Fractional Fourier transform of a higher-order cosh–Gaussian beam

A higher-order cosh–Gaussian beam is an appropriate model to describe the flattened laser beam. The fractional Fourier transform (FRFT) is applied to treat the propagation of higher-order cosh–Gaussian beams. An analytical expression for a higher-order cosh–Gaussian beam passing through a FRFT system has been derived. By using the derived expression, the properties of a higher-order cosh–Gaussian beam in the FRFT plane are graphically illustrated with numerical examples.     

Uncertain dynamic response of a deterministic elastic–plastic beam

The counterintuitive phenomenon of elastic–plastic beam dynamics was demonstrated by Symonds and Yu (ASME J Appl Mech 1985;52:517). An analytical model has been developed to explain this phenomenon from a deterministic viewpoint. However, experimental evidence in (Int J Impact Eng 1991;11(3):341; Int J Impact Eng 1991;11(4):445) showed that the response of this deterministic system is uncertain, which is studied qualitatively in the present paper based on parametric sensitive characteristics of the deterministic system and parametric uncertainty of the studied system. FEM and Monte Carlo method are applied to study this phenomenon.     

Fraunhofer diffraction of a Laguerre–Gaussian laser beam by fork-shaped grating

In this article we present a theoretical study for Fraunhofer diffraction of a Laguerre–Gaussian laser beam with zeroth radial mode number and azimuthal mode number l by a diffractive grating with embedded fork-shaped dislocations of integer order p. Analytical expressions describing the diffracted wave field amplitude and intensity distributions in the Fourier plane are deduced and analyzed. They are also followed by the vortex radii expressions.     

The M 2 factor matrix of a paraxial Laguerre–Gaussian beam

The M ² factor matrix of a paraxial Laguerre–Gaussian beam has been derived based on the combination of the M ² factor definition and introduced quantities such as the coupled beam’s half width square, the coupled M ² factor and 2×2 rotational matrix, when the beam is rotated around the propagation z-axis by an arbitrary azimuth angle α. The beam’s half widths at different distances from the beam waist along the z-axis, the M ² factors of the beam with different orders, and the track of M ² factor matrix elements versus α have been obtained by numerical simulation and analytical derivation. Results illustrate that by using the invariability of the M ² factor matrix, the beam quality of the mode with an arbitrary rotational angle can be easily evaluated. Like the M ² factor matrix of the Hermite–Gaussian mode, this M ² factor matrix is meaningful in the evaluation of the beam quality of a two-dimensional astigmatic or asymmetric laser beam.     

The kurtosis parametric characterization of the passage of a standard Hermite–Gaussian beam and an elegant Hermite–Gaussian beam through a fractional Fourier transformation system with a spherically aberrated lens

The propagation characteristics of the kurtosis parameters of a standard Hermite–Gaussian (SHG) beam and of an elegant Hermite–Gaussian (EHG) beam, each passing through a fractional Fourier transformation (FRFT) system with a spherically aberrated lens, are studied in detail. Some numerical calculations are made by introducing an efficient algorithm, based on the Collins diffraction integral formula. The resulting graphs illustrate the striking difference between ideal FRFT systems and those with a spherically aberrated lens. The kurtosis parameters of both SHG and EHG beams passing through a type I Lohmann system with a spherically aberrated lens are seen to change with the fractional order periodically and the fundamental period is 4, but for type II the fundamental period is 2. Different values of spherical aberration coefficients affect the kurtosis parameters in greatly different ways. The values of the kurtosis parameters of a SHG beam passing through either type of Lohmann system with a spherically aberrated lens are no longer equal to those of an EHG beam, even when they have the same fractional orders and the same spherical aberration coefficients.     

Mo surface enrichment on a Fe–Cr alloy by electron beam irradiation

Mo alloying has been carried out on a Fe–Cr alloy by electron beam irradiation. It is shown that a Mo surface enrichment can be obtained in the range 2–8 at %. Scanning and transmission electron microscopy, X ray microanalysis and electron energy loss spectroscopy have been used to investigate the microstructures obtained after irradiation. Mo–Cr carbide and intermetallic X phase have been identified after alloying. The intermetallic phase is preferentially formed when the Mo content increases. It is concluded from electrochemical studies and potentiostatic attacks that the active and transpassive dissolution rates are closely related to the Cr content in the ferritic phase. This work must be also regarded as a first step towards the surface preparation of 316 S. S. by Mo incorporation to 304 S. S.     

Production of a positron beam in the energy range 1–10 MeV

The setup and characteristics of a low energy positron beam at the Giessen linac are desribed. The beam energy can be varied between 1.1 and some MeV. The measured mean positron currents are in the order of some pA within a momentum bin of 0.5%.     

Microstructure and mechanical properties of a new Al–Zn–Mg–Cu alloy joints welded by laser beam

A new type of Al–Zn–Mg–Cu alloy sheets with T6 temper were welded by laser beam welding (LBW). Microstructure characteristics and mechanical properties of the joints were evaluated. Results show that grains in the heat affected zone (HAZ) exhibit an elongated shape which is almost same as the base metal (BM). A non-dendritic equiaxed grain zone (EQZ) appears along the fusion line in the fusion zone (FZ), and grains here do not appear to nucleate epitaxially from the HAZ substrate. The FZ is mainly made up of dendritic equiaxed grains whose boundaries are decorated with continuous particles, identified as the T (AlZnMgCu) phase. Obvious softening occurs in FZ and HAZ, which mainly due to the changes of nanometric precipitates. The precipitates in BM are mainly η′, while plenty of GPI zones exist in FZ and HAZ adjacent to FZ, in the HAZ farther away from FZ, η phase appears. The minimum microhardness of the joint is always obtained in FZ at different times after welding. The ultimate tensile strength of the joint is 471.1 MPa which is 69.7% of that of the BM. Samples of the tensile tests always fracture at the FZ.     

Aluminum–GFRP hybrid square tube beam reinforced by a thin composite skin layer

Ultimate bending moments and energy-absorption capability of aluminum–glass fiber reinforced plastic (GFRP) hybrid tube beams were experimentally analyzed with particular focuses on effects of thin GFRP skin layer in relation to bending deformation behavior and fracture characteristics. Various hybrid tube beams were fabricated by inserting adhesive film between prepreg and metal layers and by aligning various composite ply angles. Under 3-point bending loads, aluminum–GFRP hybrid tube beams showed characteristic fracture processes according to the lay-up kinds of the skin layer in comparison to the virgin aluminum tube beams. In particular, the hybrid tube beams having a 0.5 mm thick [0°/90°]_s skin layer showed the largest improvement in specific maximum moment (about 67%) and in specific energy-absorption (29%). Consequently, there was an optimal thickness and lay-up of the composite skin layer in creating the best performance of the hybrid tubes.     

A note on the integrability of a remarkable static Euler–Bernoulli beam equation

It has recently been shown that the fourth-order static Euler–Bernoulli ordinary differential equation, where the elastic modulus and the area moment of inertia are constants and the applied load is a function of the normal displacement, in the maximal case has three symmetries. This corresponds to the negative fractional power law y ^?5/3, and the equation has the nonsolvable algebra ${sl(2, \mathbb{R})}$ . We obtain new two- and three-parameter families of exact solutions when the equation has this symmetry algebra. This is studied via the symmetry classification of the three-parameter family of second-order ordinary differential equations that arises from the relationship among the Noether integrals. In addition, we present a complete symmetry classification of the second-order family of equations. Hence the admittance of ${sl(2, \mathbb{R})}$ remarkably allows for a three-parameter family of exact solutions for the static beam equation with load a fractional power law y ^?5/3.     

Spectral shift of a partially coherent standard or elegant Laguerre–Gaussian beam in turbulent atmosphere

Spectral changes of a partially coherent standard or elegant Laguerre–Gaussian (LG) beam propagating in turbulent atmosphere were studied numerically. Our results show that the spectral changes of a partially coherent standard or elegant LG beam in turbulent atmosphere are determined by both the structure constant of the turbulent atmosphere and the initial beam parameters. Furthermore, it is found that a partially coherent elegant LG beam is less affected by the turbulent atmosphere than a partially coherent standard LG beam from the aspect of the on-axis spectral shift, and this advantage is enhanced for small structure constant, small beam waist size, large mode orders, and large transverse coherence length. Our results will be useful in long-distance free-space optical communications.     

An analytical elastic–plastic stress analysis in a woven steel reinforced thermoplastic cantilever beam loaded uniformly

In this investigation, an elastic–plastic stress analysis is carried out in a woven steel fiber reinforced thermoplastic (LDFE, F.2.12) composite cantilever beam loaded uniformly at the upper surface. Closed form solution is found satisfying both the governing differential equation and boundary conditions. The beam is loaded by a small uniform force at the upper surface, therefore during the solution of the problem for the elastic–plastic case, the normal stress σ_y is neglected in comparison with other stress components. The orientation angle of the fibers are chosen as 0, 15, 30 and 45°. The intensity of the residual stress component of σ_x is maximum at the upper and/or lower surfaces of the beam. The intensity of the residual stress component of the shear stress τ_xy is maximum on or around the x axis.     

Flutter of a beam in supersonic flow: truncated version of Timoshenko–Ehrenfest equation is sufficient

International Journal of Mechanics and Materials in Design - This paper deals with flutter due to gas flow of a uniform and homogeneous beam with shear deformation and rotary inertia effects taken...     

Focal switch and dual focal switch of a Hermite–Gaussian beam illuminating a lens and sinusoidal amplitude grating separated system

Based on the Collins formula, analytical representation of a Hermite–Gaussian beam propagating through a lens and sinusoidal amplitude grating separated system is obtained. Focal shift occurring in such a system is studied in detail with numerical calculation examples. It is found that the focal switch and the dual focal switch can emerge depending not only on the effective Fresnel number, but also on the uniform transmittance amplitude and the spatial frequency of the grating and beam parameter of the illumination Hermite–Gaussian beam.     

Fatigue crack growth behaviour of gas tungsten arc,electron beam and laser beam welded Ti–6Al–4V alloy

The present investigation is aimed to evaluate fatigue crack growth parameters of gas tungsten arc, electron beam and laser beam welded Ti–6Al–4V titanium alloy for assessing the remaining service lives of existing structure by fracture mechanics approach. Center cracked tensile specimens were tested using a 100 kN servo hydraulic controlled fatigue testing machine under constant amplitude uniaxial tensile load. Crack growth curves were plotted and crack growth parameters (exponent and intercept) were evaluated. Fatigue crack growth behavior of welds was correlated with mechanical properties and microstructural characteristics of welds. Of the three joints, the joint fabricated by laser beam welding exhibited higher fatigue crack growth resistance due to the presence of fine lamellar microstructure in the weld metal.     

Evolution properties of the complex degree of coherence of a partially coherent Laguerre–Gaussian beam in turbulent atmosphere

Evolution properties of the complex degree of coherence of a partially coherent Laguerre–Gaussian beam (LGB) on propagation in free space and turbulent atmosphere are studied comparatively with the help of the general propagation formula for such beam. It is found that the behavior of the complex degree of coherence of a partially coherent LGB on propagation in turbulent atmosphere is much different from that in free space and is closely related to the initial beam parameters and the structure constant of the turbulent atmosphere. The distribution of the modulus of the complex degree of coherence of the partially coherent LGB finally becomes of Gaussian distribution at long propagation distance in turbulent atmosphere, and it becomes of Gaussian distribution more slowly with the increase of the mode orders, beam width and wavelength. Our results will be useful in long-distance free-space optical communications.     

Hybrid control of vibrations of a smart von Kármán plate

The suppression of large vibrations of a smart thin elastic rectangular von Kármán’s plate is considered. The plate is subjected to external disturbances and generalized control forces produced by electromechanical feedback. The considered nonlinear initial-boundary value problem is spatially discretized by means of the time spectral method. The implicit Newmark-β iterative method is employed for the time integration of the obtained system of nonlinear equations of motion. Nonlinear controllers are designed, based on a fuzzy inference system. Two numerical algorithms involving a general control of displacement/velocity and a direct control of the Fourier coefficients are proposed. The techniques have been implemented within MATLAB environment with the use of the fuzzy logic toolbox. Numerical examples are presented.     

Vectorial structure of Ince–Gaussian beam in the far field

By means of the vector angular spectrum representation of the electromagnetic beam and the method of stationary phase, the analytical vectorial structure of the Ince–Gaussian beam has been presented in the far field. The amplitude distributions of the Ince–Gaussian beam and its TE and TM terms are investigated in the far field. The extreme cases of the ellipticity parameter tending to infinity or zero are also considered. Although the vectorial structures of different Ince–Gaussian beams are apparently distinct, the ratios of the amplitude distributions of the TE and TM terms to the whole beam amplitude are independent of the parity and the values of the radial and angular elliptic mode numbers. This research reveals the abundant and interesting internal details of the Ince–Gaussian beam in the far field.