered is called the CMSM, and the MSM with both the “elastic” and the “rigid-body” modes considered is called the MMSM. From the existing literature, one finds that the CMSM is one of the effective approaches for the free vibration analysis of a “constrained” bare beam (such as the C-C, P-P, or C-F beam) carrying various CEs, where “C, P and F” denote the abbreviations of “clamped, pinned and free”, respectively. However, the CMSM is not available for that of an “unconstrained” bare beam (such as the F-F beam) carrying various CEs. For this reason, this paper presented a MMSM to solve the title problem so that one can easily obtain the natural frequencies and mode shapes for both the “rigid-body” motions and the “elastic” vibrations of the F-F loaded Timoshenko beam. The main difference between a “constrained” bare beam and an “unconstrained” one is that the free vibration responses of latter consist of the “rigid-body” motions and those of the former do not, so that the coupling effect between the “rigid-body” motions and the “elastic” vibrations” of the F-F loaded beam are not considered by using the CMSM. To confirm the correctness of the presented theory and the developed computer program for this paper, all numerical results obtained from the MMSM are compared with those obtained from the finite element method (FEM) and good agreement is achieved. Numerical examples reveal that the CMSM can provide neither any information regarding the “rigid-body” motions nor the accurate natural frequencies and mode shapes for the “elastic” vibrations of a F-F loaded beam, and all the above-mentioned drawbacks of the CMSM have been improved by the presented MMSM.

Application of genetic algorithm for crack diagnosis of a free-free aluminum beam with transverse crack subjected to axial and bending load

A beam structure in presence of crack subjected to both bending and axial load has been investigated for damage diagnosis to predict crack location and crack depth based on an optimized intelligent technique using genetic algorithms approach. Numerical with experimental investigations have been conducted in a free-free beam model to calculate natural frequencies. In this work, genetic algorithm as an optimized artificial intelligent technique has been used which employs first three relative modal parameters like natural frequencies as input data taken from numerical and experimental results to obtain optimized crack locations and crack depths as output parameters. This methodology has been found to be quite reliable for fault diagnosis by monitoring the possible variations in the relative natural frequencies of a free-free beam element in presence of crack. This paper also gives an insight into the deviation of the results of the proposed methodology from the experimental result.

Evaluation of noise in measurements with speckle shearography

Abstract Speckle shearography has the advantage of allowing non-contact, full-field and high resolution measurements. Nevertheless, due to its nature and to its sensitivity to ambient perturbations, the measurements may present high levels of noise. The evaluation of the experimental noise is needed in order to assess the degree of accuracy of the measurements. This assessment is not straightforward since the measurements will involve resolving simultaneously the phase discontinuities and the phase fluctuations produced by the noise which are 2π wrapped. The phase discontinuities can be easily removed if the phase map is smooth and the phase jumps are not greater than 2π rad. However, this procedure cannot be directly applied for resolving the random phase fluctuations produced by the noise. In this work, a method is proposed to efficiently evaluate the noise in measurements with speckle shearography whenever it is in the range [−π, π]. The limits of the proposed method are assessed by applying it to phase maps with added noise, which are obtained through the numerical simulation of speckle shearography. The results show a very good evaluation of noise with low and moderate amplitudes. Finally, the proposed method is used to evaluate the experimental noise in rotation fields of a free-free aluminum beam measured with speckle shearography.

Investigating relaxation of glassy materials based on natural vibration of beam: A comparative study of borosilicate and chalcogenide glasses

The present work investigates the validity of using the frequency and decay rate of free-free beam vibration, which are measured by the impulse excitation technique (IET), to characterize the viscoelastic properties of glass in the temperature range of glass transition. Based first on the classical Burgers model, we show that the temperature-dependent flow viscosity of borosilicate glass, calculated from the measured frequency and amplitude decay rate of the flexural vibration, agrees very well with the existing data. While for chalcogenide glass, the same calculation approach does not render the similar agreement, and the reason probably lies in non-exponential effects. To comprehend the IET data in the temperature range of glass transition, we propose a simplified theoretical framework for describing the transition from the solid-like to liquid-like viscoelastic behavior and discuss the cause of difference of the rheology behaviors of these two types of glass based on the formulation of non-exponential relaxation.

A hybrid intelligent model for crack diagnosis in a free-free aluminium beam structure

A hybrid intelligent model for crack diagnosis in a free-free aluminium beam structure

A General Analytical Formulation for the Motional Parameters of Piezoelectric MEMS Resonators.

This paper reports on a general analytical expression for the motional resistance ( ) of an arbitrary mode in a piezoelectric microelectromechanical system resonator with parallel plate electrode geometry. After applying simplifying assumptions and using analytical modes shapes, expressions for the of modes with out-of-plane flexure as the primary displacement are presented. These modes include free-free transverse beam flexure (TBF), unclamped disk flexure resonators (DFRs), and antisymmetric Lamb modes. For verification, is extracted from resonators fabricated in a lead zirconate titanate on silicon process. Predicted of TBF and DFR modes is validated using on-wafer extracted constants, analytical modal properties, and independently measured material properties.

Damage Identification Based on Curvature Mode Shape using Cubic Polynomial Regression and Chebyshev Filters

Structure Health Monitoring (SHM) has been applied in various application such as aerospace, machinery and civil structures to maintain structure’s safety and integrity. Gapped smoothing method (GSM) is most popular non-destructive identification (NDI) method due to its simplicity and did not require baseline data for comparisons. However, GSM is less accurate to detect wide size of damage in structure and cause false detection. Objective of this study is to propose a method to detect damage in structure using curvature mode shape data estimated from damaged structure and did not require data from undamaged structure. Finite element analysis (FEA) on a free-free boundary condition steel beam was carried out to demonstrate the feasibility of the proposed method that estimate undamaged curvature mode shape data using cubic polynomial regression (CPR) and Chebyshev filters (CF) methods. The results shows proposed method that used Chebyshev filters has better accuracy damage detection on wide notch compared to GSM. Although application of an interpolation and Chebyshev filters showed results with a high potential for overcoming the issue of false detection due to different notch size, however the proposed method still need refinement to better detection of different damage cases.

An investigation into the accelerometer mounting effects on signal transmissibility in modal measurements

The purpose of this study is to define a simple model and discuss the main effects due to the use of the sensors with imperfect mounting in experimental measurements. This paper presents a theoretical and experimental investigation for the effects of different mounting methods of accelerometers on signal transmissibility in modal testing. In the theoretical part, a two degree-of-freedom (2-DOF) model is used, where the first DOF accounts for the accelerometer seismic mass and Piezo-crystal and the second DOF represents the mounting interface dynamics. An experimental modal analysis is conducted on a simple steel free-free beam using impact hammer excitation. The time domain signals and frequency response functions (FRFs) are measured in the case of magnetic, wax and stud mounting. It is found that the method of mounting has a significant effect on damping rates of measured responses. Although natural frequencies have no important changes, but the quality of measured FRFs is degraded considerably.

Free Dry Vibration of Low-Aspect-Ratio Cantilever Wing: SemiAnalytical and Numerical Analysis With Experimental Verification

The free vibration of a low-aspect cantilever wing is studied by semianalytical, numerical, and experimental means. The wing is modeled as a two-way tapered, hollow Kirchhoff's plate, with the chord-wise section as symmetrical NACA0018 aerofoil. The chord length and the thickness taper from root to tip, over the span. The semianalytical approach is based on Galerkin's method, which includes the modal superposition of two orthogonal beam modeshapes (free-free beam in chord-wise direction and cantilever beam in span-wise direction). The free vibration is also studied numerically using ANSYS. The results have been compared with an experimental study, performing the dry impact hammer test. A model scale wing has been constructed from a 3-mm-thick metal sheet, with a length-scale ratio of 1:10. Comparative studies have been done among the three methods. The feasibility of modeling a low-aspect-ratio wing as a plate has been investigated. 1. Introduction Lifting surfaces commonly occur in engineering structures: fans, steam and gas turbine blades, impeller blades, wind turbine blades, helicopter fans, airplane wings, marine propeller blades, marine rudders and skegs, and other control surfaces like hydrofoils and wings in high-speed marine crafts. The lifting surface is typically a cantilever, with one edge welded to the main structure (root), and the other end free (tip). The span-wise geometry is tapered, for greater strength at the root and lighter weight at the tip. The chord-wise cross section is an aerofoil, which acts as a lifting surface to the incoming flow at varying angles of attack. The aspect ratio of the lifting surface, i.e., the ratio of the span length to the mean chord length, may vary from 0.4 to 0.6 in a turbine blade to 12 to 15 in an airplane wing/wind turbine. This work focuses on a low-aspect-ratio lifting surface, whose usual span-to-chord ratio is 1.5–2.

Natural vibration frequencies of horizontal tubes partially filled with liquid

This work presents an experimental and numerical study on the flexural vibration of horizontal circular tubes partially filled with liquid. The tube is configured as a free-free beam with attention being directed to the case of small amplitudes of transverse oscillation whereas the axial movements of the tube and liquid are disregarded. At first vertical and horizontal polarizations of the flexural tube are investigated experimentally for different amounts of filling liquid. In contrast with the empty and fully-filled tubes, it is observed that natural frequencies of the vertical and horizontal polarizations are different due to asymmetry induced by the liquid layer, which acts like an added mass. Less mass of liquid is added to the tube when oscillating horizontally; as a consequence, eigenfrequencies for the horizontal polarization are found to be greater than the case of the vertically polarized tube. A simple method to calculate the natural vibration frequencies using coefficients of added mass of liquid is proposed. It is shown that the added mass coefficient increases with the liquid's level and viscosity. At last a numerical investigation of the interaction between the liquid and the tube is carried out by solving in two-dimensions the full Navier-Stokes equations via a finite volume method, with the free-surface flow being modeled with a homogeneous multiphase Eulerian-Eulerian fluid approach. Vertical and horizontal polarizations are imposed to the tube with pressure and shear stresses being determined numerically to assess the liquid's forcing onto the tube's wall. The coefficient of added mass of liquid is then estimated by the ratio between the resulting force and the acceleration imposed to the wall. A good agreement is found between experimental and numerical results, especially for the horizontally oscillating tube. It is also shown that viscosity can noticeably affect the added mass coefficients, particularly at low filling levels.

Analyzing Free-Free Beams by Green's Functions and Fredholm Alternative Theorem

AbstractThis article deals with the analysis of free-free beams by an analytical method. The well-known Green's function method is employed, and exact solution for the problem is obtained. As a second problem, the simply supported-free beam with rotational rigid body motion is analyzed. It is initially shown that ordinary Green's functions cannot be constructed due to a mathematical contradiction. To remedy this limitation, the Fredholm Alternative Theorem is utilized. This theorem eliminates the contradiction and enables analysts to obtain modified Green's functions. The fundamental existence conditions are derived and thoroughly investigated from the structural point of view. Finally, the deflection functions of these beams are found using modified Green's functions.

Analytical solution for the elastic bending of beams lying on a linearly variable Winkler support

The present paper considers the analytical static bending response of a finite uniform free-free Euler-Bernoulli elastic beam resting on a Winkler elastic foundation with a spatially inhomogeneous stiffness coefficient. Specifically, a linear variation of the foundation coefficient is assumed. The beam-foundation system is loaded by a force and a moment applied at one beam's (top) end, a configuration which may resemble that of a foundation pile. The analytical solution of the governing Ordinary Differential Equation is derived and represented in explicit closed-form in terms of generalized hypergeometric functions. Through the derived solution, parametric analyses are carried out, by interpreting the parametric variation of the mechanical response of the beam-foundation system due to changes in its mechanical properties.

Structural damage detection using transmissibility together with hierarchical clustering analysis and similarity measure

Maintenance and repairing in actual engineering for long-term used structures, such as pipelines and bridges, make structural damage detection indispensable, as an unanticipated damage may give rise to a disaster, leading to huge economic loss. A new approach for detecting structural damage using transmissibility together with hierarchical clustering and similarity analysis is proposed in this study. Transmissibility is derived from the structural dynamic responses characterizing the structural state. First, for damage detection analysis, hierarchical clustering analysis is adopted to discriminate the damaged scenarios from an unsupervised perspective, taking transmissibility as feature for discriminating damaged patterns from undamaged ones. This is unlike directly predicting the structural damage from the indicators manifestation, as sometimes this can be vague due to the small difference between damaged scenarios and the intact baseline. For comparison reasons, cosine similarity measure and distance measure are also adopted to draw out sensitive indicators, and correspondingly, these indicators will manifest in recognizing damaged patterns from the intact baseline. Finally, for verification purposes, simulated results on a 10-floor structure and experimental tests on a free-free beam are undertaken to check the suitability of the raised approach. The results of both studies are indicative of a good performance in detecting damage that might suggest potential application in actual engineering real life.

Simulation for whole-body vibration to assess ride comfort of a low–medium speed railway vehicle

Vibration in trains constitutes one aspect of the physical environment that can cause discomfort to passengers. In order to assess the vibration, a variety of techniques have been developed and used. The general approach is to acquire acceleration at the passenger-train interface, and to then process these acceleration signals in order to calculate human comfort. However, the comfort index so calculated is independent of the seat characteristics and human parameters. Thus, a human biodynamic model with dynamic seat characteristics is necessary in order to perform true comfort analysis. The biodynamics of human subjects has been a topic of interest over the years, and a number of mathematical models have been established. However, there are only a few studies incorporating biodynamic models in railway applications. So, the present work proposes to evaluate Sperling's Ride Index for a low-medium speed railway vehicle and further to calculate root mean square acceleration of different body parts, as per the International Organization for Standardization ISO 2631 guidelines, through bond graph technique. Car body flexibility is also incorporated using modal expansion of a free-free beam. Physiological effects of the vibrations on the human body were analyzed using the criteria specified in ISO 2631.

Experimental validation of a FRF-based model updating method

In this paper, a finite element model updating method using frequency response functions is experimentally validated. The method is a sensitivity-based model updating approach which utilizes a pseudo-linear sensitivity equation. The method is robust against the adverse effects of incomplete measurement, measurement errors and modeling errors. The experimental setup consists of a free-free aluminum beam, where changes are introduced by reducing the stiffness and attaching lumped mass at certain parts of the beam. The method is applied to identify the location and amount of the changes in structural parameters. The results indicate that the location and the size of different level of changes in the structure can be properly identified by the method. In addition, a study is done on the influence of the number of impacts and sensors on the quality of the identified parameters.

Hybrid phase field simulation of dynamic crack propagation in functionally graded glass-filled epoxy

Abstract Numerical simulation of dynamic crack propagation in functionally graded glass-filled epoxy (FG) beams using a regularized variational formulation is presented. The Griffith’s theory based hybrid phase field approach for diffusive fracture is taken, which is able to accurately simulate complex behaviors of dynamic crack growth in FGMs. The FG beams under impact loads experimented by Kirugulige and Tippur (Exper. Mech. 2006; 46:269–281) are considered, taking the same configurations, material property, crack location, and other relevant assumptions. The crack paths, crack length, crack velocity, energies, etc., computed through the hybrid phase field model are numerically analyzed, and some of those results are directly compared with the experimental data. Due to lack of necessary information regarding impact loading profiles and boundary conditions in setting the tests, the simulations become difficult as an inappropriate definition of loading and boundary conditions can significantly alter the outputs of numerical solutions. This issue is important and thus is discussed. Two specific loading profiles, the constant and the linear displacement velocities, are taken into account, while free-free FG beams are considered. We show that good agreements of crack paths between the experiment and phase field approaches can be obtained. Numerical results also confirm a significant effect of elastic gradients on final crack paths. Similar to the experimental results, we also found that the crack path kinks significantly when situated on the stiffer side compared to the compliant side of the FG specimen.

Implementation of a CMOS-MEMS Filter Through a Mixed Electrical and Mechanical Coupling Scheme

We propose a novel filter coupling scheme, which combines the merits of mechanically and electrically coupled methods to enable a well-defined narrow bandwidth and a decent stopband rejection in a CMOS-MEMS bandpass filter. In the proposed design, a previously developed free-free beam arrayed resonator is employed as a basic resonant tank for the filter concept validation. By the use of several design strategies, including oxide-rich structure, arrayed design, and deep-submicrometer transducer’s gap, the constituent resonant tanks feature small motional impedance ( $R_{m}$ ), making filter termination feasible. Furthermore, we also provide the complete equivalent circuit and numerical modeling for such a fourth-order bandpass filter under the proposed coupling scheme. Compared with the traditional coupling topologies, the presented coupling concept is capable of achieving significant feedthrough suppression and precise bandwidth control, only at the cost of slight in-band loss. As a result, we successfully demonstrate a terminated CMOS-MEMS filter centered at 8.58 MHz with a narrow passband of 35 kHz and a stopband rejection greater than 20 dB. The proposed design concept of mixed coupling scheme can also be implemented in other MEMS fabrication platforms. [2015–0304]

Characterization of Root-Mean-Square Acceleration Errors in Flexible Structures Undergoing Vibration Testing

A methodology is described to evaluate the Root-Mean-Square (RMS) errors in acceleration response during the vibration testing of flexible structures like missile and rockets. Simulation studies are carried out on a free-free uniform beam to characterize the error estimations in the desired and achieved acceleration spectra at different locations of the structure. Equal and unequal desired spectra are also considered at different locations of the structure and individual error in the achieved spectrum, as well as overall error is computed. The simulation results are obtained for traversing a single input along the length of the structure with three control sensors at different locations. The variation of the force requirement at different locations of the control sensor and the normalized RMS error for each control sensor is computed. The investigations show that the force requirement is less when the control sensors are placed at the far ends of the structure. It is also observed that the force requirement is increased marginally for the increased number of control sensors, whereas the overall error between the desired and achieved spectrum is reduced with the increase in the number of control sensors.

Multimodal vibration damping of a beam with a periodic array of piezoelectric patches connected to a passive electrical network

A multimodal damping strategy is implemented by coupling a beam to its analogue electrical network. This network comes from the direct electromechanical analogy applied to a transverse lattice of point masses that represents the discrete model of a beam. The mechanical and electrical structures are connected together through an array of piezoelectric patches. A discrete and a semi-continuous model are proposed to describe the piezoelectric coupling. Both are based on the transfer matrix formulation and consider a finite number of patches. It is shown that a simple coupling condition gives a network that approximates the modal properties of the beam. A multimodal tuned mass effect is then obtained and a wide-band damping is introduced by choosing a suitable positioning for resistors in the network. The strategy and the models are experimentally validated by coupling a free-free beam to a completely passive network. A multimodal vibration reduction is observed, which proves the efficiency of the control solution and its potential in term of practical implementation.