Closed form analytical inverse solutions for Risley-prism-based beam steering systems in different configurations

Nonparaxial ray tracing through Risley prisms of four different configurations is performed to give the exact solution of the inverse problem arisen from applications of Risley prisms to free space communications. Predictions of the exact solution and the third-order theory [Appl. Opt. 50, 679 (2011)] are compared and results are shown by curves for systems using prisms of different materials. The exact solution for the problem of precision pointing is generalized to investigate the synthesis of the scan pattern, i.e., to create a desirable scan pattern on some plane perpendicular to the optical axis of the system by controlling the circular motion of the two prisms.     

Some closed-form solutions in random vibration of Bresse-Timoshenko beams

Random vibration of simply supported uniform Bresse-Timoshenko beams is considered under ‘rain-on-the-roof’ (stationary space- and time -wise ideal white noise) excitation. An approximate differential equation is used with both shear distortion and rotary inertia included, but with the term which covers the simultaneous action of these effects omitted. A closed-form solution is derived for the displacement and velocity space-time correlation function of the Bresse-Timoshenko beam with transverse damping, generalizing the corresponding result by Eringen for the classical Bernoulli-Euler beam. Closed-form solutions are also derived for beams with structural or Voigt damping mechanisms. The mean-square value of the stress diverges for both the classical Bernoulli-Euler and Bresse-Timoshenko beams with transverse damping, but converges for the beam possessing structural damping. The main finding of this study is identity of the space-time correlation functions of displacement according to the refined Bresse-Timoshenko and classical Bernoulli-Euler theories, when the joint action of rotary inertia and shear deformation is neglected for the beam under the ‘rain-on-the-roof’ excitation. This remarkable coincidence takes place for beams possessing (a) transverse viscous damping, (b) Voigt damping, and (c) combined rotary and transverse viscous damping.     

A novel formulation leading to closed-form solutions for buckling of circular plates

Summary In this paper we study axisymmetric buckling of radially graded circular plates. The flexural rigidity is considered to be a suitably varying function of the radial coordinate. The problem is posed as a semi-inverse one. The buckling mode is selected first, then the variation of the flexural rigidity consistent with the buckling mode is determined. Apparently for the first time in the literature, closed-form solutions are found. Such solutions allow the design of a circular plate whose buckling load is at least the pre-specified one. Such a design appears to find much applications in various fields of engineering.     

Some closed-form solutions in random vibration of Bernoulli-Euler beams

The only closed-form solutions for random vibration of beams are that due to Houdijk, for the tip mean-square displacement of a cantilever beam under space- and time-wise ideal white noise, and that due to Eringen for a simply-supported beam under identical excitation. In both instances, beams possessing transverse damping were treated. In the present study closed-form solutions are found for uniform, simply supported beams subjected to a stationary excitation that is white both in space and time. The beams possess either structural, Voigt or rotary damping mechanisms. Expressions are obtained for the space-time correlation functions of displacement, velocity and stress. Previously derived interesting conclusions by Crandall and Yildiz on divergence of the mean-square stress for a beam with Voigt damping, and its convergence for the beam with combined transverse and rotary damping, are confirmed. Moreover, the closed form solution is obtained for the probabilistic characteristics of a beam under a number of separate or combined dampings.     

Analytical solutions of sensitivity for pressure microsensors

Pressure microsensors are normally designed in linear operation range. In this study, analytical solutions are presented in order to offer a set of simple equations to designers and researchers to calculate and predict the sensitivity of pressure microsensors. The pressure sensitivity is proportional to the square of the ratio of diaphragm thickness to diaphragm length, but it is inversely proportional to burst pressure. We found that the analytical solutions are in good agreement with simulation by finite element method     

Large amplitude vibration analysis of composite beams: Simple closed-form solutions

Large amplitude vibration analysis of laminated composite beam with axially immovable ends is investigated with symmetric and asymmetric layup orientations by using the Rayleigh–Ritz (R–R) method. The displacement fields used in the analytical formulation are coupled by using the homogeneous governing static axial equilibrium equation of the beam. Geometric nonlinearity of von-Karman type is considered which accounts for the membrane stretching action of the beam. The simple closed-form solutions are presented for the nonlinear harmonic radian frequency as function of central amplitude of the beam using the R–R method. The nonlinear harmonic radian frequency results obtained from the closed-form solutions of the R–R method in general show good agreement with the results obtained from simple iterative finite element formulation. Furthermore, the closed-form expressions are corrected for the harmonic motion assumption from the available literature results on the existence of quadratic and cubic nonlinearity. It is interesting to note that the composite beams can result in asymmetric frequency vs. amplitude curves depending upon the nature of direction of displacement in contrast to isotropic beams which exhibit cubic nonlinearity only and leads to symmetric frequency vs. amplitude curves with respect to sign of the amplitude.     

Minimal free resolutions of some configurations of fat lines

In this paper we compute the minimal free resolution and then the Hilbert function of a \(m\) -homogeneous fat complete grid in \(\mathbb {P}^3_\mathbf {K}\) . This proves a conjecture about the minimal free resolution of these configurations of lines.     

Analytical solutions for image orientation changes in prisms

Optical prisms are most commonly employed because of their ability to output an image of different orientation relative to the input object. Previous papers have presented a systematic but numeric approximation for designing a single prism, which outputs an image with a specific image orientation. Instead of a numerical solution, an exact analytical solution for the same problem is offered. Further, how to design prism systems by using off-the-shelf right-angle prisms and roof prisms as building blocks to obtain an output image with a specific image orientation is addressed. Illustrative examples are given to verify the proposed approach and demonstrate its use.     

Analytical closed-form investigation of pwm inverter induction motor drive performance under dc bus voltage pulsation

The voltage unbalance conditions at the input rectifier stage of the AC?DC?AC rectifier-inverter fed induction motor drive is analysed. This unbalance can cause significant voltage harmonic of twice the line frequency 2f1 in the DC bus. This voltage ripple can have a degrading effect on the induction-machine performance characteristics. The authors present an analytical closed-form mathematical model and analysis of the impact of DC bus ripple voltage of the three-phase voltage source inverter with the space-vector PWM on the induction machine phase voltages, currents and torque pulsations. The analytical expressions for the voltage and current space vectors as a function of the DC bus voltage pulsation are derived. Using superposition, the separate parts of the motor currents can be determined. From the current space vectors, the torque behaviour is estimated, again as a function of DC link voltage pulsation. Next, it is shown that the DC link voltage ripple components may cause large torque pulsation. The proposed analytical method is based on the mixed p?z approach, enabling presentation of the results in lucid and closed form. To verify the effectiveness of the proposed analytical model, experimental results based on laboratory setup were obtained.     

Design and application of microstrip parallelcoupled lines employing step-impedance transmission-line compensation

The application of step-impedance transmission lines (SITLs) to improving the directivity of parallelcoupled microstrip lines is presented. A simple design procedure for two configurations based on the SITL technique is given. The application of the technique is demonstrated practically with a design at 0.9 GHz on an RF60 substrate. The directivities obtained from the measured results are improved by more than 6 and 30 dB compared with the designed uncompensated coupler. The compensation technique is then applied to the design of a coupled-line resonator and bandpass filter for suppressing harmonic responses in these circuits. Experimental results show that the second harmonic in the filter response is suppressed by 30 dB.     

Design of a microstrip filter using multiple capacitively loaded coupled lines

A method to design a microstrip filter has been proposed. The multiple capacitively loaded coupled lines are adopted for the design of the microstrip filter. The filter shows different stopband behaviour with respect to even or odd total number of resonators. The transmission zeros may be located around the lower, higher or both regions of the passband's skirt. The coupling among the capacitively loaded lines can produce another transmission zero particularly in the higher stopband. A bandpass filter and a duplexer utilising these features are designed. The equivalent model is also proposed to illustrate the filter's coupling among these capacitively loaded coupled lines. The design has been verified by experimental results     

Analytical solutions for sagittal plane waves in three-layer composites

The analytic dispersion equations for the symmetric and antisymmetric saggital plane plate modes of a three-layer composite system are presented. The composite consists of a solid isotropic plate sandwiched between two acoustically thin (<0.02 lambda) isotopic solid layers, where lambda is the acoustic wavelength. The thin layers are considered either as the mass loading or the chemical selective coating layers for the plate wave sensors. Explicit formulas which identify the contributions of the elasticity and inertia effects for the phase velocity and mass loading sensitivity of the lowest symmetric (S(0)) and antisymmetric (A(0)) mode for the case where the thickness of the composite plate is much less than lambda are obtained. The amounts by which the elasticity of the thin layer and the inertia decrease the mass loading sensitivity is found for both sensors. It is also found that the sensitivity of the A(0) mode significantly depends on the operating frequency but that of the S(0) mode does not. Specific examples are given for the case of a fused silica plate sandwiched by two thin lucite layers.     

Analytical series solutions for three-dimensional supercritical flow over topography

An analytical series solution method for three-dimensional, supercritical flow over topography is presented. Steady, nonlinear solutions are calculated for a single layer of inviscid, constant-density fluid that flows irrotationally over an obstacle that varies significantly in the x-, y- and z-directions. Accurate series solutions for the free surface and a series of stream tubes throughout the flow region are calculated to demonstrate the three-dimensional properties of the problem. These solutions provide valuable insight into the three-dimensional interactions between the fluid and obstacle which is impossible to gain from any two-dimensional model. The model is described by a Laplacian free-boundary problem with fully nonlinear boundary conditions. The solution method consists of iteratively updating the location of the free surface (on top of the fluid) using a cost function which is derived from the Bernoulli equation. Root-mean-square errors in the boundary conditions are used as convergence criteria and a measure of the accuracy of the solution. This method has been used to solve the two-dimensional version of this problem in the past. Here, we detail the extensions required for three-dimensional flow.     

Analytical solutions for unsteady free convection in porous media

Analytic solutions for two of the similarity cases identified by Johnson and Cheng (1978) for the unsteady free-convection boundary-layer flow over an impermeable vertical flat plate adjacent to a fluid saturated porous medium are given in the present paper. These are the solutions corresponding to an exponential (e sup a ² t sup) and a power-law (t ^m) variation of the surface temperature, respectively. They represent exact solutions for doubly infinite plates and approximate solutions for semi-infinite plates. In the latter cases their validity is restricted to the so-called `conduction regime' of the flow. It is shown that in the power law case, physical solutions only exist in the range m>–1 of the temperature exponent and they can be expressed in terms of Kummer's confluent hypergeometric functions. For m 0 exponentially decaying unique solutions were found, while in the range –1<m<0 both exponentially and algebraically decaying multiple solutions occur. The origin of the multiple solutions as well as the feasibility conditions of all the above mentioned solutions is discussed in detail.     

Analytical solutions of the ideal model for gradient liquid chromatography

The analytical solutions of the ideal model for gradient elution that ignores the influence of the solute concentration on the retention factor (k) were studied by using the method of characteristics for solving partial differential equations. It is found for any gradient profiles and solvent strength models used that the concentration of the solute will be discontinuous where the mobile-phase composition is. On a given characteristic curve, the product of the concentration and the retention factor is kept constant at the point where the concentration is continuous. At the point where the concentration is discontinuous, the product on the left side of this point is equal to that on the right side. We also discussed the basic equations to predict the retention time in gradient elution and introduced the injection time into them. For linear solvent strength stepwise and linear gradient elution, general expressions were proposed for the prediction and they can be used as the basis to derive others for specific gradient modes such as single linear, stepwise, and ladderlike gradients. For these modes, simple expressions to account for the band compression and the concentration change during the elution were also given.     

An approach for deriving field equations for magnetic heads of different geometrical configurations

This paper describes an approach for deriving field equations for virtually any magnetic head configuration. As an example, equations are derived for the thin-film head configuration. Results obtained from the new equations and those obtained from previously derived equations are compared to results obtained from the finite difference method. Results obtained from the new equations are also compared to data obtained from measurements on a large-scale thin-film head.     

Analytical solutions of plane structures using Mathematica

A study investigating the possibilities of using the program Mathematica to obtain analytical solutions ofplane structures is presented. Two typical applications are developed: plane frames with only bending deformations and plane trusses. In an attempt to keep the number of unknowns as low as possible, the indirect stiffness method and the force method are used to state the first and the second problems, respectively. All the formulations are developed by making use of kernel and generalized inverse matrices, which are also used to characterize the solution of overdetermined systems of equations. Two programs (PlaneFrame and PlaneTruss) were written to solve the first and second problems and the listing of the codes are given in the paper. The results obtained show that problems of a certain complexity can be solved. Copyright © 2001 John Wiley & Sons, Ltd.     

Calculation of the damping factors for slot and microstrip lines based on ferroelectric film-dielectric substrate structures

The factor of damping related to a finite conductivity of metal electrodes and losses in a ferroelectric film is calculated for slot and microstrip lines based on ferroelectric film-dielectric substrate structures.     

Analytical stress intensity factor solutions for resistance and friction stir spot welds in lap-shear specimens of different materials and thicknesses

In this paper, analytical stress intensity factor and J integral solutions for resistance and friction stir spot welds without and with gap and bend in lap-shear specimens of different materials and thicknesses are developed. The J integral and stress intensity factor solutions for spot welds are first presented in terms of the structural stresses for a strip model. Analytical structural stress solutions for spot welds without and with gap and bend in lap-shear specimens are then developed based on the closed-form structural stress solutions for a rigid inclusion in a finite thin plate subjected to various loading conditions. With the available structural stress solutions, the analytical J integral and stress intensity factor solutions can be obtained as functions of the applied load, the elastic material property parameters, and the geometric parameters of the weld and specimen. The analytical stress intensity factor solutions are selectively validated by the results of three-dimensional finite element analyses for a spot weld with ideal geometry and for a friction stir spot weld with complex geometry, gap and bend. The stress intensity factor and J integral solutions at the critical locations of spot welds in lap-shear specimens of dissimilar magnesium, aluminum and steel sheets with equal and different thicknesses are then presented in the normalized forms as functions of the ratio of the specimen width to the weld diameter. Finally, general trends and simple estimation methods of the stress intensity factor and J integral solutions at the critical locations of spot welds in lap-shear specimens of different materials and thicknesses are given for convenient engineering applications.