Straight Beam Research Articles

Buckling of lattice columns made from three-dimensional chiral mechanical metamaterials

The three-dimensional (3D) chiral mechanical metamaterials were found to exhibit unique compression-twisting coupling effect. The metamaterial will twist in addition to axial shortening when subjected to the external compressive load. For a slender structure made from 3D chiral mechanical metamaterial, global buckling may occur if the compressive load exceeds the critical value. In this work we investigated the buckling strength of the chiral lattice columns which were constructed by periodically placing the inclined straight beams in a chiral manner. Based on the Cosserat rod theory, a novel constitutive model with a new parameter governing the compression-twisting coupling was built to describe the deformation of 3D chiral metamaterial. A semi-analytical homogenization method was proposed to connect the stiffness parameters of arbitrary sized lattice column to the properties of the unit cell. The constitutive model together with the homogenization method well interpreted and quantified the size dependency of the chirality. The fourth-order governing equations of buckling were developed and solved analytically to predict the critical buckling load of 3D chiral metamaterial. The effects of chirality on the buckling strength and buckling mode were revealed.

ANALYTICAL SOLUTIONS FOR FORCED VIBRATIONS OF TIMOSHENKO CURVED BEAM BY MEANS OF GREEN’S FUNCTIONS

This paper derives analytical solutions for the forced vibrations of Timoshenko curved beams and establishes the vibration equation of Timoshenko curved beams by analyzing the equilibrium equation for the intersection of curved beams. Green’s functions of Timoshenko curved beams are solved for different boundary conditions using the separation of variables and Laplace transform. Two characteristic parameters are introduced to measure damping effects on beam vibrations. Numerical calculations are conducted to validate analytical solutions, and the effects of various related physical parameters are investigated. The results show that by setting the radius R to infinity, it can be simplified to the Timoshenko straight beam vibration model, and on this basis, if the shear correction factor κ is set to infinity, it can be reduced to the Prescott straight beam vibration model. Finally, the moment of inertia γ is set to 0, which can be reduced to the Bernoulli-Euler straight beam vibration model. Numerical calculations are performed to validate the solutions.

Examination of erection safety of precast beam assembly using a slanted beam section based on assembly test

ABSTRACT Assembling precast frames subjected to heavy construction loads may require a significant crane operation time when they were assembled by conventional steel joints having a straight beam cut. This study proposed a method that will provide a facile and rapid assembly of heavy precast frames, having the temporary bolting be established as noncritical path. Lower flanges of wide flange beam installed at end of precast beams were partially removed to prevent bottom flange of steel beam from running into flanges of L-shaped guide pockets when steel beam was inserted in L-shaped guide pockets. Permanent connections are made by bolting L-shaped guide pockets. Stable behavior was observed during an erection test of the proposed connection. This study also aimed to investigate an assembly safety of steel beam-column joints with a skewed beam section based on a nonlinear finite element analysis. Strain and stress analysis also demonstrated that stable structural behavior of joint elements was ensured during an assembly of precast frames.

Natural Frequencies and Band Gaps of Periodically Corrugated Beams

Abstract Structures with geometric periodicity can present interesting dynamic properties like stop and pass frequency bands. In this case, the geometric periodicity has the effect of filtering the propagating waves in the structure, in a similar way to that of phononic crystals and metamaterials (non-homogeneous materials). Hence, by adopting such structures, we can design systems that present dynamic characteristics of interest, e.g., with minimum dynamic response in a given frequency range with large bandwidth. In the present work, we show that corrugated beams also present the dynamic properties of periodic structures due to their periodic geometry only (no need of changing mass or material properties along the beam). Two types of corrugated beams are studied analytically: beams with curved bumps of constant radii and beams with bumps composed of straight segments. The results show that, as we change the proportions of the bump, the natural frequencies change and tend to form large band gaps in the frequency spectrum of the beam. Such shifting of the natural frequencies is related to the coupling between longitudinal and transverse waves in the curved beam. The results also show that it is possible to predict the position and the limits of the first band gap (at least) as a function of the fundamental frequency of the straight beam (without bumps), irrespective of the total length of the corrugated beam.

Design and characterization of a polymer electrothermal microgripper with a polynomial flexure for efficient operation and studies of moisture effect on negative deflection

Microgrippers are important devices for micro-manipulation in various applications in automation, microrobotics, material science, and biological studies. There has been a significant interest in improving the performance and reliability of microgrippers by developing efficient designs and actuation methods, and in reducing the cost of fabrication by utilizing polymer materials. This paper presents the design and characterization of an SU-8 based electrothermal microgripper with embedded Au thin film heaters. The design combines a Z-shape chevron actuator and a polynomial-shape flexure to convert the driving force into rotational movement of the microgripper arm. The results of electro-thermo-mechanical simulation using COMSOL Multiphysics software tool have shown that the optimized polynomial design is 5–10% more efficient in operation as compared to the straight beam design due to better stress distribution. The microgripper was capable of producing a gripping stroke of ~ 80 µm at 0.4 V driving voltage, the corresponding heater temperature was 120 °C. The characteristic behavior of negative deflection in SU-8 based microgrippers was observed at low actuation temperatures, resulting in several microns of displacement in the direction opposite to the expected movement. This effect was found to be associated with moisture absorption from the ambient environment by the polymer structure. Further tests under controlled humidity conditions were carried out and showed that the negative deflection is due to the moisture absorption effect in the SU-8 structures. It is shown that the microgripper is capable of performing pick-and-place of micro-objects using spherical glass micro-beads, therefore demonstrating the potential for applications in microassembly.

Dust and gas in the central region of NGC 1316 (Fornax A)

Context. The early-type galaxy NGC 1316, associated with the radio source Fornax A, hosts about 107 M⊙ of dust within a central radius of 5 kpc. These prominent dust structures are believed to have an external origin, which is also a popular interpretation for other dusty early-type galaxies. Moreover, it has been long known that ionised gas is present in NGC 1316, but to date there has been a lack of detailed investigation. Aims. Our aim is to understand the nature of ionised gas and dust in NGC 1316 and to offer an interpretation for the origin of the dust. Methods. We use archival Hubble Space Telescope/Advanced Camera for Surveys data to construct colour maps that delineate the dust pattern in detail, and we compare these data with maps constructed with data from the Multi Unit Spectroscopic Explorer (MUSE) instrument of the Very Large Telescope at the European Southern Observatory. Twelve MUSE pointings in wide field mode form a mosaic of the central 3.3 × 2.4 arcmin2. We use the tool PyParadise to fit the stellar population. We use the residual emission lines and the residual interstellar absorption NaI D-lines, and we measure line strengths, the velocity field, and the velocity dispersion field. Results. The emission lines resemble low-ionisation nuclear emission-line region lines, with [NII] being the strongest line everywhere. Ionising sources are plausibly the post-asymptotic giant branch stars of the old or intermediate-age stellar population. There is a striking match between the dust structures, ionised gas, and atomic gas distributions, the last of which is manifested by interstellar absorption residuals of the stellar NaI D-lines. In the dust-free regions, the interstellar NaI D-lines appear in emission, which is indicative of a galactic wind. The velocity field of the ionised gas (and thus of the dust) is characterised by small-scale turbulent movements that indicate short lifetimes. At the very centre, a bipolar velocity field of the ionised gas is observed, which we interpret as an outflow. The low-velocity part is associated with dust. We identify a strongly inclined gaseous dusty disc along the major axis of NGC 1316. A straight beam of ionised gas with a length of about 4 kpc emanates from the centre. Conclusions. The dust in NGC 1316 has different origins. Our findings are strongly suggestive of a dusty outflow that is curved along the line-of-sight. Nuclear outflows may be important dust-producing machines in galaxies. Another dusty gaseous component forms a disc that we identify as the predecessor of a central dust lane.

In-Plane Free Vibration Analysis of a Scimitar-Type Rotating Curved Beam Using the Adomian Modified Decomposition Method

Free in-plane vibrations of a scimitar-type nonprismatic rotating curved beam, with a variable cross-section and increasing sweep along the leading edge, are calculated using an innovative, efficient and accurate solver called the Adomian modified decomposition method (AMDM). The equation of motion includes the axial force resulting from centrifugal stiffening, and the boundary conditions imposed are those of a cantilever beam, i.e., clamped-free and simple-free. The AMDM allows the governing differential equation to become a recursive algebraic equation suitable for symbolic computation, and, after additional simple mathematical operations, the natural frequencies and corresponding closed-form series solution of the mode shapes are obtained simultaneously. Two main advantages of the application of the AMDM are its fast convergence rate to a solution and its high degree of accuracy. The design shape parameters of the beam, such as transitioning from a straight beam pattern to a curved beam pattern, are investigated. The accuracy of the model is investigated using previously reported investigations and using an innovative error analysis procedure.

Design and Simulation Analysis of A Z Axis Microactuator with Low Mode Cross-Talk

ABSTRACTFor a vibration system, the best designed spring is compliant to a desired vibration mode while it is robust to other undesired modes. There are several types of spring design for displacing the proofmass along the x and y axes, however, very few designs of spring compliant to the z axis are introduced. Therefore, we propose a z axis microactuator in which the suspending spring is designed so that it is only compliant to vibration along the z axis. The suspending spring consists of straight beam stages mechanically coupled with each other via frames which are symmetrically designed around a center plate. The operation characteristics of the microactuator is investigated by theoretical expresses and numerical simulation. The frequency split between the z axis mode and undesired modes can obtain more than 45%. The operation frequency can be modified in a wide range, from 68 kHz to 400 kHz, by changing the dimensional parameters of spring beams. The spring beams can be lengthened to increase displacement in the z axis while the mode cross-talk is still suppressed. Compared to the previously reported researches, the current microactuator shows robustness to undesired vibration modes, which is potential for integration in low mode cross-talk multi-axis micro-stages and low-noise mechanical sensors.

Free and Forced Vibration of Coupled Beam Systems Resting on Variable Viscoelastic Foundations

This paper presents a modified variational method for free and forced vibration analysis of coupled beam systems resting on various viscoelastic foundations. Non-uniform as well as uniform curved and straight Timoshenko beam components are considered in the coupled beam system. Using proper coordinate transformations, interactions among the beam components of the coupled beam system are accommodated by combining Lagrange multiplier method and least-square weighted residual method. Interface potential energy for various boundary conditions including the elastic ones is simultaneously formulated. Thus, the proposed method allows flexible choice of the admissible functions, regardless of the boundary conditions. Based on the proposed energy method, Winkler, Pasternak or even variable foundations distributed in a parabolic or sinusoidal manner can be easily introduced into the coupled beam systems. Two kinds of damping, namely the proportional and viscous damping, are also employed to model the energy dissipation of the viscoelastic foundations. Corresponding finite element (FE) simulations are performed where possible and good agreement is observed. Thus, great efficiency and accuracy of the present approach are demonstrated for free, steady-state and transient vibration of the coupled beam systems. The influences of the parameters of the variable viscoelastic foundations on the dynamic properties of the coupled beam system are also examined.

On Design and Analysis of Electrostatic Arch Micro-Tweezers

Abstract This article provides criteria for the design of electrostatic arch micro-tweezers. The tweezers can be operated in two modes: a traditional quasi-static mode where a direct current voltage commands the tweezers arms along a trajectory to manipulate objects and dynamic mode where a harmonic signal commands release or characterization of objects. While the arms are rigid and move in tandem in the static mode, this is not guaranteed in the dynamic mode. To satisfy this, we carried out modal analysis of the tweezers using a finite element model (FEM) and a reduced-order model (ROM). The results show that the arms kinetic and potential energies divide the beam span into a middle sub-span between the arms and two outer sub-spans and result in significant changes in the relative compliance of the sub-spans. The changes in the platform compliance place limitation on the tweezers dynamic operation, such that only the first symmetrical mode shape of the tweezers satisfies the design criteria. We also investigate the adequacy of an ROM using straight unbuckled beam mode shapes as basis functions to represent the tweezers response by comparing its results with those of FEM. A five-mode ROM is found adequate to represent small motions in the vicinity of the tweezers initial curvature. It is inadequate for larger motions involving snap-though motions between the initial and counter curvatures. To capture larger motions, ROM should be improved by incorporating higher order straight beam modes or using the actual tweezers modes.

Vibration and snapping of a self-contacted beam under prescribed end rotations

An initially straight beam is first bent into a circular configuration. Equal rotation angles of opposite direction are then prescribed at the two ends to bend the beam into self contact and eventually snap. We conduct a vibration analysis based on an Eulerian formulation to determine the natural frequencies and the stability of the bent beam. The theory predicts that the rotated beam snaps sideways at the bifurcation point. In experiments, however, the self-contacted beam passes far beyond the bifurcation point and snaps symmetrically by squeezing itself past the center of the two end points to the other side. By looking into the mode shapes of the unstable mode at the bifurcation point, it is believed that the predicted sideway snapping may be prevented by the sliding friction between the contact surfaces, which is not included in the theory. Instead, the beam snaps in a second unstable mode which involves rolling between contact surfaces. We then propose an imperfection analysis in which the bent beam is slanted to one side by a small angle in its initial configuration. In an experiment with a slant angle of 3∘, the deformation follows the load-deflection curve of the imperfect model and snaps sideways near the limit point. This experimental result may be considered as an auxuliary evidence of the existence of the bifurcation point and the associated sideway snapping phenomenon in the perfect model.

Mechanical modeling and analysis of V-shaped LUM mechanical drift

The flexible clamping component is used to support the stator of linear ultrasonic motors (LUM). It distinctly improves the vibration characteristics of the motor and simplifies the structure, nevertheless brings the mechanical drift phenomenon. In order to improve the positional accuracy and structural stability of the V-shaped LUM, this paper studies the mechanical drift mechanism of the LUM with clamping component. Mechanical model of the motor with flexible clamp which occur mechanical drift is established, and the mechanism as well as control methods are analyzed. Based on the model, the mechanical drift experiments of clamping components with different stiffness are carried out. The experimental results show that the mechanical drift is obvious when the stiffness of the two flexible clamps are different, while the mechanical drift hardly occur when using the clamping components with tremendous tangential stiffness. Therefore, a kind of straight beam clamping LUM is proposed. The research indicate that the new motor has little mechanical drift, better running performance and higher structural stability, which can be used in the precision positioning of the mechanical devices.

Influence of spatially varying material properties on the bimoment normal and shear stresses by warping torsion of FGM beams

In this contribution, which is an extension of author’s research [1,2], an influence of the in two and three directions varying material properties on non-uniform torsion of the Functionally Graded Material (FGM) thin-walled beams is originally investigated. Based on the semi-analytical solution of the fourth order differential equation for non-uniform torsion, the local finite element equations of the twisted FGM beam are presented, considering the non-uniform torsion with effect of warping and secondary deformations due to the angle of twist. The warping part of the first derivative of the twist angle caused by the bimoment is considered as an additional degree of freedom at the beam nodes. The Multi-Layers Method (MLM) [3], and the Reference Beam Method (RBM) [4] are extended for homogenization of the spatially varying material properties and relevant stiffnesses in the real beam onto effective longitudinally varying ones in the homogenized beam for the load case of non-uniform torsion.The focus of the numerical investigation, with consideration of the warping and Deformation Effect due to the Secondary Torsional Moment (STMDE), is on elastostatic analysis of straight FGM beams with rectangular hollow cross-sections. The influence of the variation of the material properties with respect to two or three directions on the angle of twist and the bimoment normal and torsional shear stresses is investigated. The obtained results are compared with the ones calculated by a very fine mesh of standard solid finite elements. This FGM warping non-uniform torsion beam finite element can be used not only in torsional analysis of general thin-walled beams but also in the design of FGM sensors and actuators in mechatronics.

Forced vibration analysis of composite-geometrically exact elliptical cone helices via mixed FEM

In this pioneering study, the cross-sectional warping included transient response and normal/shear stress components of composite elliptical and elliptical cone helices over exact axis geometry are investigated using a mixed FEM. The transient analysis is performed using the Newmark time integration algorithm with or without the amplitude decay factor. The constitutive equations of composite curved rods are derived from three-dimensional elasticity theory. A displacement-type finite element formulation computing the warping-included torsional rigidity is incorporated with the mixed finite element formulation. The curvatures and displacement-type finite elements are used to estimate the normal and shear stress distributions on the respective cross-sections. The maximum normal/shear stresses of a composite straight beam are compared with the literature. An excellent agreement is obtained for the results of an exact elliptical cone helix under dynamic loads compared to the results of 3D solid finite elements. During the implementation of the time integration scheme, the first and second time derivatives of forces and moments are preserved, and their time histories are discussed. Finally, the influences of helix geometry, lamination, and the ratios of material constants on the transient response besides the stresses are investigated. All the numerical examples in this paper are original for the literature.

Behavior of RC stepped beams with different configurations: An experimental and numerical study

AbstractIn the current research, an experimental program, consisting of one straight beam and seven stepped beams (SBs), is to be carried out. Furthermore, a numerical validation is to be applied using finite element analysis. The effect of the additional vertical and horizontal stirrups, the diagonal bars, the configuration of the tension bars, and the inclination angle of the stepped joint on the behavior of the SB is to be examined. Moreover, the influence of the width and the drop of the stepped joint on the behavior of the SB is to be induced. Compared with the straight beam, the ratios of the ultimate load decrease of the SB ranged from −64.23% to 0.00%. On the one hand, as the joint drop increases, the ultimate load of the SB decreases. On the other hand, when the joint width increases, the ultimate load of the SB increases. Many proposed techniques in reinforcing and dimensions of the stepped part have increased the flexural capacity. Proposed equations of the flexural capacity of the SB are to be induced after examination, taking into consideration the required parameters of the stepped beams, such as the compressive strength of the concrete, the tension and the compression steel, the dimension of the beam section, the horizontal stirrups, the diagonal bars, or the width and the drop of the stepped joint.

Vibration of cylindrical shells with embedded annular acoustic black holes using the Rayleigh-Ritz method with Gaussian basis functions

The numerical simulation of beams and plates with embedded acoustic black holes (ABHs) is computationally demanding because of the very thin thickness attained at the ABH central area. Semi-analytical approaches relying on the Rayleigh-Ritz method with wavelet or Gaussian basis functions have thus revealed as an accurate and fast alternative to determine the ABH vibration field in parametric studies. To date however, the vast majority of works on ABHs have only dealt with ABH indentations on straight beams and flat plates. It would be also worth exploring the feasibility of ABHs to control the vibrations of curved shells, typically found in aerospace and naval structures. In this work, we address this issue and extend the Gaussian expansion method (GEM) to characterize annular ABHs embedded on cylindrical shells. First, we show how the GEM can be modified to make Gaussian shape functions satisfy periodic boundary conditions in the circumferential direction of the cylinder. The GEM is then used to determine the vibration field of the ABH cylindrical shell and gets validated by comparison with finite element simulations. A thorough analysis of the performance of the annular ABH follows, which stresses the differences with the behavior of ABHs on flat surfaces. In particular, we show the influence that waves propagating in the circumferential direction have on the operational frequency range of the ABH. The effects of the viscoelastic layer and the inclusion of longitudinal stiffeners to strengthen the cylinder rigidity are also analyzed by means of the proposed GEM approach. This work broadens previous semi-analytical methods to start investigating the ABH effect on curved structures.

Vibration of the Biomass Boiler Tube Excited with Impact of the Cleaning Device

In boilers with biomass fuel, a significant problem is caused due to the slag layer formed from the unburned particles during combustion. In the paper, a tube cleaning method from slag is proposed. The method is based on the impact effect of the end of the tube with the aim to produce vibration for slag elimination. The tube is modeled as a clamped-free nonlinear oscillatory system. The initial impact of the tube causes vibrations. The mathematical model of the system is a nonlinear partial differential equation with zero initial deflection. To obtain the ordinary differential equations, the Galerkin method is applied. By discretizing the equation into a finite degree of freedom system, using the undamped linear mode shapes of the straight beam as basic functions, the reduced order model, consisting of ordinary differential equations in time, is obtained. The ordinary time equations are analytically solved by adopting the Krylov–Bogoliubov procedure. Special cases of nonlinear differential equations are investigated. In the paper, the influence of the nonlinear parameters and initial conditions on the vibration properties of the tube is obtained. We use the procedure developed in the paper and the analytical results for computation of the impact parameters of the cleaning device.

Nonlinear Buckling Analysis of Thin-Walled Box Beams considering Shear Lag

In this work, a general geometric nonlinear model of straight thin-walled box beams (STBBs) under combined eccentric and axial loads is established. In order to accurately reflect the behavior of STBB, the additional shear lag warping is added to enrich the displacement field. It is necessary to define the section shape function to describe the local section deformation. Therefore, extension, bending, torsion, distortion, and shear lag effects are expressed by the generalized coordinate method. Based on the stability of transverse unconstrained box beam theory, meaningful higher-order solutions can be obtained by defining a set of coupled deformation modes. The equilibrium equation is discretized by the Galerkin method, and the Newton–Raphson incremental method is used to derive and solve the nonlinear governing equations. On this basis, the analytical expression of stiffness matrix is established. For solving the stability problem, the effectiveness of the proposed method is verified by comparing the calculation results of shell element (Ansys) with other theories. Numerical examples even show that the proposed method can not only get the influence of shear lag but also obtain the variation of lateral buckling of the beam model.

A generalized pseudo-rigid-body PPRR model for both straight and circular beams in compliant mechanisms

The pseudo-rigid-body (PRB) models have been widely applied for modeling of straight beams in compliant mechanisms. In this paper, a generalized PRB PPRR model with two mutually perpendicular linear springs is established for both straight and circular beams in compliant mechanisms. In the developed model, the static equations of the deflected beams are derived from Bernoulli’s equation. The characteristic parameters of the developed model are identified by the reasonably organized particle swarm optimization (PSO) algorithm. A simulation experiment of tip locus of a circular beam is conducted to evaluate the developed model. The simulation results show the developed model has higher accuracy than the PRB 2R, 3R, and RRHR model. In addition, two numerical examples are further implemented to demonstrate the effectiveness of the developed model.

Boundary tracking control of flexible beams for transferring motions

Although flexible beams for transmitting both translational and rotational large motions are used in practice such as ocean drill pipes, their control has not been considered. This paper develops boundary feedback controllers to stabilise these beams at their reference configurations. Exact nonlinear partial differential equations governing motion of the beams in three-dimensional space are derived and used in the control design. The designed controllers guarantee globally practically asymptotically stability of the beam motions at the reference states (i.e. positions and rotations of a straight beam moving axially with a desired velocity and rotating around its axial axis with a desired velocity). In the control design and analysis of well-posedness and stability, we utilise different transformations between the earth-fixed and body-fixed coordinates, Sobolev embeddings, and a Lyapunov-type theorem developed for a class of evolution systems in Hilbert space. Simulation results are also included to illustrate the effectiveness of the proposed control design.