Order Beam Research Articles

Modal analysis of the FGM beams with effect of the shear correction function

In the proposed contribution the effect of the shear correction function is originally studied and evaluated in modal analysis of the functionally graded material (FGM) beams. Spatially continuous variations of the material properties are considered. The shear correction function is calculated from the shear strain energy equation including spatial Poisson′s ratio variations. The equations of the homogenized FGM beam free vibration and their solution is presented including the shear correction function. Four coupled differential equations are derived and used in the modal analysis of beams with polynomial continuous longitudinal and transversal variations of material properties. Further, 2nd order beam effects and longitudinal varying elastic beam foundations are considered. The influence of using an average shear correction factor is evaluated through numerical experiments. Additionally, the longitudinal eigenfrequencies are calculated. Continuum solutions using solid finite elements are taken as a reference for comparison purposes.

Acoustic radiation force and radiation torque on Rayleigh particles

In this work, the acoustic radiation force and radiation torque exerted by an arbitrary shaped wave on a spherical particle in the Rayleigh approximation (i.e. the incident wavelength is much smaller than the particle dimensions) are discussed. The host fluid in which the particle is suspended is assumed to be inviscid. Expressions for the force and the torque are obtained in terms of the incident acoustic fields, namely pressure and particle velocity. As it turns out, the obtained radiation force expression represents a generalization of Gor'kov's formula [Sov. Phys. Dokl. 6, 773-775 (1962)]. Moreover, the radiation torque can be expressed in terms of the incident Reynolds' stress tensor. The method is applied to calculate both radiation force and radiation torque produced by Bessel beams. It is demonstrated that only the first-order Bessel vortex beam generates radiation torque on a particle placed in the beam's axis. In addition, results involving off-axial particles and Bessel beams of different order are illustrated.

Measurement of adhesive shear properties by short beam shear test based on higher order beam theory

This paper refers to the measurement of the shear properties of adhesive bonding by a new beam theory using the short beam shear (SBS) test configuration. A novel higher-order sandwich beam theory has been developed to analyze the adhesive bonded beam that consists of two adhered laminates and a single layer of adhesive in between. The closed form analytical solution for the SBS test model of the adhesively bonded beam is obtained in terms of deflection and stress distribution. The present theory has been used for calculating the adhesive shear modulus from the structural compliance. The initiation of stiffness degradation for the short beam shear test model was used as the critical load value for deriving the adhesive shear strength. A finite element model is built for validating the present model, and to evaluate its suitability for measuring adhesive shear properties. The present theory shows better accuracy for measuring the shear modulus than existing theories for both thin and thick adhesive layers. The measured strength values are more accurate than those obtained from the single lap joint shear test model. This theory can be used for adhesive materials with linear elastic deformation behavior.

Linear stability analysis for travelling waves of second order in time PDE's

We study travelling waves φc of second order in time PDE's . The linear stability analysis for these models is reduced to the question of the stability of quadratic pencils in the form , where .If is a self-adjoint operator, with a simple negative eigenvalue and a simple eigenvalue at zero, then we completely characterize the linear stability of φc. More precisely, we introduce an explicitly computable index , so that the wave φc is stable if and only if . The results are applicable both in the periodic case and in the whole line case.The method of proof involves a delicate analysis of a function , associated with , whose positive zeros are exactly the positive (unstable) eigenvalues of the pencil . We would like to emphasize that the function is not the Evans function for the problem, but rather a new object that we define herein, which fits the situation rather well.As an application, we consider three classical models—the ‘good’ Boussinesq equation, the Klein–Gordon–Zakharov (KGZ) system and the fourth order beam equation. In the whole line case, for the Boussinesq case and the KGZ system (and as a direct application of the main results), we compute explicitly the set of speeds which give rise to linearly stable travelling waves (and for all powers of p in the case of Boussinesq). This result is new for the KGZ system, while it generalizes the results of Alexander et al (2012, personal communication) and Alexander and Sachs (1995 Nonlinear World 2 471–507), which apply to the case p = 2. For the beam equation, we provide an implicit formula (depending only on the function , which works for all p and for both the periodic and the whole line cases.Our results complement (and exactly match, whenever they exist) the results of a long line of investigation regarding the related notion of orbital stability of the same waves. Informally, we have found that in all the examples that we have looked at, our theory applies, whenever the Grillakis–Shatah–Strauss (GSS) theory applies. We believe that the results in this paper (or a variation thereof) will enable the linear stability analysis as well as asymptotic stability analysis for most models in the form .

Finite element simulation of dense wire packings

A finite element program is presented to simulate the process of packing and coiling elastic wires in two- and three-dimensional confining cavities. The wire is represented by third order beam elements and embedded into a corotational formulation to capture the geometric nonlinearity resulting from large rotations and deformations. The hyperbolic equations of motion are integrated in time using two different integration methods from the Newmark family: an implicit iterative Newton-Raphson line search solver, and an explicit predictor-corrector scheme, both with adaptive time stepping. These two approaches reveal fundamentally different suitability for the problem of strongly self-interacting bodies found in densely packed cavities. Generalizing the spherical confinement symmetry investigated in recent studies, the packing of a wire in hard ellipsoidal cavities is simulated in the frictionless elastic limit. Evidence is given that packings in oblate spheroids and scalene ellipsoids are energetically preferred to spheres.

The effect of transverse normal strain in contact of an orthotropic beam pressed against a circular surface

The contact problem of a straight orthotropic beam pressed onto a rigid circular surface is considered using beam theories that account for transverse shear and transverse normal deformations. The circular nature of the rigid surface emphasizes the difference between Euler Bernoulli theory behavior, where point loads develop at the edge of contact, and the higher order theories that predict non-singular pressure distributions. While Timoshenko beam theory is the simplest theory that addresses this behavior, the prediction of a maximum value of pressure at the edge of contact contradicts the elasticity theory result that contact pressure must drop to zero. Transverse normal strain is therefore introduced, both to study this fundamental discrepancy and to include an important effect in many contact problems. To investigate this effect, higher order beam theories that account for both constant and linear transverse normal strain through the beam thickness are derived using the principle of virtual work. The resulting orthotropic beam theories depend on the bending stiffness (EI), shear stiffness (GA), axial stiffness (EA1) and transverse normal stiffness (EA2), which are independent stiffness parameters that can differ by orders of magnitude. The above mentioned contact problem is then solved analytically for these theories, along with the Timoshenko beam model which assumes zero transverse normal strain. The results for different orthotropic materials show that inclusion of transverse normal deformation has a significant effect on the contact pressure solution. Furthermore, the solution using higher order beam theories encompasses the two extremes of a Hertz-like contact pressure when the half contact length is smaller than the thickness of the beam, and the Timoshenko beam theory case when the half contact length is much larger than the thickness. Concerning the behavior of the pressure at the edge of contact, adherence to the boundary conditions required by the principle of virtual work, shows that while the pressure does tend to zero, it does not become zero unless artificially enforced. In this regard the solution for the case of linear strain is better than that for constant strain. All beam solutions are validated with plane elasticity solutions obtained using the commercial finite element software ABAQUS.

An Influence Analysis of Second-Order Effect to the Vibration Control of Piezoelectric Beam by the Spline Finite Point Method

The field functions of the spline finite point (SFP) method were constructed by the linear combination of B-spline basis function, and the higher-precision results would be obtained with less discrete nodes by the SFP method. In this paper, based on Reddy's third order beam theory, a motion equation was developed by the SFP method to analyze the first five natural frequencies of piezoelectric laminated beam under different axial forces. The influence of second-order effect caused by the axial force on vibration control was discussed based on the modal control theory and the Linear Quadratic Regulator (LQR) optimal control method. It can be concluded that the SFP method is suitable for the dynamic analysis of piezoelectric beam which needs less computational cost and has high accuracy. The vibration of the structure can be effectively inhibited by the LQR method and modal control theory. And the axial force has significant impact on the natural frequencies and control voltage of piezoelectric beam.

A new two-dimensional refined higher order theory for in-plane free vibrations of sandwich and composite thick curved beams with flexible cores

Closed-form formulations of two-dimensional refined higher order beam theories and higher order beam theories for the free vibration analysis of simply supported cross-ply laminated composite and sandwich curved beams are presented. The present refined higher order beam theory analysis incorporates a trapezoidal shape factor that arises due to the fact that the stresses through the thickness of the beam are integrated over a curved surface. Therefore, the present theory refines other higher order beam theories established hitherto for free vibration analysis of strongly thick circular curved beams. The stress–strain relationship is derived from an orthotropic lamina in a three-dimensional state of stress. Numerical results are presented for the natural frequencies of laminated composite and sandwich thin/thick straight beams and shallow/deep and moderately/strongly thick curved beams. The closed-form solutions presented herein are compared with the available exact two-dimensional elasticity and analytical solutions in the available literature and excellent agreement is obtained.

Analysis of composite beams with longitudinal and transverse partial interactions using higher order beam theory

A new one dimensional finite element model based on a higher order beam theory is presented for the analysis of composite beams taking into account the effect of longitudinal as well as vertical partial interaction between the adjacent layers. The proposed method models the transverse shear deformation of the beam components in a refined manner. A third order variation of the axial displacement of the fibres over the beam depth is taken to have a parabolic variation of shear stress which is also made zero at the beam top and bottom surfaces. In the proposed FE model, there is no need of incorporating any shear correction factor and the model is free from shear locking problem. In addition to correctly predicting the global responses of the beam, the model can predict better distribution of stresses than the existing models based on Euler–Bernoulli or Timoshenko beam theory. Many new results are presented as there is no published result on the present problem based on higher order beam theory.

Accurate measurement of orthogonality of equal-period, two-dimensional gratings by an interferometric method

A two-dimensional grating can be used as a key component in planar encoders for measuring two-dimensional displacements or calibrating coordinate measuring machines. Ideally the two main periodic directions of the grating, the directions along which a translation by a grating period leaves the two-dimensional pattern indistinguishable from the untranslated one, should be perfectly orthogonal; any deviation from orthogonality causes cross-talk errors and necessitates system calibration. We present a method to measure the orthogonality, or the non-orthogonality angle of two-dimensional gratings precisely. This method uses interference fringes generated by diffracted beams of different orders to align the grating's periodic directions, and applies a new measurement strategy to directly measure the non-orthogonality angle by an autocollimator. Compared with traditional optical diffractometry, its angular position alignment is of higher sensitivity and its angle measurement is of lower uncertainty, and the measurement uncertainty is reduced. Orthogonalities of four gratings were measured and the standard uncertainty was 0.28 arcsec. The results agree well with the measurement results of optical diffractometry.

Analysis of Thin-Walled Straight Beams with Generally Shaped Closed Sections Using Numerically Determined Sectional Deformation Functions

This investigation presents one-dimensional static and eigenvalue analyses of thin-walled straight beams with generally shaped closed single-cell or multicell sections. For accurate beam analysis, sectional warping and distortional deformations should be considered in addition to the standard Timoshenko displacement field, but it is difficult to obtain the deformation functions analytically for arbitrarily shaped sections. Thus, a numerical method is proposed to obtain sectional deformations for any arbitrarily shaped sections. Once the deformations are identified, they can be integrated over a cross section to yield one-dimensional higher order beam equations. For the numerical determination, the cross section of a thin-walled beam is modeled as a beam frame, where the warping and distortional deformation functions of the section are identified as the eigenmodes of the frame model; the lowest few energy mode sets of in-planar and out-of-planar modes are selected as the distortional and warping deformation functions, respectively. The validity of this approach is checked by comparing the present results with shell finite-element results. For numerical tests, several thin-walled closed sections, including those with flanges or varying wall thicknesses, are considered. The effect of the number of selected warping and distortion sets on solution convergence is also investigated.

Measuring the rotation rates of superpositions of higher-order Bessel beams

Experimental measurements are reported of the rotation rates of superpositions of higher-order Bessel beams. Digitally generated phase masks of two annular rings, were imprinted on a spatial light modulator and used to obtain superpositions of higher-order Bessel beams of the same order but of opposite topological charge. Such a superposition field carries on average zero orbital angular momentum, yet exhibits a rotation in the intensity pattern: the resultant field rotates at a constant rate about the optical axis as it propagates. The rotation rates of the generated fields were measured for different orders and for various values of the difference between the wave-vectors of the superimposing beams, and are shown to be in good agreement with that predicted theoretically.

Analysis of composite beams with partial shear interactions using a higher order beam theory

A new finite element model based on a higher order beam theory is presented for the analysis of composite beams. The proposed model takes into account the effect of partial shear interaction between the adjacent layers as well as transverse shear deformation of the beam. A third order variation of the axial displacement of the fibres over the beam depth is taken to have a parabolic variation of shear stress which is also made zero at the beam top and bottom surfaces. In the proposed FE model, there is no need of incorporating any shear correction factor and the model is free from shear locking problem. In addition to correctly predicting the global responses of the beam, the model can predict better distribution of stresses than the existing models based on Euler–Bernoulli or Timoshenko beam theory. The proposed finite element model is validated by comparing the results with those available in literature. Many new results are presented for future references as there is no published result on composite beams based on higher order beam theory.

Flexural Strength and Stiffness of High Strength Concrete Beams with Exposed Main Steel

In a number of previous publications, results were reported for a series of extensive and carefully conducted tests on large scale reinforced concrete (R.C.) beams with various extents of loss of concrete cover and exposure of main reinforcement along their spans, with such areas of simulated damage being located within their regions which are dominated by either shear or flexure. These tests on R.C. beams made with normal strength concrete have covered a wide range of first order beam design parameters, with their results used to verify the generality of various theoretical models. In the present paper, much attention will be devoted to various structural characteristics (such as ultimate strength, flexural stiffness, etc.) of similar damaged R.C. beams with the proviso that, instead of the previously used normal strength concrete, the beams are made with high strength concrete. No such results (for high strength R.C. beams) have previously been reported in the public domain.

Modal analysis of the FGM-beams with continuous transversal symmetric and longitudinal variation of material properties with effect of large axial force

In this contribution we deal with modelling and simulation of a free vibration of the 2D functionally graded material (FGM) beams with continuous spatial variation of material properties. The fourth-order differential equation of the second order beam theory has been presented which was used in modal analysis with effect of large axial force. The continuous variation of the effective elasticity modulus and mass density can be caused by continuous variation of both the volume fraction and material properties of the FGM constituents in the transversal and longitudinal direction. The polynomial variation of these parameters has been considered. Homogenization of the varying material properties and the calculation of other parameters have been done by two methods. In the first one, the extended mixture rules and laminate theory have been used, where the real composite beam has been transformed to the multilayer beam. In the second one, the direct integration method has been used. Not only the shear force deformation effect and the effect of consistent mass distribution and mass moment of inertia but also the effect of large axial force has been taken into account. Numerical experiments have been done concerning the calculation of the eigenfrequencies and eigenmodes of chosen FGM beams. Effect of division fineness of the beam to the layers in transversal direction on the solution results has been evaluated. The solution results obtained by the both methods have been discussed and compared with the ones obtained using very fine meshes of 2D solid elements of a FEM commercial program.

Fourier Cosine Differential Quadrature Method for Beam and Plate Problems

In this paper, we combined the Fourier cosine series and differential quadrature method (DQM) in barycentric form to develop a new method (FCDQM), which is applied to the 1D fourth order beam problem and the 2D thin isotropic plate problems. Furthermore, we solved the complex boundary conditions on irregular domains with DQM directly. The numerical results illustrate the stability, validity and good accuracy of the method in treating this class of engineering problems.

Influence of the beam profile formulation when modeling fiber-guided laser welding

During deep penetration laser welding, the focused laser beam determines the vapor capillary, called keyhole, and in turn the whole process physics. Beside spot diameter and Rayleigh length, the beam profile is another important but hardly explored part of the focused laser beam. The focusing of fiber-guided Yb:fiber, Nd:YAG or diode laser beams creates the complex situation that the beam has a top-hat profile in the focal plane but toward the far field transforms to a Gaussian beam. Such power density distribution was measured for a focused high power Yb:fiber laser beam and then approached by three different beam formulations. The beam formulations were then applied to model the keyhole shape during laser welding. Although a second order beam model approached the measured beam significantly more accurately, the first order Gaussian beam was similarly suitable to predict the keyhole shape as long as the central beam domains do not interact with the material, which occurs only for low focal plane positions. A hypothetical top-hat beam would cause a different, steeper keyhole shape. Consequently, a Gaussian beam is still a suitable formulation for a wide range of welding parameters.

Transverse (lateral) instantaneous force of an acoustical first-order Bessel vortex beam centered on a rigid sphere

In a recent report [F.G. Mitri, Z.E.A. Fellah, Ultrasonics 51 (2011) 719–724], it has been found that the instantaneous axial force (i.e. acting along the axis of wave propagation) of a Bessel acoustic beam centered on a sphere is only determined for the fundamental order (i.e. m = 0) but vanishes when the beam is of vortex type (i.e. m > 0, where m is the order (or helicity) of the beam). It has also been recognized that for circularly symmetric beams (such as Bessel beams of integer order), the transverse (lateral) instantaneous force should vanish as required by symmetry. Nevertheless, in this commentary, the present analysis unexpectedly reveals the existence of a transverse instantaneous force on a rigid sphere centered on the axis of a Bessel vortex beam of unit magnitude order (i.e. | m| = 1) not reported in [F.G. Mitri, Z.E.A. Fellah, Ultrasonics 51 (2011) 719–724]. The presence of the transverse instantaneous force components of a first-order Bessel vortex beam results from mathematical anti-symmetry in the surface integrals, but vanishes for the fundamental ( m = 0) and higher-order Bessel (vortex) beams (i.e. | m| > 1). Here, closed-form solutions for the instantaneous force components are obtained and examples for the transverse components for progressive waves are computed for a fixed and a movable rigid sphere. The results show that only the dipole ( n = 1) mode in the scattering contributes to the instantaneous force components, as well as how the transverse instantaneous force per unit cross-sectional surface varies versus the dimensionless frequency ka ( k is the wave number in the fluid medium and a is the sphere’s radius), and the half-cone angle β of the beam. Moreover, the velocity of the movable sphere is evaluated based on the concept of mechanical impedance. The proposed analysis may be of interest in the analysis of transverse instantaneous forces on spherical particles for particle manipulation and rotation in drug delivery and other biomedical or industrial applications.

Linear diffraction grating interferometer with high alignment tolerance and high accuracy

We present an innovative structure of a linear diffraction grating interferometer as a long stroke and nanometer resolution displacement sensor for any linear stage. The principle of this diffractive interferometer is based on the phase information encoded by the ±1st order beams diffracted by a holographic grating. Properly interfering these two beams leads to modulation similar to a Doppler frequency shift that can be translated to displacement measurements via phase decoding. A self-compensation structure is developed to improve the alignment tolerance. LightTool analysis shows that this new structure is completely immune to alignment errors of offset, standoff, yaw, and roll. The tolerance of the pitch is also acceptable for most installation conditions. In order to compact the structure and improve the signal quality, a new optical bonding technology by mechanical fixture is presented so that the miniature optics can be permanently bonded together without an air gap in between. For the output waveform signals, a software module is developed for fast real-time pulse counting and phase subdivision. A laser interferometer HP5529A is employed to test the repeatability of the whole system. Experimental data show that within 15 mm travel length, the repeatability is within 15 nm.

Zonal wavefront sensor with reduced number of rows in the detector array

In this paper, we describe a zonal wavefront sensor in which the photodetector array can have a smaller number of rows. The test wavefront is incident on a two-dimensional array of diffraction gratings followed by a single focusing lens. The periodicity and the orientation of the grating rulings of each grating can be chosen such that the +1 order beam from the gratings forms an array of focal spots in the detector plane. We show that by using a square array of zones, it is possible to generate an array of +1 order focal spots having a smaller number of rows, thus reducing the height of the required detector array. The phase profile of the test wavefront can be estimated by measuring the displacements of the +1 order focal spots for the test wavefront relative to the +1 order focal spots for a plane reference wavefront. The narrower width of the photodetector array can offer several advantages, such as a faster frame rate of the wavefront sensor, a reduced amount of cross talk between the nearby detector zones, and a decrease in the maximum thermal noise. We also present experimental results of a proof-of-concept experimental arrangement using the proposed wavefront sensing scheme.