Convergence of the Ritz Method

1995 ◽  
Vol 48 (11S) ◽  
pp. S90-S95 ◽  
Author(s):  
Arthur W. Leissa ◽  
Samir M. Shihada
Keyword(s):  
Eigenvalue Problems ◽  
Ritz Method ◽  
Equation Of Motion ◽  
Upper Bounds ◽  
Free Vibrations ◽  
Bending Moments ◽  
Shear Forces ◽  
Buckling Loads ◽  
Derivatives Of ◽  
Admissible Functions

The Ritz method is widely used for the solution of problems in structural mechanics, especially eigenvalue problems where the free vibration frequencies or buckling loads are sought. It is well-known that the method yields upper bounds for these eigenvalues, and that convergence to exact eigenvalues will occur if proper admissible functions are used to represent the displacements (eigenfunctions). However, little is known about the convergence of the derivatives of the eigenfunctions. In this paper the method is studied for the problem of free vibrations of a cantilever beam. Convergences of the eigenfunctions and their second and third derivatives (i.e., bending moments and shear forces) are examined, as well as convergence to satisfy the differential equation of motion (which involves fourth derivatives). Another study examines the effects of deliberately omitting one term from the set of admissible displacement functions which would otherwise be complete. It is found that in such cases, when orthogonal polynomials are used to represent the displacement, one eigenvalue (such as the lowest frequency) may be completely missed, and that the others will converge incorrectly. When ordinary polynomials are used, correct convergence is obtained even with a missing term.

The Rayleigh-Ritz Method With Quasi-Comparison Functions in Nonself-Adjoint Problems

1993 ◽  
Vol 115 (3) ◽  
pp. 280-284 ◽  
Author(s):  
P. Hagedorn
Keyword(s):  
Multibody Systems ◽  
Eigenvalue Problems ◽  
Ritz Method ◽  
Continuous Linear ◽  
Linear Elastic ◽  
Comparison Functions ◽  
Elastic Systems ◽  
Nonlinear Motions ◽  
Admissible Functions

In the determination of the first eigenmodes of continuous linear elastic systems the Rayleigh-Ritz method is often used. It is also very useful in the discretization of the elastic members of multibody systems undergoing large nonlinear motions. Recently the concept of quasi-comparison functions has been introduced for the Rayleigh-Ritz discretization in self-adjoint eigenvalue problems, where it may lead to a considerable improvement of the convergence when compared with other classes of admissible functions. In this paper it is shown with a simple example that a similar phenomenon also holds for nonself-adjoint problems. Since the exact solutions are known, precise information on the errors can be given.

scholarly journals Buckling and Free Vibrations of Nanoplates – Comparison of Nonlocal Strain and Stress Approaches

2019 ◽  
Vol 9 (7) ◽  
pp. 1409 ◽  
Author(s):  
Małgorzata Chwał ◽  
Aleksander Muc
Keyword(s):  
Eigenvalue Problems ◽  
Scale Effects ◽  
Ritz Method ◽  
Shear Deformation Theory ◽  
Free Vibration Analysis ◽  
Free Vibrations ◽  
Small Scale ◽  
Governing Equations ◽  
First Order ◽  
Fundamental Equations

The buckling and free vibrations of rectangular nanoplates are considered in this paper. The refined continuum transverse shear deformation theory (third and first order) is introduced to formulate the fundamental equations of the nanoplate. In addition, the analysis involves the nonlocal strain and stress theories of elasticity to take into account the small-scale effects encountered in nanostructures/nanocomposites. Hamilton’s principle is used to establish the governing equations of the nanoplate. The Rayleigh-Ritz method is proposed to solve eigenvalue problems dealing with the buckling and free vibration analysis of the nanoplates considered. Some examples are presented to investigate and illustrate the effects of various formulations.

Vibration and Buckling Analysis of Internally Cracked Square Plates by the MLS-Ritz Approach

2018 ◽  
Vol 18 (09) ◽  
pp. 1850105 ◽  
Author(s):  
C. S. Huang ◽  
M. C. Lee ◽  
M. J. Chang
Keyword(s):  
Plane Stress ◽  
Plate Theory ◽  
Ritz Method ◽  
Internal Crack ◽  
Vibration Frequencies ◽  
Buckling Loads ◽  
Finite Element Software ◽  
Stress Resultant ◽  
Out Of Plane ◽  
Admissible Functions

Based on the classical thin plate theory, this paper proposes new sets of enriched basis functions for in-plane and out-of-plane displacements of square plates that can yield admissible functions for the Ritz method using the moving least-squares (MLS) approach. These admissible functions display the discontinuities of displacement and slope across a crack; give the correct singularity order for the stress resultants at the crack tips; and enhance the Ritz method’s ability to recognize the existence of an internal crack in a plate. To confirm the validity of the proposed approach, convergence studies were performed on the buckling loads and vibration frequencies of plates with central horizontal cracks, and the results obtained agree closely with the published ones as well as those generated by the commercial finite element software. To demonstrate the importance of including all the in-plane stress resultant components in the analysis, the effects of different in-plane stress resultant components on the buckling loads and vibration frequencies of simply supported center-cracked square plates under uniaxial uniform loading were investigated. The present approach was further employed to study the effects of cracks’ lengths, orientations, and locations on the buckling loads and frequencies of cracked square plates under different boundary conditions and in-plane loading conditions.

Large Deflection of a Rectangular Plate Resting on a Pasternak-Type Elastic Foundation

1977 ◽  
Vol 44 (3) ◽  
pp. 509-511 ◽  
Author(s):  
P. K. Ghosh
Keyword(s):  
Rectangular Plate ◽  
Elastic Foundation ◽  
Large Deflection ◽  
Lateral Load ◽  
Bending Moments ◽  
Shear Forces ◽  
Simply Supported ◽  
Base Parameters ◽  
Square Plate

The problem of large deflection of a rectangular plate resting on a Pasternak-type foundation and subjected to a uniform lateral load has been investigated by utilizing the linearized equation of plates due to H. M. Berger. The solutions derived and based on the effect of the two base parameters have been carried to practical conclusions by presenting graphs for bending moments and shear forces for a square plate with all edges simply supported.

Free Vibrations of Thick, Complete Conical Shells of Revolution From a Three-Dimensional Theory

2005 ◽  
Vol 72 (5) ◽  
pp. 797-800 ◽  
Author(s):  
Jae-Hoon Kang ◽  
Arthur W. Leissa
Keyword(s):  
Ritz Method ◽  
Three Dimensional ◽  
Axial Direction ◽  
Free Vibrations ◽  
Mode Shapes ◽  
Vibration Frequencies ◽  
Shells Of Revolution ◽  
Dimensional Theory ◽  
Conical Shells ◽  
Cylindrical Coordinate

A three-dimensional (3D) method of analysis is presented for determining the free vibration frequencies and mode shapes of thick, complete (not truncated) conical shells of revolution in which the bottom edges are normal to the midsurface of the shells based upon the circular cylindrical coordinate system using the Ritz method. Comparisons are made between the frequencies and the corresponding mode shapes of the conical shells from the authors' former analysis with bottom edges parallel to the axial direction and the present analysis with the edges normal to shell midsurfaces.

Higher order Wirtinger inequalities

1980 ◽  
Vol 85 (1-2) ◽  
pp. 87-110 ◽  
Author(s):  
Kurt Kreith ◽  
Charles A. Swanson
Keyword(s):  
Boundary Conditions ◽  
Quadratic Forms ◽  
Differential Operators ◽  
Conjugate Point ◽  
Wirtinger Inequality ◽  
Even Order ◽  
Linear Differential ◽  
Natural Boundary Conditions ◽  
Derivatives Of ◽  
Admissible Functions

SynopsisWirtinger-type inequalities of order n are inequalities between quadratic forms involving derivatives of order k ≦ n of admissible functions in an interval (a, b). Several methods for establishing these inequalities are investigated, leading to improvements of classical results as well as systematic generation of new ones. A Wirtinger inequality for Hamiltonian systems is obtained in which standard regularity hypotheses are weakened and singular intervals are permitted, and this is employed to generalize standard inequalities for linear differential operators of even order. In particular second order inequalities of Beesack's type are developed, in which the admissible functions satisfy only the null boundary conditions at the endpoints of [a, b] and b does not exceed the first systems conjugate point (a) of a. Another approach is presented involving the standard minimization theory of quadratic forms and the theory of “natural boundary conditions”. Finally, inequalities of order n + k are described in terms of (n, n)-disconjugacy of associated 2nth order differential operators.

Analysis of In-Plane Modal Characteristics of an Annular Disk With Multiple Point Supports

2009 ◽  
Author(s):  
S. Bashmal ◽  
R. Bhat ◽  
S. Rakheja
Keyword(s):  
Finite Element ◽  
Orthogonal Polynomials ◽  
Ritz Method ◽  
Element Model ◽  
Free Vibrations ◽  
Mode Shapes ◽  
Multiple Point ◽  
Annular Disk ◽  
Free Boundary Conditions ◽  
Boundary Characteristic

In-plane free vibrations of an isotropic, elastic annular disk constrained at some points on the inner and outer boundaries are investigated. The presented study is relevant to various practical problems including disks clamped by bolts along the inner and outer edges or the railway wheel vibrations. The boundary characteristic orthogonal polynomials are employed in the Rayleigh-Ritz method to obtain the frequency parameters and the associated mode shapes. The boundary characteristic orthogonal polynomials are generated for the free boundary conditions of the disk while artificial springs are used to realize clamped conditions at discrete points. The frequency parameters for different point constraint conditions are evaluated and compared with those computed from a finite element model to demonstrate the validity of the proposed method. The computed mode shapes are presented for a disk with different point constraints at the inner and outer boundaries to demonstrate the free in-plane vibration behavior of the disk. Results show that addition of point supports causes some of the modes to split into two different frequencies with different mode shapes. The effects of different orientations of multiple point supports on the frequency parameters and mode shapes are also discussed.

scholarly journals Determine of the wave equation in the task of electrical oscillations

2018 ◽  
Vol 184 ◽  
pp. 01023
Author(s):  
Gordana V. Jelić ◽  
Vladica Stanojević ◽  
Dragana Radosavljević
Keyword(s):  
Wave Equation ◽  
Initial Conditions ◽  
Mathematical Physics ◽  
Equation Of Motion ◽  
Independent Variables ◽  
Equations Of Mathematical Physics ◽  
Basic Equations ◽  
Derivatives Of ◽  
Two Independent Variables ◽  
Transverse Oscillations

One of the basic equations of mathematical physics (for instance function of two independent variables) is the differential equation with partial derivatives of the second order (3). This equation is called the wave equation, and is provided when considering the process of transverse oscillations of wire, longitudinal oscillations of rod, electrical oscillations in a conductor, torsional vibration at waves, etc… The paper shows how to form the equation (3) which is the equation of motion of each point of wire with abscissa x in time t during its oscillation. It is also shown how to determine the equation (3) in the task of electrical oscillations in a conductor. Then equation (3) is determined, and this solution satisfies the boundary and initial conditions.

Bending Moments and Shear Forces in Ships Sailing in Irregular Waves

1981 ◽  
Vol 25 (04) ◽  
pp. 243-251
Author(s):  
J. Juncher Jensen ◽  
P. Terndrup Pedersen
Keyword(s):  
Linear Part ◽  
Vertical Motion ◽  
Bending Moment ◽  
Irregular Waves ◽  
Bending Moments ◽  
Shear Forces ◽  
Strip Theory ◽  
Wave Heights ◽  
Exciting Forces ◽  
Wave Induced

This paper presents some results concerning the vertical response of two different ships sailing in regular and irregular waves. One ship is a containership with a relatively small block coefficient and with some bow flare while the other ship is a tanker with a large block coefficient. The wave-induced loads are calculated using a second-order strip theory, derived by a perturbational procedure in which the linear part is identical to the usual strip theory. The additional quadratic terms are determined by taking into account the nonlinearities of the exiting waves, the nonvertical sides of the ship, and, finally, the variations of the hydrodynamic forces during the vertical motion of the ship. The flexibility of the hull is also taken into account. The numerical results show that for the containership a substantial increase in bending moments and shear forces is caused by the quadratic terms. The results also show that for both ships the effect of the hull flexibility (springing) is a fair increase of the variance of the wave-induced midship bending moment. For the tanker the springing is due mainly to exciting forces which are linear with respect to wave heights whereas for the containership the nonlinear exciting forces are of importance.

scholarly journals Independent coordinate coupling method for vibration analysis of a functionally graded carbon nanotube–reinforced plate with central hole

2019 ◽  
Vol 11 (8) ◽  
pp. 168781401987292 ◽  
Author(s):  
Yan Guo ◽  
Yanan Jiang ◽  
Bin Huang
Keyword(s):  
Carbon Nanotube ◽  
Volume Fraction ◽  
Ritz Method ◽  
Matching Condition ◽  
Functionally Graded ◽  
Central Hole ◽  
Lagrange’S Equation ◽  
Reinforced Plate ◽  
Distribution Type ◽  
Admissible Functions

In this article, the free vibration of a functionally graded carbon nanotube–reinforced plate with central hole is investigated by means of the independent coordinates-based Rayleigh–Ritz method. For the proposed method, the kinematic and potential energies are substituted into Lagrange’s equation in order to obtain the equation of motion. However, the total energies are computed by the difference of energies between the hole domain and the plate domain. By applying the displacement matching condition at the hole domain, two coordinate systems are coupled. For the Rayleigh–Ritz method, the mode shape functions of uniform beams are assumed as admissible functions. By this method, convergent results can be obtained with certain number of terms of admissible functions. The present results clearly reflect the effects of the carbon nanotube distribution type, carbon nanotube volume fraction, hole size, and boundary condition on the nondimensional natural frequencies. The provided results show that the present method is efficient in studying the vibration problems of functionally graded carbon nanotube–reinforced plate with central hole.

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