A New Global Spatial Discretization Method for Calculating Dynamic Responses of Two-Dimensional Continuous Systems With Application to a Rectangular Kirchhoff Plate

2017 ◽  
Vol 140 (1) ◽  
Author(s):  
K. Wu ◽  
W. D. Zhu
Keyword(s):  
Boundary Conditions ◽  
Continuous System ◽  
Dynamic Responses ◽  
Kirchhoff Plate ◽  
Continuous Systems ◽  
Discretization Method ◽  
Two Dimensional ◽  
Spatial Discretization ◽  
Arbitrary Boundary ◽  
Discretization Methods

A new global spatial discretization method (NGSDM) is developed to accurately calculate natural frequencies and dynamic responses of two-dimensional (2D) continuous systems such as membranes and Kirchhoff plates. The transverse displacement of a 2D continuous system is separated into a 2D internal term and a 2D boundary-induced term; the latter is interpolated from one-dimensional (1D) boundary functions that are further divided into 1D internal terms and 1D boundary-induced terms. The 2D and 1D internal terms are chosen to satisfy prescribed boundary conditions, and the 2D and 1D boundary-induced terms use additional degrees-of-freedom (DOFs) at boundaries to ensure satisfaction of all the boundary conditions. A general formulation of the method that can achieve uniform convergence is established for a 2D continuous system with an arbitrary domain shape and arbitrary boundary conditions, and it is elaborated in detail for a general rectangular Kirchhoff plate. An example of a rectangular Kirchhoff plate that has three simply supported boundaries and one free boundary with an attached Euler–Bernoulli beam is investigated using the developed method and results are compared with those from other global and local spatial discretization methods. Advantages of the new method over local spatial discretization methods are much fewer DOFs and much less computational effort, and those over the assumed modes method (AMM) are better numerical property, a faster calculation speed, and much higher accuracy in calculation of bending moments and transverse shearing forces that are related to high-order spatial derivatives of the displacement of the plate with an edge beam.

A New Global Spatial Discretization Method for Two-Dimensional Continuous Systems

2017 ◽  
Author(s):  
K. Wu ◽  
W. D. Zhu
Keyword(s):  
Boundary Conditions ◽  
Degrees Of Freedom ◽  
Bending Moment ◽  
Continuous System ◽  
Dynamic Responses ◽  
Continuous Systems ◽  
Two Dimensional ◽  
Spatial Discretization ◽  
One Dimensional ◽  
Dimensional Boundary

A new global spatial discretization method is developed to accurately calculate natural frequencies and dynamic responses of two-dimensional continuous systems such as membranes and Kirchhoff plates. The transverse displacement of a two-dimensional continuous system is separated into a two-dimensional internal term and a two-dimensional boundary-induced term; the latter is interpolated from one-dimensional boundary functions that are further divided into one-dimensional internal terms and one-dimensional boundary-induced terms. The two- and one-dimensional internal terms are chosen to satisfy predetermined boundary conditions, and the two- and one-dimensional boundary-induced terms use additional degrees of freedom at boundaries to ensure satisfaction of all boundary conditions. A general formulation of the method that can achieve uniform convergence is established for a two-dimensional continuous system with an arbitrary domain shape and arbitrary boundary conditions, and it is elaborated in detail for a general rectangular Kirchhoff plate. An example of a rectangular Kirchhoff plate that has three simply-supported boundaries and one free boundary with an attached Euler-Bernoulli beam is investigated using the developed method and results are compared with those from other global and local spatial discretization methods. Natural frequencies and dynamic responses that include the displacement, the velocity, rotational angles, a bending moment, and a transverse shearing force are calculated using both the developed method and the assumed modes method, and compared with results from the finite element method and the finite difference method, respectively. Advantages of the new method over local spatial discretization methods are much fewer degrees of freedom and much less computational effort, and those over the assumed modes method are better numerical property, a faster calculation speed, and much higher accuracy in calculation of the bending moment and the transverse shearing force that are related to high-order spatial derivatives of the displacement of the plate with an edge beam.

Dynamic Analysis of a Rotating Planar Timoshenko Beam Using an Accurate Global Spatial Discretization Method

2019 ◽  
Author(s):  
W. Fan ◽  
W. D. Zhu ◽  
H. Zhu
Keyword(s):  
Boundary Conditions ◽  
Dynamic Analysis ◽  
Timoshenko Beam ◽  
Slope Angle ◽  
Dynamic Responses ◽  
Discretization Method ◽  
Spatial Discretization ◽  
Dependent Variables ◽  
Spatial Derivatives ◽  
Shear Strains

Abstract A new formulation is developed for dynamic analysis of a rotating planar Timoshenko beam. The configuration of Timoshenko beam is described using its slope angle and axial and shear strains; hence, the shear locking problem can be naturally avoided. While six boundary conditions are needed for choices of trial functions of three dependent variables, there are only four boundary conditions that can be determined and two boundary conditions are undetermined. An accurate global spatial discretization method is used, where dependent variables are divided into internal and boundary-induced terms. Internal terms only need to satisfy homogeneous boundary conditions, which can be easily chosen as trigonometric functions. Boundary-induced terms are interpolated using dependent variables at boundaries that are taken as generalized coordinates. When the hub rotates at a constant angular velocity, nonlinear governing equations can be linearized for vibration analysis. Frequency veering and mode shift phenomena occur. Nonlinear dynamic responses of the system are then calculated and compared with those from the commercial software ADAMS, and they are in good agreement. Axial and shear strains of the beam and their spatial derivatives are also calculated. Since trial functions in the assumed modes method cannot satisfy undetermined boundary conditions, inaccurate results of strains and their spatial derivatives are obtained using the assumed modes method. Hence, use of the accurate global spatial discretization method in the current formulation is essential.

Free vibration analysis of three-layer thin cylindrical shell with variable thickness two-dimensional FGM middle layer under arbitrary boundary conditions

2021 ◽  
pp. 109963622110204
Author(s):  
Xue-Yang Miao ◽  
Chao-Feng Li ◽  
Yu-Lin Jiang ◽  
Zi-Xuan Zhang
Keyword(s):  
Cylindrical Shell ◽  
Boundary Conditions ◽  
Volume Fraction ◽  
Ritz Method ◽  
Admissible Function ◽  
Free Vibration Analysis ◽  
Middle Layer ◽  
Two Dimensional ◽  
Arbitrary Boundary ◽  
Arbitrary Boundary Conditions

In this paper, a unified method is developed to analyze free vibrations of the three-layer functionally graded cylindrical shell with non-uniform thickness. The middle layer is composed of two-dimensional functionally gradient materials (2D-FGMs), whose thickness is set as a function of smooth continuity. Four sets of artificial springs are assigned at the ends of the shells to satisfy the arbitrary boundary conditions. The Sanders’ shell theory is used to obtain the strain and curvature-displacement relations. Furthermore, the Chebyshev polynomials are selected as the admissible function to improve computational efficiency, and the equation of motion is derived by the Rayleigh–Ritz method. The effects of spring stiffness, volume fraction indexes, configuration on of shell, and the change in thickness of the middle layer on the modal characteristics of the new structural shell are also analyzed.

Nonlinear Normal Modes of a Continuous System With Quadratic Nonlinearities

1995 ◽  
Vol 117 (2) ◽  
pp. 199-205 ◽  
Author(s):  
A. H. Nayfeh ◽  
S. A. Nayfeh
Keyword(s):  
Boundary Conditions ◽  
Multiple Scales ◽  
Method Of Multiple Scales ◽  
Normal Modes ◽  
Continuous System ◽  
Nonlinear Normal Modes ◽  
Continuous Systems ◽  
Quadratic Nonlinearities ◽  
Nonlinear Normal ◽  
Quadratic And Cubic Nonlinearities

We use two approaches to determine the nonlinear modes and natural frequencies of a simply supported Euler-Bernoulli beam resting on an elastic foundation with distributed quadratic and cubic nonlinearities. In the first approach, we use the method of multiple scales to treat the governing partial-differential equation and boundary conditions directly. In the second approach, we use a Galerkin procedure to discretize the system and then determine the normal modes from the discretized equations by using the method of multiple scales and the invariant manifold approach. Whereas one- and two-mode discretizations produce erroneous results for continuous systems with quadratic and cubic nonlinearities, all methods, in the present case, produce the same results because the discretization is carried out by using a complete set of basis functions that satisfy the boundary conditions.

A Hybrid Approach for the Modal Analysis of Continuous Systems With Localized Nonlinear Constraints

2010 ◽  
Author(s):  
M. R. Brake
Keyword(s):  
Boundary Conditions ◽  
Modal Analysis ◽  
Piecewise Linear ◽  
Hybrid Approach ◽  
Constitutive Relationship ◽  
Leaf Spring ◽  
Continuous Systems ◽  
Superposition Method ◽  
Arbitrary Boundary ◽  
Linear Force

The analysis of continuous systems with nonlinearities in their domain have previously been limited to either numerical approaches, or analytical methods that are constrained in the parameter space, boundary conditions, or order of the system. The present analysis develops a robust method for studying continuous systems with arbitrary boundary conditions and nonlinearities using the assumption that the nonlinear constraint can be modeled with a piecewise-linear force-deflection constitutive relationship. Under this assumption, a superposition method is used to generate homogeneous boundary conditions, and modal analysis is used to find the displacement of the system in each state of the piecewise-linear nonlinearity. In order to map across each nonlinearity in the piecewise-linear force-deflection profile, a variational calculus approach is taken that minimizes the L2 energy norm between the previous and current states. To illustrate this method, a leaf spring coupled with a connector pin immersed in a viscous fluid is modeled as a beam with a piecewise-linear constraint. From the results of the convergence and parameter studies, a high correlation between the finite-time Lyapunov exponents and the contact time per period of the excitation is observed. The parameter studies also indicate that when the system’s parameters are changed in order to reduce the magnitude of the velocity impact between the leaf spring and connector pin, the extent of the regions over which a chaotic response is observed increases.

Maximum amplitudes in transient resonance of distributed-parameter systems / Amplitudy maksymalne w rezonansie przejściowym układów o parametrach rozłożonych

2012 ◽  
Vol 57 (3) ◽  
pp. 657-665 ◽  
Author(s):  
Jerzy Michalczyk
Keyword(s):  
Energy Balance ◽  
Kinetic Energy ◽  
Continuous System ◽  
Distributed Parameter ◽  
Engineering Practice ◽  
Continuous Systems ◽  
Two Dimensional ◽  
Approximate Methods ◽  
Unbalanced Rotor ◽  
Transient Resonance

Abstract The application of the kinetic energy balance for the estimation maximum amplitudes of continuous systems in the transient resonance excited by the free coasting of unbalanced rotor or piston machines placed on the continuous system - was proposed in the study. The exact as well as the approximate methods were shown. For the typical one- and two-dimensional systems the calculation formulae, useful for the engineering practice, were given.

scholarly journals The theory of Rayleigh's principle as applied to continuous systems

1928 ◽  
Vol 119 (782) ◽  
pp. 276-293 ◽  
Keyword(s):  
Approximate Solution ◽  
Boundary Conditions ◽  
Degrees Of Freedom ◽  
Continuous System ◽  
Accurate Solution ◽  
Successive Approximations ◽  
Continuous Systems ◽  
Constrained Motion ◽  
Mean Values ◽  
Kinetic And Potential Energies

The object of this paper is to examine and extend a method frequently applied by Lord Rayleigh to the calculation of the frequency of the gravest mode of a vibrating system. In this method no attempt is made to obtain an accurate solution of the differential equations of vibration in any normal mode, but any fairly simple function satisfying the boundary conditions is adopted as an approximate solution. This solution will involve the time only through a harmonic factor such as sin 2 π vt , so that the mean value of the kinetic and potential energies of the system may be calculated—always on the supposition that frictionless constraints are applied so that the displacements of the system follow the law of the approximate solution adopted. By equating these mean values of the kinetic and potential energies there is obtained an expression for v , the frequency of the constrained motion. According to Rayleigh’s principle, this value of the frequency is always in excess of the natural frequency of the gravest mode of vibration. Moreover, it is usually found that almost any differentiable function satisfying the boundary conditions of the problem may be made the basis of a calculation yielding a close approximation to the fundamental frequency. Although these results follow easily enough from the Lagrangian method of treating the oscillations of a system possessing only a finite number of degrees of freedom, they do not appear to have been investigated in the far more important case of a continuous system. In this paper a closer examination of Rayleigh’s principle is made by considering a series of successive approximations to the accurate solutions of the problems proposed, and a method is devised for obtaining an upper bound to the error involved in the approximate values of the frequency. Rayleigh’s method is also extended to the calculation of the frequency of the first overtone, and analogous methods are given for more general problems of the computation of “characteristic numbers.”

Separated Solution Procedure for Bending of Circular Plates With Circular Holes

1991 ◽  
Vol 44 (11S) ◽  
pp. S27-S35 ◽  
Author(s):  
M. D. Bird ◽  
C. R. Steele
Keyword(s):  
Boundary Conditions ◽  
Processing Time ◽  
Solution Procedure ◽  
Trigonometric Fourier Series ◽  
Two Dimensional ◽  
Circular Plates ◽  
Arbitrary Boundary ◽  
Circular Holes ◽  
Circular Domains ◽  
Harmonic Equation

A separated solution procedure is presented for the two-dimensional bi-harmonic equation on circular domains with circular holes and arbitrary boundary conditions. The solutions use the traditional trigonometric Fourier series on the boundaries with a power series decay into the domain. The interaction of the boundaries is expressed simply and exactly resulting in quick processing time. The only simplification made is the use of a finite number of terms in the boundary conditions.

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