Frequencies Of Flexural Vibration Research Articles

Free flexural vibrations of standard wide-flange H-beams with consideration of the shear effect

The subject of the paper is a simply supported standard wide-flange H-beam. Cross sections of this beam is analytically described as a three-layer structure. The shear effect in its succesive layers is taking into account with consideration of the classical shear stress formula called Zhuravsky shear stress. Based on Hamilton’s principle, two differential equations of motion are obtained. These equations are analytically solved and the fundamental natural frequency of flexural vibration for this beam is derived. Examplary calculations are carried out for selected five I-beams.

Effects of Coating Aggregates on Dynamic Properties of Concrete by Impact Resonance Method

Concrete structures may be subjected to dynamic loadings like earthquakes, impact, and vehicular loads. These loads may cause considerable damage to infrastructures and shorten their life span. To reduce the effects, improving concrete’s dynamic properties is important. This can be done using granular elastic components and fibers in the production of concrete. This research explores the influence of using coated aggregates on the physical, mechanical, and dynamic properties of concrete. Experimental investigation on concrete with epoxy, epoxy-sand, and epoxy-crumb rubber as a partial replacement of coarse aggregates at different volume fractions ranging from 5% to 20% was conducted. Furthermore, the dynamic modulus of elasticity of concrete with coated aggregate was determined by measuring the resonant frequencies of flexural vibrations in a prismatic beam using an impactor (hammer). Results indicate that partial replacement of epoxy-crumb rubber coated aggregates in concrete shows a reduction in mechanical proprieties. However, significant improvements in the mechanical and dynamic properties of concrete were observed by the partial replacement of coarse aggregates with epoxy and epoxy-sand coated aggregates. Compressive strength and dynamic elastic modulus were enhanced by 12% and 10%, respectively, when concrete with 15% epoxy-sand coated was used. The results showed that concrete specimens using epoxy and epoxy-sand coated as a partial replacement for coarse aggregate have higher mechanical properties as compared to those of concrete specimens with epoxy-crumb rubber-coated aggregates. Moreover, the results showed that the calculated values of the dynamic modulus of concrete using the empirical relations proposed by Popovics and Hansen were overestimated as compared to the experimental values.

Direct measurement of negative Poisson ratios of auxetic materials via the impulse excitation technique (IET)

The direct measurement of Poisson ratios via the impulse excitation technique (IET) is based on determining the frequency ratio between the resonant frequencies of flexural and torsional vibrations of circular discs. However, the direct measurement of negative Poisson ratios, which occur in auxetic materials, has been impeded until now by the absence of numerical values of Martinček’s solution for frequency ratios smaller than 1.35, because the relevant ASTM standards have not considered the possibility of negative Poisson ratios so far. Here we provide this missing link by reconsidering Martinček’s solution and presenting the necessary values in the forms of graphs and fit relations from which the Poisson ratio, including negative ones (down to −0.273), can be determined from frequency ratios down to 1.1 for a wide range of specimen aspect ratios (down to 4).

Quantifying Shallow-Depth Concrete Delamination using Impact-Echo

Corrosion-induced concrete deck delamination is quite common in bridges. Locating these embedded delaminations is important to assess the extent of the damage, the remaining member capacity and any necessary rehabilitation. The impact-echo (IE) is a simple and straightforward non-destructive testing (NDT) technique by which the depth and extent of concrete delamination may be estimated. It is effective in detecting the location and determining the depth of relatively deep delaminations in concrete. Delaminations that occur near the concrete surface can also be detected by the IE method. However, the exact depth cannot be quantified due to difficulties in analysing the flexural mode that dominates the vibration response over the corresponding delaminated region. This study developed an IE-based procedure to quantitatively estimate the depth of shallow depth delaminations in concrete slab members. Four slab specimens were prepared, each with three artificial delaminations with varying shapes and sizes at shallow depths. The frequency contour maps showed good agreement with the actual location of the delaminations. The perimeter-to-depth ratio of the delaminated region can be used to analyze the flexural mode vibration frequency. Two equations are proposed that relate the depth of arbitrarily shaped delaminations to the flexural mode vibration frequency measured over the shallow-depth delaminations. Keywords: impact echo, concrete delamination, concrete slabs, flexural mode vibration, non-destructive evaluation (NDE)

Damage Detection in Free–Free Glass Fiber Fabric Composite Beams by measuring Flexural and Longitudinal Vibrations

This paper presents the experimental investigation results of the use of the vibration method by impulse excitation of free flexural and longitudinal vibrations of composite materials. The purpose of the study is to establish the sensitivity of the method used for defect detection and localization. To realize the objective, rectangular notch-type defects were simulated at different distances and depths. The influence of the location and depth of the artificial cracks on the dynamic properties of the beams was investigated by measuring the natural frequencies of flexural and longitudinal vibrations. The conducted experiments show a change in the dynamic characteristics of the beam depending on the dimensions and location of the defects.

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.

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.

Investigation of vortex-induced vibration of a cable-stayed bridge without backstays based on wind tunnel tests

Cable-stayed bridges without backstays located in the abrupt valley are sensitive to wind-induced structural vibrations. The understanding of vortex-induced vibration for specific aerodynamic configuration with glass deck is of great importance for structure design. This paper thus aims to investigate the vortex-induced vibration of a cable-stayed bridges without backstays through the segmental model test and scaled full-aeroelastic model wind tunnel test. The results indicate that the bridge presents the vertical vortex-induced vibrations at the low wind speed whereas the maximum wind velocity causing the torsional vortex-induced vibrations increases to 41 m/s, which is much larger than that of the traditional bridge. The vertical and torsional vortex-induced vibrations in the smooth flow are observed at the positive wind angle of attack, owing to the responses for the bridge deck present a sharp single-frequency amplitude when the frequency of vortex-shedding is close to the flexural vibration frequency. The turbulence can effectively suppress the vortex-induced vibrations and only some small peaks are visible in the turbulent flow. The vortex-induced response of the bridge deck is significantly alleviated with increasing of damping ratio. Finally, the vortex-induced vibrations in the uniform flow disappear and the wind-induced responses are impaired with the ventilated railing in place of the airtight railing because the flow can go through the railing. It can provide a designing basis for engineering reference of cable-stayed bridges without backstays.

Band Gaps Characteristics Analysis of Periodic Oscillator Coupled Damping Beam.

The vibration of the periodic oscillator coupled damping beam model is reduced through the band gaps designing method, which can be applied in equivalent engineering structures. In this paper, the flexural wave dispersion relations of the infinite long periodic oscillator coupled damping beam were calculated using the reverberation-ray matrix method combined with the Bloch theorem. The flexural wave vibration frequency response function of the finite long periodic oscillator coupled damping beam was carried out using the finite element method. The flexural wave vibration band gaps occur in the infinite long periodic oscillator coupled damping beam model in both the analytical and numerical results. In these band gaps, flexural waves’ propagation is prohibited, and flexural vibration is significantly suppressed. Furthermore, the effects of structure and material parameters on the flexural wave vibration band gaps characteristics are studied. Thus, the structural vibration reduction design can be realized by adjusting the related parameters of the periodic coupled damping beam structures and the equivalent 2D periodic stiffened plate structures.

Laser processing of metal foam - A review

Abstract Some important properties of the metal foams that make them attractive materials for various purposes are their excellent stiffness to weight ratio (light-weight structure), ability to take up strain by crushing at controlled pressure (energy and impact absorption), order of magnitude higher damping capacity as compared to the solids (mechanical damping), higher natural flexural vibration frequencies (vibration control), reasonable sound absorbing capacity accompanied by durability and fire resistance (acoustic absorption), and, high area of contact accompanied by high thermal conductivity (heat exchanger). Often foam is joined or connected to cover sheets of the same metal and the whole part together is called a foam sandwich. Among all the metal foams aluminium metal foam has found the widest and diverse application. This article covers important recent literature on laser processing viz. laser forming, laser welding, laser cutting and laser additive manufacturing of metal foams. Prior to this, a link is established between foam/foam sandwich and laser processing techniques stating the limitations of other relatively more conventional processing techniques. Future research direction regarding different laser processing techniques using metal foams has also been outlined.

Determination of Elastic Properties of Beech Plywood by Analytical, Experimental and Numerical Methods

This research article examines the application of various methods to determine the effective elastic properties of beech veneer-wood composites. Using laminate theory, the theoretically calculated effective values of the in-plane and out-of-plane modulus of elasticity as well as shear modulus are compared with the values determined from the natural frequencies of flexural, torsional and longitudinal vibrations of samples having different orientations and numbers of composite layers. The samples are also modelled using the finite element method, and their natural frequencies are calculated by the modal analysis. Research has shown that the laminate theory, which is well established and applied in the world of synthetic composites, can also be applied to beech plywood composites, where the theoretically calculated effective values can be up to 15% higher. Similarly, due to the higher calculated effective elastic properties, higher natural frequencies of flexural, torsional and longitudinal vibrations are also calculated by the finite element method.

Effects of the Variational Consistency and Variational Inconsistency on Free Flexural Vibration Frequencies of Simply-supported Rectangular Isotropic Shear Deformable Beams

In order to derive governing differential equations of a displacement-based shear deformation beam theory, one can either utilise the variationally consistent approach (VCA) or the variationally inconsistent approach (VIA). In the VCA, variational principles are utilised to derive beam governing equations. Whereas in the VIA, beam gross / integrated two-dimensional equilibrium equations are utilised to derive beam governing equations. Shimpi et al. [2] have proposed the single variable shear deformation beam theory (SVSDBT) which is based on the VIA. Whereas, Shimpi et al. [1] have proposed the two variable shear deformation beam theory (TVSDBT) which is based on the VCA. In the SVSDBT and TVSDBT, the beam axial displacement and beam transverse displacement consist of bending and shearing components. In both theories, bending components do not take part in the cross-sectional shearing force, and shearing components do not take part in the cross-sectional bending moment. Both theories utilise linear strain-displacement relations. Displacement functions of both theories give rise to the beam transverse shear strain (and hence to the beam transverse shear stress) which varies quadratically through the beam thickness and maintains transverse shear stress-free beam surface conditions. For the case of dynamics, the SVSDBT has only one governing equation, and the TVSDBT has two governing equations which are inertially coupled. The aim of this paper is to present the comparison study on free flexural vibration frequencies of simply-supported rectangular isotropic shear deformable beams obtained by utilising the SVSDBT and TVSDBT. These frequencies are obtained for various values of the beam thickness-to-length ratio and beam vibration mode parameter. This comparison shows that for both theories, the VCA and VIA have a minimal effect on resulting beam free flexural vibration frequencies.

Determination of the Mass-Flow Rate of Blood in a Blood Vessel Using Natural Frequencies of Flexural Vibrations

The natural flexural vibrations of a blood vessel with moving blood are investigated. The vessel is subjected to the action of tensile force and pressure in it. The forces of inertia of the blood vessel, as well as the Coriolis and centrifugal forces induced by the motion of blood, are taken into account. The wave numbers are determined, and the frequency equation is found using the boundary conditions. Two frequencies of flexural vibrations can determine the speed parameter, the relative mass of blood per unit length of the blood vessel, and, as a consequence, the mass-flow rate of blood through the blood vessel. The results obtained can be used for the acoustic method of determining the blood speed, the relative blood mass per unit length of the blood vessel, and the mass-flow rate of blood through it.

Vibration-Based Experimental Identification of the Elastic Moduli Using Plate Specimens of the Olive Tree

Mechanical parameters of the olive wood plate have been computed by data inversion of vibrational experimental tests. A numerical-experimental method has allowed the evaluation of the two transverse shear moduli and the four in-plane moduli of a thick orthotropic olive tree plate. Therefore, the natural flexural vibration frequencies of olive trees plates have been evaluated by the impulse technique. For our purposes, we define the objective function as the difference between the numerical computation data and the experimental ones. The Levenberg–Marquardt algorithm was chosen as optimization strategy in order to minimize the matching error: the evaluation of the objective function has required a complete finite element simulation by using the ANSYS code. As input, we have used the uniaxial test data results obtained from the olive plates. The converged elastic moduli with n = 10 natural modes were E1 = 14.8 GPa, E2 = 1.04 GPa, G12 = 4.45 GPa, G23 = 4.02 GPa, G13 = 4.75 GPa, ν12 = 0.42, and ν13 = 0.42. The relative root mean square (RMS) errors between the experimental frequencies and the computed one is 9.40%. Then, it has been possible to obtain a good agreement between the measured and calculated frequencies. Therefore, it has been found that for plates of moderate thickness the reliability of the estimated values of the transverse shear moduli is good.

Flexural vibrations of the pipeline on elastic supports with moving fluid

In the work investigated the flexural vibrations of the pipeline. Parts of the pipeline on both sides of the sagging section have elastic supports. It is assumed that a constant longitudinal force acts along the neutral line. An incompressible fluid flows through the pipe at a constant average speed. The influence of internal pressure in the pipe on these oscillations is taken into account. The direct problem of determining the eigenfrequencies of flexural vibrations of the pipeline by the Kirchhoff model using Ferrari formulas is solved. The frequency spectrum is determined depending on the fluid pressure, the elasticity of the supports, the velocity of the fluid through the pipe. Particular and limiting cases are considered, for example, when the stiffness of the supports is very large and when they are very small. Graphs of the dependence of the first and second eigenfrequencies on the velocity of the transported liquid at different values of the liquid density parameter are constructed. It is shown that with the growth of the velocity parameter there is a decrease in the natural frequencies of flexural vibrations of the pipeline, and the faster the higher the density parameter of the liquid. It is determined that with an increase in the mass of the liquid per unit length of the pipeline there is a decrease in the natural frequencies of bending vibrations of the pipe. It is found that with the increase in the mass flow through the pipe, the natural frequencies of bending oscillations also decrease. It is confirmed that the frequencies of flexural vibrations of the pipeline are the same for the cases of pipe fastening “rigid fixing — rigid fixing” and “free end — free end”. The results of the study will contribute to the development of methods of acoustic diagnostics and non-destructive testing and can find technical application for monitoring and diagnostics of pipeline systems.

THE STUDY OF ACOUSTIC CHARACTERISTICS OF THE PERFORATED AND HOMOGENOUS PLATES WITH SIMILAR GEOMETRICAL DIMENSIONS

The resonance frequencies of the flexural vibrations, input vibration excitability and acoustic radiation of the homogeneous and perforated plates were investigated. It is established that the average reduction range of the lower resonant frequency of flexural vibrations of the tested plates with the holes virtually coincides with the predictive estimate. The levels of the input vibration excitability of the perforated plate at the lower resonant frequencies exceeded the levels at the corresponding frequencies of the homogeneous plates greater than the calculated value. The levels of resonance acoustic radiation of the perforated plate were significantly less than of the homogeneous one.

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.

Analysis of natural frequency of flexural vibrations of a single-span beam with the consideration of Timoshenko effect

This paper presents general solution of boundary value problem for constant cross-section Timoshenko beams with four typical boundary conditions. The authors have taken into consideration rotational inertia and shear strain by using the theory of influence by Cauchy function and characteristic series. The boundary value problem of transverse vibration has been formulated and solved. The characteristic equations considering the exact bending theory have been obtained for four cases: the clamped boundary conditions; a simply supported beam and clamped on the other side; a simply supported beam; a cantilever beam. The obtained estimators of fundamental natural frequency take into account mass and elastic characteristics of beams and Timoshenko effect. The results of calculations prove high convergence of the estimators to the exact values which were calculated by Timoshenko who used Bessel functions. Characteristic series having an alternating sign power series show good convergence. As it is shown in the paper, the error lower than 5% was obtained after taking into account only two first significant terms of the series. It was proved that neglecting the Timoshenko effect in case of short beams of rectangular section with the ratio of their length to their height equal 6 leads to the errors of calculated natural frequency: 5%÷12%.

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.

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.