Flexural Vibration Research Articles

Tapered rainbow metabeam for wideband multimode acoustic blocking based on quadruple-mode resonators

We construct a rainbow metamaterial for multimode sound blocking over a broad range of sub-kHz frequencies in the form of a tapered rectangular cross section beam of machined cells based on elements that can, on average, simultaneously attenuate the majority of the possible elastic-wave polarizations. Using aluminum, we construct a five-cell structure containing sub-wavelength planar resonators with interconnected ribs, which couple to compressional, in-plane shear, flexural, and torsional vibrations. Backed up by numerical simulations, we verify that this tapered structure can, on average, strongly attenuate acoustic modes over the frequency range of the combined metamaterial bandgaps, that is over a frequency range representing ∼50% around ∼0.7 kHz. Applications include vibration isolation.

Metamaterial beams with graded two-stage inertial amplification and elastic foundation

In this study, a novel local resonant metamaterial (LRM) beam with elastic foundation is proposed by introducing grounded local resonators based on the two-stage inertial amplification structure with adjustable large inertia. The complex band structure is calculated by using the extended plane wave expansion method, and it is found that bandgaps of the novel LRM beam are lower and wider than that of the existing inertant LRM beam. The formation mechanisms of such bandgaps are revealed, and the effects of structural parameters on bandgap characteristics are discussed. Numerical analysis and experimental measurement of vibration transmission characteristics of finite beam structure verify the accuracy of the theoretical results. Furthermore, three kinds of bandgap widening methods based on the structure tunability of local resonator are proposed by setting graded structural parameters. The regularities of bandgap variation with different parameter spacings are given, and the effectiveness of the method is verified by numerical analysis of the finite graded LRM beam under different boundary conditions. These results are expected to provide certain guidance for low and ultra-low frequency broadband flexural waves and vibration control.

Control of the surface profile of powder using the flexural vibration of a V-shaped plate

In the pharmaceutical field, a technique to level the surface of powdered medicines is important in the dispensing process because the flatness of the powder surface affects the packaging accuracy directly. This paper investigates a method for leveling of the surface profile of powder using ultrasound vibration on a plate. The system used comprises a V-shaped plate and two ultrasound transducers, and the plate configuration required was determined from the results of simulations performed by finite element analysis. The resonant longitudinal vibration of the transducers generated the resonant flexural vibration mode of the plate at 27.4 kHz, which resulted in transportation and leveling of the powder. The powder was aggregated at the nodal positions of the flexural standing wave, and a correlation was observed between the sound pressure distribution over the plate and the surface profile of the powder. The powder leveling accuracy was investigated by varying the driving phase difference between the transducers, and it was found that a smaller standing wave ratio for the flexural vibration produced higher leveling accuracy. When the input voltage to the transducers was increased, the leveling time decreased and the leveling accuracy improved; the highest leveling accuracy obtained was 2.2 mm at 130 Vpp.

Broadband low-frequency flexural wave attenuation in beam-type metastructures with double-sides inertial amplified resonators

We present the theoretical investigation on the flexural wave propagation and vibration attenuation characteristics in a Euler–Bernoulli beam-type metastructure with double-sides inertial amplified resonators. Based on Bloch theory, the dispersion relation and propagation characteristics of flexural waves are calculated by the spectral element method. The influences of system parameters on the flexural wave attenuation of the proposed beam-type metastructures are analyzed. Results show that broadband low-frequency flexural wave attenuation can be achieved by the proposed double-sides beam-type metastructures. The tunability of flexural wave band gap and vibration suppression performance significantly enhances. In addition, the low-frequency flexural wave band gap can be effectively modulated and optimized by changing the system parameters.

Development of an analytical model for the flexural vibration of fish bone active camber structures with truncated, variable thickness partitions

The need for smooth, gapless mechanisms that offer superior aerodynamic performance has been the prime mover for a wide range of studies on the morphing concepts from the early years of the twenty-first century. In light of the importance of novel analytic means for inspecting such structures, this article describes a comprehensive dynamic model for the flexural vibration of a special type of morphing structure known as fish bone active camber (FishBAC). In view of the shortcomings of previous models, the developed procedure considers the variable thickness of the central spine of FishBAC as well as the possibility of arbitrary truncations in its numerous partitions to offer more accurate results. A general idea is first proposed based on the assumptions of classical plate theory for such partitioned structures. The equations are derived using the energy method and the assumption of artificial springs to satisfy the continuity of shear force and bending moment along the connecting lines between partitions. A robust Rayleigh-Ritz method along with a set of fast-converging admissible functions is presented to solve the governing equations of motion. Combined with the power of Mathematica and validated by other analytic models and finite element codes, this technique can be used to deal with a plethora of similar systems while paving the path for the dynamic analysis of FishBAC built around various airfoils.

Application of a Modified Differential Quadrature Finite Element Method to Flexural Vibrations of Composite Laminates with Arbitrary Elastic Boundaries

This paper formulates a modified differential quadrature finite element method (DQFEM) by a combination of the standard DQFEM and the virtual boundary spring technique, which makes it easy to implement arbitrary elastic restraints by assigning reasonable values to the boundary spring stiffnesses. This new formulated method can offer a unified solution for flexural vibrations of composite laminates subjected to general elastic boundary combinations including all the classical cases. The influences of the number of Gauss–Lobatto nodes and the boundary spring stiffnesses on the convergence characteristics of natural frequencies are investigated, and some conclusions are drawn in terms of the minimum number of unilateral nodes required to generate convergent solutions and the optimal values of the boundary spring stiffnesses to simulate classical boundaries. Numerical examples are performed for composite laminates under various classical boundary conditions. Excellent accuracy, numerical stability, and reliability of the present method are demonstrated by comparisons with available exact and numerical solutions in open literatures. Additionally, for elastically constrained composite laminates, which are beyond the scope of most existing approaches, numerous new results obtained by the present method may serve as reference values for other research.

Varifocal Concave-Convex Lens Using Viscoelastic Gel and Ultrasound Vibration.

A varifocal concave-convex lens using ultrasound and transparent viscoelastic gel is reported. The configuration of the lens is simple and thin, consisting of four pieces of a piezoelectric ultrasound transducer, a glass disk, and a transparent silicone gel film. It uses a combination of the ultrasound resonant flexural standing- and traveling-wave modes excited by in-phase and four-phase drives so that the lens can change its shape to both concave and convex by switching the resonance mode with the same structure. The acoustic radiation force (ARF) originated from the resonant flexural vibration modes changed the surface profile of the gel. Convex and concave deformation were generated at the center of the lens at the resonance frequencies of 38 and 60 kHz, respectively, indicating that a varifocal concave-convex lens could be fabricated by controlling the driving frequency, voltage amplitude, and phase differences among the ultrasound transducers. The deformational displacement on the lens surface and the change in the focal length increased with the input voltage amplitude. The optical microscopic images observed through the lens were enlarged 1.28× (reduced 0.92× ) in the convex (concave) mode with 20 [Formula: see text]. The response time for focusing and the temperature stability under operation were evaluated. By switching the resonance vibration modes of the lens through the input signals to multiple ultrasound transducers, the variable-focus function with both concave and convex lenses was achieved in the same configuration.

Electromagnetic and piezoelectric time-varying units for the control of flexural vibrations of thin-walled structures

This paper presents two parallel experimental studies on time-varying semi-active piezoelectric and electromagnetic units for the control of low-frequency flexural vibrations of distributed structures excited by stationary stochastic disturbances. The studies show that the resulting sweeping vibration absorbers effectively reduce the vibration of the hosting structures over the frequency range of the sweeps without the need of a precise tuning, which would require an accurate system identification of the flexural response of the structures. This result suggests that the two semi-active control systems can effectively work under changes of the physical properties of the hosting structures due to operation conditions.

Delamination Detection in Concrete Plates using a Laser Ablation Impact Echo Technique

This study investigates the detection of delaminations in concrete plates using non-contact laser ablation instead of the conventional hammer excitation. Tests were performed on two specimens of which one had an artificial delamination. 5 mm to 15 mm diameter steel tip hammers were used as a reference source and compared to a 7 ns pulsed 1064 nm Nd:YAG laser with 150 mJ pulse energy. Signals were recorded by surface- mounted accelerometers and in air by microphones and a laser Doppler vibrometer. Frequencies up to 150 kHz are excited with a broad frequency spectrum and little energy compared to the narrower frequency spectra of the hammers. The characteristic thickness resonance frequency of the intact plates can be accurately detected by the hammer excitation. The laser can excite low frequency flexural vibration modes over a shallow delamination at 3 cm depth, but does not excite the thickness resonance frequency. The low frequency flexural vibrations are verified by numerical natural frequency analysis.

A higher-order plate theory for the analysis of vibrations of thick orthotropic laminates

In this contribution, flexural vibrations of linear elastic laminates composed of thick orthotropic layers, such as cross-laminated timber, are addressed. For efficient computation, an equivalent single-layer plate theory with eight kinematic degrees of freedom is derived, both in terms of equations of motion at the continuum level and in terms of a finite element representation. The validity of the plate theory is demonstrated by comparing natural frequencies of a simply supported plate over a wide frequency range for which an analytical solution is available. Furthermore, the influence of individual material properties on the accuracy of the plate theory is investigated, demonstrating its broad applicability. The influence of material orientation on the accuracy of the plate theory is examined by comparative finite element simulations. It is shown that, for cross-ply laminates, the plate theory is valid for elements oriented at any angle to the material principal axes.

On the stochastic vibration analysis of the geometrically nonlinear graded cellular curved panels with material stochasticity

In this paper, an effort was made to study the free flexural vibration analysis of the geometrically nonlinear graded cellular curved panels in a stochastic domain. The nonlinear finite element formulations were developed using von Karman nonlinearity hypotheses based on higher-order shear deformation theory, including Lame's parameter. The first-order perturbation technique (FOPT) was employed to assess the stochastic vibration behavior of the graded cellular panels considering randomness in the material properties. The graded cellular curved panels are considered to have three distinguished porosity distributions. The properties of the graded cellular curved panels are continuously varying in the thickness direction as a function of porosity coefficient and mass density. The influences of material stochasticity, porosity coefficient, porosity distribution type, and geometric parameters have been included in the formulations. The detailed convergence and comparison studies were conducted to demonstrate the accuracy and efficiency of the present formulation and highlight the importance of the present study in stochastic environments. The various numerical examples for the second-order statistics (i.e., mean and standard deviation) of the linear frequency parameters (λ) and nonlinear frequencies of the graded cellular structures have been presented which may be considered as the guidelines for future studies in this area of research.

Free and forced small flexural vibrations of slightly curved slender composite beams with interlayer slip

This paper presents a beam theory for analyzing the dynamic bending response of slender slightly curved composite beams whose layers are flexibly connected and therefore subject to interlayer slip. The equations of motion and boundary conditions are derived using Hamilton’s principle, assuming separately for each layer the applicability of Euler–Bernoulli theory and a linear elastic relationship between the interlayer slip and the shear traction. For the problem of a three-layer slightly curved single-span beam with symmetric layer arrangement and soft-hinged bearings, analytical expressions for the natural frequencies and the eigenfunctions are derived. For the arbitrarily supported two-layer beam, on the other hand, a numerical solution scheme of the combined initial boundary value problem is presented. Several examples show how important it is to consider even very small deviations from the straight beam axis in the prediction of the dynamic response for slender beams with interlayer slip, in particular when all supports are immovable. The comparison of the beam solutions with the results of much more expensive FE analyses based on plane stress elasticity proves the accuracy of the presented theory.

Double Sub-Resonance Mechanism in Torsional–Flexural Vibrations of a Double-Track Short Bridge Under a Moving Train

Train-induced resonance of a railway bridge occurs when the train speed coincides with the primary resonance speed ([Formula: see text]) of the bridge, from which we can determine the sub-resonance speed as ([Formula: see text]). The primary resonance speed of a short bridge is generally much higher than the operation speeds of current high-speed trains but the corresponding sub-resonance speeds probably lie in the operation speed range. Such a resonance scenario can be observed from the vibration of a double-track bridge subjected to an eccentrically moving train, in which the deck vibration is combined by the vertical-flexural and torsional vibration modes of the bridge. Once the two modal sub-resonance speeds coincide with each other, a double sub-resonance will be developed on the bridge. In this study, an iteration-based train–bridge interaction finite element procedure will be presented to demonstrate the double resonance phenomenon. As expected, the double resonance may bring about a dramatic amplification on deck vibration. Such an excessive vibration is harmful to ballast stability and track maintenance of railway bridges.

A Novel Model of Ultrasonic Fatigue Test in Pure Bending.

The very high cycle fatigue (VHCF) failure of in-service components is mainly caused by the vibration of thin-wall elements at a high frequency. In this work, a novel model of ultrasonic fatigue test was developed to test thin-wall material in bending up to VHCF with an accelerated frequency. The theoretical principle and finite element analysis were introduced for designing a sample that resonated at the frequency of 20 kHz in flexural vibration. In the advantage of the second-order flexural vibration, the gauge section of the sample was in the pure bending condition which prevented the intricate stress condition for thin-wall material as in the root of cantilever or the contact point of three points bending. Moreover, combining the constraint and the loading contact in one small section significantly reduced heating that originated from the friction at an ultrasonic frequency. Both strain gauge and deflection angle methods were applied to verify the controlling of stress amplitude. The fractography observation on Ti6Al4V samples indicated that the characterized fracture obtained from the novel model was the same as that from the conventional bending test.

Development and practical application of locally resonant metamaterials for attenuation of noise and flexural vibration of floors in residential buildings

In this paper, a new locally resonant metamaterial (LRM) is proposed to attenuate noise and vibration of floors in residential buildings. The band gap behavior of LRMs has recently attracted attention in the field of noise and vibration absorption. However, no research has been reported in the literature on the practical application of LRMs in buildings and their generalization to civil engineering developments. The LRMs were designed and applied to a floor in an under-construction residential building to perform an in-depth investigation of vibro-acoustic control of the floor impact sound. Experiments were conducted to perform a multifaceted analysis of the band gap behavior of a host structure coupled with LRMs, as well as the effects of attaching different number of LRMs and mode response of the host structure. The experimental results indicated that the acceleration and sound pressure levels of the LRM-coupled structure was attenuated within the band gaps, and the band gap of the host structure that was coupled with LRMs widened to the upper and lower regions of the local resonance frequency.

Variable stiffness composite laminated beams - nonlinear free flexural vibration behavior using a sinusoidal based shear flexible structural theory accounting for Poisson’s effect

In the present work, formulation for the nonlinear free vibrational behavior of laminated variable stiffness composite beams with curvilinear fibers is made incorporating sinusoidal function for representing the transverse shear flexibility, geometrical nonlinearity, and accounting for Poisson’s effect through the constitutive equation for analyzing beams with general composite lay-up. Applying Hamilton’s principle combining with a finite element approach based on three-noded C1 beam element, the governing equations are formed through matrices. The solutions for the developed governing equations are evaluated iteratively by introducing eigenvalue analysis and the results are viewed through the frequency–amplitude relationship. A large number of design parameters like curvilinear fiber angles in a layer, number of layers, lay-up sequence, thickness ratio, etc. is assumed to visualize the nonlinear free vibration characteristics of VSCL beams. Also, for the practical situation, the influence of thermal environment and restraint elastically against beam ends rotation, to accommodate other than classical boundary conditions, hybrid beam (consisting of constant and variable stiffness layers) on the non-linear free vibrational behavior is presented.

Reflection of waves in a waveguide from a boundary with nonlinear stiffness: application to axial and flexural vibrations

The reflection of time-harmonic waves in a waveguide with a nonlinear boundary stiffness is considered with applications to rods and beams. Incident waves at frequencies that are multiples of a fundamental frequency give rise to reflected propagating and nearfield waves at the same frequencies. An infinite set of equations is developed for the reflection coefficients, which depend on the amplitudes and phases of the incident waves. Nonlinear boundary conditions are applied, and equations is truncated by using the harmonic balance method and solved numerically. The case of zero linear boundary stiffness, i.e. essential nonlinearity, is studied. First, the case where there is only one incident wave is considered. An approximate solution is found when retaining only two reflected waves. Numerical examples are presented, energy being seen to leak into the higher harmonics. The minimum magnitudes of the reflection coefficients of axial and flexural vibrational waves at the fundamental frequency and the maximum energy that can leak into the higher harmonics are determined. Accuracy and convergence when retaining different numbers of reflected harmonics are illustrated. The case of two incident waves is then considered. Multiple incident waves affect the leakage of energy to higher harmonics and can have a significant effect on the reflection coefficient for the fundamental harmonic. With some parameters, a much lower reflection coefficient is obtained for the wave at the fundamental frequency as compared to the case of one incident wave. It is seen that with two incident flexural waves, the reflection coefficients can be multi-valued for certain values of the system parameters. A numerical study is performed to show the region of multiple solutions.

Steady State and Transient Vibration Analysis of an Exponentially Graded Rotor Bearing System Having a Slant Crack

The dynamic behaviour of a slant-cracked exponentially graded (EG) rotor-bearing system has been investigated using the finite element method for flexural vibrations. A two nodded EG rotor element has been developed based on the Timoshenko beam theory. Local flexibility coefficients (LFCs) of a slant-cracked EG shaft element have been derived using fracture mechanics concepts to develop the stiffness matrix of a cracked EG element. The steady-state and transient vibration responses of cracked and uncracked rotor systems have been simulated using the Houbolt time marching method. When a crack is present in the shaft, the subharmonic frequency peaks are centred on operating speed in the steady-state frequency responses, whereas on critical speed in the transient frequency responses at an interval frequency corresponding to the torsional frequency. It has been found that the crack parameters such as crack depth and location, temperature gradients and torsional frequencies have a significant influence on natural frequencies and dynamic responses, which could be implemented for efficient rotor crack detection methodologies.

Analytical solution for vibration and buckling of cylindrical sandwich panels with improved FG metal foam core

In this research paper, a novel higher-order porosity distribution is proposed, and free vibration and buckling behavior of cylindrical sandwich panels with functionally graded metal foam core are investigated. It is assumed that the sandwich panel is made of two thin face metal sheets and a moderately thick functionally graded metal foam core. The dynamic governing equations are obtained based on the third-order shear deformation theory and utilizing the Hamilton principle. It is assumed that the voids are distributed non-uniformly along the thickness of the functionally graded metal foam core. The elastic modulus of the core depends on the square of the relative density, thus is changing nonlinearly through the thickness of the core. The Navier technique is implemented to derive closed form solutions for the natural frequencies and the buckling load of the simply supported cylindrical sandwich panels. The accuracy and effectiveness of the proposed model are verified by comparison with previous researches. Results revealed that the presented higher-order porosity distribution can increase the buckling load and fundamental natural frequency by 10% and 5% in comparison to the conventional porosity distributions, respectively. In addition, it is shown that if the flexural vibration is of interest, ignoring the in-plane and rotary inertias, which can significantly reduce the computational complexity, does not have any considerable effect on the natural frequencies.

Application of the Higher-Order Hamilton Approach to the Nonlinear Free Vibrations Analysis of Porous FG Nano-Beams in a Hygrothermal Environment Based on a Local/Nonlocal Stress Gradient Model of Elasticity.

Nonlinear transverse free vibrations of porous functionally-graded (FG) Bernoulli–Euler nanobeams in hygrothermal environments through the local/nonlocal stress gradient theory of elasticity were studied. By using the Galerkin method, the governing equations were reduced to a nonlinear ordinary differential equation. The closed form analytical solution of the nonlinear natural flexural frequency was then established using the higher-order Hamiltonian approach to nonlinear oscillators. A numerical investigation was developed to analyze the influence of different parameters both on the thermo-elastic material properties and the structural response, such as material gradient index, porosity volume fraction, nonlocal parameter, gradient length parameter, mixture parameter, and the amplitude of the nonlinear oscillator on the nonlinear flexural vibrations of metal–ceramic FG porous Bernoulli–Euler nano-beams.