Non-uniform Beam Research Articles

Investigation on dynamics of rotating wind turbine blade using transferred differential transformation method

The wind turbine blade is commonly treated as a non-uniform beam because of its high aspect ratio. The bend-twist coupled vibrations of the wind turbine blade occur when there exist the misalignments among the shear center, neutral axis and pressure center in the blade cross-section. The differential governing equations of motion for the rotating wind turbine blade are derived by Hamilton principle and solved by the transferred differential transformation method (TDTM). Due to the rotations, three terms, namely the tension, centrifugal and gyroscopic forces, are distinctively summarized and taken into account in the proposed model. The natural frequencies obtained by the TDTM are compared with both experimental and numerical simulations to validate the effectiveness of the proposed solving method. The influences of the tension, centrifugal and gyroscopic forces on the natural frequencies are investigated for the wind turbine blade. The mode veering phenomenon is found and explained by using the complex mode decomposition theory. The phase differences among the flap-wise, edge-wise and torsional motions are also explored by observing the whirling orbits at the blade tip.

Non-homogeneous cross section variation enhanced flexoelectric coupling in semiconductor beams and its application in charge carrier redistribution

A pure bending semiconductor beam that exhibits a non-homogeneous cross section variation enhanced flexoelectric coupling effect is investigated in this paper from the point of view of theoretical analysis and numerical calculations. According to the macroscopic theory of flexoelectric semiconductors, a one-dimensional bending model for a simple-supported beam with variable cross section is developed. The semi-analytical method, which is composed of cosine series and supplementary functions, is introduced to solve the partial differential equations with variable coefficients. Before the analysis, the convergence and correctness of the semi-analytical method are demonstrated systematically. The calculated results show that both the electric potential and the charge carrier distribute non-homogeneously in the entire beam under a pair of symmetric bending moments. It is revealed that more carriers, a larger electric potential, and more charges tend to concentrate in the zone near the ends. After introducing a dimensionless carrier density, this paper systematically investigates the non-homogeneous cross section variation effect on the flexoelectric coupling for charge carrier redistribution. It is indicated that more perturbation carriers in a non-uniform beam are produced by the enhanced flexoelectric coupling. The demonstration of this concept could be potential guidance for designing new semiconductor devices.

Earthquake‐resistant subsystem stiffness demand estimate of frame–core tube structures with mass and stiffness nonuniformities

SummaryTo develop reasonable structural schemes of frame–core tube structures in the preliminary seismic design, a stiffness estimation of the main subsystems, that is, core tube, frame, and outrigger, is provided through the rocking model (RM) and flexural–shear coupled model (FSM). A nonuniform RM (RM‐MS) is proposed with the dual nonuniform distributions of mass and subsystem stiffness corresponding to those of the nonuniform FSM (FSM‐MS). The dynamic properties of the RM‐MS are semianalytically solved with the general solution for the free vibration of the nonuniform flexural beam and the rotational stiffness matrix of outriggers. The relation between the FSM‐MS and RM‐MS is established based on the first two vibration periods and nonlinear fitting and extended to present a subsystem stiffness estimation strategy with target maximum interstory drift ratios (IDRmax) and overturning moment distribution ratios (RM,B). The feasibility and effectiveness of the strategy are fully demonstrated by three cases with different target RM,B. RM‐MS effectively controls structural lateral stiffness of FSM‐MS in advance. With the target RM,B beyond the feasible range, it is necessary to adjust the first two vibration periods estimated by FSM‐MS. Compared with the axial rigidity of frame columns, the flexural rigidity of outriggers shows higher priority to increase RM,B.

Meshfree Galerkin Method for a Rotating Non-Uniform Rayleigh Beam with Refinement of Radial Basis Functions

The rotating Rayleigh beam problem is solved with meshfree method where the radial basis functions are explored. Numbers of basis functions have been used for meshfree methods which also include radial basis function. In this paper, the Gaussian radial basis function and multiquadrics radial basis functions are combined to get the new basis function which provides accuracy for higher natural frequencies. The radial basis functions satisfy the Kronecker delta property and it is easy to apply the essential boundary conditions. The Galerkin method is used for weak formulation. The matrices have been derived. The results are obtained for Gaussian radial basis function and new basis function. The results show more accurate values for fourth and fifth natural frequency with new basis function where only six nodes are used within the subdomain of trial and test function. The results are also obtained with conventional finite element method where forty, two node elements are considered. Also, the results are obtained for rotating Euler-Bernoulli beam to observe the difference in results with rotating Rayleigh beam.

Multi-field coupling characteristics of photovoltaic cell under non-uniform laser beam irradiance

The technical barrier of Laser wireless power transmission (LWPT) mainly lies in the development of a high-power laser emitter and high conversion efficiency of the receiver. Under the middle/far transmission range, the receiving heat flux presents a fluctuating, random ununiform distribution which will aggravate the interactions of the optical-thermal-electrical field on solar cells and decrease the energy transfer efficiency. The current study constructed a multi-field coupled model of a single solar cell under ununiform heat flux distribution. An LWPT experimental platform has been built and tested to verify the modeling parameters of I-V relation formula changing with optical intensity and temperature. Based on the corrected parameters, the optical, thermal heat transfer, and electrical characteristics have been solved by the finite element method. It was found that the voltage, current and temperature profiles on PV cell are apparently influenced by the changing of beam irradiance, and enhancing the cooling of PV cell can dramatically increase electrical conversion efficiency for the LWPT systems.

Computing Natural Frequencies and Mode Shapes of an Axially Moving Nonuniform Beam

Abstract The partial differential equation of motion of an axially moving beam with spatially varying geometric, mass, and material properties has been derived. Using the theory of linear time-varying systems and numerical optimization, a general algorithm has been developed to compute complex eigenvalues/natural frequencies, mode shapes, and the critical speed for stability. Numerical results from the new method are presented for beams with spatially varying rectangular cross sections with sinusoidal variation in thickness and sine-squared variation in width. They are also compared to those from the Galerkin method. It has been found that critical speed of the beam can be significantly reduced by nonuniformity in a beam's cross section.

A new method for estimating the bending stiffness curve of non-uniform Euler-Bernoulli beams using static deflection data

The static Euler-Bernoulli beam is a simplified model from linear elasticity theory for small deformations, with which the stresses and deflections of slender beams can be determined very well. The simplicity of Euler-Bernoulli beam theory makes it still an important tool in the sciences, particularly in construction and mechanical engineering. The behavior and the deflection of a static Euler-Bernoulli beam is completely given by the bending stiffness curve. The bending stiffness curve of a real beam structure can be determined using measured data from a bending test. However, the determination of the bending stiffness curve for non-uniform beams leads to a so-called inverse problem, which is ill-posed. In this work, a robust and computationally effective method for determining the bending stiffness curve of a non-uniform Euler-Bernoulli beam based on measured data from a static bending test and the calculus of variations is presented. With the calculus of variations, the complex optimization problem of the inverse problem is transformed into a boundary value problem, which is then solved by collocation with lower computational costs. In addition, the new method does not require the second derivative of the beam deflection and the problem of determining the bending stiffness at the boundaries does not arise, as with the traditional method. The new method to determine the bending stiffness curve of non-uniform beams is demonstrated in several test examples.

Vibration analysis of Non-uniform axially functionally graded beam resting on Pasternak foundation

Non-uniform beam with lengthwise material variation resting on Pasternak foundation is investigated for free vibration. A beam with its material properties and thickness varying linearly along its length is selected for investigation. Euler-Bernoulli beam theory along with Rayleigh-Ritz method is employed for the formulation and the equation of motion is derived by using Hamilton’s principle. Beams with two end conditions i.e., clamped on both ends and simply-supported on both ends are considered for investigation. The results of the current work are validated with established results in literature and new results which illustrate the effect various parameters have on the dynamic response of the beam are furnished. The study revealed that, the material variation, thickness variation and the stiffness of Winkler and shear layer significantly influence the dynamic characteristics of functionally graded beam with non-uniformity.

Static stability analysis of tapered Timoshenko sandwich beam resting on Pasternak foundation with thermal gradient

The static stability investigation of a Timoshenko sandwich beam with varying thickness and supported on a Pasternak foundation with varying spring stiffness with axial loading at the ends is performed considering thermal gradient. The complete mathematical modelling of the non-uniform Timoshenko sandwich beam is expressed with the help of energy principle given by Hamilton. The static stability analysis of the system is analysed for pinned-pinned (P-P) boundary condition. The plots presenting the static stability of the Timoshenko sandwich beam are achieved and analysed for various values of system and geometric parameters using MATLAB and are illustrated over a series of figures. The system static stability is improved due to shear parameter, the ratio of shear layer thickness to beam length and the ratio of modulus of rigidity of supporting shear layer and modulus of elasticity of top layer. An increment in the taper parameter and thermal gradient parameter degrades the static stability.

Concept of mathematical modeling of lumbar and thoracic spine based on elastic beam theory

Modeling of the human spine is important for establishing quantitative assessment methods of low back pain (LBP) caused by manual material handling (MMH). This study developed mathematical models of the lumbar and thoracic spine based on the elastic beam theory. Two types of models, uniform and non-uniform, were derived. First, to simplify the Bernoulli-Euler beam theory, three assumptions were introduced for the uniform beam model: (i) the vertebrae and intervertebral disc were considered as a composite material, (ii) constant cross-sectional area, and (iii) conversion of muscular strength to bending moment. However, on fitting to the actual spine position data from the literature, the uniform beam model was found to be unable to reproduce the spine shape, particularly in the lumbar region. Therefore, the non-uniform beam model was derived by removing assumption (ii), which does not agree with the actual human spine profile. Consequently, the spine was modeled as a trapezoid from around S1 to C7 in the sagittal plane, by adjusting the proportionality factor of trapezoid via tapering or widening, and the non-uniform beam model was found to better reproduce the spine shape, including the lumbar region. The model has practicability, such as estimation of spine shape; however, the proportionality factor that was used to reproduce the spine shape did not agree with the actual human spine profile; further theoretical development has been left as future work.

Simple Derivation of the Nonuniform Beam Filling (NUBF) Bias Formula in Spaceborne Doppler Radar Measurements

The nonuniform beam filling (NUBF) bias in Doppler velocity measurements from a spaceborne precipitation radar is investigated. A simple derivation of the already-known formula that relates the bias to the along-track gradient of measured radar reflectivity in dBZ is shown. From the formula, the uncertainty of bias correction due to discretization is discussed. The error is shown to be proportional to the square of the spacing for discretization, and is related to the third moment of the distribution of the radar reflectivity convolved with the antenna pattern. The effectiveness of averaging is also shown from a simulation.

Application of Nonuniform Beam Filling (NUBF) Doppler Velocity Error Correction on Airborne Radar Measurements

Non-uniform beam filling (NUBF) within an atmospheric radar sampling volume can cause significant Doppler velocity errors when the radar is located on a fast-moving platform. A case of strong NUBF errors was identified in ER-2 X-band Radar (EXRAD) data from the Investigation of Microphysics and Precipitation for Atlantic Coast-Threatening Snowstorms (IMPACTS) 2020 field campaign. A basic NUBF correction was employed using an estimate of the along-track radar reflectivity gradient. The relationship is verified empirically using co-located Doppler velocity measurements from the higher-resolution Cloud Radar System (CRS).

Clarifying duplicated electromagnetic characteristics for 220-GHz two-beam extended interaction oscillator

In this paper, we clarify the duplicated electromagnetic characteristics between the proposed two-beam overmoded multi-gap structure published in the paper titled “Tractable resonant structure with two nonuniform beams for a high-power 0.22-THz extended interaction oscillator” [Bi et al., IEEE Electron Device Lett. 42(6), 931–934 (2021)] and conventional multi-gap structures from the physical concept point of view. The replacement of the TE10 mode by the TE30 mode used in the interaction gaps provides the basis for the duplicated electromagnetic field distributions and frequencies between the two structures. The concept of magnetic coupling is proposed to support the resonant condition that supports duplication of the TE30 field and the frequency of the single-gap structure into the periodic standing-wave field of the multi-gap structure. With this concept, the perturbation of different axial phase shifts across one period to the standing-wave field of the TE30-0 mode is analyzed to clarify the profiles of the dispersion curves of the TE30 mode in the overmoded structure and the TE10 mode in the conventional structure. To demonstrate the profiles, a 220-GHz overmoded and conventional structure with nine gaps is designed, and their dispersion curves are obtained by using eigenmode simulations. In addition, the effect of the difference in the boundary conditions of the geometries of these two structures on the magnetic field distribution is analyzed to demonstrate the duplication of the ohmic losses between the two structures. The physical analysis in the duplicated electromagnetic characteristics lays a foundation for the promising design of the overmoded structure with two beams for high frequency extended interaction oscillators.

STUDY ON VIBRATION RESPONSE OF A NON-UNIFORM BEAM WITH NONLINEAR BOUNDARY CONDITION

Forced vibration of non-uniform beam with nonlinear boundary condition is studied in this paper by proposing an iterative model combining Adomian Decomposition Method and modal analysis. An exponentially tapered beam with a hardening nonlinearity spring boundary is simulated as a case study. The model accuracy is proved by comparing iteration results and analysis solutions with linear and weakly nonlinear boundary conditions. Sin-weep nonlinear frequency spectrum is then obtained by the proposed model. The influence of boundary nonlinearity on the vibration response of non-uniform beam is analyzed. And the effect of different excitation amplitudes on nonlinearity in the vibration response is studied. The mathematical model and numerical solutions proposed in this paper can be used to solve and analysis broad vibration problems on general non-uniform beams with different nonlinear boundary conditionsunder various excitations.

Flexural wave control via the profile modulation of non-uniform Timoshenko beams

A methodology to control the flexural wave propagation via modulating the beam profile is presented in this paper. Firstly, the power series method is utilized to theoretically solve the flexural waves in a non-uniform Timoshenko beam with arbitrarily contoured profiles. It is demonstrated that this method is effective for most of inhomogeneous beams as long as the convergence criterion about the beam thickness is satisfied in advance. After validation by comparing the theoretical results with those from the finite element analysis for different thickness variations, lenses are designed by using the quadratic thickness variation pattern with the aim of focusing the wave emitted from a point source on one or two particular positions. Additionally, according to the generalized Snell's law, the thickness-induced phase modulation of the flexural wave is proved from the views of theoretical analysis and numerical simulations, based on which lenses for focusing a plane flexural wave are also realized. It is illustrated through systematic analysis in the frequency domain that all of the lenses designed can exhibit good performances with the evidently increased energy at actual focal positions and the focusing sizes smaller than one working wavelength, which has great potentials in versatile engineering applications.

Dynamic stability of CNTs-reinforced non-uniform composite beams under axial excitation loading

This paper presents an analytical study on the dynamic stability of carbon nanotubes (CNTs) reinforced functionally graded composite beams when subjected to axial excitation loading. The composite beams are functionally graded using a I-type CNTs reinforcement. The section properties of the non-uniform functionally graded composite beams are assessed using Halpin–Tsai model. The analysis of dynamic stability of the composite beams is performed by using Bolotin’s method. Analytical expressions of determining free vibration frequency, critical buckling load, and excitation frequency of the non-uniform functionally graded composite beams are derived, in which both the shear deformation and rotary inertia effects are considered. The study demonstrates that the dynamic stability of the beam can be improved significantly when it is functionally graded using the I-type CNTs reinforcement.

Full-aperture chirped-pulse grating compression with a non-uniform beam

A novel chirped-pulse compression scheme is proposed to pursue higher pulse peak power of chirped-pulse amplifiers based on state-of-the-art gratings. It is characterized by the spectral non-uniformity of the beam cross section as well as by the uniform size of the beam projections on the four compression gratings. The spatial distribution of spectra is well designed so that the beam can remain generally collimated in the compressor and the gratings can be employed with a full work area. The maximum possible output pulse energy can therefore be increased within the damage threshold limit. A detailed design is developed for a scheduled 50-PW optical parametric chirped-pulse amplifier. Results of the numerical simulation on the focused output show that the spot size and pulse duration in the focal plane are close to those of conventional grating compression. Further, it is a promising method for upgrading existing chirped-pulse amplifiers.

Large amplitude vibration analysis of a non-uniform beam under arbitrary boundary conditions based on a constrained variational modeling method

Large amplitude vibration analysis of a non-uniform beam under arbitrary boundary conditions based on a constrained variational modeling method

LEO Satellite Multibeam Coverage Area Division and Beamforming Method

The low-orbit satellite has a large communication angle, resulting in a large difference in path loss at different angles. The existing multibeam coverage area division method cannot meet the coverage requirement of the effective isotropic radiated power (EIRP) in these large-angle areas. Therefore, this letter proposes a method for dividing the beam coverage area from the outside to the inside. Combined with the subarray multibeam forming method, it achieves the optimal coverage of isoflux multiple beams to the ground with unchanged output power of the transmitter, which greatly improves the overall efficiency of existing beamforming methods. At the meantime, a frequency reuse color separation algorithm based on the depth first search is proposed to reduce the mutual interference among nonuniform multiple beams. As an example, 52 beams with 12-color frequency reuse are designed to cover ±55°. The edge EIRP of a single beam is no less than 25.5 dBW and the carrier-to-interference ratio is no less than 16 dB.

Closed-form dynamic stiffness formulation for exact modal analysis of tapered and functionally graded beams and their assemblies

The paper proposes a closed-form dynamic stiffness (DS) formulation for exact transverse free vibration analysis of tapered and/or functionally graded beams based on Euler–Bernoulli theory. The novelties lie in both the DS formulation and the solution technique. For the formulation, the developed DS is applicable to a wide range of non-uniform beams whose bending stiffness and linear density are assumed to be polynomial functions of position. This fills a gap of existing closed-form DS element library which is generally limited to linearly tapered/functionally graded beams. For the solution technique, an elegant and efficient J0 count of tapered element is proposed to apply the Wittrick-Williams (WW) algorithm most effectively. The investigation sheds lights on the so-called J0 count challenge of the algorithm for other DS elements. The above two novelties make exact and highly efficient modal analysis possible for a wide range of tapered and/or functionally graded beams, without resorting to series solution, numerical integrations or refined mesh discretization. Results for a particular case show excellent agreement with published results. Moreover, we investigate the effects of the taper/functional gradient rate/index and boundary conditions on the free vibration behaviour. Benchmark solutions are provided for individual beams as well as beam assemblies.