Flexural Frequencies Research Articles

Design and experimental validation of a metamaterial solution for improved noise and vibration behavior of pipes

This paper investigates a metamaterial solution for efficient vibration attenuation and acoustic radiation reduction of an aluminum pipe. To this end, using unit cell predictions, locally resonant structures are designed to have a pronounced flexural resonance frequency at the vicinity of a dominant vibration mode of the pipe. A direct approach of the Bloch-Floquet theorem is adopted to provide the dispersion relation representing wave motion in an infinite metamaterial pipe. Using these wave dispersion relations, the frequency range of the stopband zone created by the metamaterial solution is predicted. The dynamic behavior of the finite counterpart is predicted using the Finite Element Method (FEM). The resonant structures are produced from polymethyl methacrylate (PMMA) panels and are added to the host structure. In order to properly characterize both the vibrational behavior of the metamaterial pipe and the acoustic radiation from its wall, impact tests using roving hammer technique is performed on the pipe and both accelerations and acoustic pressures are measured at different locations. The experimental results show a pronounced stopband zone created by the addition of a few rows of resonant structures. Moreover, comparisons between the measurements and numerical predictions show a good agreement.

Experimental Substantiation of Using Acoustic Noise in Above-Ground Pipeline Diagnostics

The full-scale experiments on acoustic noise recording on the surface of an above-ground pipeline are carried out on operating heating main. The tests were performed in the pipeline branches with different style attachment between pipe and support—rigid (the pipe is welded to the support) and flexible (the freely supported heat-insulated pipe). The experiments show that collection of recorded noise amplitude spectra makes it possible to determine natural frequencies and forms of flexural coincident waves generated by noise in the pipeline spans. Both frequencies and forms of the waves depend on the style of the pipe attachment at the span ends, which may be used in diagnostics of pipeline branches by acoustic noise to detect damaged stiffness of the pipe-support attachment and/or instability of the supports. The computer modeling using the finite element method yields flexural wave frequencies similar to the experiment results. The distributions of nodes and antinodes of flexural coincident waves along pipeline spans at different style pipe-support attachments qualitatively agree with the earlier lab test data.

Tuning forks with optimized geometries for quartz-enhanced photoacoustic spectroscopy.

We report on the design, realization, and performance of novel quartz tuning forks (QTFs) optimized for quartz-enhanced photoacoustic spectroscopy (QEPAS). Starting from a QTF geometry designed to provide a fundamental flexural in-plane vibrational mode resonance frequency of ~16 kHz, with a quality factor of 15,000 at atmospheric pressure, two novel geometries have been realized: a QTF with T-shaped prongs and a QTF with prongs having rectangular grooves carved on both surface sides. The QTF with grooves showed the lowest electrical resistance, while the T-shaped prongs QTF provided the best photoacoustic response in terms of signal-to-noise ratio (SNR). When acoustically coupled with a pair of micro-resonator tubes, the T-shaped QTF provides a SNR enhancement of a factor of 60 with respect to the bare QTF, which represents a record value for mid-infrared QEPAS sensing.

Numerical and experimental investigation on the vibro-impact responses analysis of shrouded blade

A finite element model of shrouded blades is established to analyze the vibration responses due to the impact between adjacent blades. In the finite element model, the blade and the shroud are simulated using beam element and lumped mass, respectively. On the basis of the finite element model, a beam–beam impact model is developed to study the vibro-impact responses of shrouded blades. By comparing the natural frequencies and vibration responses, the finite element model is verified by an analytical model. In addition, a test rig is also set up to compare the results obtained from simulation and experiment, and the influences of different parameters such as shroud gap, excitation amplitude, and excitation frequency on the vibro-impact response are analyzed by finite element model and experiment. The results obtained from finite element model and experimental model indicate that the impacts between adjacent shrouded blades weaken with the increase of the shroud gaps and excitation frequency, and the vibro-impact responses between shrouded blades become strong with the increasing excitation amplitude. The results also show that the second-order flexural vibration of blade is easier to be excited and the super-harmonic resonance phenomenon closed to the second-order flexural natural frequency is more significant than that at other flexural natural frequencies. The reason is that the position of the external force applied on shrouded blade accords with its second-order flexural vibration mode.

Dynamic characteristic study of composite box beam with corrugated webs considering interface slip and shear deformation

In order to study the effect of interface slip and shear deformation on the natural vibration of composite box girders with corrugated webs (CBBCW), based on the Hamilton principle, the effects of the slip, shear deformation and moment of inertia of CBBCW have been investigated. Further, the flexural vibration differential equations and natural boundary conditions of the CBBCW have been deduced. The formula for the flexural natural vibration frequency of CBBCW has been developed. The influencing rule of the interface slip stiffness and web shear deformation on the flexural natural vibration frequency of the CBBCW has been analyzed. The results of this study show that the analytical results are in good agreement with the results of the ANSYS finite element calculation, and some significant conclusions can be applied to the engineering design. The interface slip stiffness has some effect on the flexural natural vibration frequency of the CBBCW, and the maximum interface slip effect is up to 13.7%. The shear deformation has a significant effect on the natural vibration properties of the CBBCW, and the shear deformation effect increases significantly with the increase of the order of flexural natural vibration frequency. The formula has been developed further as compared with the earlier developed calculation theory for the flexural natural vibration of CBBCW. It can also provide a reference for future study of dynamic characteristics of CBBCW.

Flexural cracking-induced acoustic emission peak frequency shift in railway prestressed concrete sleepers

A novel technique for acoustic emission-based condition monitoring of railway prestressed concrete sleepers under flexural loading is established. The evolution of peak frequency of emissions under increasing loads is studied using data from four emission sensors. It is found that the bulk of emitted peak frequencies are clustered in three bands - [150–300] kHz, [300–460] kHz and [500–800] kHz in all cases of full-scale sleepers tested. Only slight variations of band ranges are observed. A correspondence of acoustic emission counts to a particular peak frequency is established. Not all emissions are damage-induced – most of them have relatively small number of counts, suggesting that those are due to random noise. Thus, it is proposed to filter out the non-significant peak frequencies with the least number of counts by applying a universal threshold rule. The largest proportion of emission counts corresponds to mid-span of the sleepers. It is shown that other acoustic emission sources exhibit a nearly linear shift in maximum values of peak frequency with increasing distance from this largest concentration of acoustic emission events. This novel insight into condition-based acoustic emissions is critical to develop a suitable and efficient technique for monitoring safety-critical railway sleepers and bearers located in a discreet and remote area. It will truly enable preventative, predictive and condition-based track maintenance for railway industry, minimizing cost and environmental impacts.

A vibration analysis of a cracked micro-cantilever in an atomic force microscope by using transfer matrix method

In this paper, the effects of crack size and its location have been investigated on the free vibration of an atomic force microscopy (AFM) cantilever applying transfer matrix method. By modeling the crack as a torsional spring and considering the boundary conditions at the contact point with sample's surface, the AFM cantilever vibration behavior has been formulated. Afterwards, the characteristic equation has been derived applying the transfer matrix. At the end, the effects of crack size and its location have been investigated on the flexural resonant frequency and sensitivity of the AFM cantilever. The results indicate that the frequency and sensitivity would be maximum when the crack is approximately in the middle of the cantilever. Growing the crack size can result in the variations of either resonant frequency or sensitivity especially for more stiff samples.

Free vibrations of elastic beams by modified nonlocal strain gradient theory

Size-dependent axial and flexural free vibrations of Bernoulli–Euler nano-beams are investigated by the modified nonlocal strain gradient elasticity model presented in (Barretta & Marotti de Sciarra, 2018). The corresponding elastodynamic problem, with the natural constitutive boundary conditions, is solved by an effective analytical methodology. Axial and flexural fundamental frequencies are determined for cantilever and fully-clamped nano-beams. Effects of nonlocal and gradient scale parameters on fundamental frequencies are examined and compared with those obtained by Eringen's local/nonlocal mixture model. New benchmarks are found for vibrations of beams. The adopted nonlocal strain gradient model, with the appropriate constitutive boundary conditions, is capable of capturing both softening and stiffening dynamical responses. Accordingly, it provides an advantageous approach for design and optimization of a wide range of nano-scaled beam-like components of Nano-Electro-Mechanical-Systems (NEMS).

Damped vibration of a graphene sheet using a higher-order nonlocal strain-gradient Kirchhoff plate model

A higher-order nonlocal strain-gradient model is presented for the damped vibration analysis of single-layer graphene sheets (SLGSs) in hygrothermal environment. Based on Kirchhoff plate theory in conjunction with a higher-order (bi-Helmholtz) nonlocal strain gradient theory, the equations of motion are obtained using Hamilton's principle. The higher-order nonlocal strain gradient theory has lower- and higher-order nonlocal parameters and a material characteristic parameter. The presented model can reasonably interpret the softening effects of the SLGS, and indicates a reasonably good match with the experimental flexural frequencies. Finally, the roles of viscous and structural damping coefficients, small-scale parameters, hygrothermal environment and elastic foundation on the vibrational responses of SLGSs are studied in detail.

Unmanned aerial vehicle-based sounding of subsurface concrete defects.

A sounding technique that uses an unmanned aerial vehicle (UAV) equipped with two microphones can detect subsurface concrete defects. Use of flexural vibration frequency as a basis for defect depth estimation is evaluated. While many non-destructive tests for concrete can detect depth, current UAV-based inspection methods like optical and thermal imaging are typically limited to two-dimensional subsurface defect information. Acoustic signals from sounding and UAV noise are known to exist in similar frequency ranges. Accordingly, three noise reduction measures for this sounding technique are assessed. Given adequate distance between the microphones and UAV, a two microphone signal subtraction technique is slightly effective for some noise, but a spectral noise gating procedure is shown to substantially decrease noise in the frequency range of interest.

An analytical method for free vibration of multi cracked and stepped nonlocal nanobeams based on wave approach

Various discontinuities and boundary conditions affect natural frequencies of nanobeams. In this paper, free lateral vibration of Euler-Bernoulli nanobeam with multiple discontinuities is studied. The governing equations are developed by using Eringen's nonlocal elasticity theory. Cracks and steps are considered as discontinuities. Based on wave approach, vibrations take into account as moving waves along the structure. Waves propagate through waveguide and are reflected and transmitted at the discontinuities and boundaries. The propagation, reflection and transmission matrices are derived for discontinuities and boundaries. Cracks are modelled by a massless torsional spring with infinitesimal length and steps are considered as two connected nanobeams with different cross-sectional areas and mechanical properties. Boundary is formed by combining a torsional and a translational spring. Also buckyball (as a lumped mass) at the tip of the beam is considered as a boundary condition. Using propagation and appropriated reflection and transmission matrices for each case, leads to an analytical comprehensive succinct method so-called wave approach to analyse the free lateral vibration of nanobeams. The flexural natural frequencies are derived analytically for cracked or/and stepped nanobeams with ordinary boundary conditions or buckyball at the tip. The effects of crack severity, changing ratio of cross-sectional area as step, cracks and steps location, mass of buckyball and small-scale parameter on natural frequencies are deliberated. This approach is demonstrated with a number of examples that can be used as benchmarks for other works. The results are compared with other methods.

Unconventional lattice dynamics in few-layer h-BN and indium iodide crystals* *Project supported by the Major Program of Aerospace Advanced Manufacturing Technology Research Foundation from NSFC and CASC, China (Grant No. U1537204), the National Key Research and Development Program of China (Grant No. 2017YFA0206301), and the National Natural Science Foundation of China (Grant No. 51702146).

Unusual quadratic dispersion of flexural vibrational mode and red-shift of Raman shift of in-plane mode with increasing layer-number are quite common and interesting in low-dimensional materials, but their physical origins still remain open questions. Combining ab initio density functional theory calculations with the empirical force-constant model, we study the lattice dynamics of two typical two-dimensional (2D) systems, few-layer h-BN and indium iodide (InI). We found that the unusual quadratic dispersion of flexural mode frequency on wave vector may be comprehended based on the competition between atomic interactions of different neighbors. Long-range interaction plays an essential role in determining the dynamic stability of the 2D systems. The frequency red-shift of in-plane Raman-active mode from monolayer to bulk arises mainly from the reduced long-range interaction due to the increasing screening effect.

Influence of Local Surface Damage on the Natural Frequencies of the Higher Modes of Flexural Vibration of Cantilever Rods

The operating life of gas turbine engines is, by and large, dependent on the reliability of compressor rotor blades that are subjected to a complex set of forces of a different nature during their operation, and in particular, to mechanical damage resulting in severe accidents and material losses. The ingress of foreign objects into the air-gas channel of the engine is one of the causes giving rise to blade damage. As a consequence, various marks, dents, dimples, etc., occur on rotor blades changing the designed geometry of the blade airfoil and their natural vibration frequency spectrum and beginning to act as stress concentrators, thus reducing the vibration resistance of blades. The known investigations on the vibration of damaged mechanical systems, including also compressor rotor blades, have an insufficient amount of data on the formation of the spectrum of their natural frequencies typical of their vibration modes with consideration of the influence of a combined change in the elastic and inertia characteristics. The paper deals with a computational and experimental investigation on the influence of local surface damage on the spectrum of natural flexural vibration frequencies of a cantilever rod with a constant cross section as a simplest model of the compressor rotor blade. The regularities in the variation of the natural frequencies of the first to fourth flexural vibration modes of the rods of different flexibility with the geometrical parameters of the notch simulating damage, such as the position along the length and its depth, are presented. The distribution of variations in the natural frequencies of the rods is found to correspond to the location of the nodes of their vibration mode under investigation. The reduction in the frequency of the damaged rod occurs independently of the vibration mode and the depth of the notch when it is located near the rod attachment, while being more significant with the depth of the notch and less pronounced for the higher vibration modes as compared to the first one, which is attributable to the variation in the rod stiffness. The equality of the natural vibration frequencies of the damaged and undamaged rods is observed at a certain position of the notch along the length irrespective of the vibration mode. As the notch approaches the free end of the rod, the natural vibration frequencies become somewhat higher than those for the undamaged rod, since in this case they are more sensitive to the variation in the inertia properties of the rod due to the presence of damage. With the decreasing flexibility, the variation in the natural frequencies of the investigated vibration modes of the damaged rod increases for the same value of the relative damage depth. The results of the performed computations are in a good agreement with the experimental testing data of the rods.

Vibration of high speed helical geared shaft systems mounted on rigid bearings

Analytical model for linear vibration analysis of a pair of meshing helical gears mounted on compliant spinning parallel shafts is developed. The gears are modeled as rigid disks while the shafts are modeled as flexible rotating cantilever beams. The rotational speed is high such that the gyroscopic effect is non-negligible. Hamilton’s principle is used to obtain the non-dimensional governing equations. Longitudinal and rotational mesh stiffnesses and corresponding elastic energies are incorporated in the total strain energy formulation. Extended operator formulation is used to express the system equations in a consolidated form and later, discretized using Galerkin method. Gyroscopic system structure is apparent in the system equations. Natural frequency sensitivity to various system parameters such as the rotational speed and mesh stiffnesses is determined. Response to loaded static transmission error at the mesh is obtained by performing modal analysis after reducing the system to a first order form. The study shows that gyroscopic effect is present even for short lengths of the shafts. Natural frequency veering between flexural frequencies is present at low speeds and rare at high speeds. For coupled frequencies, response of the system is found to be approximated by modal superposition of smaller number of modes.

A method of broadening the bandwidth by tuning the proof mass in a piezoelectric energy harvesting cantilever

We propose and demonstrate a new method for broadening the bandwidth of a piezoelectric energy harvesting cantilever by tuning a proof mass. Our approach is to make the bandwidth broad by decreasing the difference between two consecutive flexural resonance frequencies of the cantilever. The prototype broadband energy harvesting device consists of a cantilever with double piezoelectric patch and a tuned proof mass, which is composed of two different materials: aluminium and brass. We tuned resonance frequencies of the device based on the optimal design framework. In order to prove the effectiveness of the proposed device, prototypes of two cantilevers, one with a tuned proof mass and the other with a conventional proof mass, were manufactured and the same bimorph cantilever were used in prototypes. Performances of the manufactured prototypes were evaluated and compared. It is observed that the proposed device shows a 426.6 % increase in bandwidth at the same output level and 508.5 % increase in power compared to the conventional device.

Nonlinear bending vibration of a prestressed thick plate

In this paper, the flexural vibration frequency in the antisymmetric mode of a thick plate as a function of the amplitude of the vibration and the axial force applied is investigated. With this aim, the theory of geometrically nonlinear deformation of second order and an optimized three-dimensional Ritz method are used. The plate is homogeneous, elastically linear, free from any constraints, and subjected to axial forces uniformly distributed on two of its opposite sides. Several approaches are discussed. First, the problem based on finite stress and infinitesimal strains is solved. Second, the deformation energy is assumed as the energy in the initial state plus the vibration energy of small or large amplitude. Third, without assumptions about the size of the deformation and of the vibration amplitude, the theory of nonlinear deformation is employed. Finally, numerical calculations for free vibration are compared with experimental results, including their systematic uncertainties.

Possible Areas of Splitting the Flexural Frequency Spectrum of Shells with Asymmetric Shape Imperfections

The influence of initial deviations from the ideal circular shape on the dynamic characteristics of the thin circular cylindrical shell is investigated. The emergence of new zones of splitting the flexural frequency spectrum of imperfect shells is detected.

Free flexural vibration studies on laminated composite skew plates

This paper presents studies made on fundamental flexural frequencies of isotropic and laminated composite skew plates with simply supported and clamped boundary conditions. The fundamental flexural frequencies have been obtained using finite element, the accuracy of which is verified against literature values. The effects of the skew angle, aspect ratio and length-tothickness- ratio on the fundamental frequency of isotropic skew plates are presented. In addition, the effect of parameters such as skew angle, fiber orientation angle, numbers of layers in the laminate and laminate sequence on the fundamental frequencies of antisymmetric composite laminates have also been presented. It is found that the CQUAD8 element yields better results than the CQUAD4 element in the said study of free flexural vibration. The frequencies are found to increase with the skew angle. When the number of layers in the laminate is large, the variation of frequency with the number of layers is not appreciable.Keywords: Skew Plates, Antisymmetric Laminates, Free Flexural Vibration, Fundamental Frequency Coefficient, Frequency Ratio

Damping Capacity of Coated Plane VT6 Titanium Alloy Samples

The damping capacity of a VT6 titanium alloy sample is shown to depend on the thickness of a new designed Al–Ni–Y alloy coating during vibrodynamic tests at the first flexural mode resonance frequency. The coating 20–100 μm thick is found to decrease the vibrostresses in the weakest section of the sample by 9–31% at a temperature of 20°C and by 40.6% at a temperature of 400°C and a coating thickness of 60 μm. The dependences of the damping capacity of the alloy–coating composition on high-temperature holding of coated samples in an atmospheric furnace (t = 400°C, τ = 500 h), the salt fog chamber conditions (t = 35°C, τ = 3 months), and the action of an abrasive flux (quartz sand, average particle fraction of 400 μm, particle velocity of 100 m/s) are studied.

Linear flexural natural frequencies and stability analysis of spinning Rayleigh beams: application to clamped-clamped beams.

In the present paper 3D bending linear free vibrations of spinning Rayleigh beams are investigated. Four linear models, that differ in the linearization process, are studied. A focus on analytical computation of natural frequencies for a broad range of boundary conditions is highlighted. Then, the conditions of occurrence of divergence and flutter instabilities are determined. Finally, a case study consisting of a clamped-clamped Rayleigh beam is studied. It is found that the free vibrations destabilization process depends on the used linearization approach.