Rectangular Planform Research Articles

Proof of convergence for a set of admissible functions for the Rayleigh–Ritz analysis of beams and plates and shells of rectangular planform

This work presents a discussion on the characteristics of sets of admissible functions to be used in the Rayleigh–Ritz method (RRM). Of particular interest are sets that can lead to converged results when penalty terms are added to model constraints and interconnection of elements in vibration and buckling problems of beams, as well as plates and shells of rectangular planform. The discussion includes the use of polynomials, trigonometric functions and a combination of both. In the past, several sets of admissible functions that have a limit on the number of terms that can be included in the solution without producing ill-conditioning were used. On the other hand, a combination of trigonometric and low order polynomials have been found to produce accurate results without ill-conditioning for any number of terms and any number of penalty parameters that can be accommodated by the computer memory.

Levy-Type Boundary Fourier Analysis of Thick Clamped Hyperbolic-Paraboloidal Cross-Ply Panels

An analytical solution to the problem of deformation of a finite dimensional general cross-ply, thick doubly curved panel with negative Gaussian curvature and of rectangular planform, modeled using a higher-order shear deformation theory, is presented. A solution methodology, based on a boundary-discontinuous, generalized double Fourier series approach is used to solve a system of five highly coupled linear partial differential equations, generated by the higher-order shear deformation-theory-based laminated shell analysis, with the C4-type rigidly clamped boundary condition prescribed on two opposite edges, whereas the remaining two edges are subjected to the SS3-type constraint. Hitherto unavailable important numerical results presented include sensitivity of the predicted response quantities of interest to applied loading, shell geometry (negative or positive Gaussian curvature), edge clamping constraints, and lamination and thickness effects, as well as their possible interactions. Comparison with their nonnegative Gaussian curvature counterparts demonstrates the effect of negative Gaussian curvature on the deflections and moments of these doubly curved panels.

English

In this paper, we present a new numerical treatment of the lifting surface integral equation (LSIE) in the case of rectangular wing planform. The kernel of this equation possesses a strong singularity in Hadamard sense. First the equation is transformed into one containing weaker singularities, of Cauchy-type, and then the 2D singular integral is discretized by the aid of the Gauss- type quadrature formulae. Thus the problem is reduced to a finite system of linear algebraic equations. The numerical simulation reveals a very good agreement, in terms of jump pressure over the wing and aerodynamic coefficients, with the exact solution in the case of the low aspect ratio wing and also with other numerical solutions.

Dynamic Analysis of Tapered Plates Based on Higher Order Beam Theory

Modal solutions of plates with uniformly varying cross section using unified beam theory are presented. The results are given in the form of Euler-Bernoulli, Timoshenko and quasi 3D solutions. Numerical results for cantilever and CFCF supported rectangular planform plates are presented. Different types of modes, i.e. axial, bending and torsional modes are observed. The frequency values are in good agreement with 3D finite element results as well as published literature. Due to uniform taper in plate cross section, bending vibration modes become asymmetric along the longitudinal axis of the structure. Further, it can also be noticed that the vibration behavior of thick tapered plates is characterized by the appearance of significant number of axial and torsional modes at lower frequency values.

Vortex development on pitching plates with lunate and truncate planforms

AbstractThe three-dimensional flow field and instantaneous forces are measured on pitching rectangular, lunate and truncate planforms of aspect-ratio four. The leading-edge vortex on the rectangular planform is compressed as it grows, and subsequently forms an arch-shaped vortex. For the lunate and truncate planforms, which both have identical spanwise leading-edge curvature but differ in planform area, outboard-directed convection of vorticity, rather than vortex stretching, mitigates arch-vortex formation. The vortical near wake that is formed by the planforms with spanwise leading-edge curvature is found to be strongly correlated with a favourable lift-to-drag ratio during the force-relaxation phase.

Structural dynamics modeling of rectangular rotor blades

This study investigates 1-D and 3-D methods for modeling helicopter rotor blade structural dynamics to better understand the accuracy of current structural modeling based on 1-D beam theory. Natural frequencies are calculated at various rotor angular speeds for a large variety of blades ranging from simple isotropic beams to a realistic composite blade. The blade shape is limited to rectangular planforms, but various lengths are considered. 1-D beam analysis is conducted using the RCAS rotorcraft comprehensive analysis with 2-D cross-sectional properties calculated from VABS. 3-D finite element analysis is based on the commercial code MSC/Marc. Accuracy of both 1-D and 3-D analyses have been assessed through analysis of discretization errors that originate from insufficiently refined meshing. There is very good agreement between 1-D and 3-D predictions for the eight lowest modes of a large variety of blades, when there is no coupling between modes of different nature (flap, torsion) induced by materials and when the blade length is greater than ten times chord. Effects of blade length for isotropic and composite beams with no coupling between modes are similarly predicted by 1-D and 3-D analyses, except for torsion frequency, where 1-D analysis closely follows classical beam theory. With the presence of flap–torsion coupling between modes, the two approaches differ on prediction of the torsion-dominant frequency and significantly on the flap-dominant frequency.

Effect of delamination on nonlinear transient response of composite shells in hygrothermal environments

Purpose – The purpose of this paper is to study the nonlinear forced vibration responses of delaminated composite shells in hygrothermal environments. Design/methodology/approach – Finite element method using an eight-noded C0 continuity, isoparametric quadrilateral element is employed. The theoretical formulations are based on the first order shear deformation theory and von Kármán type nonlinear kinematics. For modeling the delamination, multipoint constraint algorithm is incorporated in the finite element code. Findings – The effect of delaminations on the nonlinear transient response of delaminated composite shells is dependent not only on the size but also on the location of the delaminations and hygrothermal environments. Research limitations/implications – The present study is limited to cylindrical and spherical shells having rectangular planform containing single delamination. Studies on different shell forms having non rectangular planforms containing multiple delaminations can be taken up for future research. Originality/value – Nonlinear transient response of delaminated shells in hygrothermal environments is studied for the first time. It will assist researchers of nonlinear dynamic behavior of elastic systems.

Roll Stall for Low-Aspect-Ratio Wings

The longitudinal aerodynamics of micro aerial vehicles, although not fully understood, have been the subject of several studies in recent years; however, little work has been done to investigate the lateral loading. In this experiment, flat-plate wings with rectangular planforms of aspect ratios , 1, 1.5, and 3 and tapered planforms of , 0.5, and 0.25 were placed in the wind tunnel at a Reynolds numbers of . Angle-of-attack sweeps were performed at sideslip angles of , , , , 20 and , and the side force, yaw moment, and roll moment were measured. Although the side force and yaw moment coefficients ( and ) were typically negligible, the roll moment coefficient was found to increase linearly with angle of attack before stalling in a manner reminiscent of a lift curve. This “roll stall”, which has not previously been observed for micro-aerial-vehicle-type wings, is attributed to the upstream tip vortex creating an additional lift component due to its impact on the spanwise variation of effective angle of attack in addition to the force generated by its own low-pressure core. As the downstream tip vortex is convected away from the wing, a net moment is created. Computations of the roll-stability derivative indicate that a micro aerial vehicle in equilibrium flight conditions may experience magnitudes of at the upper limit of, or even above, the range of considered to represent good handling qualities in aircraft.

Flexural response of doubly curved laminated composite shells

In the present work, analytical solutions for laminated composite doubly curved panels on rectangular plan form undergoing small deformations and subjected to uniformly distributed transverse load have been obtained. The problem is formulated using first order shear deformation theory. The spatial descretization of the linear differential equations is carried out using fast converging finite double Chebyshev series. The effect of panel thickness, curvature, boundary conditions, lamination scheme as well as material property on the static response of panel has been investigated in detail.

An experimental investigation into cavitation behaviour and pressure characteristics of alternative blade sections for propellers

During the final quarter of the last century considerable efforts have been spent to reduce the hull pressure fluctuations caused by unsteady propeller cavitation. This has resulted in further changes in propeller design characteristics including increased skew, tip unloading and introduction of “New Blade Sections” (NBS) designed on the basis of the so-called Eppler code. An experimental study was carried out to investigate flow characteristics of alternative two-dimensional (2-D) blade sections of rectangular planform, one of which was the New Blade Section (NBS) developed in Newcastle University and other was based on the well-known National Advisory Committee for Aeronautics (NACA) section. The experiments comprised the cavitation observations and the measurements of the local velocity distribution around the blade sections by using a 2-D Laser Doppler Anemometry (LDA) system. Analysis of the cavitation tests demonstrated that the two blade sections presented very similar bucket shapes with virtually no width at the bottom but relatively favourable buckets arms at the suction and pressure sides for the NACA section. Similarly, pressure analysis of the sections displayed a slightly larger value for the NBS pressure peak. The comparative overall pressure distributions around the sections suggested that the NBS might be more susceptible to cavitation than the NACA section. This can be closely related to the fundamental shape of the NBS with very fine leading edge. Therefore a further investigation into the modification of the leading edge should be considered to improve the cavitation behaviour of the NBS.

Evaluation of Comparative Performance of Three Wind Turbine Rotors

This study presents the performance evaluation of wind turbine implementing the mathematical model based on Blade Element Momentum (BEM) theory. The investigation concentrates on the comparison of power producing characteristics of three types of rotors for a horizontal axis wind turbine. The rotors considered have rectangular plan form shape with sections fabricated from aerofoil sections of NACA families. The three families of NACA airfoils used in this investigation are NACA4424, NACA23024 and NACA62-206. The lift and drag characteristics, of the rotors were experimentally obtained using 250 mm span and 60 mm chord rotor blades in an Armfield 300×300 mm subsonic, suction type and closed jet wind tunnel. The wind speeds used in the investigation were in the range of 8 to 11 m/s, which is the range of wind speeds found near the coastal belt and the mountainous parts of Saudi Arabia. Three bladed rotor subscale models of the wind turbines rotors having 250 mm rotor diameter, 100 mm blade span and 30 mm chord were used for testing rotational behavior of the wind turbines in a Plinth 600×600 mm blow down wind tunnel facility. The parameters used for the comparison in this study include variations in setting angle, diameters, tip speed ratio, taper, wind speed and angle of attack. It is observed that rotors with NACA4424 aerofoil sectioned blades produce more power at lower wind speeds and lower angles of attack where as rotors with NACA23024 aerofoil sectioned blades generate better performance at higher wind speeds and higher angles of attack. Further the taper of the rotors produced more decrease in the mass of the rotors than the decrease power produced by these. The reduction in mass could drastically decrease the inertia of the rotors which could result in higher rotational motions. Results depict the dependence of power produced on the Cd/Cl characteristics of the rotors.

Plasma Control for a Maneuvering Low-Aspect-Ratio Wing at Low Reynolds Number

Abstract Plasma-based flow control was explored as a means of enhancing the performance of a maneuvering flat-plate wing. For this purpose, a numerical investigation was conducted via large-eddy simulation (LES). The wing has a rectangular planform, a thickness to chord ratio of 0.016, and an aspect ratio of 2.0. Computations were carried out at a chord-based Reynolds number of 20,000, such that the configuration and flow conditions are typical of those commonly utilized in a small unmanned air system (UAS). Solutions were obtained to the Navier–Stokes equations, that were augmented by source terms used to represent body forces imparted by plasma actuators on the fluid. A simple phenomenological model provided these body forces resulting from the electric field generated by the plasma. The numerical method is based upon a high-fidelity time-implicit scheme and an implicit LES approach, which were applied to obtain solutions on an overset mesh system. Specific maneuvers considered in the investigation all began at 0 deg angle of attack, and consisted of a pitch-up and return, a pitch-up and hold, and a pitch-up to 60 deg. The maximum angle of attack for the first two maneuvers was 35 deg, which is well above that for static stall. Two different pitch rates were imposed for each of the specified motions. In control situations, a plasma actuator was distributed in the spanwise direction along the wing leading edge, or extended in the chordwise direction along the wing tip. Control solutions were compared with baseline results without actuation in order to assess the benefits of flow control and to determine its effectiveness. In all cases, it was found that plasma control can appreciably improve the time integrated lift over the duration of the maneuvers. The wing-tip actuator could achieve up to a 40% increase in the integrated lift, above that of the baseline value.

Higher order theory based Fourier analysis of cross-ply plates and doubly curved panels

An analytical solution to the static analysis of a general cross-ply finite-dimensional doubly curved panel of rectangular plan-form, modeled using a higher order shear deformation theory, is presented. A solution methodology, based on boundary-discontinuous generalized double Fourier series approach is used to solve a system of five highly coupled linear partial differential equations, generated by the higher order-based laminated shell analysis, with the fully clamped type 4 (C4) boundary condition prescribed on two opposite edges, while the remaining two edges are subjected to the simply supported type 4 (SS4) constraint. The numerical accuracy of the solution is ascertained by studying the convergence characteristics of the deflection and moment of a cross-ply spherical panel, and also by comparison with the available results in literature. Additional comparisons were made with the finite element counterparts using commercially available software for distributed load. Hitherto unavailable important numerical results presented include sensitivity of the predicted response quantities of interest to shell geometry (cylindrical and spherical), lamination, lamina material property, thickness and membrane effects as well as their interactions.

Plasma flow control simulations of a low-Reynolds number low-aspect-ratio wing

Large-eddy simulation (LES) is employed to investigate use of plasma-based control in order to improve aerodynamic performance of a flat-plate wing. The wing has a rectangular planform, a thickness to chord ratio of 0.016, and an aspect ratio of 2.0. Computations are carried out at a chord-based Reynolds number of 20,000, such that the configuration and flow conditions are typical of those commonly utilized in micro air vehicle (MAV) applications. Solutions are obtained to the Navier–Stokes equations, that were augmented by source terms used to represent body forces imparted by plasma actuators on the fluid. A simple phenomenological model provided these body forces resulting from the electric field generated by the plasma. The numerical method is based upon a high-fidelity time-implicit scheme and an implicit LES approach, which are applied to obtain solutions on an overset mesh system. The control strategy explores plasma actuators that are distributed in the spanwise direction along the wing leading edge, and actuators extending in the chordwise direction along the wing tip. The investigation considers plasma-induced forces that are imposed in either the co-flowing or counter-flow directions for the former arrangement, and directed inboard or outboard for the latter. Both continuous and pulsed operation of actuators are simulated, and the magnitude of the plasma force is varied in order to establish the most efficient means of control. Computations are carried out at angles of attack that are below (8.0deg) and above (25.0deg) the stall value. Control solutions are compared with baseline results without actuation to determine the most beneficial control strategies. It was found that actuation is effective only for sufficiently large values of the plasma force. A favorable comparison is shown with the limited available experimental data.

Vibration Analysis of Curved Shell using B-spline Wavelet on the Interval (BSWI) Finite Elements Method and General Shell Theory

The implementation of the B-spline Wavelet on the Interval (BSWI) for curved shell elements with rectangular planform is presented in this paper. By aid of the general shell theory, cylinder shells, doubly-curved shallow shells and hy- perbolic paraboloidal shells BSWI elements are formulated. Instead of traditional polynomial interpolation, scaling functions at certain scale have been adopted to form the shape functions and construct wavelet-based elements. Because of the good character of BSWI scaling functions, the BSWI curved shell elements com- bine the accuracy of wavelet-based elements approximation and the character of B-spline functions for structural analysis. Different from the flat shell elements, the curved shell elements obtain a better geometrical fitting property in idealiz- ing the practical curved structures. This paper focuses on the dynamic analysis of shell. The study covers wide combinations of boundaries such as cantilever, sim- ply supported and clamped boundary. Numerical results have been established to validate the efficiency and accuracy of the presented elements through comparison with published data from the open literature and some commercial finite element method software.

Thermoelastic free vibration and dynamic behaviour of an FGM doubly curved panel via the analytical hybrid Laplace–Fourier transformation

The paper presents an analysis of functionally graded material doubly curved panels with rectangular planform under the action of thermal and mechanical loads. Based on the first-order shear deformation theory of modified Sanders assumptions, five coupled partially differential equations (PDEs) are established as equations of motion. Each thermo-mechanical property of the shell follows the power law distribution across the thickness, except Poisson’s ratio, which is kept constant through the panel. Assuming that four edges of the shell-panel are simply supported, a Navier-based solution is adopted to reduce the PDEs into time-dependent ODEs. Applying the Laplace transformation, the equations of motion are transformed into the Laplace domain. With the aid of analytical Laplace inverse method, solutions of stresses, strains, and displacements are obtained in time domain and expressed in explicit phrases. Dynamic, free vibration, and thermo-mechanical bending analysis of the panel is carried out for various geometries. Obtained results are validated with the well-known available data reported in the literature.

Static and dynamic analysis of an FGM doubly curved panel resting on the Pasternak-type elastic foundation

The static, dynamic, and free vibration analysis of a functionally graded material (FGM) doubly curved panel are investigated analytically in the present paper. The FGM Panel is originated from a rectangular planform and its principle curvatures are considered to be constant. All mechanical properties of the FGM panel are assumed to vary continuously through the thickness according to a power law formulation except Poisson’s ratio, which is kept constant. A Pasternak-type elastic foundation containing damping effects is considered to be in contact with the panel during deformation. The elastic foundation reacts in both compression and tension. Equations of motion are established based on the first order shear deformation and the modified Sanders shell theories. Following the Navier type solution, the established equations are reduced to time-dependent ordinary differential equations. Using the Laplace transform, the time-dependency of the problem is eliminated. The solutions are obtained analytically in the Laplace domain and then are inverted to the time domain following an analytical procedure. Finally, the analytical results are verified with those reported in the literature.

Low-Reynolds-Number Effects in Passive Stall Control Using Sinusoidal Leading Edges

The objective of the presentwork is to investigate the application of a sinusoidal leading edge to the design ofmicro air vehicles. Wind-tunnel tests of wings with low aspect ratios of 1 and 1.5, rectangular planforms, and five distinct leading edges [four sinusoidal leading edges and one baseline (straight) leading edge for each aspect ratio] have been conducted. The Reynolds numbers of 70,000 and 140,000 have been analyzed. For the higher Reynolds number, a proper combination of amplitude andwavelength can lead to a substantial increase in lift for angles of attack greater than the baseline stall angle. Maximum lift coefficient gains of the order of 45% were achieved by combining both large amplitude and large wavelength. At the lower Reynolds number, the benefits can be extended to low angles of attack, leading to a dramatic increase in the range of operation. The results depend strongly on the aspect ratio.

Detailed Observations of Interactions of Wingtip Vortices in Close-Formation Flight

The interactions of wingtip vortices in close-formation flight of two wings were examined experimentally. Both wings consisting of a NACA 23012 airfoil section and a rectangular planform of an aspect ratio of 5 were set in echelon formation in a wind tunnel. The lift-to-drag ratio increased the most effectively when the two wings were overlapped with each other by 5% of the wingspan with zero vertical offset for the present wing configuration. Flow visualizations and particle image velocimetry measurements showed that the trailing vortex of the lead wing collided at a slightly inboard point of the trail wing and was divided into two streamwise vortices in the most effective overlap condition. It was also observed that the tip vortices of the lead and trail wings interacted closely with each other to form a pair of counter-rotating vortices for smaller overlap conditions less than 5 % with zero vertical offset. For such small overlap cases, circulation of the trailing vortex of the trail wing was reduced markedly by influences of the trailing vortex of the lead wing.

Flow structure on finite-span wings due to pitch-up motion

AbstractThe flow structure on low-aspect-ratio wings arising from pitch-up motion is addressed via a technique of particle image velocimetry. The objectives are to: determine the onset and evolution of the three-dimensional leading-edge vortex; provide complementary interpretations of the vortex structure in terms of streamlines, projections of spanwise and surface-normal vorticity, and surfaces of constant values of the second invariant of the velocity gradient tensor (iso-$Q$ surfaces); and to characterize the effect of wing planform (rectangular versus elliptical) on this vortex structure. The pitch-up motion of the wing (plate) is from 0 to $4{5}^{\ensuremath{\circ} } $ over a time span corresponding to four convective time scales, and the Reynolds number based on chord is 10 000. Volumes of constant magnitude of the second invariant of the velocity gradient tensor are interpreted in conjunction with three-dimensional streamline patterns and vorticity projections in orthogonal directions. The wing motion gives rise to ordered vortical structures along its wing surface. In contrast to development of the classical two-dimensional leading-edge vortex, the flow pattern evolves to a strongly three-dimensional form at high angle of attack. The state of the vortex system, after attainment of maximum angle of attack, has a similar form for extreme configurations of wing planform. Near the plane of symmetry, a large-scale region of predominantly spanwise vorticity dominates. Away from the plane of symmetry, the flow is dominated by two extensive regions of surface-normal vorticity, i.e. swirl patterns parallel to the wing surface. This similar state of the vortex structure is, however, preceded by different sequences of events that depend on the magnitude of the spanwise velocity within the developing vortex from the leading edge of the wing. Spanwise velocity of the order of one-half the free stream velocity, which is oriented towards the plane of symmetry of the wing, results in regions of surface-normal vorticity. In contrast, if negligible spanwise velocity occurs within the developing leading-edge vortex, onset of the regions of surface-normal vorticity occurs near the tips of the wing. These extremes of large and insignificant spanwise velocity within the leading-edge vortex are induced respectively on rectangular and elliptical planforms.