Flexural Vibration Research Articles

Torsional Vibration Tests of Extruded Polystyrene with Improved Accuracy in Determining the Shear Modulus.

Recently, extruded polystyrene (XPS) foam has been used as a component of construction materials; therefore, it is important to characterize its mechanical properties, including shear modulus. Despite the importance, it is often difficult to determine the shear modulus accurately by using many of the conventional methods; therefore, it is desirable to establish another method to measure the shear modulus with a high accuracy. Among various methods, torsional vibration test is advantageous because it can be performed easily under the pure shear stress condition in the test sample and both the in-plane and out-of-plane shear moduli can be obtained. However, it is difficult to find any examples performing the torsional vibration tests. In this study, the in-plane and out-of-plane shear moduli of XPS were determined through torsional vibration tests using samples of various widths. In addition, the shear moduli were also determined through flexural vibration tests and compared with those obtained from the torsional vibration tests. In the torsional vibration tests, the anisotropy in these shear moduli became an obstacle, and the in-plane shear modulus determined using a single sample was often dependent on the width/thickness ratio of the sample. In this condition, the coefficient of variation of the in-plane shear modulus value was often close to 10%. However, when using data obtained from the samples with various width/thickness ratios, both the in-plane and out-of-plane shear moduli could be obtained while reducing the abovementioned dependence. Additionally, the coefficients of variation were restricted to those below 2% and 7% for the in-plane and out-of-plane shear moduli, respectively, and these values were obviously lower than those obtained from the flexural vibration tests (approximately 20%). In the proposed method, both the in-plane and out-of-plane shear moduli can be obtained accurately without using any numerical analyses, which are often required in the standardized methods to improve the accuracy. Thus, for accurate measurement of both types of shear moduli of XPS, we recommend performing torsional vibration tests using a range of samples of various width/thickness ratios.

Flexural vibration properties of particle board using acrylic emulsion adhesives with differential glass transition temperature

Vibration characteristics of wood-based materials are essential parameters in considering the indoor environment of a building. This study investigates the effect of the physical properties of water-based adhesives on vibration characteristics of wood-based materials. Adhesive films and particle boards (PB) were prepared from acrylic emulsion adhesives (AE) with differential glass transition (Tg), and loss tangent of dynamic viscoelasticity (tanδD) and flexural vibration property (tanδF) were compared. The tanδD master curve and tanδF showed a similar tendency regarding frequency dependence. The apparent activation energy (ΔH) calculated from the shift factor of the master curve tends to be different depending on the Tg of AE and was especially high for AE with Tg = 13 °C along with blended AE. The dynamic Young’s modulus (EF) calculated from the flexural vibration tests showed higher values for AE at Tg = 13 °C, 41 °C, and 90 °C than that of urea–formaldehyde adhesive (UF).

An analytical model for flexural vibration of composite beams with shear slip based on third order deformation kinematics

In this paper, an analytical solution is derived for accurately predicting the free and forced vibration responses due to time varying as well as moving loads of two layered composite beams which consist of two different materials. For this purpose, the third order deformation kinematics is used in the proposed model, and it allows to take a parabolic (third order) variation of the longitudinal displacement along the beam depth for the two material layers. The partial shear interaction between the two material layers produced by the deformability of shear connectors joining these layers in the form of shear slip at their interface is modelled using distributed shear springs along the beam length. Hamilton’s principle is applied to derive the governing equations of the dynamic system which are solved analytically using a Navier type solution technique. Moreover, a two-dimensional (2D) finite element model is built up in ABAQUS for the validation of the proposed analytical model. Some parametric studies are conducted to investigate the effect of shear deformation on the forced vibration response of a two-layered composite beam partial with different shear stiffness, span-to-depth ratio and elastic-to-shear modulus ratio.

Flexural vibration of multistep rotating Timoshenko shafts using hybrid modeling and optimization techniques

In this paper, the flexural vibration of uniform and non-uniform rotating shafts based on the Timoshenko beam theory is investigated. Considering shear deformation and gyroscopic effects to build an exact framework of a solution, the fourth-order differential equation of vibration is solved by an analytical method. The overall transfer matrix of the system formed by a series of flexible distributed in addition to lumped elements is achieved. The results obtained by the distributed lumped modeling technique (DLMT) are verified with other techniques. The damped frequencies obtained by the hybrid modeling method for a multistep gas turbine rotor system using the Euler–Bernoulli and Timoshenko beam theories are compared with the results of the transfer matrix method (TMM). It is shown that the presented method provides highly accurate results, while with no compromise, it can be simply and effectively employed for complex systems. It is also shown how the Hooke and Jeeves and the ant colony optimization (ACO) besides the direct enumeration method can be applied to obtain the damped natural frequencies of complicated vibrational systems such as the gas turbine rotors. The comparison between the frequency and damping ratio values obtained by the optimization and direct enumeration methods shows less than 1% error for different cases.

Dynamic modulus of elasticity in flexural vibration tests of Pinus sylvestris sawn timber obtained with fundamental resonant frequency and overtones

Abstract The use of resonant frequency is presently one of the most used non-destructive testing (NDT) methods to predict the mechanical properties of sawn timber. In most cases, only the first vibration mode is used to estimate the static modulus of elasticity (MOES). The use of other vibration modes (overtones) can improve the predictive capacity of this NDT. In this study, the dynamic modulus of elasticity (MOEdyn) obtained from the fundamental flexural resonant frequency and overtones of 38 structural size samples (2000 × 100 × 70 mm3) of Pinus sylvestris were analysed. The study was complemented with 16 small size samples (1000 × 20 × 8.7 mm3) of the same species to know the shear effect in samples with a very high length-to-depth ratio. For the structural size samples, a 6% decrease in the MOEdyn was detected each time the vibration mode increased. Using the second or third vibration mode offered lower errors in the prediction of the MOES than the fundamental resonant frequency. The lowest MOES prediction errors were obtained combining various vibration modes through power regression. This regression turned out to be a simple alternative to avoid the shear effect in the calculation of the MOEdyn.

Analytical study on the dynamic responses of a sheet-pile groin subjected to transient lateral impulses

A novel sheet-pile groin structure consisting of row piles and a tied sheet has been introduced to resist the impact of tidal bores in recent years. However, the dynamic responses of the structure under tidal bore loads have not been carefully studied before. This paper proposed an analytical solution for the dynamic responses of a sheet-pile groin subjected to transient lateral excitation. To take the flexural and longitudinal vibrations into consideration, the piles and the sheet are modeled by Timoshenko beam (TB) theory and a one-dimensional rod, respectively. The soil around the piles is simulated by the dynamic Winkler foundation model. The excitation is considered by a lateral concentrated point load acting on the front pile. On this basis, the governing equations are constructed in the time domain and solved in the frequency domain, while the time-domain responses are obtained using the inverse Fourier transform. The accuracy of the presented analytical solution is validated by comparison with the results obtained from finite element method (FEM) simulation. The effects of the properties of the sheet, piles and soil on the dynamic responses are investigated through a comprehensive parametric study.

Finite element formulation for the dynamic analysis of shear deformable thin-walled beams

The present study develops a finite element formulation for the natural vibration and dynamic analysis of thin-walled members with asymmetric cross-sections. The formulation captures shear deformation within the middle surface, global and local warping, viscous damping, translational and rotatory inertial effects, and inertial coupling. The formulation starts with the extended variational form of Hamilton’s principle to account for non-conservative damping effects, in conjunction with a special interpolation scheme aimed at circumventing shear locking. The resulting finite element has two nodes with seven degrees of freedom per node. The discretized equations of motion are integrated using the unconditionally stable Newmark integration scheme with constant average acceleration. For members with doubly symmetric sections, comparisons with shell solutions show that the formulation reliably predicts the natural frequencies and vibration modes associated with flexure/shear and torsion/warping responses. Further comparisons with other non-shear deformable thin-walled beam solutions show that shear deformation effects gain significance in shorter spans, higher vibration modes, and in flexural vibration about the major axis. For monosymmetric and asymmetric sections, comparisons with benchmark solutions showcase the ability of the formulation to capture the inertial coupling arising between the flexure/shear and torsion/warping responses, while keeping the computational cost to a minimum. Comparisons to shell-based solutions demonstrate the ability of the solution to accurately capture the damped time history response of members with doubly symmetric, monosymmetric, and asymmetric sections subjected to harmonic, ramped, and blast-like forces. A separate treatment is provided for degenerate cross-sections with a vanishing global warping function.

High temperature thermophysical properties of spark plasma sintered tungsten carbide

Fast Breeder Test Reactor (FBTR) is a unique Indian fast reactor and irradiation facility for materials testing. As a part of its life extension activities, high density tungsten mono carbide (α-WC) is considered as a candidate material for the lower axial neutron shielding of the grid plate. Spark plasma sintering technique is used to synthesize high density (99.5%) tungsten carbide pellets under optimized conditions without a binder phase and subjected to high temperature thermal property characterization. The dilatational strain, thermal diffusivity as well as the resonance frequency of flexural vibration is measured as a function of temperature and the corresponding values of linear thermal expansion, thermal conductivity and the dynamic Young's modulus are estimated. The measured values are analyzed in combination with quasi-harmonic Debye-Grüneisen model to obtain a comprehensive database on vibrational thermodynamic properties from 0 to 1273 K. The observed Debye temperature for tungsten carbide under ambient conditions is 630 K which gradually decreases to 586 K at 1273 K. The estimated thermal Grüneisen parameter varied in the range of 1.41–1.45.

Experimental analysis of tip vibrations at higher eigenmodes of QPlus sensors for atomic force microscopy

QPlus sensors are non-contact atomic force microscope probes constructed from a quartz tuning fork and a tungsten wire with an electrochemically etched tip. These probes are self-sensing and offer an atomic-scale spatial resolution. Therefore, qPlus sensors are routinely used to visualize the chemical structure of adsorbed organic molecules via the so-called bond imaging technique. This is achieved by functionalizing the AFM tip with a single CO molecule and exciting the sensor at the first vertical cantilever resonance mode. Recent work using higher-order resonance modes has also resolved the chemical structure of single organic molecules. However, in these experiments, the image contrast can differ significantly from the conventional bond imaging contrast, which was suspected to be caused by unknown vibrations of the tip. This work investigates the source of these artefacts by using a combination of mechanical simulation and laser vibrometry to characterize a range of sensors with different tip wire geometries. The results show that increased tip mass and length cause increased torsional rotation of the tuning fork beam due to the off-center mounting of the tip wire, and increased flexural vibration of the tip. These undesirable motions cause lateral deflection of the probe tip as it approaches the sample, which is rationalized to be the cause of the different image contrast. The results also provide a guide for future probe development to reduce these issues.

Numerical analysis of passive toroidal tuned liquid column dampers for the vibration control of monopile wind turbines using FVM and FEM

The toroidal tuned liquid column damper (TTLCD) is a recently developed multidirectional vibration control device. This paper investigates the TTLCD for the vibration control of monopile fixed-bottom offshore wind turbines. The TTLCD with its circular versatile configuration can match the geometrical shape of wind towers and be tuned to control wind, wave and earthquake induced flexural tower vibrations. An effective numerical framework accounting for the liquid flow and sloshing as well as two-way fluid–structure interaction is developed for the TTLCD-structure system. Its accuracy is validated by comparing the numerical results with experimental ones. The design procedures of the TTLCD for monopile wind turbines are illustrated in detail. The control performance of the TTLCD under wind, wave and seismic loads is presented. The results show that the TTLCD can control the structural responses in both fore-aft and side-side directions simultaneously. Furthermore, it is revealed that for wind-induced vibrations, the control performance relies on the wind velocities and the related frequency components.

Flexural wave bandgap properties in metamaterial dual-beam structure

In this paper, a metamaterial dual-beam structure is firstly constructed by periodically adding pillars between two identical homogeneous beams. By simplifying the pillar as an equivalent spring-mass-spring system, another metamaterial dual-beam structure carrying periodic spring-mass-spring resonators is then established. This equivalent representation method is validated through both a vibration transmission analysis and a band structure analysis. Two bands corresponding to the antisymmetric and symmetric flexural vibration modes respectively open a flexural narrow bandgap in the low frequency range. Then, the improved structure with high-static-low-dynamic stiffness (HSLDS) resonators is proposed in the need of widening the bandgap and lowering the beginning frequency. Results show that with the decrease of the proportion of the residual stiffness, the beginning frequency of the bandgap can be reduced, while the ending frequency keeps unchanged. This results in a relative bandgap as high as 1.465, which is about 4.4 times of the original structure.

Thermally Induced Flexural Vibration Problem of Inhomogeneous Rectangular Plates: When Temperature Change at the Bottom Surface Is Kept at Zero

Herein, a thermally induced flexural vibration problem of inhomogeneous rectangular plates in which the material constants are given as a power of the coordinate in the thickness direction is considered. Assuming that the top surface of the plate uniformly receives cyclic heat supply and that the bottom surface is kept at zero temperature change, a problem of 1D heat conduction in an unsteady state in the inhomogeneous plate is analyzed theoretically. Analytical solutions of temperature change, deflection, and stresses in the plate with simply supported edges in which power exponents in the material constants are free from any constraints are derived. Performing numerical calculations, an effect of inhomogeneity in the specific heat capacity on amplitude and phase shift in temperature change inside of plate is clarified. To evaluate the influence of inhomogeneity in the material constants on the inertia effect in the thermoelastic response of the plate, amplification factors are defined as ratios of maximum amplitudes in dynamic solutions of deflection and stresses to those in quasistatic solutions. It is clarified that amplification factors in deflection and stresses of the plate are significantly affected by inhomogeneity in the Young's modulus of elasticity and the mass density.

Energy harvesting efficiency of unimorph piezoelectric acoustic black hole cantilever shunted by resistive and inductive circuits

A unimorph piezoelectric cantilever equipped with an Acoustic Black Hole (ABH) termination is designed for broadband energy harvesting. The ABH termination, with its tapered region, induces a focusing of the flexural vibrations which can be used to increase the efficiency of an energy harvesting device. A modal-based analytical model is presented, providing an explicit form of the electro-mechanical coupling for each beam eigenmode. Closed-form expressions for the coupled mechanical response and electrical outputs are obtained, allowing one to draw out a complete parametric study to optimize the device. The optimization procedure is conducted following two steps: first, optimal location and dimensions of a single piezoelectric patch are achieved by maximizing the modal electro-mechanical coupling factor (MEMCF) for each structural mode. Thanks to the proposed analytical approach, it is clearly shown that by putting the piezoelectric patch at the maximum of the strain field in the tapered termination, and by adjusting its length in accordance with the focalization created by the ABH effect, the ABH cantilever produces much higher MEMCFs over a wide frequency range and thus outperforms those of a uniform beam. Second, optimization of the shunted circuit is comprehensively performed for a circuit with only resistance, or both resistance and inductance, in series or in parallel. Analytical results show that the key design rule resides in matching the time scale of the circuit with that of the forcing frequency. Addition of the inductance allows enhancing the performance, but on a narrow frequency band. Finally, broadband advantages can be further obtained by considering multiple piezoelectric patches, in which the optimum is obtained when the shunted circuit in each patch is tuned targeting an eigenmode of the ABH beam.

Notch-type damage influence on the frequency of the principal mode of the composite cantilever beam flexural vibrations

The results of calculation investigations aimed at the determination of the influence mechanisms of the surface damage in the form of a rectangular notch and fibers’ orientation on the frequency of the principal mode of the composite cantilever beam flexural vibrations are presented. The three-dimensional finite element model of the beam made of the unidirectional carbon-fiber-reinforced plastic with the local surface rectangular notch-type damage has been developed using the effective modulus concept to determine the composite material elastic properties. Its adequacy is corroborated via the comparison of the determination results of the principal frequency of flexural vibrations of the intact beam as a function of the degree of anisotropy in composite material with the known experimental data. In this paper has been found the mechanisms of influence of the damage parameters (notch depth and its location along the beam length) and the degree of anisotropy of the elastic properties in composite material, which is defined by the angle of the fibers, on the variation of the frequency of the principal mode of flexural vibrations. It is established that there is such a notch location along the beam length for its specified depth and width, the so-called transition point, where the natural frequency of vibrations of the damaged and intact beams are equal independent of the angle of the fibers. This fact indicates that there is a simultaneous damage effect on both the variation of flexural stiffness of the beam and its inertia properties. The location of the transition point depends on the notch depth and fibers’ orientation. With the larger distance of the notch from the transition point to the free end, the principal frequency of vibrations of the damaged beam becomes larger as compared with that one for the intact beam.

2Pb4-4 Sound field between an object and a thin ultrasound touchless sensor using flexural vibration

2Pb4-4 Sound field between an object and a thin ultrasound touchless sensor using flexural vibration

ДОСЛІДЖЕННЯ ВПЛИВУ ПІДВИЩЕНОЇ ТЕМПЕРАТУРИ НА ДИСИПАТИВНІ ВЛАСТИВОСТІ ОДНОСПРЯМОВАНОГО ВУГЛЕПЛАСТИКУ

An experimental method for determination of composite materials dissipative properties, the main characteristic of which is the minimization of external energy losses and ensuring uniform distribution of deformations along the specimen working part under the pure bending condition has been developed. The scheme of the modernized experimental setup is presented. Its mechanical system represents a closed circuit in which the energy in material specimen under flexural vibrations is dissipated. Automated testing and registration of vibration characteristics were carried out by the method of attenuating vibrations at different temperatures. Based on the results for a set of experimental studies on the determination of the dissipative properties of unidirectional CFRP specimens under the pure bending conditions, the dependencies of the logarithmic decrement of vibrations on the relativedeformation in the range (2...12)∙10–4for temperatures from 300 K to 423 K was established.To analyze the effect of temperature on the dissipative properties of the composite, a parameter is proposed that is determined by the increment of the logarithmic decrement of vibrations (LDV) before deformation. It is shown that it is a more informative indicator of the structural integrity of carbon fiber samples, even with a relatively small violation of it due to heating and simultaneous cyclic deformation, compared to the relative growth of the LDV and the drop in the vibration frequency.The obtained experimental data and the presented method of investigation the dissipative properties of materials can be used as a directory data in the design of new models of equipment and in the development of mathematical models of composite structural elements operating under dynamic loads and elevated temperatures.

Highly Sensitive Temperature Sensor Based on Coupled-Beam AlN-on-Si MEMS Resonators Operating in Out-of-Plane Flexural Vibration Modes.

This paper reports a type of highly sensitive temperature sensor utilizing AlN-on-Si resonators with coupled-beam structures of double- and triple-ended-tuning-fork (D/TETF). For both resonators, the out-of-plane flexural mode is adopted as it favors the effect of thermal mismatch between the composite layers inherent to the AlN-on-Si structure and thus helps attain a large temperature coefficient of resonant frequency (TCF). The analytical model to calculate TCF values of D/TETF AlN-on-Si resonators is provided, which agrees well with the finite-element simulation and experimental results. The resonant temperature sensor is built by closing the loop of the AlN-on-Si resonator, a transimpedance amplifier, a low-pass filter, and a phase shifter to form an oscillator, the output frequency of which shifts proportionally to the ambient temperature. The measured sensitivities of the temperature sensors using D/TETF resonators are better than -1000 ppm/°C in the temperature range of 25°C~60°C, showing great potential to fulfill the on-chip temperature compensation scheme for cofabricated sensors.

Reduction of Flexure Vibration for HDD Suspension by Damper

Reduction of Flexure Vibration for HDD Suspension by Damper

Parallel Measurements of Vibrational Modes in a Few-Layer Graphene Nanomechanical Resonator Using Software-Defined Radio Dongles

Software-defined radio dongles are small and inexpensive receivers well known to amateur radio enthusiasts. When connected to an antenna, they enable monitoring of a wide range of the radio spectrum by conditioning the input signal and transferring a downconverted version of it to a personal computer for software processing. Here, we employ a composite of two such dongles, interfaced with codes written in MATLAB and GNU Radio, as a measuring instrument to study the flexural vibrations of a few-layer graphene nanomechanical resonator. Instead of an antenna, we connect the dongles to the split output of a photodetector used to detect vibrations optically. We first perform a quantitative analysis of the dynamics of the first vibrational mode. We then measure the response of the first two vibrational modes in parallel. To illustrate our technique, we detect changes in the vibrational amplitude of both modes induced by periodic strain modulation with a delay of $\approx1$ ms between measurements. Last, we show that our software-based instrument can be employed to demodulate human voice encoded in the vibrations of our resonator. For parallel measurements of several frequency channels, and provided that the input signal is not too weak, our composite system may offer an alternative to the use of multiple lock-in amplifiers or multiple spectrum analyzers, with the distinct advantage of being cost-effective per frequency channel.

Diagnostics of the presence of a fatigue crack in the compressor blade using the spectrum of vibration amplitudes at the superharmonic resonance

In this study, the solution of the forced vibration response of a structural element with a fatigue crack was carried out using the finite element method to determine the influence of its presence on the flexural forced vibration behaviour of the compressor blade airfoil at the superharmonic resonance of the order 1/2. The blade airfoil with a low twisted angle was used as an object of investigation to perform the computational analysis. Its vibrations in the plane of minimum stiffness were excited by the kinematic displacement of root edge elements. The fatigue crack was modelled as a mathematical cut. Two locations of the crack were investigated – on the leading edge and convex side of the blade airfoil. The nonlinearity due to the intermittent contact of the crack surfaces, which is caused by the opening and closing of the crack during each vibration cycle, was taken into account by solving the contact problem. To quantify this kind of nonlinear dynamic behaviour, the vibration diagnostic parameter was defined as the displacement amplitude ratio of the dominant harmonics at the superharmonic resonance of the order 1/2. Based on the results of the calculations it has been found that regardless of the crack location, the ratio nature is the same for all vibration axes. However, with vibrations in the plane of minimum stiffness, the crack on the convex side of the airfoil has an opening mode propagation, which makes it possible to fix its location due to a sharp change in the ratio of the amplitudes of the dominant harmonics along the corresponding axis.