Ritz Solution Research Articles

形状初期不整が軸圧円筒殼の座屈性状に及ぼす影響

Buckling of axially compressed cylinders is studied using a fully nonlinear Ritz solution procedure. An axisymmetric geometric initial imperfection mode is chosen as to be of the inverse mode to the linear axisymmetric deflection mode, and is shown to increase the buckling load carrying capacity. Adopting non-axisymmetric imperfections, the loss of load carrying capacity and membrane energy occurs. The relationship between these nonlinear results and a reduced stiffness criterion is discussed.

A variational formulation and ritz solution of the radiative equilibrium of a plane slab

Radiative equilibrium problem for an absorbing emitting medium between plane parallel diffuse surfaces can be expressed by the Fredholm equation of the second kind. In this paper a variational formulation of the problem is obtained in terms of a functional. Dimensionless temperatures and heat fluxes for various optical thicknesses are then solved by the Ritz method. Results are expressed in terms of a third order polynomial with two coefficients and are very close to the exact solution of Heaslet and Warming.

Global/local analysis of composite plates with cutouts

The study focuses on the development of a simple and accurate global/local method for calculating the static response of stepped, simply-supported, isotropic and composite plates with circular and elliptical cutouts. The approach primarily involves two steps. In the first step a global approach, the Ritz method, is used to calculate the response of the structure. Displacement based Ritz functions for the plate without the cutout are augmented with a perturbation function, which is accurate for uniform thickness plates only, to account for the cutout. The Ritz solution does not accurately satisfy the natural boundary conditions at the cut-out boundary, nor does it accurately model the discontinuities caused by abrupt thickness changes. Therefore, a second step, local in nature is taken in which a small area in the vicinity of the hole and encompassing other points of singularities is discretized using a fine finite element mesh. The displacement boundary conditions for the local region are obtained from the global Ritz analysis. The chosen perturbation function is reliable for circular cutout in uniform plates, therefore elliptical cutouts were suitably transformed to circular shapes using conformal mapping. The methodology is then applied to the analysis of composite plates, and its usefulness successfully proved in such cases. The proposed approach resulted in accurate prediction of stresses, with considerable savings in CPU time and data storage for composite flat panels.

Solution of elastoplastic problems of the non-axially symmetric deformation of bodies of revolution

A numerical method of solving spatial problems involving homogeneous isotropic bodies of revolution which obey the equations of flow theory and are under a non-axially symmetric load is developed. The method makes use of an algorithm in which a step is made in the load and iterations are carried out on this step. The Ritz method is used to solve the elastic problem at each iteration. In this method, an expansion in a system of trigonometric functions along a peripheral direction and with respect to the coordinates in the meridian plane is used, that is, a two-dimensional finite-element approximation. It is proved that the iterative process converges in the case of an isotropically hardening body which obeys the associated flow law subject to the Mises plasticity condition. Sufficient conditions for the Ritz solution to converge, on iteration, to the exact solution are also obtained both in the case of external and internal problems. The method is used to calculate a preloaded elastoplastic half-space with a blind hole as the load applied to the lateral surface of the hole is removed. The problem simulates the process of boring a hole in a body which is used to determine the residual stresses in it. A comparison is made between the results obtained and the results of the solution of the same problem in an elastic formulation which is used in practice at the present time to determine residual stresses.

Comments on ‘‘Comparisons of vibration frequencies for rods and beams from one-dimensional and three-dimensional analyses’’ [J. Acoust. Soc. Am. 98, 2122–2135 (1995)

Leissa and So use accurate solutions generated by the Rayleigh–Ritz procedure as a basis of comparison for the one-dimensional theories for longitudinal, torsional, and bending vibrations of rods and beams. They considered both the free–free and fixed–free cases. In the past I have considered similar comparisons, for the free–free case, based on a series solution. A comparison of my series solution with the authors Rayleigh–Ritz solution showed some disagreements. Those disagreements are tabulated. The authors did not use the best shear coefficient in their comparison, nor did they compare the Pickett solution which they reference. Comparisons using Timoshenko’s shear coefficient and Pickett’s solution are also presented in this comment.

Comments on ‘‘Accurate vibration frequencies of circular cylinders from three-dimensional analysis’’ [J. Acoust. Soc. Am. 98, 2136–2141 (1995)

Leissa and So presented accurate frequencies for solid cylinders using the Rayleigh–Ritz method for both a completely traction-free solid cylinder and a solid cylinder which is fixed at one end. They compared their frequencies for the traction-free case with my series solution in their Tables VI and VII. The comparison does not demonstrate the accuracy which can be achieved with the series solution. A table demonstrating the convergence of the series solution is presented in this Comment. Discrepancies were found between the Rayleigh–Ritz solutions and the converged series solution for some of the frequencies in Table VI. Further discrepancies were found in some of the other results. All discrepancies are tabulated for the traction-free case.

H∞ control for flutter suppression of a laminated plate with self-sensing actuators

Active flutter suppression system of a composite plate wing model is designed using a reduced order model. The analysis for a laminated composite wing with segmented piezoelectric sensor/actuator pairs is conducted by the Ritz solution technique. Unsteady aerodynamic forces calculated by doublet lattice method are approximated as the transfer functions of the Laplace variable by the minimum state method. Among the aerodynamic states obtained from rational function approximation, only one aerodynamic state is included in the plant model for feedback purpose. The neglected aerodynamic states are regarded as modeling error. The control system uses the integrated and collocated piezoelectric self-sensing actuator pairs so as to prohibit the non-minimum phase model and the spillover due to the unmodeled dynamics. Based on the mixed-sensitivityH∞ control method, the control parameters are determined. Using a simple wing model, the performance of the controlled system is shown in the frequency and time domain respectively. The electric current and the power requirement for aeroelastic control are also predicted.

FORMULATION AND EXPERIMENTAL VERIFICATION OF A PNEUMATIC FINITE ELEMENT

Many engineering structures completely surround and enclose gas filled volumes. The enclosed gas adds additional stiffness to the surrounding structure. This paper shows how to account for this effect by means of an augmented virtual work principle. The additional term augmenting the virtual work equilibrium statement for the structure is the virtual boundary work done by the pressure of the enclosed gas. The augmented equations are discretized using standard finite element methods, and the additional terms are discussed. The resulting ‘pneumatic’ finite element is shown to be analogous to regular structural finite elements. To assess the accuracy and efficiency and also to illustrate the applicability of the present formulation, a series of four examples was selected. In two of the examples, the behaviour of the end cap of a partially filled plastic food product container is studied. The numerical results using the pneumatic element compare well with an alternative Rayleigh–Ritz solution of the end cap behaviour. The other two examples represent the behaviour of a double bellows air spring shock absorber under static isothermal and dynamic adiabatic conditions. For the static isothermal case, an experimental study was performed with results in good agreement with the pneumatic element solutions. For the dynamic adiabatic loading case, the dynamic stiffness of the air spring was predicted using the pneumatic element. The numerical results agree with experimental data published in an air spring application guide. The examples illustrate that the pneumatic element formulation can be applied to the large deflection analysis of structures that enclose gas filled volumes.

Optimal sizing and placement of piezo-actuators for active flutter suppression

The purpose of this work is to design an active control system for flutter suppression of a laminated plate wing model using segmented piezo-actuators. We describe investigations pertaining to the optimal size, thickness and locations of piezo-actuators on a laminated plate-wing structure for flutter suppression. The analysis for the laminated composite wing model is conducted by the Ritz solution technique, which represents the displacement on the plate in terms of power series in spanwise and chordwise directions. The active control system design for flutter suppression requires the equation of motion to be expressed in a linear time-invariant state-space form. The doublet-lattice method is used to compute unsteady aerodynamic forces, which are approximated as the transfer functions of the Laplace variable by the minimum-state method combined with an optimization technique. To design the control system, linear quadratic regulator theory with output feedback is considered in this study. The feedback control gains are obtained by solving coupled nonlinear matrix equations via numerical optimization routines. The optimal geometry of piezo-actuators which minimizes the control performance index is determined by the optimization technique referred to as the sequential linear programming method. The numerical result shows a substantial saving in control effort compared with the initial model.

On the vibration of annular, circular and sectorial plates with cut-outs or on partial supports

Natural frequency parameters are given for the free vibration of annular, circular and sectorial plates which may be supported along only parts of their boundaries or may have sectorial cut-outs. The results are obtained by using a Ritz solution previously presented for the study of the vibration of cracked circular or annular plates and which makes use of a minimum number of sectorial elements joined together by means of artificial springs, the stiffnesses of which are permitted to become very high, thus satisfying the required continuity conditions.

Shear buckling of simply supported skew Mindlin plates

Based on the Reissner Mindlin shear deformation plate theory and applying the pb-2 Rayleigh Ritz solution procedure, this note presents critical shear load factors (for R and S shear loadings) for simply supported skew Mindlin plates of various apect ratios, skew angle, and thickness length ratios

Natural frequencies of circular and annular plates with radial or circumferential cracks

A Ritz solution is presented for the determination of the natural frequencies of free vibration of circular and annular plates with radial or circumferential cracks or slits through the full thickness. The approach uses a minimum number of sectorial plate elements which are joined together by means of artificial springs, the stiffness of which is permitted to become very high in order to satisfy the required continuity conditions. Numerical results are given for several plates for different crack positions and lengths. Through convergence studies and comparison with the few results available from the literature, the validity and accuracy of the solution is demonstrated.

Direct Ritz method for random seismic response for non-uniform beams

Based on a fast and accurate method for the stationary random seismic response analysis for discretized structures(Lin 1992, Lin et al. 1992), a Ritz method for dealing with such responses of continuous systems in developed. This method is studied quantitatively, using cantilever shear beams for simplicity and clarity. The process can be naturally extended to deal with various boundary conditions as well as non-uniform Bernoulli-Euler beams, or even Timoshenko beams. Algorithms for both proportionally and non-proportionally damped responses are described. For all of such damping cases, it is not necessary to solve for the natural vibrations of the beams. The solution procedure is very simple, and equally efficient for a white or a non-white ground excitation spectrum. Two examples are given where various power spectral density functions, variances, covariances and second spectral moments of displacement, internal force response, and their derivatives are calculated and analyses. Some Ritz solutions are compared with "exact" CQC solutions.

Sensitivity Analysis of Aeroelastic Response of a Wing in Transonic Flow

A sensitivity study is carried out for changes in shape parameters, namely, aspect ratio, surface area of the wing, taper ratio, and sweep. The structural model is based on a Ritz solution technique with wing deflections represented by Chebyshev polynomials. A recently developed compressible aerodynamic state-space model is used to represent the unsteady aerodynamic airloads on the wing. The aeroelastic equations are solved as an eigenvalue problem to determine the critical speed of the wing. The derivatives of the flutter and divergence speeds with respect to the shape parameters are calculated by both analytical and finite difference methods

Induced strain actuation of isotropic and anisotropic plates

The development and experimental verification of the induced strain actuation of plate components of an intelligent structure is presented. Equations relating the actuation strains, created by induced strain actuators, to the strains induced in the actuator/substrate system are derived for isotropic and anisotropic plates. Plate strain energy relations are also developed. Several exact solutions are found for simple actuator/substrate systems, and a general procedure for solving the strain energy equations with a Rayleigh-Ritz procedure is formulated. Approximate Ritz solutions lead to both an understanding of the system design parameters, and to detailed models of cantilever plate systems. Simple test articles were used to verify the accuracy of the basic induced strain actuator/substrate system models, and cantilever plate test articles were built and tested to verify the ability of the models to predict the strains induced in systems with extensive stiffness coupling and complicated boundary conditions.

A stress concentration due to edge damage in a rectangular boron-epoxy sheet

The stress field in a square sheet of epoxy-boron composite with damaged fibers is determined for the case of applied uniform normal stress longitudinal to the fibers. The damage model assumes that damaged fibers are exactly as effective as the matrix and that the distribution of undamaged fibers can be given by polynomial. A Ritz solution for the displacements is obtained and used to investigate the stresses. A stress concentration is found and described. The energy release rate curve for a family of progressive damage configurations is exhibited.

Buckling Behavior of Pressure Loaded Cylindrical Panels

Buckling of pressure loaded cylindrical panels is investigated using a fully nonlinear Ritz solution procedure and a classical bifurcation analysis of an idealized model. The dialectic between these two contrasting approaches is shown to produce greater understanding than if either one of them were to be used alone. Apart from illuminating the specific buckling behavior of the pressure loaded panel and demonstrating the lower boundedness of the reduced stiffness analytical procedure, the paper suggests that a similar interplay between classical and numerical analysis could be useful in elucidating behavior and providing design analysis procedures for many other shell buckling problems.

Generation of load dependent Ritz transformation vectors in structural dynamics

The evaluation of linear dynamic response analysis of large structures by vector superposition requires, in its traditional formulation, the solution of a large and expensive eigenvalue problem. A method of solution based on a Ritz transformation to a reduced system of generalized coordinates using load dependent vectors generated from the spatial distribution of the dynamic loads is shown to maintain the high expected accuracy of modern computer analysis and significantly reduces the execution time over eigensolution procedures. New computational variants to generate load dependent vectors are presented and error norms are developed to control the convergence characteristics of load dependent Ritz solutions. Numerical applications on simple structural systems are used to show the relative efficiency of the proposed solution procedures.

Lanczos method for heat conduction analysis

AbstractAn analysis technique that uses the Lanczos vectors to construct a Reyleigh–Ritz solution for the transient heat conduction problem is presented. The Lanczos vectors are derived from the heat supply vector and incorporate all the information peculiar to a given problem. They are orthonormal with respect to the heat capacitance matrix and are generated with a minimum of computational effort. Furthermore, an error bound is derived that can be used to terminate the Lanczos algorithm as soon as the number of vectors required to obtain a desired degree of accuracy has been generated. A summary of the algorithm is presented and the accuracy and efficiency of the method is illustrated using a numerical example.

Equivalent plate analysis of aircraft wing box structures with general planform geometry

A new equilvalent plate analysis formulation is described which is capable of modeling aircraft wing structures with a general planform such as cranked wing boxes. Multiple trapezoidal segments are used to represent such planforms. A Ritz solution technique is used in conjunction with global displacement functions which encompass all the segments. This Ritz solution procedure is implemented efficiently into a computer program so that it can be used by rigorous optimization algorithms for application in early preliminary design. A direct method to interface this structural analysis procedure with aerodynamic programs for use in aeroelastic calculations is described. This equivalent plate analysis procedure is used to calculate the static deflections and stresses and vibration frequencies and modes of an example wing configuration. The numerical results are compared with results from a finite element model of the same configuration to illustrate typical levels of accuracy and computation times resulting from use of this procedure.