http://www.hindawi.comThe latest articles from Hindawi Publishing Corporation© 2010, Hindawi Publishing Corporation. All rights reserved.http://www.hindawi.com/journals/mpe/2009/242396.htmlAn analytical approach for spin-stabilized satellites attitude propagation is presented, considering the influence of the residual magnetic torque and eddy currents torque. It is assumed two approaches to examine the influence of external torques acting during the motion of the satellite, with the Earth's magnetic field described by the quadripole model. In the first approach is included only the residual magnetic torque in the motion equations, with the satellites in circular or elliptical orbit. In the second approach only the eddy currents torque is analyzed, with the satellite in circular orbit. The inclusion of these torques on the dynamic equations of spin stabilized satellites yields the conditions to derive an analytical solution. The solutions show that residual torque does not affect the spin velocity magnitude, contributing only for the precession and the drift of the spacecraft's spin axis and the eddy currents torque causes an exponential decay of the angular velocity magnitude. Numerical simulations performed with data of the Brazilian Satellites (SCD1 and SCD2) show the period that analytical solution can be used to the attitude propagation, within the dispersion range of the attitude determination system performance of Satellite Control Center of Brazil National Research Institute.Roberta Veloso Garcia, Maria Cecília F. P. S. Zanardi, and Hélio Koiti Kuga© 2010, Hindawi Publishing Corporation. All rights reserved.http://www.hindawi.com/journals/mpe/2009/185965.htmlA shear flow motivated by relatively moving half-planes is theoretically studied in this paper. Either the mass influx or the mass efflux is allowed on the boundary. This flow is called the extended Stokes' problems. Traditionally, exact solutions to the Stokes' problems can be readily obtained by directly applying the integral transforms to the momentum equation and the associated boundary and initial conditions. However, it fails to solve the extended Stokes' problems by using the integral-transform method only. The reason for this difficulty is that the inverse transform cannot be reduced to a simpler form. To this end, several crucial mathematical techniques have to be involved together with the integral transforms to acquire the exact solutions. Moreover, new dimensionless parameters are defined to describe the flow phenomena more clearly. On the basis of the exact solutions derived in this paper, it is found that the mass influx on the boundary hastens the development of the flow, and the mass efflux retards the energy transferred from the plate to the far-field fluid.Chi-Min Liu© 2010, Hindawi Publishing Corporation. All rights reserved.http://www.hindawi.com/journals/mpe/2010/570136.htmlThis article deals with the application of the periodized harmonic wavelets for solution of integral equations and eigenvalue problems. The solution is searched as a series of products of wavelet coefficients and wavelets. The absolute error for a general case of the wavelet approximation was analytically estimated.Carlo Cattani and Aleksey Kudreyko© 2010, Hindawi Publishing Corporation. All rights reserved.http://www.hindawi.com/journals/mpe/2009/410243.htmlThe techniques of K-means algorithm and Gaussian Markov random field model are integrated to provide a Gaussian Markov random field model (GMRFM) feature which can describe the texture information of different pixel colors in an image. Based on this feature, an image retrieval method is also provided to seek the database images most similar to a given query image. In this paper, a genetic-based parameter detector is presented to decide the fittest parameters used by the proposed image retrieval method, as well. The experimental results manifested that the image retrieval method is insensitive to the rotation, translation, distortion, noise, scale, hue, light, and contrast variations, especially distortion, hue, and contrast variations.Meng-Hsiun Tsai, Yung-Kuan Chan, Jiun-Shiang Wang, Shu-Wei Guo, and Jiunn-Lin Wu© 2010, Hindawi Publishing Corporation. All rights reserved.http://www.hindawi.com/journals/mpe/2009/826908.htmlWe investigate local robust stability of fuzzy neural networks (FNNs) with time-varying and S-type distributed delays. We derive some sufficient conditions for local robust stability of equilibrium points and estimate attracting domains of equilibrium points except unstable equilibrium points. Our results not only show local robust stability of equilibrium points but also allow much broader application for fuzzy neural network with or without delays. An example is given to illustrate the effectiveness of our results.Kaihong Zhao and Yongkun Li© 2010, Hindawi Publishing Corporation. All rights reserved.http://www.hindawi.com/journals/mpe/2009/758046.htmlThe problem of transient, laminar, MHD double-diffusive free convection over a permeable vertical plate embedded in Darcy and non-Darcy porous medium is numerically investigated. Nonsimilarity solutions are obtained for constant wall temperature and concentration with a specified power law of mass flux parameter. The effects of the magnetic parameter, the inertial coefficient, Lewis number, the buoyancy ratio, and the lateral mass flux on heat and mass transfer coefficients are presented and discussed.Mohammed Q. Al-Odat, Tariq A. Al-Azab, M. Al-Hasan, and B. A. Shannak© 2010, Hindawi Publishing Corporation. All rights reserved.http://www.hindawi.com/journals/mpe/2010/845984.htmlWe present a numerical technique based on the coupling of boundary and finite element methods for the steady Oseen equations in an unbounded plane domain. The present paper deals with the implementation of the coupled program in the two-dimensional case. Computational results are given for a particular problem which can be seen as a good test case for the accuracy of the method.Minfu Feng, Yinnian He, and Ruiting Ren© 2010, Hindawi Publishing Corporation. All rights reserved.http://www.hindawi.com/journals/mpe/2010/782457.htmlLinear and nonlinear static feedback controls are implemented on a nonlinear aeroelastic system that consists of a rigid airfoil supported by nonlinear springs in the pitch and plunge directions and subjected to nonlinear aerodynamic loads. The normal form is used to investigate the Hopf bifurcation that occurs as the freestream velocity is increased and to analytically predict the amplitude and frequency of the ensuing limit cycle oscillations (LCO). It is shown that linear control can be used to delay the flutter onset and reduce the LCO amplitude. Yet, its required gains remain a function of the speed. On the other hand, nonlinear control can be effciently implemented to convert any subcritical Hopf bifurcation into a supercritical one and to significantly reduce the LCO amplitude.M. Ghommem, A. H. Nayfeh, and M. R. Hajj© 2010, Hindawi Publishing Corporation. All rights reserved.http://www.hindawi.com/journals/mpe/2009/147326.htmlThis work presents a (Data Reception Network) DRN software investigation considering simulated conditions inserting purposely errors into the Doppler measurements, satellites ephemeris, and time stamp, to characterize the geographical location software (GEOLOC) developed by Sousa (2000) and Sousa et al. (2003). The extension of reception stations in Brazilian territory can result in more precise locations if the network is considered in the GEOLOC. The results and analyses were first obtained considering the ground stations separately, to characterize their effects in the geographical location (GL) result. Six conditions were investigated: ideal simulated conditions, random and bias errors in the Doppler measurements, errors in the satellite ephemeris, and errors in the time stamp in order to investigate the DRN importance to get more accurate locations; an analysis was performed considering the random errors of 1 Hz in the Doppler measurements. The results are quite satisfactory and also show good compatibility between the simulator and the GEOLOC using the DRN.Cristina Tobler de Sousa, Rodolpho Vilhena de Moraes, and Hélio Koiti Kuga© 2010, Hindawi Publishing Corporation. All rights reserved.http://www.hindawi.com/journals/mpe/2009/396267.htmlWe show the expressions of the gravitational potential of homogeneous bodies with well-defined simple geometric shapes to study the phase space of trajectories around these bodies. The potentials of the rectangular and triangular plates are presented. With these expressions we study the phase space of trajectories of a point of mass around the plates, using the Poincaré surface of section technique. We determined the location and the size of the stable and collision regions in the phase space, and the identification of some resonances. This work is the first and an important step for others studies, considering 3D bodies. The study of the behavior of a point of mass orbiting around these plates (2D), near their corners, can be used as a parameter to understand the influence of the gravitational potential when the particle is close to an irregular surface, such as large craters and ridges.A. A. Silva, O. C. Winter, and A. F. B. A. Prado© 2010, Hindawi Publishing Corporation. All rights reserved.http://www.hindawi.com/journals/mpe/2009/829703.htmlThe static buckling of a cylindrical shell of a four-lobed cross section of variable thickness subjected to non-uniform circumferentially compressive loads is investigated based on the thin-shell theory. Modal displacements of the shell can be described by trigonometric functions, and Fourier's approach is used to separate the variables. The governing equations of the shell are reduced to eight first-order differential equations with variable coefficients in the circumferential coordinate, and by using the transfer matrix of the shell, these equations can be written in a matrix differential equation. The transfer matrix is derived from the nonlinear differential equations of the cylindrical shells by introducing the trigonometric series in the longitudinal direction and applying a numerical integration in the circumferential direction. The transfer matrix approach is used to get the critical buckling loads and the buckling deformations for symmetrical and antisymmetrical shells. Computed results indicate the sensitivity of the critical loads and corresponding buckling modes to the thickness variation of cross section and the radius variation at lobed corners of the shell.Mousa Khalifa Ahmed© 2010, Hindawi Publishing Corporation. All rights reserved.http://www.hindawi.com/journals/mpe/2010/560429.htmlThis paper discusses the estimation of autocorrelation function (ACF) of fractional Gaussian noise (fGn) with long-range dependence (LRD). A variance bound of ACF estimation of one block of fGn with LRD for a given value of the Hurst parameter (H) is given. The present bound provides a guideline to require the block size to guarantee that the variance of ACF estimation of one block of fGn with LRD for a given H value does not exceed the predetermined variance bound regardless of the start point of the block. In addition, the present result implies that the error of ACF estimation of a block of fGn with LRD depends only on the number of data points within the sample and not on the actual sample length in time. For a given block size, the error is found to be larger for fGn with stronger LRD than that with weaker LRD.Ming Li and Wei Zhao© 2010, Hindawi Publishing Corporation. All rights reserved.http://www.hindawi.com/journals/mpe/2009/965603.htmlThe unsteady heat, mass, and fluid transfer over a horizontal stretching sheet has been numerically investigated. Using a similarity transformation the governing time-dependent boundary layer equations for the momentum, heat, and mass transfer were reduced to a sets of ordinary differential equations. These set of ordinary differential equations were then solved using the Chebyshev pseudo-spectral collocation method, and a parametric analysis was carried out. The study observed, among other observations that the local Sherwood number increases as the values of the stretching parameter A and the Schmidt number Sc increase. Also the fluid temperature was found to be significantly reduced by increases in the values of the Prandtl number Pr, the unsteadiness parameter A, and the radiation parameter R. The velocity and concentration profiles were found to be reduced by increasing values of the unsteadiness parameter A.Stanford Shateyi and Sandile Sydney Motsa© 2010, Hindawi Publishing Corporation. All rights reserved.http://www.hindawi.com/journals/mpe/2009/187874.htmlProtection of the sensitive data is an important issue because of the fast development of applications that need exchange of the secret information over the Internet. Secret sharing is an idea proposed by Shamir and Blakley separately with different implementations in 1979. Lin and Tsai proposed a method that uses Steganography to create meaningful shares by using Shamir's secret sharing scheme in 2004. In recent years, researchers work to remove some of the weaknesses of this method. However, all of these methods need cover images four times bigger than the secret image. This arises two problems: increased storage and bandwidth need for shares. We used cover images with the same size as the secret image by using both Blakley's secret sharing approach and Steganography. Therefore, we achieved reduced storage and transmission bandwidth for shares. Besides, the proposed method creates meaningful shares by using Steganography instead of noise-like shares, different from other studies that use Blakley's approach.Mustafa Ulutas, Vasif V. Nabiyev, and Guzin Ulutas© 2010, Hindawi Publishing Corporation. All rights reserved.http://www.hindawi.com/journals/mpe/2009/859678.htmlOne of the key issues in a reentry risk analysis is the calculation of the aerodynamic coefficients. This paper presents a methodology to obtain these coefficients and couple it to a code that computes re-entry trajectories considering six degrees of freedom. To evaluate the different flight conditions encountered during the natural re-entry of conical objects, the Euler Equations for gasdynamics flows are used. A new scheme TVD (Total Variation Diminishing) is incorporated to a finite volume unstructured cell-centred formulation, for application to three-dimensional Euler flows. Finally, numerical results are obtained for a conical body at different attack angles and Mach. With these results, the calculation of the trajectories during atmospheric re-entry is completed.J. P. Saldia, A. Cimino, W. Schulz, S. Elaskar, and A. Costa© 2010, Hindawi Publishing Corporation. All rights reserved.http://www.hindawi.com/journals/mpe/2009/856138.htmlThe current Brazilian System of Environmental Data Collection is composed of several satellites (SCD-1 and 2, CBERS-2 and 2B), Data Collection Platforms (DCPs) spread mostly over the Brazilian territory, and ground reception stations located in Cuiabá and Alcântara. An essential functionality offered to the users is the geographic location of these DCPs. The location is computed by the in-house developed “GEOLOC” program which processes the onboard measured Doppler shifts suffered by the signal transmitted by the DCPs. These data are relayed and stored on ground when the satellite passes over the receiving stations. Another important input data to GEOLOC are the orbit ephemeris of the satellite corresponding to the Doppler data. In this work, the impact on the geographic location accuracy when using orbit ephemeris which can be obtained through several sources is assessed. First, this evaluation is performed by computer simulation of the Doppler data, corresponding to real existing satellite passes. Then real Doppler data are used to assess the performance of the location system. The results indicate that the use of precise ephemeris can improve the performance of GEOLOC by reducing the location errors, and such conclusion can then be extended to similar location systems.Claudia C. Celestino, Cristina T. Sousa, Wilson Yamaguti, and Helio Koiti Kuga© 2010, Hindawi Publishing Corporation. All rights reserved.http://www.hindawi.com/journals/mpe/2010/734340.htmlThe fuzzy Lyapunov method is investigated for use with a class of interconnected fuzzy systems. The interconnected fuzzy systems consist of J interconnected fuzzy subsystems, and the stability analysis is based on Lyapunov functions. Based on traditional Lyapunov stability theory, we further propose a fuzzy Lyapunov method for the stability analysis of interconnected fuzzy systems. The fuzzy Lyapunov function is defined in fuzzy blending quadratic Lyapunov functions. Some stability conditions are derived through the use of fuzzy Lyapunov functions to ensure that the interconnected fuzzy systems are asymptotically stable. Common solutions can be obtained by solving a set of linear matrix inequalities (LMIs) that are numerically feasible. Finally, simulations are performed in order to verify the effectiveness of the proposed stability conditions in this paper.Ken Yeh and Cheng-Wu Chen© 2010, Hindawi Publishing Corporation. All rights reserved.http://www.hindawi.com/journals/mpe/2009/937327.htmlWe consider the problem of a two-dimensional rectangular cavity in a PEC half plane covered by layers of material with uniform thickness. The rectangular geometry allows for an application of Fourier methods to solve the problem. The paper will also discuss how to compute the far field scattering once the solution is found. The fast mode matching approximation technique developed by Morgan and Schwering will be applied to this layered cavity problem setting. Numerical results will show that using the Fourier solution on the layered problem combined with Morgan and Schwering's technique can produce good approximations while using much less computation time than would be required for the entire solution.John L. Fleming and Jessica Moser© 2010, Hindawi Publishing Corporation. All rights reserved.http://www.hindawi.com/journals/mpe/2010/586718.htmlThis paper presents a new wavelet interpolation Galerkin method for the numerical simulation of MEMS devices under the effect of squeeze film damping. Both trial and weight functions are a class of interpolating functions generated by autocorrelation of the usual compactly supported Daubechies scaling functions. To the best of our knowledge, this is the first time that wavelets have been used as basis functions for solving the PDEs of MEMS devices. As opposed to the previous wavelet-based methods that are all limited in one energy domain, the MEMS devices in the paper involve two coupled energy domains. Two typical electrically actuated micro devices with squeeze film damping effect are examined respectively to illustrate the new wavelet interpolation Galerkin method. Simulation results show that the results of the wavelet interpolation Galerkin method match the experimental data better than that of the finite difference method by about 10%.Pu Li and Yuming Fang© 2010, Hindawi Publishing Corporation. All rights reserved.http://www.hindawi.com/journals/mpe/2009/925276.htmlA computational technique based on asymptotic analysis for solving singularly perturbed ODE systems involving a small parameter ϵ is considered. The focus is on second-order systems, but the procedure is also applicable for first-order systems. Both initial value and boundary value problems will be solved. The application of the method is considered over the entire time domain for a wide range of ϵ and the resulting approximation is compared with the direct numerical solution. The convection-diffusion problem from fluid mechanics and the telegraph equation from electrical engineering are considered.N. Parumasur, P. Singh, and V. Singh© 2010, Hindawi Publishing Corporation. All rights reserved.http://www.hindawi.com/journals/mpe/2010/191898.htmlThis paper gives a new approach for modeling the fluid-structure interaction of servovalve component-actuator. The analyzed valve is a precision flow control valve-jet pipe electrohydraulic servovalve. The positioning of an actuator depends upon the flow rate from control ports, in turn depends on the spool position. Theoretical investigation is made for No-load condition and Load condition for an actuator. These are used in finite element modeling of an actuator. The fluid-structure-interaction (FSI) is established between the piston and the fluid cavities at the piston end. The fluid cavities were modeled with special purpose hydrostatic fluid elements while the piston is modeled with brick elements. The finite element method is used to simulate the variation of cavity pressure, cavity volume, mass flow rate, and the actuator velocity. The finite element analysis is extended to study the system's linearized response to harmonic excitation using direct solution steady-state dynamics. It was observed from the analysis that the natural frequency of the actuator depends upon the position of the piston in the cylinder. This is a close match with theoretical and simulation results. The effect of bulk modulus is also presented in the paper.Somashekhar S. Hiremath and M. Singaperumal© 2010, Hindawi Publishing Corporation. All rights reserved.http://www.hindawi.com/journals/mpe/2009/794578.htmlOne of the largest sources of error in positioning and navigation with GNSS is the ionosphere, and the associated error is directly proportional to the TEC and inversely proportional to the square of the signal frequency that propagates through the ionosphere. The equatorial region, especially in Brazil, is where the highest spatial and temporal value variations of the TEC are seen, and where these various features of the ionosphere, such as the equatorial anomaly and scintillation, can be found. Thus, the development and assessments of ionospheric models are important. In this paper, the quality of the TEC was evaluated, as well as the systematic error in the L1 carrier and the inter-frequency biases of satellites and receivers estimated with the Mod_Ion, observable from GPS and integration with the GLONASS, collected with dual frequency receivers.Paulo de Oliveira Camargo© 2010, Hindawi Publishing Corporation. All rights reserved.http://www.hindawi.com/journals/mpe/2010/120403.htmlAn analytical solution is derived to describe the wave-induced flow field and surge motion of a deformable platform structure controlled with fuzzy controllers in an oceanic environment. In the controller design procedure, a parallel distributed compensation (PDC) scheme is utilized to construct a global fuzzy logic controller by blending all local state feedback controllers. The Lyapunov method is used to carry out stability analysis of a real system structure. The corresponding boundary value problems are then incorporated into scattering and radiation problems. These are analytically solved, based on the separation of variables, to obtain a series of solutions showing the harmonic incident wave motion and surge motion. The dependence of the wave-induced flow field and its resonant frequency on wave characteristics and structural properties including platform width, thickness and mass can thus be drawn with a parametric approach. The wave-induced displacement of the surge motion is determined from these mathematical models. The vibration of the floating structure and mechanical motion caused by the wave force are also discussed analytically based on fuzzy logic theory and the mathematical framework to find the decay in amplitude of the surge motion in the tension leg platform (TLP) system. The expected effects of the damping in amplitude of the surge motion due to the control force on the structural response are obvious.Cheng-Wu Chen© 2010, Hindawi Publishing Corporation. All rights reserved.http://www.hindawi.com/journals/mpe/2009/510759.htmlThrough the averaged equations we revisit theoretical and numerical aspects of the strong resonance that increases the eccentricity of the disposed objects of GPS and Galileo Systems. A simple view of the phase space of the problem shows that the resonance does not depend on the semi-major axis. Therefore, usual strategies of changing altitude (raising perigee) do not work. In this problem we search for a set of initial conditions such that the deactivated satellites or upper-stages remain at least for 250 years without penetrating in the orbits of the operational satellites. In the case that Moon's perturbation is not significant, we can identify, in the phase space, the regions where eccentricity reaches maximum and minimum values so that possible risks of collision can be avoided. This is done semi-analytically through the averaged system of the problem. Guided by this idea, we numerically found the (ω,Ω) values of the real unaveraged problem. In particular, for the Galileo case, the theoretical results predicted in the averaged system are in good agreement with numerical results. We also show that initial inclination of the Moon plays an important role in the search of these conditions.Diogo Merguizo Sanchez, Tadashi Yokoyama, Pedro Ivo de Oliveira Brasil, and Ricardo Reis Cordeiro© 2010, Hindawi Publishing Corporation. All rights reserved.http://www.hindawi.com/journals/mpe/2009/467865.htmlThe New Horizons project is currently in progress, but until recently years ago, a mission to Neptune-Triton System predicted in the mid term 2008–2013 was one of the priorities of NASA's Solar System Exploration theme. Any way, it is important to increase our knowledge on the dynamics of the inner satellites of Neptune, since according to some authors, this was a key question in the mission to Neptune-Triton system. In a previous work, we presented the expansion of the disturbing function for the dynamics of this system. Here we derive the averaged classical equations of the precession of the equator for this problem. The highly inclined and retrograde orbit of Triton makes this problem very unusual. Therefore, the classical truncations in the inclinations are not acceptable, so that the precession equations must be obtained in closed form for the inclination. With a significant mass and due to its distance from Neptune, which is continuously decreasing, Triton should exert in the future, important precession on Neptune's equator. The effects of this precession on the inner satellites are shown, including some resonant cases predicted in the future. Although Triton's orbit is almost circular, no expansion in the eccentricity is needed.Clair do Nascimento and Tadashi Yokoyama© 2010, Hindawi Publishing Corporation. All rights reserved.http://www.hindawi.com/journals/mpe/2009/690547.htmlWe apply He's polynomials coupled with the diagonal Padé approximants for solving various singular and nonsingular boundary value problems which arise in engineering and applied sciences. The diagonal Padé approximants prove to be very useful for the understanding of physical behavior of the solution. Numerical results reveal the complete reliability of the proposed combination.Syed Tauseef Mohyud-Din and Ahmet Yildirim© 2010, Hindawi Publishing Corporation. All rights reserved.http://www.hindawi.com/journals/mpe/2009/824385.htmlThe solution of the fractional hyperbolic partial differential equation is obtained by means of the variational iteration method. Our numerical results are compared with those obtained by the modified Gauss elimination method. Our results reveal that the technique introduced here is very effective, convenient, and quite accurate to one-dimensional fractional hyperbolic partial differential equations. Application of variational iteration technique to this problem has shown the rapid convergence of the sequence constructed by this method to the exact solution.Fadime Dal© 2010, Hindawi Publishing Corporation. All rights reserved.http://www.hindawi.com/journals/mpe/2009/857871.htmlWe use antagonistic stochastic games and fluctuation analysis to examine a single-server queue with bulk input and secondary work during server's multiple vacations. When the buffer contents become exhausted the server leaves the system to perform some diagnostic service of a minimum of L jobs clustered in packets of random sizes (event A). The server is not supposed to stay longer than T units of time (event B). The server returns to the system when A or B occurs, whichever comes first. On the other hand, he may not break service of a packet in a middle even if A or B occurs. Furthermore, the server waits for batches of customers to arrive if upon his return the queue is still empty. We obtain a compact and explicit form functional for the queueing process in equilibrium.Najeeb Al-Matar and Jewgeni H. Dshalalow© 2010, Hindawi Publishing Corporation. All rights reserved.http://www.hindawi.com/journals/mpe/2009/507370.htmlNonlinearities in spacecraft attitude determination problem has been studied intensively during the past decades. Traditionally, multiplicative extended Kalman filter_MEKF_algorithm has been a good solution for most nominal space missions. But in recent years, advances in space missions deserve a revisit of the issue. Though there exist a variety of advanced nonlinear filtering algorithms, most of them are prohibited for actual onboard implementation because of their overload computational complexity. In this paper, we address this difficulty by developing a new algorithm framework based on the marginal filtering principle, which requires only 4 sigma points to give a complete 6-state attitude and angular rate estimation. Moreover, a new strategy for sigma point construction is also developed to further increase the efficiency and numerical accuracy. Incorporating the presented framework and novel sigma points, we proposed a new, nonlinear attitude and rate estimator, namely, the Marginal Geometric Sigma Point Filter. The new algorithm is of the same precision as traditional unscented Kalman filters, while keeping a significantly lower computational complexity, even when compared to the reduced sigma point algorithms. In fact, it has truly rivaled the efficiency of MEKF, even when simple closed-form solutions are involved in the latter.Chunshi Fan and Zheng You© 2010, Hindawi Publishing Corporation. All rights reserved.http://www.hindawi.com/journals/mpe/2009/841017.htmlOptimality conditions are derived for the robust optimal design of beams subject to a combination of uncertain and deterministic transverse and boundary loads using a variational min-max approach. The potential energy of the beam is maximized to compute the worst case loading and minimized to determine the optimal cross-sectional shape which results in coupled nonlinear differential equations for the unknown functions except for the case of a variable width beam. The uncertain component of the transverse load acting on the beam is not known a priori resulting in load uncertainty subject only to an norm constraint. Similarly the optimal area function is subject to a volume constraint leading to an isoperimetric variational problem. The min-max approach leads to robust optimal designs which are not susceptible to unexpected load variations as it occurs under operational conditions. The solution methodology is illustrated for the variable width beam by obtaining analytical results for several cases. The efficiency of the optimal designs is computed with respect to a uniform beam under worst case loading taking the maximum deflection as the quantity for comparison. It is observed that the optimal shapes are more than 70% efficient for the examples given in this study.Ismail Kucuk, Sarp Adali, and Ibrahim Sadek© 2010, Hindawi Publishing Corporation. All rights reserved.