Flexural Oscillations Research Articles

On the stability of prestressed beams undergoing nonlinear flexural free oscillations

We study the nonlinear free undamped motions of a hinged-hinged beam exhibiting geometric stretching-induced nonlinearity and arbitrary initial conditions. We treat the governing integral-partial-differential equation of motion as an infinite dimensional Hamiltonian system. We analytically obtain a quantitative Birkhoff Normal Form via a nonlinear coordinate transformation that yields the reduced (modulation) equations describing the free oscillations to within a certain nonlinear order with an estimate of the reminder. The obtained solutions provide a very precise description of small amplitude oscillations over large time scales. The analytical optimization of the involved estimates yields time stability results obtained for plausible values of the physical quantities and of the perturbation parameter. The role played by internal resonances in determining the time stability of the solution is highlighted and discussed. We show that initial conditions with a finite number of eigenfunctions yield bounded solutions living on invariant subspaces of the involved modes at all times. Conversely, initial conditions comprising the full (infinite) spectrum of eigenfunctions provide solutions for which time stability for all times cannot be stated.

Dynamics of a wall-mounted cantilever plate under low Reynolds number transverse flow in a two-dimensional channel

The present work numerically investigates the dynamics of an elastic two-dimensional cantilever plate fixed at the bottom wall of a channel carrying flow using an open-source multi-physics computational fluid dynamics solver, SU2. Chief non-dimensional parameters, viz., Cauchy number (Ca), channel height, and mass ratio, are explored to predict the structural response of the plate interacting with the laminar parabolic profile in the channel at relatively low Reynolds numbers (Re=20−120). For a steady inflow, we show the existence of two distinctive modes of plate flexural oscillations, namely, F1 and F2, where the plate attains self-sustained periodic oscillations close to its first and second natural frequencies, respectively, for discrete ranges of Ca and three static modes, namely, S1, S2, and S3 for the other ranges of Ca in which steady-state configuration is obtained. The physical reasons underpinning the flow-induced oscillations and static shapes are examined using scaling arguments. F1 oscillations are shown to be vortex-induced oscillations, which get suppressed at low enough channel height, owing to higher viscous dissipation. Additionally, the window of F1 zone was found to shift to lower Ca with an increase in the mass ratio. Increasing the Reynolds number was found to cause the F1 zone to diminish in size, and beyond a critical Reynolds number, F1 was completely suppressed. On the other hand, F2 oscillations, which are shown to be induced by an unsteady drag force, are found to exist throughout the range of Re considered in the study.

Use of Magnetostrictive Actuators for Wave Energy Conversion with Improvised Structures

This paper presents work on a wave energy device with an on-board power take-off based on a magnetostrictively actuated deformable structure. Such devices potentially could be used in low-cost, short-term expeditionary operations. The paper discusses an analytical model that describes the heave oscillations of a buoy with two inclined, overhanging beams with magnetostrictive strips affixed to them. This work comprises the first steps toward an analytical model that would enable potential users to obtain quick power estimates at the planning stage. Here, the fully nonlinear magneto-mechanical-electrical constitutive relations are linearized about a desirable operating point, and a coupled dynamic model is derived using a variational formulation that includes buoy heave, flexural oscillations of the two beams, and the voltage response of the magnetostrictive strips. Energy conversion performance in wind-sea-dominated Pierson–Moskowitz spectra is found to be modest. However, present results also indicate that performance could be improved with suitable mechanical modifications.

Benefits of controlled inclination for contactless transport by squeeze-film levitation

Developed in this paper is a theoretical description of the fluid flow involved in contactless transport systems that operate using squeeze-film levitation. Regular perturbation methods are employed to solve the appropriate Reynolds equation that governs the viscous, compressible flow of air in the slender film separating the oscillator and the levitated object. The resulting reduced formulation allows efficient computation of the time-averaged levitation force and moment induced by fluid pressure, as well as the accompanying quasistatic thrust force that accounts additionally for shear stresses. Investigated, in particular, is the possibility of combining two distinct methods of thrust generation that have been experimentally demonstrated in previous studies – (i) inclination of the levitated body and (ii) generation of asymmetrical flexural deformations, such as travelling waves, on the oscillator surface – the latter of which is shown to allow a transition from the typically repulsive levitation force to one that is attractive. Computations reveal that systematic control of the inclination angle can provide significant performance benefits for squeeze-film transport systems. In the case of attractive levitation, the amount of improvement that can be obtained appears to correlate closely with the degree of lateral asymmetry exhibited by the flexural oscillations.

Analysis of flexural oscillations of the shaft of saw cylinder on the elastic bearing supports of the fiber cleaner

The results of analysis of flexural oscillations of the shaft of saw cylinder on the elastic bearing supports of the fiber cleaner are presented in the article. A design diagram, a mathematical model of flexural oscillations of the shaft of saw cylinder on the elastic bearing supports of the fiber cleaner are given. Analytical and numerical solutions of the problem of oscillations of the saw cylinder shaft are given. The main parameters of the system are substantiated.

Вплив імпульсних рульових двигунів на динаміку твердопаливної ракети

The object of this study is a rocket with a solid propellant engine launched from a vehicle launcher. The current work studies the oscillations of a rocket airframe in flight under the influence of impulses of the steering motors and the effectiveness of flight control along the trajectory by the steering motors located in the head of the solid rocket. Analytical and numerical methods for determining the frequencies of free vibrations of a rigid body and analytical methods for integrating differential equations of motion are applied. A mathematical model of rocket flexural oscillations has been constructed. Methods for determining the modes and frequencies of natural oscillations have been analyzed. Three lower frequencies and modes of natural oscillations of the rocket were calculated. The discrepancy between the results of the calculation by the computer program and the analytical calculation of the first approximation is 3%. The calculation of oscillations of a rocket with a caliber of 30 mm and a length of 7655 mm showed that torsional and longitudinal oscillations have relatively high frequencies and have little effect on the stability and controllability of the rocket flight. The vibrations of the rocket airframe, which has a gas-dynamic flight control system with the help of rudders and a gas-jet one with the help of steering engines, are studied. Steering motors are small-sized disposable solid propellant motors, which are located in five rows around the perimeter of the rocket airframe at the head. Methods for calculating the forced oscillations of a rocket during the operation of steering engines have been developed. Under the action of the control pulse, transverse oscillations of the rocket airframe y are excited and simultaneously, the rocket rotates as a solid body around the center of mass. Airframe oscillations quickly decay, and rotation changes the pitch or slip angle. To stop the rotation and leave some required angle of inclination or azimuth of the trajectory, a second impulse of force must be given by the steering motor mounted on the opposite side of the hull. The dependence of the change in pitch and slip angles on the duration of time intervals between control pulses was studied. The analysis of the dependence obtained shows that, although the weight of the fuel charge is approximately half the weight of the rocket, the controllability by the steering engines is equally effective at any fuel consumption.

Ballistic dynamics of flexural thermal movements in a nanomembrane revealed with subatomic resolution.

Flexural oscillations of freestanding films, nanomembranes, and nanowires are attracting growing attention for their importance to the fundamental physical and optical properties and device applications of two-dimensional and nanostructured (meta)materials. Here, we report on the observation of short–time scale ballistic motion in the flexural mode of a nanomembrane cantilever, driven by thermal fluctuation of flexural phonons, including measurements of ballistic velocities and displacements performed with subatomic resolution, using a free electron edge-scattering technique. Within intervals <10 μs, the membrane moves ballistically at a constant velocity, typically ~300 μm/s, while Brownian-like dynamics emerge for longer observation periods. Access to the ballistic regime provides verification of the equipartition theorem and Maxwell-Boltzmann statistics for flexural modes and can be used in fast thermometry and mass sensing during atomic absorption/desorption processes on the membrane.

Torsional and lateral eigenmode oscillations for atomic resolution imaging of HOPG in air under ambient conditions

Combined in-plane and out-of-plane multifrequency atomic force microscopy techniques have been demonstrated to be important tools to decipher spatial differences of sample surfaces at the atomic scale. The analysis of physical properties perpendicular to the sample surface is routinely achieved from flexural cantilever oscillations, whereas the interpretation of in-plane sample properties via force microscopy is still challenging. Besides the torsional oscillation, there is the additional option to exploit the lateral oscillation of the cantilever for in-plane surface analysis. In this study, we used different multifrequency force microscopy approaches to attain better understanding of the interactions between a super-sharp tip and an HOPG surface focusing on the discrimination between friction and shear forces. We found that the lateral eigenmode is suitable for the determination of the shear modulus whereas the torsional eigenmode provides information on local friction forces between tip and sample. Based on the results, we propose that the full set of elastic constants of graphite can be determined from combined in-plane and out-of-plane multifrequency atomic force microscopy if ultrasmall amplitudes and high force constants are used.

Electrostatic actuation and charge sensing in piezoelectric nanomechanical resonators with a two-dimensional electron gas

The features of electrostatic actuation are experimentally studied in nanomechanical resonators based on AlGaAs/GaAs heterostructures with a two-dimensional electron gas. First, it is found that, when the gate voltage is varied, the oscillation amplitude reaches its minimum and the resonant frequency reaches its maximum at different non-zero voltages. Using numerical modeling, it is shown that this feature can be explained by the influence of the bound charge induced due to the piezoelectric effect in the resonator performing flexural oscillations. Second, the amplitude as a function of the gate voltage does not reach zero at its minimum. A short illumination makes the minimal amplitude much closer to zero. These facts can be explained by the influence of the poorly conducting edges of a two-dimensional electron gas. The obtained results show that the measurement of the specific gate voltages corresponding to the minimal amplitude and maximal resonant frequency can be used for sensing the electrical charge induced by the illumination.

Non-uniform along thickness spin excitations in magnetic vortex-state nanodots

We summarize our experimental findings in the arrays of Ni80Fe20 circular nanodots with diameter 300 nm and thickness 20 nm ≤ L ≤ 100 nm, probed by broadband ferromagnetic resonance spectroscopy in the absence of external magnetic field. Spin excitation modes related to the vortex core gyrotropic dynamics were observed in the gigahertz frequency range. Micromagnetic simulations revealed that they are flexure oscillations of the vortex core string with n = 0, 1, 2 nodes along the dot thickness. It was found that for L &amp;gt; 70 nm the intensity of more complicated n = 1 vortex gyrotropic mode is unexpectedly higher than the one of the lowest n = 0 gyrotropic mode. This behavior was clarified on the basis of the inhomogeneous vortex mode phase profiles extracted from micromagnetic simulations and calculated analytically. Precise measurements of the dependence of resonance frequency of the vortex n = 0 mode on the dot thickness demonstrated a clear maximum around L = 70 nm, that was theoretically explained by introducing a vortex mass, which is a result of the vortex distortion due to interaction with spin waves having azimuthal indices m = ±1. Finally, several azimuthal spin-wave modes having curled structure at the dot top and bottom faces were found in the spectrum of the dots with thicknesses L ≥ 40 nm.

Finite amplitude torsional oscillations of shape-morphing plates immersed in viscous fluids

In this paper, we study torsional oscillations of a cross section of a thin plate submerged in a quiescent, Newtonian, incompressible, and viscous fluid. The plate is subjected to a prescribed shape-morphing deformation in phase with the rigid oscillation. The problem is completely described by three nondimensional parameters indicating oscillation frequency and amplitude and intensity of the shape-morphing deformation. We conduct a parametric study to investigate the possibility of controlling hydrodynamic moments and power dissipation through an active time-varying shape-morphing strategy. The problem is studied in both the linear and nonlinear flow regimes, by employing the boundary element method and direct numerical simulations via computational fluid dynamics methods, respectively. Investigation of flow physics demonstrates that, similarly to what is observed for the case of flexural oscillations, the shape-morphing strategy is effective in modulating vortex shedding in torsional oscillations. The results show that hydrodynamic power dissipation can be minimized and hydrodynamic moments can be controlled through an optimal imposed shape-morphing deformation. Findings from this study are directly applicable to torsional oscillation-based underwater energy harvesting or sensing and actuation systems, where control of hydrodynamic moments and reduction of hydrodynamic power losses are necessary for optimal device operation.

Poroelastic Relaxation in Thermally Cracked and Fluid‐Saturated Glass

AbstractTo test theoretical models of modulus dispersion and dissipation in fluid‐saturated rocks, we have investigated the broadband mechanical properties of four thermally cracked glass specimens of simple microstructure with complementary forced‐oscillation (0.004–100 Hz) and ultrasonic techniques (~1 MHz). Strong pressure dependence of moduli (bulk, Young's, and shear), axial strain, and ultrasonic wave speeds for dry conditions attests to essentially complete crack closure at a confining pressure of 15 MPa—consistent with ambient‐pressure crack aspect ratios ≤2 × 10−4. Oscillation of the confining pressure reveals bulk modulus dispersion and a corresponding dissipation peak, near 0.002 Hz only at the lowest effective pressure (2.5 MPa)—attributed to the transition with increasing frequency from the drained to saturated‐isobaric regime. The observations are consistent with Biot‐Gassmann's theory, with dispersion and dissipation adequately represented by Zener model. Above the draining frequency, axial forced‐oscillation tests show dispersion of Young's modulus and Poisson's ratio, and an associated broad dissipation peak centered near 0.3 Hz, thought to reflect local “squirt” flow and adequately modeled with a continuous distribution of relaxation times over two decades. Observations of Young's and shear moduli dispersion and dissipation from complementary flexural and torsional oscillation measurements for differential pressure ≤10 MPa provide supporting evidence of the transition with increasing frequency from the saturated‐isobaric to the saturated‐isolated regime—also probed by ultrasonic technique. These findings validate predictions from theoretical models of dispersion in cracked media and emphasize need for caution in the seismological application of laboratory ultrasonic data for cracked media.

РОЗРОБКА МЕТОДИКИ ВИЗНАЧЕННЯ ЗАПАСУ МІЦНОСТІ БОЙОВИХ ЛІТАКІВ НА ОСНОВІ АНАЛІЗУ ДИНАМІЧНИХ ХАРАКТЕРИСТИК З УРАХУВАННЯМ ЕКСПЛУАТАЦІЙНИХ ФАКТОРІВ

The article considers the issues of diagnostics of the technical condition of cantilevered aircraft structures based on the dynamic response analysis. This task provides for determining the real technical condition of the damaged structure and, as a result, the reserve factor. The main criterion is to use as little information as possible to obtain a precise objective assessment. It is suggested to use the structural dynamics being the most informative as diagnostic features. These are natural oscillation frequencies (torsional and flexural) and zero-twist axis position (according to the data from volitionally undamaged structures taken as reference). The methodology has been patented. As a technical solution, the methodology provides for frequency tests with dysfunction of flexural and torsional oscillation frequencies under which a minimum frequency of natural oscillations corresponding to the minimum residual strength shall be set for each type of aircraft, namely, for its part - wings, fins (cantilevered structures of the glider). The increased number of analysed natural oscillation frequencies increases the reliability of the residual strength estimation. Based on this, the developed and produced equipment and monitor equipment as a mobile version (which are also patented), can be effectively used when basing aircraft in the combat theatre. Based on the provisions of the NIAS similar monitoring system can be implemented in military units of the Air Force of the Armed Forces of Ukraine during the operation of aircraft according to the technical condition.

Theoretical and Experimental Researches of Spring Damping Flexural Oscillations for Beam Structures

One of the main problems in the design of rigid bus in open switchgears is the problem of structural stabilization under the action of various dynamic loads. The implementation of rigid bus with large spans of tube showed a real risk of wind resonance for cylindrical tube structures (aeroelastic self-oscillations). This phenomenon is dangerous due to the fact that at low speeds of the wind flow, intensive oscillations of the tubes occur in the vertical plane, adding to the static loads an essential dynamic component. Its level is comparable with loads from its own weight of structures and can, in combination with other loads, will be the cause of stresses close to the limiting ones by the 1st group of limiting states. The problem of reducing the level of structural oscillations in many cases is associated with the need to increase rigidity and reduce the material consumption of structures, however, it is important to fulfil the technological requirements of the operating conditions and protect people from the harmful effects of oscillations. The article highlights some of the main ways to damp oscillations of rigid bus structures. Is given a mathematical model of the simplest single-mass spring inertial dynamic oscillation damper. Experimental studies have been carried out on the joint work of the rigid bus structure with the spring damper. Has been confirmed the effectiveness of spring dampers using, both outside and inside the tube-bus.

Nested Bloch waves in elastic structures with configurational forces.

Small axial and flexural oscillations are analysed for a periodic and infinite structure, constrained by sliding sleeves and composed of elastic beams. A nested Bloch-Floquet technique is introduced to treat the nonlinear coupling between longitudinal and transverse displacements induced by the configurational forces generated at the sliding sleeve ends. The action of configurational forces is shown to play an important role from two perspectives. First, the band gap structure for purely longitudinal vibration is broken so that axial propagation may occur at frequencies that are forbidden in the absence of a transverse oscillation and, second, a flexural oscillation may induce axial resonance, a situation in which the longitudinal vibrations tend to become unbounded. The presented results disclose the possibility of exploiting configurational forces in the design of mechanical devices towards longitudinal actuation from flexural vibrations of small amplitude at given frequency. This article is part of the theme issue 'Modelling of dynamic phenomena and localization in structured media (part 1)'.

Longitudinal Displacement of a Fuel Rod Nonfixed in the Support Grid under the Conditions of Hydromechanical Vibration of VVER-440 Reactor Fuel Assemblies

Experiments aimed at studying the effect of fuel rod vibration in a turbulent coolant flow has on the effort of moving and longitudinal displacement of a fuel rod in the system of VVER-440 reactor fuel assembly (FA) spacer grids (SGs) for the case when the fuel rod is not fixed in the lower support are carried out. It has been found that the fuel rod longitudinal displacement force becomes essentially (by approximately 30%) higher in a coolant flow than it is in stagnant water. The fuel rod’s longitudinal displacement force decreases with an increase in the water flow velocity and, hence, with an increase in the fuel rod bundle vibration intensity. At the water flow velocity equal to approximately 3.8 m/s, which is close to the its nominal value in the VVER-440 reactor FAs, the displacement force decreases by approximately 10% as compared with its value at the velocity equal to 0.8 m/s. The regularities relating to the change with time in the fuel rod’s longitudinal displacement speed and acceleration testify that the force resisting its motion in the spacer grid system has a periodic pattern. During vibration (flexural oscillations) of a fuel rod bundle, the stiffness in the fuel rod–SG cell system changes its value, and the axes of SG cells shift relative to one another for the characteristic period of one vibration cycle. It is particularly the above-mentioned effects connected with fuel rod bundle vibration that lead to a growth in the fuel rod’s longitudinal displacement force. The results from the accomplished calculated assessments show that the algebraic sum of the Archimedean force, gravity force, force due to pressure loss across the fuel bundle, and inertial forces with the water velocity values up to 4.81 m/s do not exceed 6% of the minimal fuel rod longitudinal displacement force. In turn, this means that the system of honeycomb-type spacer grids reliably secures the VVER-440 reactor fuel rods that are not fixed in the FA lower support grid.

Inelastic Spin‐Wave Scattering by Bloch Domain Wall Flexure Oscillations

The calculations of the inelastic spin wave scattering by flexure vibrations of the Bloch domain wall (Winter's magnons) in thin magnetic films are presented. The approach is based on the interaction of the propagating spin waves with the dynamical emergent electromagnetic field generated by the moving inhomogeneous magnetization texture (domain wall). The probability of the spin wave scattering for the Winter's magnon emission and absorption processes essentially rises with the spin wave scattering angle increase up to 90°. The angular dependence of the scattering probability is essentially stronger for the magnon absorption processes that allow distinguishing these elementary emission/absorption processes experimentally.

Permeability and seismic-frequency elasticity of cracked glass

With the objective of improved understanding of frequency-dependent properties of cracked and fluid-saturated media, permeability and forced oscillation measurements are being undertaken on dry and fluid-saturated cracked glass media. In particular, we address some of the issues outstanding from a prior intensive study across a wide range of frequency. Our preliminary results demonstrate the potential of improved methods for the measurement of the very low permeabilities of such cracked glass media, the feasibility of torsional forced oscillation measurements at longer oscillation periods and lower differential pressures, and the benefits of improved alignment of our specimen assembly for complementary flexural oscillation tests. Such complementary flexural oscillation tests on glass-rod specimens are expected to provide further insight into the relevant fluid-flow regimes.

On the warping of the extreme ends of a beam under flexural oscillations

In this paper, we explain why beyond a certain critical frequency the extreme ends of a beam under flexural oscillations remain sometimes plane and in other cases become warped. We have seen this to be the case when the flexural oscillations of a beam are studied experimentally. Using the plane stress elastodynamic theory as well as a more precise calculation based on a 3D finite element method, we show for the first time that when the transverse displacement has a derivative equal to zero near the beam extremes, the ends can indeed become warped. In addition it is shown, from an energetic point of view, that warped modes result from a constructive spatial interference of the two components of flexural oscillations while flat modes result from a destructive interference of these two components.

Resonant phenomena of elastic bodies that perform bending and torsion vibrations

The method of study of the influence of torsional oscillations of one-dimensional models of nonlinear elastic bodies, along which moves with a constant velocity continuous flow of inelastic homogeneous medium, into bending, is developed. It is believed that information on torsional oscillations is known from empirical studies. Based on the latter, the refined model of the dynamics of the process of the investigated object is constructed. The latter is a boundary value problem for nonlinear nonautonomous differential equations with partial derivatives. The imposed restrictions on power factors and the main parameters of torsional oscillations allow for the analytical study of the dynamics of the process to use the basic ideas of the asymptotic integration of equations with partial derivatives. With their help, we obtain a two-parameter set of solutions that describe the determinant parameters of bending vibrations of an elastic body. It is established that for the considered elastic body there can be resonance oscillations, which are caused not only by external factors, but also by internal – torsional oscillations. Regarding the law of the change in the basic parameters of the dynamics of the bending motion of an elastic body, its rotation around the vertical axis reduces the frequency of its own flexural oscillations of the body, and even small torsional oscillations cause an additional periodic action on the transverse. In connection with the above bending vibrations of the elastic body, which performs complex oscillations (torsion and bending), resonances are possible both at the frequency of the external periodic perturbation and at the frequencies of the torsional oscillations (internal resonances). The amplitude of the transition through the resonance: a) at the basic frequency of external perturbation takes less value for elastic bodies of greater flexural rigidity and for higher values of the relative motion of the medium; b) at the frequency of torsional oscillations for larger values of the angular velocity takes more importance; c) with “fast” transition through resonance at the frequency of external or internal perturbation is less than with “slow”. The obtained results can serve as the basis for the choice of operating parameters of elastic elements of machines that carry out complex oscillations.