vibration of twisted and curved cylindrical panels with variable thickness abstract a numerical procedure with an exact strain displacement relationship of twisted and curved cylindrical panels having variable thickness derived by considering the green strain tensor on general shell theory is presented using the principle of virtual work and the rayleigh ritz method with algebraic polynomials as in plane and transverse displacement functions the accuracy and applicability of the procedure are verified by comparing the present results with previous experimental and theoretical results for several panels the effects of variation ratio of thickness in chordwise and lengthwise directions twist and curvature both in two directions aforementioned on vibrations of cylindrical panels are studied in detail and typical vibration mode shapes are plotted to demonstrate the effects

free vibration of rotating ring stiffened cylindrical shells with non uniform stiffener distribution in this research the free vibration analysis of simply supported rotating cylindrical shells with circumferential stiffeners i e rings with non uniform stiffeners eccentricity and non uniform stiffeners spacing distribution is investigated ritz method is applied while stiffeners are treated as discrete elements in strain energy formulation by adopting sander s theorem stretching and bending characteristics of shells are considered also stretching bending and wrapping effects of stiffeners are investigated the translational inertia in three directions for shell and stiffeners and rotary inertia for stiffeners are considered the effects of initial hoop tension centrifugal and coriolis forces due to the rotation of shell are studied the polynomial functions are used for ritz functions natural frequency results for rings with uniform spacing and constant eccentricity have been compared with analytical and experimental results of other researchers which showed good agreement fortunately the agreements of the presented analytical results with the experimental values are better than the analytical values at constant total mass of stiffeners the effects of non uniform eccentricity distribution and non uniform rings spacing distribution separately and simultaneously on natural frequencies are investigated moreover the influence of rotating speed on natural frequencies for the so called non uniform stiffeners distribution is studied

exact penalization of mathematical programs with equilibrium constraints we study theoretical and computational aspects of an exact penalization approach to mathematical programs with equilibrium constraints mpecs in the first part we prove that a mangasarian fromovitz type condition ensures the existence of a stable local error bound at the root of a real valued nonnegative piecewise smooth function a specification to nonsmooth formulations of equilibrium constraints e g complementarity conditions or normal equations provides conditions which guarantee the existence of a nonsmooth exact penalty function for mpecs in the second part we study a trust region minimization method for a class of composite nonsmooth functions which comprises exact penalty functions arising from mpecs we prove a global convergence result for the general method and incorporate a penalty update rule a further specification results in an sqp trust region method for mpecs based on an ell 1 penalty function

semi infinite programming theory methods and applications starting from a number of motivating and abundant applications in 2 including control of robots eigenvalue computations mechanical stress of materials and statistical design the authors describe a class of optimization problems which are referred to as semi infinite because their constraints bound functions of a finite number of variables on a whole region in 3 5 first and second order optimality conditions are derived for general nonlinear problems as well as a procedure for reducing the problem locally to one with only finitely many constraints another main effort for achieving simplification is through duality in 6 there algebraic properties of finite linear programming are brought to bear on duality theory in semi infinite programming section 7 treats numerical methods based on either discretization or local reduction with the emphasis on the design of superlinearly convergent sqp type methods taking this differentiable point of view this paper can be considered to be complementar

rank 4 finite dimensional nichols algebras of diagonal type in positive characteristic abstract nichols algebras are fundamental objects in the construction of quantized enveloping algebras and in the classification of pointed hopf algebras by lifting method of andruskiewitsch and schneider arithmetic root systems are invariants of nichols algebras of diagonal type with a certain finiteness property in the present paper all rank 4 nichols algebras of diagonal type with a finite arithmetic root system over fields of arbitrary characteristic are classified our proof uses the classification of the finite arithmetic root systems of rank 4

a ritz vibration analysis of doubly curved rectangular shallow shells using a refined first order theory integral expressions for strain and kinetic energies in vibration analysis of shear deformable doubly curved shallow shells are presented although the formulation follows the first order shear deformation theory the consideration of lame parameters for the transverse shear strain through shell thickness which have been hitherto neglected by other researchers shows linear distribution functions instead of constant values if the lame parameters were dropped finite higher order two dimensional orthogonal polynomials pb 2 functions previously used in thin shell analyses 3 6 have been generalized from three to five degrees to account for the additional rotation fields of a moderately thick shell these global shape functions satisfy the kinematic boundary conditions at the outset the excellent performance and versatility of the computational methodology of the pb 2 ritz method are illustrated in representative numerical simulations the resultant matrix size is far smaller than other discretization methods and yet accurate solutions can be obtained the validity of the results is verified through direct comparisons the discrepancy of various formulations with and without the lame parameters is presented and discussed

computing a global optimal solution to a design centering problem in this paper we present a method for solving a special three dimensional design centering problem arising in diamond manufacturing find inside a given not necessarily convex polyhedral rough stone the largest diamond of prescribed shape and orientation this problem can be formulated as the one of finding a global maximum of a difference of two convex functions over 3 and can be solved efficiently by using a global optimization algorithm provided that the objective function of the maximization problem can be easily evaluated here we prove that with the information available on the rough stone and on the reference diamond evaluating the objective function at a pointx amounts to computing the distance with respect to a minkowski gauge fromx to a finite number of planes we propose a method for finding these planes and we report some numerical results

vibration of pretwisted cantilever shallow conical shells abstract this paper presents a mathematical model to investigate the effects of initial twist on the vibratory characteristics of cantilever shallow conical shells the energy functional is minimized according to the ritz procedure to arrive at the governing eigenvalue equation a set of orthogonally generated two dimensional polynomials associated with a basic function which accounts for the boundary expressions and constraints is introduced to approximate the in plane and transverse displacement amplitude functions the complete procedure has been automated to compute the vibration frequencies and mode shapes for exemplary problems in the numerical experiments the convergence of eigenvalues is confirmed by increasing the degrees of polynomials employed in the admissible shape functions to enhance the existing literature a set of first known frequency parameters is presented the paper highlights the important effects of angle of twist on the vibration frequencies and mode shapes of conical shells the fundamental physical frequency decreases monotonically for a longer conical shell the result shows that an increase in the angle of twist does not ensure higher torsional stiffness for a conical shell which is in contradiction with previous observation for a pretwisted beam or plate the symmetry of modes is absent when the angle of twist is non zero

free vibration analysis of symmetric laminated skew plates by discrete singular convolution technique based on first order shear deformation theory in this study free vibration of laminated skew plates was investigated discrete singular convolution dsc method is used for numerical solution of vibration problems the straight sided quadrilateral domain is mapped into a square domain in the computational space using a four node element by using the geometric transformation typical results are presented for different geometric parameters and boundary conditions it is concluded from the results that the skew angle have considerable influence on the variations of the frequencies with fibre orientation angle and number of layers in the laminate the results obtained by dsc method are compared with those obtained by analytical and numerical approaches it is shown that reasonable accurate results are obtained present work also indicates that the method of dsc is a promising and potential approach for computational mechanics copyright 2009 john wiley sons ltd

on the use of pb 2 rayleigh ritz method for free flexural vibration of triangular plates with curved internal supports this paper presents a set of comprehensive and accurate results for the free flexural vibration of triangular plates of arbitrary boundary conditions with without curved internal supports these results are presented for ten distinct cases involving the combinations of free simply supported or clamped edge conditions in the present study a simple and straightforward pb 2 rayleigh ritz method is employed to determine the first few natural frequencies for the triangular plates with a wide range of aspect ratios i e a b 2 5 1 2 1 2 3 and 2 a set of pb 2 ritz functions consisting of the product of a two dimensional polynomial function and a basic function is used the basic function is defined by the product of 1 equations of the prescribed continuous piecewise boundary each raised to the power of 0 1 or 2 corresponding to a free simply supported or clamped edge and 2 equations of the internal supports the members of the generated set of functions satisfy automaticaly all the kinematic boundary conditions of the plate at the outset the present results for some cases are compared with available published values from the open literature

a rayleigh ritz approach to transverse vibration of isotropic and anisotropic trapezoidal plates using orthogonal plate functions abstract to overcome the lack of a more general method for free vibration analysis of trapezoidal plates a computationally efficient and highly accurate rayleigh ritz approach with the newly developed orthogonal plate functions is proposed to solve these problems with any combination of clamped simply supported and free edge support conditions the deflection of the plate is approximated by a set of two dimensional orthogonal plate functions generated using the gram schmidt procedure which expresses the entire plate domain into two implicitly related variables in the present paper the effects of the fibre orientation on the vibrational behaviour of the plates are considered the numerical results for isotropic and anisotropic trapezoidal plates are presented where possible the numerical results are verified with other existing values in the literature

a pb 2 ritz formulation for flexural vibration of shallow cylindrical shells of rectangular planform abstract this paper presents a free flexural vibration analysis of shallow cylindrical shells of rectangular planform with arbitrary combinations of free simply supported and clamped boundaries the pb 2 ritz method previously used in the plate analysis by one of the authors is now extended to shallow shell studies the ritz approach along with the in plane and transverse displacements assumed in the form of orthogonal polynomials is used to formulate the problems numerical results for the vibration frequencies for wide ranges of aspect ratio shallowness ratio and thickness ratio are presented several selected contour plots of the mode shapes for six different combinations of boundary conditions are presented as usual the accuracy of the present method is illustrated by convergence tests and comparison with the well established results from the literature

on the mathematical foundations of nondifferentiable optimization in engineering design it is shown by example that a large class of engineering design problems can be transcribed into the form of a canonical optimization problem with inequality constraints involving mar functions such problems are commonly referred to as semi infinite optimization problems the bulk of this paper is devoted to the development of a mathematical theory for the construction of first order nondifferentiable optimization algorithms related to phase i phase ii methods of feasible directions which solve these semi infinite optimization problems the applicability of the theory is illustrated with examples that are relevant to engineering design

the adaptive convexification algorithm a feasible point method for semi infinite programming we present a new numerical solution method for semi infinite optimization problems its main idea is to adaptively construct convex relaxations of the lower level problem replace the relaxed lower level problems equivalently by their karush kuhn tucker conditions and solve the resulting mathematical programs with complementarity constraints this approximation produces feasible iterates for the original problem the convex relaxations are constructed with ideas from the alpha bb method of global optimization the necessary upper bounds for second derivatives of functions on box domains can be determined using the techniques of interval arithmetic where our algorithm already works if only one such bound is available for the problem we show convergence of stationary points of the approximating problems to a stationary point of the original semi infinite problem within arbitrarily given tolerances numerical examples from chebyshev approximation and design centering illustrate the performance of the method

vibratory characteristics of pretwisted cantilever trapezoids of unsymmetric laminates treated in this paper is a theoretical investigation on the vibratory characteristics of twisted unsymmetric laminates an approach based on the global energy principle is developed to examine the effects of initial twist upon the vibration response of cantilevered unsymmetric laminates of a trapezoidal planform it employs the ritz extremum energy principle with a set of admissible pb 2 shape functions these kinematically orientated pb 2 shape functions are composed of the product of 1 mathematically complete two dimensional orthogonally generated polynomials and 2 a basic function to ensure satisfaction of the geometric boundary conditions numerical convergence of the eigenvalues is established by increasing the degree of polynomial of the shape functions results covering wide ranges of angles of twist and fiber orientation for 2 ply 4 ply and 8 ply unsymmetric graphite epoxy laminates are presented several findings with respect to these results are highlighted and conclusions are drawn finally a set of vibration mode shapes illustrating the effects of aspect ratio chord ratio angles of twist and fiber orientation is also included

solving semi infinite optimization problems with interior point techniques we introduce a new numerical solution method for semi infinite optimization problems with convex lower level problems the method is based on a reformulation of the semi infinite problem as a stackelberg game and the use of regularized nonlinear complementarity problem functions this approach leads to central path conditions for the lower level problems where for a given path parameter a smooth nonlinear finite optimization problem has to be solved the solution of the semi infinite optimization problem then amounts to driving the path parameter to zero we show convergence properties of the method and give a number of numerical examples from design centering and from robust optimization where actually so called generalized semi infinite optimization problems are solved the presented method is easy to implement and in our examples it works well for dimensions of the semi infinite index set at least up to 150

vibratory characteristics of cantilevered rectangular shallow shells of variable thickness in this paper a comprehensive study on the modeling of vibratory response of variable thickness cantilevered shallow cylindrical shells of rectangular planform is carried out through the principle of minimum total energy the equations of stretching and bending strain energies kinetic energy and the associated boundary conditions of the shells are derived the present model is developed based on the ritz formulation with the assumption of sets of mathematicall y complete admissible two dimensional polynomials to approximate the in plane and transverse displacement amplitude functions a basic function is introduced in the approximate method to enforce automatic satisfaction of the kinematic boundary conditions sets of comprehensive reasonably accurate vibration frequencies of the shallow shells are presented for wide ranges of aspect ratio thickness variation ratios and shallowness ratio these results where possible are verified by comparing with other established experimental and theoretical solutions few selected contour plots for the first known mode shapes of these cylindrical shells of variable thickness are included

global solution of semi infinite programs optimization problems involving a finite number of decision variables and an infinite number of constraints are referred to as semi infinite programs sips existing numerical methods for solving nonlinear sips make strong assumptions on the properties of the feasible set e g convexity and or regularity or solve a discretized approximation which only guarantees a lower bound to the true solution value of the sip here a general deterministic algorithm for solving semi infinite programs to guaranteed global optimality is presented a branch and bound bb the lower bounding problem is formulated as a convex relaxation of a discretized approximation to the sip the sip b b algorithm is shown to converge finitely to optimality when the subdivision and discretization procedures used to formulate the node subproblems are known to retain certain convergence characteristics other than the properties assumed by globally convergent b b methods for finitely constrained nlps this sip algorithm makes only one additional assumption for every minimizer x of the sip there exists a sequence of slater points xn for which cf section 5 4 numerical results for test problems in the sip literature are presented the exclusion test and a modified upper bounding problem are also investigated as heuristic approaches for alleviating the computational cost of solving a nonlinear sip to guaranteed global optimality

a global continuum ritz formulation for flexural vibration of pretwisted trapezoidal plates with one edge built in abstract the development of a continuum ritz model for twisted plate vibration on the basis of minimization of the energy functional is presented this method involves the use of a set of orthogonally generated pb 2 shape functions to approximate the transverse and in plane deflections in this paper a two dimensional recurrence formula is proposed to generate this set of functions during the orthogonalization process a basic function is introduced to each transverse and in plane displacement to ensure the automatic satisfaction of the essential boundary conditions the whole process can be easily implemented numerically and the accuracy of the method can be achieved by increasing the number of terms used in the polynomial functions following the above procedures a governing eigenvalue equation is derived this equation is solved to determine the vibration response of trapezoidal plates with initial twist in order to verify the validity and reliability of this method in addition to the convergence study the present results are compared with the available values from the literature where possible

transverse vibration of trapezoidal plates of variable thickness unsymmetric trapezoids abstract a set of simple two dimensional polynomial functions is employed as the admissible displacement function in the rayleigh ritz energy approach for the free transverse vibration analysis of symmetric trapezoidal plates with linearly varying thickness the admissible function consists the product of i a two dimensional polynomial function and ii a basic function the basic function is defined by the product of the equations of the prescribed continuous piecewise boundary shape each raised to the power of 0 1 or 2 corresponding to a free simply supported or clamped edge respectively the set of functions generated ensures the satisfaction of all the kinematic boundary conditions at the outset the proposed method is applied to solve several symmetric trapezoidal plates with different combinations of boundary conditions and variable thickness the results for some cases are compared with the available published values from the open literature these new results may serve as benchmark data for the development of other numerical methods

free vibration analysis of a twisted beam using the dynamic stiffness method abstract an exact dynamics stiffness matrix is developed and subsequently used for free vibration analysis of a twisted beam whose flexural displacements are coupled in two planes first the governing differential equations of motion of the twisted beam undergoing free natural vibration are derived using hamilton s principle next the general solutions of these equations are obtained when the oscillatory motion of the beam is harmonic this is followed by application of boundary conditions for displacements and forces which essentially leads to the formation of the dynamics stiffness matrix of the twisted beam relating harmonically varying forces with harmonically varying displacements at its ends the resulting dynamic stiffness matrix is used in connection with the wittrick williams algorithm to compute natural frequencies and mode shapes of a twisted beam with cantilever end condition these are compared with previously published results to confirm the accuracy of the method and some conclusions are drawn