Stepped Beam Research Articles

A Variational Formulation to Find Finite Element Bending, Buckling and Vibration Equations of Nonlocal Timoshenko Beams

A variational approach is developed to obtain bending, buckling and vibration finite element equations of nonlocal Timoshenko beams in this study. The reason for using the finite element method in this research is to investigate the behavior of nano-beams with complex geometry, material property and different boundary conditions. Weak forms of governing equations are derived, and the nonlocal differential elasticity theory is used to find the finite element formulation of nonlocal Timoshenko beams. In deriving the weak formulations, it is seen that it is impossible to construct the quadratic functional form due to non-symmetric bilinear property. Using the developed concepts and formulations, the bending and buckling of nonlocal Timoshenko beams with four classical boundary conditions are analyzed and the obtained results are compared with those reported in the literature. In order to show the capabilities of the proposed formulation in comparison with exact methods, the simply supported stepped nonlocal Timoshenko beam is selected and bending and buckling analyses are performed as well.

An analysis of stepped trapezoidal-shaped microcantilever beams for MEMS-based devices

Microcantilever beams are the most widely used mechanical elements in the design and fabrication of MEMS/NEMS-based sensors and actuators. In this work, we have proposed a new microcantilever beam design based on a stepped trapezoidal-shaped microcantilever. Single-, double-, triple- and quadruple-stepped trapezoidal-shaped microcantilever beams along with conventional rectangular-shaped microcantilever beams were analysed experimentally, numerically and analytically. The microcantilever beams were fabricated from silicon dioxide material using wet bulk micromachining in 25 wt% TMAH. The length, width and thickness of the microcantilever beams were fixed at 200, 40 and 0.96 µm, respectively. A laser vibrometer was utilized to measure the resonance frequency and Q-factor of the microcantilever beams in vacuum as well as in ambient conditions. Furthermore, finite element analysis software, ANSYS, was employed to numerically analyse the resonance frequency, maximum deflection and torsional end rotation of all the microcantilever beam designs. The analytical and numerical resonance frequencies are found to be in good agreement with the experimental resonance frequencies. In the stepped trapezoidal-shaped microcantilever beams with an increasing number of steps, the Q-factor, maximum deflection and torsional end rotation were improved, whereas the resonance frequency was slightly reduced. Nevertheless, the resonance frequency is higher than the basic rectangular-shaped microcantilever beam. The observed quality factor, maximum deflection and torsional end rotation for a quadruple-stepped trapezoidal-shaped microcantilever are 38%, 41% and 52%, respectively, which are higher than those of conventional rectangular-shaped microcantilever beams. Furthermore, for an applied concentrated mass of 1 picogram on the cantilever surface, a greater shift in frequency is obtained for all the stepped trapezoidal-shaped microcantilever beam designs compared to the conventional rectangular microcantilever beam.

ELASTOPLASTIC BEND OF ROUND STEEL BEAM. REPORT 1. SPRINGBACK COEFFICIENT

The round steel beam is widely used in metallurgy, mechanical engineering, construction and is one of the major players in the engine industry. Having the excellent anti-corrosion properties, combined with a remarkable strength, the round steel beam is often indispensable in the production of the various mechanical machines and devices. The cylindrical springs for the railway and motor transport are made from the round beam with the help of the special bending machines. Billets from the round beam are also used in the metallurgy at the manufacture of seamless pipes for the oil and gas industry. The rollers of the sheet-straightening machines and rolling mills in metallurgy have the form of stepped round beam. The steel construction armature is made from the round beam and is close to it by geometric dimensions. The main foreign producers of continuous-casting-billets machines for production of round-cross-section blanks are SMS-Demag (Germany), Danieli (Italy), SMS Concast (Switzerland) and Siemens VAI (Austria). The modern production of round steel beam has place on many Russian metallurgical plants, for example, on JSC “Chusovskoy metallurgical plant”, PJSC “Chelyabinsk metallurgical plant”, JSC “Volzhsky pipe plant”, OJSC “Nizhneserginsky metizno-metallurgical plant”, JSC “Chepetsky mechanical plant”, PJSC “Seversky pipe plant” and PJSC “Taganrog metallurgical plant”. In manufacture of articles from round beam and under their exploitation, they often have an elastic or elastoplastic deformation of bending or have a complex deformation of torsion with bending. The analytical method for determining residual curvature of round steel beam under elastoplastic bend is proposed in this paper. The calculations allow us to determine residual curvature of round beam after bending and the bending moments of beam’s cross section at bending depending on the beam’ radius, elastic modulus, yield stress and hardening modulus of beam’s metal. The research results can be widely used at engineering and metallurgical plants.

Surface crack and sand inrush disaster induced by high-strength mining: example from the Shendong coal field, China

Sand inrush disaster and ground destruction induced by high-strength mining in the Shendong coal field seriously threaten the normal operation of the mine and cause significant property losses and environmental disruption. The physical simulation experiment demonstrate that the roof of high-strength mining working face can be regarded as a “step beam” structure and broken by sliding instability. The vertical damage state of overlying strata is summarized into three types: slightly, severely and very severely damage. On the basis of in situ data of the working face with the mining height greater than 3 m, the prediction formulas of the caved and fractured zone heights are given. The vertical damage types of working faces 22407 and 22402 are analyzed. Owing to the sliding instability of the roof and the thin bedrock, the surface stepped crack has become widely distributed above the high-strength mining working face. The sand inrush of working face 22402 can be interpreted by the very severely damaged of overburden and the thick aeolian sand aquifer. This work can be used to improve the understanding of mining-induced disaster and establish a disaster prediction model.

Elastoplastic Flexure of Round Steel Beams. 1. Springback Coefficient

Round steel beams are widely used in metallurgy, manufacturing, and construction. It is often irreplaceable in the production of machines and mechanical equipment on account of its excellent corrosion resistance and remarkable strength. Cylindrical springs for road and railroad vehicles are manufactured from round beams by means of special bending machines. Round beams also serve as blanks in the production of seamless pipe for the oil and gas industry. Stepped round beams are used as rollers in rolling and straightening mills. Steel rebar is manufactured from round beams of similar size. Outside Russia, the main producers of continuous-casting machines for the production of round billet are SMS Demag (Germany), Danieli (Italy), SMS Concast (Switzerland), and Siemens VAI (Austria). Russian producers include AO Chusovskii Metallurgicheskii Zavod, PAO Chelyabinskii Metallurgicheskii Kombinat, AO Volzhskii Trunbnyi Zavod, OAO Nizhneserginskii Metizno-Metallurgicheskii Zavod, AO Chepetskii Mekhanicheskii Zavod, PAO Severskii Trubnyi Zavod, and PAO Taganrogskii Metallurgicheskii Zavod. In the manufacture of parts from round beams and their use, they often experience elastic or elastoplastic flexure or complex torsion and flexure. In the present work, we propose an analytical method of calculating the residual curvature of round steel beams in elastoplastic flexure. In the calculations, the residual curvature of the round beam after flexure and the bending moments of the beam cross section in flexure are determined as a function of the beam radius, the Young’s modulus, the yield point, and the hardening modulus of the steel. The results may be widely used at manufacturing and metallurgical plants.

Axial-bending coupled vibration analysis of an axially-loaded stepped multi-layered beam with arbitrary boundary conditions

This paper presents an analytical method for solving both the free and forced vibration of an axially loaded stepped laminated beam with general layers and arbitrary boundary conditions based on the first-order shear deformation theory (FSDT). The beam is divided into appropriate segments and the spectral element matrix is employed to formulate the solutions for each segment. The transfer matrix method is adopted to analytically combine the solutions and boundary conditions of each segment. The solutions to the free and forced vibration of the beam are then obtained by solving a 6 × 6 matrix equation. The results obtained by the proposed method show good agreement with those by the finite element method (FEM). Parametric studies are presented to study the effect of beam parameters on the dynamic characteristics of the beams.

Variations of flutter mechanism of a span-morphing wing involving rigid-body motions

The present paper aims to reveal the significance of rigid-body motions for the flutter mechanism of a span-morphing wing model. The inclusion of rigid-body motions into aeroelastic formulation and flutter analysis is presented. A state-space aeroelastic equation combining the dynamics of stepped Euler-Bernoulli beam with unsteady strip aerodynamic theory is developed by quasi-static modeling. Using a numerical example, variations of flutter mechanism from the bending-torsional flutter to the body-freedom flutter are observed as the span increases. In addition, effects of some dimensionless parameters on the variations of flutter mechanism are investigated. The investigated parameters belonging to the fuselage have limited influences on the bending-torsional flutter but a significant impact on the body-freedom flutter.

Receptance coupling model for variable dynamics in fixed-free thin rib machining

An analytical approach for predicting thin rib, fixed-free beam dynamics with varying geometries is presented. The proposed method uses receptance coupling substructure analysis (RCSA) to predict the stepped thickness beam receptances (or frequency response functions) at the fixed-free beam’s free end and at the change in thickness. The assembly receptances are calculated by rigidly connecting receptances that describe the individual stepped beam sections, where the section receptances are computed using the Timoshenko beam model. Comparisons with finite element calculations are completed to verify the technique. It is observed that the RCSA predictions agree closely with finite element results. Experiments are also performed, where the stepped beam thickness is changed by multiple machining passes, and receptance measurements are carried out between passes. The RCSA predictions are compared to experimental results for natural frequency and stiffness. Agreement in natural frequency to within two percent is reported.

Pull-in Voltage Analysis of Electrostatically Actuated Stepped Micro-cantilever Beam

Pull-in Voltage Analysis of Electrostatically Actuated Stepped Micro-cantilever Beam

Stability and vibrations control of a stepped beam using piezoelectric actuation

The objects of this studies are the stability and transversal vibrations of the system composed of three segments, where in the centre part of the system two piezoelectric patches are perfectly bonded to the top and bottom surface of the host beam. The system is kinematically loaded as a result of prescribed displacement of one or both end supports. For the analysis purposes three different beam end supports have been taken into consideration, which prevent longitudinal displacements i.e. clamped-clamped, clamped-pinned and pinned-pinned. This type of beam loading not only affect its natural vibration frequencies but also the system’s stability. By introducing the electric field to the piezo patches, depending on its vector direction, in-plane stretching or compressive residual force may be induced. Presented results show that piezo actuation can significantly modify both the critical buckling force and the vibration frequency.

Wind Turbine Blade Fault Detection Using Wavelet Power Spectrum and Experimental Modal Analysis

The dynamic behavior of modern multi-Megawatt wind turbines has become an important design consideration. The power generated by wind turbine depends mainly on the interaction between the blades and the flowing wind. Therefore, the major aspects related to the operational reliability of a wind turbine are the aerodynamics characteristics and integrity of the blades. This paper demonstrates the results of the application of wavelet transform combined with experimental modal analysis as a tool for wind turbine blade condition monitoring. An experimental model analysis with different vibrational excitations performed for healthy and cracked cantilever stepped beam, which approximate the wind turbine blade when attached to the wind turbine hub. The beam response signals acquired for three different locations of vibration sensor. The cantilever beam with breathing crack response exhibits a non-stationary vibrational signal because of crack open-close cycles during the oscillation of the beam. The wavelet transform as a time-frequency signal processing technique is an effective tool to extract the characteristics of the non-stationary signals; therefore, used in this paper to analyze the cracked beam vibrational signals. The wavelet power spectrum (WPS) calculated for different beam health conditions and for different locations of the vibration sensors. The results reveal the effectiveness of the wavelet power spectrum over the traditional FFT spectrum to identify the existence of the crack in the wind turbine blade. Moreover, a mathematical model, which correlates the variation of the blade natural frequencies with the crack severity and location, is presented and validated with an experimental modal analysis data.

Analysis on Nonlinear Vibrations of a Stepped Beam Considering Effects of Axial Inertia Force

Analysis on Nonlinear Vibrations of a Stepped Beam Considering Effects of Axial Inertia Force

A 28-GHz 32-Element TRX Phased-Array IC With Concurrent Dual-Polarized Operation and Orthogonal Phase and Gain Control for 5G Communications

This paper presents the first reported 28-GHz phased-array IC for 5G communications. Implemented in 130-nm SiGe BiCMOS, the IC includes 32 TRX elements and features concurrent independent beams in two polarizations in either TX or RX operation. Circuit techniques to enable precise beam steering, orthogonal phase and amplitude control at each front end, and independent tapering and beam steering at the array level are presented. A TX/RX switch design is introduced which minimizes TX path loss resulting in 13.5 dBm/16 dBm Op1dB/Psat per front end with >20% peak power added efficiency of the power amplifier (including switch and off-mode LNA) while maintaining a 6 dB noise figure in the low noise amplifier (including switch and off-mode PA). Comprehensive on-wafer measurement results for the IC across multiple samples and temperature variation are presented. A package with four ICs and 64 dual-polarized antennas provides eight 16-element or two 64-element concurrent beams with 1.4°/step beam steering (<0.6° rms error) across a ±50° steering range without requiring calibration. A maximum saturated effective isotropic radiated power of 54 dBm is measured in the broadside direction for each polarization. Tapering control without requiring calibration achieves up to 20-dB sidelobe rejection without affecting the main lobe direction.

A universal parameterized gradient-based method for photon beam field size determination.

To propose a universal, parameterized gradient-based method (PGM) for radiation field size determination. The PGM locates the beam profile's edge by parameterizing its penumbra region with a modified sigmoid function where the inflection point can be determined in a closed form. The parametrization was validated with filter-flattened (FF), flattening-filter-free (FFF) and wedged profiles measured on two Elekta linac models (Synergy and Versa HD). Gamma analysis with the delta dose function set to zero was used to quantitatively assess the parameterization accuracy. Field sizes of FF beams were calculated with the PGM and the full width at half maximum (FWHM) methods for comparison. To assess the consistency of the PGM and the FWHM method with geometric scaling across different depths, the calculated field size at a reference depth was scaled to other depths and compared with the field sizes calculated from the measured profiles. The method was also validated against a maximum-slope method (MSM) with wedge and FFF profiles. We also evaluated the robustness of the three methods with respect to measurement noise, varying scanning step sizes, detector characteristics, and beam energy/modality. Small distance-to-agreement (0.02±0.02mm) between the measured and parameterized penumbra region was observed for all profiles. The differences between the field sizes calculated with the FWHM method and the PGM were consistent (0.9±0.3mm), with the FWHM method yielding larger values. With geometrical scaling, the PGM and the FWHM method produced maximum differences of 0.26 and 1.16mm, respectively. For wedge and FFF beams, the mean differences relative to FF fields were 0.15±0.09mm and 0.57±0.91mm for the PGM and the MSM, respectively. The PGM was also found to produce more consistent results than the FWHM method and the MSM when measurement noise, scanning step size, detector characteristics, and beam energy/modality changed. The proposed PGM is universally applicable to all beam modalities (FF, wedge and FFF) for accurate field size determination. Compared to the FWHM and the MSM, it is more robust to variations in measurement condition and detection system.

Frequency equation using new set of fundamental solutions with application on the free vibration of Timoshenko beams with intermediate rigid or elastic span

Stepped beams are crucial power transmission components in many mechanical engineering systems. These beams may be dynamically analyzed using the stepped Timoshenko model or the rigid mass model. In this paper, a new set of fundamental solutions is derived in order to normalize the Timoshenko beam equation at the origin of the coordinates. This set of solutions is used to derive the frequency equation of both stepped and rigid mass models. The validity ranges of these models were investigated by comparing the modal frequency results of both models. In addition, selected cases were compared using mode shape analysis. Three different models with classical end conditions are considered through this work. These are free–free, pinned–pinned and clamped–free beam configurations. The numerical results of the current work show that increasing the intermediate diameter ratio and decreasing the length ratio of the rigid mass results in decreasing the percentage deviation between the rigid mass model results and the elastic model results.

Exact vibration modes of multiple-stepped beams with arbitrary steps and supports using elemental impedance method

A novel numerical method has been developed and applied to the prediction of vibration modes of general stepped beams with arbitrary steps and general elastic supports. Each beam section is modelled as an element with input impedance at one end and output impedance at the other. These impedances are then coupled when the beam sections are joined to form an overall stepped beam structure. General elastic supports are modelled and their effects on impedance are examined. The method is theoretically exact assuming each beam section can be modelled as Euler-Bernoulli beam and is computationally very efficient since it does not involve any matrix operations of large dimension. Vibration modes of a number of stepped beams of different configurations have been computed using the proposed method and are compared with existing results in literature. Experimental investigations have also been carried out to validate the practical usefulness of the method. The method is also ideally suited for forced vibration applications since it not only establishes vibration modes, but also frequency response functions. In addition, a general purpose software has since been developed based on the method which can be very useful for structural dynamics design of general stepped beam structures.

Analytical solutions for fixed-free beam dynamics in thin rib machining

Two different analytical approaches for predicting thin rib, fixed-free beam dynamics with varying geometries are presented. The first approach uses the Rayleigh method to determine the effective mass for the fundamental bending mode of the stepped thickness beams and Castigliano’s theorem to calculate the stiffness both at the beam’s free end and at the change in thickness. The second method uses receptance coupling substructure analysis (RCSA) to predict the beam receptances (or frequency response functions) at the same two locations by rigidly connecting receptances that describe the individual stepped beam sections, where the receptances are derived from the Timoshenko beam model. Comparisons with finite element calculations are completed to verify the two techniques. It is observed that the RCSA predictions agree more closely with finite element results. Experiments are also performed, where the stepped beam thickness is changed by multiple machining passes, and receptance measurements are carried out between passes. The RCSA predictions are compared to experimental results for natural frequency and stiffness. Agreement in natural frequency to within a few percent is reported.

A Quad-Cantilevered Plate micro-sensor for intracranial pressure measurement.

This paper proposes a new design for pressure-sensing micro-plate platform to bring higher sensitivity to a pressure sensor based on piezoresistive MEMS sensing mechanism. The proposed design is composed of a suspended plate having four stepped cantilever beams connected to its corners, and thus defined as Quad-Cantilevered Plate (QCP). Finite element analysis was performed to determine the optimal design for sensitivity and structural stability under a range of applied forces. Furthermore, a piezoresistive analysis was performed to calculate sensor sensitivity. Both the maximum stress and the change in resistance of the piezoresistor associated with the QCP were found to be higher compared to previously published designs, and linearly related to the applied pressure as desired. Therefore, the QCP demonstrates greater sensitivity, and could be potentially used as an efficient pressure sensor for intracranial pressure measurement.

Bending vibrations of stepped rods

Manufacturing stepped rods with segments of sizes proportional to stresses induced in operation is often used to reduce the material consumption in various fields of technology, for example, in the aircraft industry, where the requirements for the weight of structural elements are high. The vibration problems of continuous systems, i.e. systems which masses are considered distributed, are close to the resistance of materials and elasticity theory problems. They are described by partial differential equations. In this case, we consider a homogeneous isotropic material, obeying Hooke’s law. Of all the vibration problems of continuous systems, the transverse vibration problems of shafts and beams is of greatest practical importance. The simplest examples of vibrations of prismatic rods were studied in the 18th century in works on acoustics. But before solving problems of practical importance, the problems of stepped beams, it had taken another two hundred years and the development of approximate methods of solving differential equations. The paper presents a solution to the problem of determining the fundamental frequencies of bending vibrations of two-stepped rods with various boundary conditions, using the approximate Lagrange-Ritz method. The calculation error does not exceed 2.6 %. The fundamental frequency of vibration is defined considering different lengths and stiffness ratios of stepped rod segments. The obtained results can be used in solving practical problems in various fields of technology.

Multiple filamentation and control of properties of self-guided super Gaussian laser beam

Spatial evolution of super Gaussian beam in Chalcogenide glass (Kerr medium) is investigated for input power in the range 15Pcr to 180Pcr using Nonlinear Schrödinger Equation. Solutions are obtained by numerical simulation with split step beam propagation method and are analysed to study the effect of input noise level on the process of multiple filamentation in super Gaussian beam, where emphasis is on the control of properties of the self-guided super Gaussian laser beam by externally induced input noise. Control of properties of self–guided laser beam is a necessity in generation of tailored filaments.