Ritz Approximation Research Articles

Modeling Techniques for Coupled, Rotating Beams

AbstractBeams are parts of many industrial applications, like robot links, rolls in paper industry and turbo charger. In this work, a rotordynamical problem, the powertrain for a mill stand, is under consideration. Torsional and bending vibrations are used to describe the dynamical behavior. There are several methods for deriving the dynamical equations of motion. In this paper, the Projection Equation, a synthetical method, is used, leading to partial differential equations for the distributed parameter system. A simplification can be done by using the Ritz approximation method. This method requires the fulfillment of the geometric boundary conditions. For our example, a combination of rigid body modes and elastic modes is chosen. Also models for the gear box system and bearings are included. The solutions for the overall example are nonlinear ordinary differential equations which can be integrated numerically. The system is excited by constant torques and forces. Simulation results for this elastic multibody system are presented. (© 2010 Wiley‐VCH Verlag GmbH & Co. KGaA, Weinheim)

Compressional stability behavior of composite plates with through-the-width and embedded delaminations by using first order shear deformation theory

In this paper, the compressive, i.e. bucking and post-buckling, behavior of composite laminates with through-the-width and embedded delaminations is investigated analytically. The analytical method is based on the first order shear deformation theory (FSDT) and its formulation is developed on the basis of the Rayleigh–Ritz approximation technique. In the case of the multiple delaminations, the contact phenomenon has significant effects on the compressional behavior of the laminates. Thus, the contact between the delaminated sublaminates is also considered in the presented FSDT analytical method. The method can handle both local buckling of the delaminated sublaminates and global buckling of the whole plate. Also, the three-dimensional finite element analysis is performed by using ANSYS5.4 general purpose commercial software, and the results are compared with those obtained by the analytical model. The agreement between the results is very good.

Buckling of a thin-walled cylindrical shell with foam core under axial compression

This paper investigates the effect of partially filled foam core on the behavior of buckling in a thin-walled cylindrical shell. Previous studies have focused on 100% infill or simply used a fixed thickness of foam core. However, this may not be the optimum arrangement in terms of design. To further investigate this, a theoretical analysis is carried out using the Rayleigh–Ritz approximation, and a new formula is proposed to predict the critical buckling stress of an infill ranging from 0% up to 100% rigid. The proposed formula agrees well with works reported in the literature. It also shows that filling a foam core in a thin-walled cylindrical shell can enhance its resistance to buckling failure. Meanwhile, a simplified formula is provided to the practicing engineer. The paper concludes that an excessive increase in foam core thickness beyond 10% of outer radius is inefficient due to extra cost and weight.

Stability design criteria for steel column splices

The current work examines the buckling behaviour of an axially loaded framed spliced steel column. The general behaviour is characterized using a polynomial Rayleigh–Ritz approximation substituted into the potential energy function, in combination with the method of Lagrange multipliers, and the critical load is found. Analysis is run within an algebraic manipulator and checked against results from the classical equilibrium methodology. A set of parametric studies is implemented. Results are evaluated with respect to the following variables: (i) splice location and rotational stiffness, (ii) change in the column section serial size and (iii) column end-restraints stiffness coefficients. Simple relationships are developed for use as a basic stiffness design criterion for column splices.

Flexural vibration of non-uniform beams having double-edge breathing cracks

Flexural vibration of non-uniform Rayleigh beams having single-edge and double-edge cracks is presented in this paper. Asymmetric double-edge cracks are formed as thin transverse slots with different depths at the same location of opposite surfaces. The cracks are modelled as breathing since the bending of the beam makes the cracks open and close in accordance with the direction of external moments. The presented crack model is used for single-edge cracks and double-edge cracks having different depth combinations. The energy method is used in the vibration analysis of the cracked beams. The consumed energy caused by the cracks opening and closing is obtained along the beam's length together with the contribution of tensile and compressive stress fields that come into existence during the bending. The total energy is evaluated for the Rayleigh–Ritz approximation method in analysing the vibration of the beam. Examples are presented on simply supported beams having uniform width and cantilever beams which are tapered. Good agreements are obtained when the results from the present method are compared with the results of Chondros et al. and the results of the commercial finite element program, Ansys ©. The effects of breathing in addition to crack depth's asymmetry and crack positions on the natural frequency ratios are presented in graphics.

Rupture nucleation on an interface with a power-law relation between stress and displacement discontinuity

We consider rupture initiation and instability on a displacement-weakening interface. It is assumed to follow a power-law relation between a component of displacement discontinuity (whether tensile opening in mode I or shear slippage in modes II or III) and the reduction from peak strength of a corresponding component of stress (normal or shear stress) on the interface. That is, the stress decrease from peak strength, as the interface discontinuity develops, is assumed to be proportional to displacement-discontinuity to some exponent n > 0. The study is done in the 2D context of plane or anti-plane strain, for an initially coherent interface which is subjected to a locally peaked “loading” stress which increases quasi-statically in time. We seek to establish the instability point, when no further quasi-static solution exists for growth of the ruptured zone along the interface, so that dynamic rupture ensues. We have previously addressed the case of linear displacement-weakening (n = 1), and proven the remarkable result that for an unbounded solid, the length of the displacement-weakening zone along the interface at instability is universal, in the sense of being independent of the detailed spatial distribution of the locally peaked loading stress. Present results show that such universality does not apply when n differs from 1. Also, if n < 2/3, there is no phase of initially quasi-static enlargement of the rupturing zone; instead instability will occur as soon as the maximum value of the loading stress reaches the peak strength. We first employ an energy approach to give a Rayleigh–Ritz approximation for the dependence of quasi-static rupture length and maximum displacement-discontinuity on the loading stress distribution of a quadratic form. Results, depending on curvature of the loading distribution, show that qualitative features of the displacement-discontinuity development are significantly controlled by n, with the transition noted at n = 2/3. Predictions of the simple energy approach are in reasonable quantitative agreement with full numerical solutions and give qualitative features correctly.

Quadratic Interpolation and Rayleigh–Ritz Methods for Bifurcation Coefficients

In this article we study the estimation of bifurcation coefficients in nonlinear branching problems by means of Rayleigh–Ritz approximation to the eigenvectors of the corresponding linearized problem. It is essential that the approximations converge in a norm of sufficient strength to render the nonlinearities continuous. Quadratic interpolation between Hilbert spaces is used to seek sharp rate of convergence results for bifurcation coefficients. Examples from ordinary and partial differential problems are presented.

Vibration analysis of non-uniform beams having multiple edge cracks along the beam's height

Bending vibration of non-uniform rectangular beams with multiple edge cracks along the beam's height is investigated. These cracks are called height-edge cracks in this paper. The energy based method is used for defining the vibration of height-edge cracked beams. The opening form of the height-edge crack is determined when the external moment is assumed to be applied for stretching the beam's width. Strain energy increase is obtained by calculating the strain change at the stretched surface by taking into account the effect of angular displacement of the beam due to the bending. The Rayleigh–Ritz approximation method is used in the analysis. The cases of multiple cracks are analysed in the method by using the approach based on the definition of strain disturbance variation along the beam. Examples are presented on a fixed–fixed beam and several cantilever beams having different taper factors. When the results are compared with the results of a commercial finite element program, good agreement is obtained. The effects of taper factors, boundaries and positions of cracks on the natural frequency ratios are presented in graphics.

Standard input data for FEM–MBS coupling: importing alternative model reduction methods into SIMPACK

Various research areas in the field of vehicle modelling, structural mechanics, engine dynamics, microelectromechanical systems (MEMS), etc. require the utilization of both multibody system formalism (MBS) and finite element method (FEM) in order to sufficiently capture the model's dynamics. The FEM–MBS coupling is accomplished by reducing the dimension of the FE-modelled part and then importing it into an MBS-code for further simulation. When using commercial FEM (Nastran, ANSYS, etc.) as well as MBS (SIMPACK) software packages the necessary standard input data (SID) file is needed for the coupling procedure (FEMBS interface). A problem arises by the restriction that both commercial FEM and MBS codes support only two condensation methods (Guyan reduction and component mode synthesis (CMS)), thus disabling the direct application of any other reduction approach (e.g. from the field of control theory) that actually could be better. In this article, the theoretical background of an implemented FEM–MBS interface (MORPACK) is presented allowing the application of any kind of reduction method for FE-modelled structures and furthermore their import (Ritz approximation) into SIMPACK via the SID file generation. A benchmark problem (UIC60-rail) is used in order to capture in SIMPACK the discrepancy between the standardized CMS and the Krylov subspace method (KSM), as one of the alternatives offered by the interface.

Analytical Modeling of Circular Glare Laminated Plates under Lateral Indentation

In this paper analytical solutions are derived to predict the static response of thin circular clamped GLARE fibre-metal laminated plates under the action of a lateral hemispherical indentor. The load-indentation curve is calculated along with the first failure load and deflection due to glass-epoxy tensile fracture. The Ritz method is employed with one, two and three-parameter Ritz approximation functions. The derived formulas are applied to GLARE 2-2/1-0.3 and to GLARE 3-3/2-0.4 circular plates with various diameters. The results converge satisfactorily in all examined cases. The calculated load-indentation curve and the first failure agree well with published experimental data for the case of a GLARE 2-2/1-0.3 plate with a radius of 40 mm (failure load within 7% and failure deflection within 3%). The same load-indentation curves are also calculated using ANSYS and by comparison to FEM results the validity of the analytical model is further verified. No analytical solution of this problem is known to the authors.

Vibration analysis of multiple-cracked non-uniform beams

This paper presents the energy-based method for the vibration identification of non-uniform Euler–Bernoulli beams having multiple open cracks. The method includes significant modifications for the energy-based method presented by Yang et al. [Crack identification in vibrating beams using the energy method, Journal of Sound and Vibration 244 (2) (2001) 339–357.] The distribution of the energy consumed is determined by taking into account not only the strain change at the cracked beam surface as in general but also the considerable effect of the stress field caused by the angular displacement of the beam due to bending. The Rayleigh–Ritz approximation method is used in the analysis. The method is adapted to the cases of multiple cracks with an approach based on the definition of strain disturbance variation along the beam. Examples are presented on cantilever beams having different truncation factors. When the results are compared with a commercial finite element program and with the results of Zheng and Fan [Natural frequencies of a non-uniform beam with multiple cracks via modified Fourier series, Journal of Sound and Vibration 242 (4) (2001) 701–717], good agreements are obtained. The effects of truncation factors and positions of cracks on the natural frequency ratios are presented in graphics.

Wave scattering by an axisymmetric ice floe of varying thickness

The problem of water wave scattering by a circular ice floe, floating in fluid of finite depth, is formulated and solved numerically. Unlike previous investigations of such situations, here we allow the thickness of the floe (and the fluid depth) to vary axisymmetrically and also incorporate a realistic non-zero draught. A numerical approximation to the solution of this problem is obtained to an arbitrary degree of accuracy by combining a Rayleigh–Ritz approximation of the vertical motion with an appropriate variational principle. This numerical solution procedure builds upon the work of Bennets et al. (2007, J. Fluid Mech., 579, 413–443). As part of the numerical formulation, we utilize a Fourier cosine expansion of the azimuthal motion, resulting in a system of ordinary differential equations to solve in the radial coordinate for each azimuthal mode. The displayed results concentrate on the response of the floe rather than the scattered wave field and show that the effects of introducing the new features of varying floe thickness and a realistic draught are significant.

Postbuckling behavior of composite plates with through-the-width delaminations

Fiber-reinforced composite materials have been increasingly used over the past few decades in a variety of applications in which a fairly high ratio of stiffness/strength to weight is required. However, these materials are prone to a wide range of defects and damage that can cause significant reductions in stiffness and strength. Delamination is one of the most common failure modes in composite materials which affect the overall stiffness of the structure. In this paper, the compressive behavior of composite laminates with through-the-width delaminations is investigated analytically. The analytical method is based on the CLPT theory, and its formulation is developed on the basis of the Rayleigh–Ritz approximation technique to analyze the bucking and postbuckling behavior of delaminated laminates. The method can handle both local buckling of the delaminated sublaminate and global buckling of the whole plate. Also the three-dimensional finite element analysis is performed by using ANSYS5.4 general purpose commercial software, and the results are compared with those obtained by the analytical model. The agreement between the results is very good.

Thermal-stress analysis of plates with variable fiber spacing

Plane-stress problems of a square composite plate with variable fiber spacing under a uniform thermal loading are investigated. The problem with two edges perpendicular to the fiber direction of the plate, being itself constrained from normal displacement, is first solved analytically. It is then analyzed by the energy method together with the Rayleigh–Ritz approximation, and the computed results are verified by the analytical solution showing very good agreement. Another problem, with two edges parallel to the fiber direction, the plate again being constrained from normal displacement, which is more complex and cannot be solved analytically, is tackled by the afore-mentioned numerical procedure. To investigate the effect of the variation of fiber spacing on the distributions of the displacement-component fields and the equivalent stress field, two types of variation in fiber spacing are assumed in solving the second problem.

On the one-dimensional modeling of camber bending deformations in active anisotropic slender structures

The paper presents a one-dimensional model for anisotropic active slender structures that captures arbitrary cross-sectional deformations. The 1-D geometrically-nonlinear static problem is derived by an asymptotic reduction process from the equations of 3-D electroelasticity. In addition to the conventional (bending–extension–shear–twist) beam strain measures, it includes a Ritz approximation to account for arbitrary deformation shapes of the finite-size cross-sections. As a particular case, closed-form analytical expressions are derived for the linear static equilibrium of a composite thin strip with surface-mounted piezoelectric actuators. This solution is based on a boundary-layer approximation to the static equilibrium equations in regions where Saint-Venant’s principle for elastic bodies cannot be applied and includes camber bending deformations to account for the local bimoments induced by the distributed actuation in a finite-width strip.

Nonlinear vibrations of a ropeway system with moving passenger cabins

AbstractThe dynamic continuous model of a carrying rope for circulating bicable aerial ropeway is formulated. To describe nonlinear in‐plane vibrations excited by moving masses of passenger cabins a closed form model with Green‐Lagrange deformation is developed. The equations of motion of the system are derived on the basis of Lagrange equations with Ritz approximation of cable displacements applied. Numerical example of linear and nonlinear cable vibrations is presented. (© 2008 WILEY‐VCH Verlag GmbH &amp; Co. KGaA, Weinheim)

Variational principle for the dislocation equation in lattice theory

In the lattice theory of dislocation, the dislocation equation including the discreteness effect is a nonlinear integro-differential equation. It can hardly be solved. In this paper, the variational principle for the dislocation equation is presented. By using the Ritz approximation method, the variational solution is obtained, which is the same as the Peierls–Nabarro solution when the discreteness effect terms were ignored and the accurate value of the variational parameter is obtained when the discreteness effect terms were taken into account. As a natural generalization, a modified form of the sinusoidal law of force is proposed. It is found that the fine structure of the dislocation core can be predicted theoretically as long as the parameter in the modified law of force is selected properly.

On the Minimization of Maximum Transient Energy Growth

The problem of minimizing the maximum transient energy growth is considered. This problem has importance in some fluid flow control problems and other classes of nonlinear systems. Conditions for the existence of static controllers that ensure strict dissipativity of the transient energy are established and an explicit parametrization of all such controllers is provided. It also is shown that by means of a Q-parametrization, the problem of minimizing the maximum transient energy growth can be posed as a convex optimization problem that can be solved by means of a Ritz approximation of the free parameter. By considering the transient energy growth at an appropriate sequence of discrete time points, the minimal maximum transient energy growth problem can be posed as a semidefinite program. The theoretical developments are demonstrated on a numerical example.

Computation of eigenvalues and solutions of regular Sturm–Liouville problems using Haar wavelets

The paper presents a novel method for the computation of eigenvalues and solutions of Sturm–Liouville eigenvalue problems (SLEPs) using truncated Haar wavelet series. This is an extension of the technique proposed by Hsiao to solve discretized version of variational problems via Haar wavelets. The proposed method aims to cover a wider class of problems, by applying it to historically important and a very useful class of boundary value problems, thereby enhancing its applicability. To demonstrate the effectiveness and efficiency of the method various celebrated Sturm–Liouville problems are analyzed for their eigenvalues and solutions. Also, eigensystems are investigated for their asymptotic and oscillatory behavior. The proposed scheme, unlike the conventional numerical schemes, such as Rayleigh quotient and Rayleigh–Ritz approximation, gives eigenpairs simultaneously and provides upper and lower estimates of the smallest eigenvalue, and it is found to have quadratic convergence with increase in resolution.

Optimal control of nonholonomic motion planning for a free-falling cat

The nonholonomic motion planning of a free-falling cat is investigated. Nonholonomicity arises in a free-falling cat subject to nonintegrable angle velocity constraints or nonintegrable conservation laws. When the total angular momentum is zero, the motion equation of a free-falling cat is established based on the model of two symmetric rigid bodies and conservation of angular momentum. The control of system can be converted to the problem of nonholonomic motion planning for a free-falling cat. Based on Ritz approximation theory, the Gauss-Newton method for motion planning by a falling cat is proposed. The effectiveness of the numerical algorithm is demonstrated through simulation on model of a free-falling cat.