Solution of Impact-Induced Flexural Waves in a Circular Ring by the Method of Characteristics

1998 ◽  
Vol 65 (3) ◽  
pp. 569-579 ◽  
Author(s):  
V. P. W. Shim ◽  
S. E. Quah
Keyword(s):  
Numerical Scheme ◽  
Method Of Characteristics ◽  
Equations Of Motion ◽  
Beam Theory ◽  
Circular Ring ◽  
Higher Order ◽  
Flexural Waves ◽  
The Method Of Characteristics ◽  
Finite Difference Equations ◽  
Generalized Displacements

A study of elastic wave propagation in a curved beam (circular ring) is presented. The governing equations of motion are formulated in two forms based on Timoshenko beam theory. Solutions are obtained using the method of characteristics, whereby a numerical scheme employing higher-order interpolation is proposed for the finite difference equations. Results obtained are verified by experiments; it is found that use of the higher-order numerical scheme improves correlation with experimental results. Comparison of the relative accuracy between the two mathematical formulations—one in terms of generalized forces and velocities and the other in terms of generalized displacements—shows the former to be mathematically simpler and to yield more accurate results.

In-Plane Free Vibration of Curved Beams Using Finite Element Method

2007 ◽  
Author(s):  
F. Yang ◽  
R. Sedaghati ◽  
E. Esmailzadeh
Keyword(s):  
Finite Element Method ◽  
Finite Element ◽  
Equations Of Motion ◽  
Beam Theory ◽  
Central Line ◽  
Circular Ring ◽  
Curved Beam ◽  
The Finite Element Method ◽  
Curved Beams ◽  
Element Method

Curved beam-type structures have many applications in engineering area. Due to the initial curvature of the central line, it is complicated to develop and solve the equations of motion by taking into account the extensibility of the curve axis and the influences of the shear deformation and the rotary inertia. In this study the finite element method is utilized to study the curved beam with arbitrary geometry. The curved beam is modeled using the Timoshenko beam theory and the circular ring model. The governing equation of motion is derived using the Extended-Hamilton principle and numerically solved by the finite element method. A parametric sensitive study for the natural frequencies has been performed and compared with those reported in the literature in order to demonstrate the accuracy of the analysis.

scholarly journals Problems of massive gravities

2015 ◽  
Vol 30 (03n04) ◽  
pp. 1540006 ◽  
Author(s):  
S. Deser ◽  
K. Izumi ◽  
Y. C. Ong ◽  
A. Waldron
Keyword(s):  
Method Of Characteristics ◽  
Equations Of Motion ◽  
Propagation Speed ◽  
Massive Gravity ◽  
Parameter Range ◽  
Explicit Solutions ◽  
Matter Coupling ◽  
The Method Of Characteristics ◽  
Superluminal Signals

The method of characteristics is a key tool for studying consistency of equations of motion; it allows issues such as predictability, maximal propagation speed, superluminality, unitarity and acausality to be addressed without requiring explicit solutions. We review this method and its application to massive gravity (mGR) theories to show the limitations of these models' physical viability: Among their problems are loss of unique evolution, superluminal signals, matter coupling inconsistencies and micro-acausality (propagation of signals around local closed time-like curves (CTCs)/closed causal curves (CCCs)). We extend previous no-go results to the entire three-parameter range of mGR theories. It is also argued that bimetric models suffer a similar fate.

A Theoretical Analysis of Dynamic Elastic Response of a Circular Ring to Water Waves

2007 ◽  
Vol 129 (3) ◽  
pp. 211-218 ◽  
Author(s):  
G. H. Dong ◽  
S. H. Hao ◽  
Z. Zong ◽  
Y. N. Zheng
Keyword(s):  
Elastic Deformation ◽  
Water Waves ◽  
Equations Of Motion ◽  
Beam Theory ◽  
Fish Farming ◽  
Circular Ring ◽  
Open Ocean ◽  
Elastic Response ◽  
Superposition Method ◽  
Dominant Form

Fish farming in the open ocean is becoming a dominant form of fishery aquaculture due to the dramatic drop of fishery resources in near shore. Deep water anti-stormy fish cages are the most important tool for fish farming in the open ocean. The floatation ring, as a pivotal component of the fish cage, undergoes large deformations and it has been found that the floatation ring oscillates when moving in water waves. In the most serious cases, its deformation is so large it can be damaged. The floatation ring is simplified as a free circular ring, and the governing equations of motion and deformation of the ring can be built up according to the force equilibrium and curved beam theory. The numerical calculation is carried out in terms of the modal superposition method. The motion and the deformation of the ring are analyzed, and two corresponding equations in which the elastic deformations are neglected and considered are given, respectively. By comparing the results of the above two equations, we get the resonant frequencies and the frequency scope influenced by the elastic deformation. It is concluded that the influence of the deformation of the ring is very important for the oscillation of the ring, particularly the former three modes of the elastic deformation which cannot be neglected.

The method of characteristics for steady supersonic rotational flow in three dimensions

1956 ◽  
Vol 1 (4) ◽  
pp. 409-423 ◽  
Author(s):  
Maurice Holt
Keyword(s):  
Supersonic Flow ◽  
Method Of Characteristics ◽  
Equations Of Motion ◽  
Three Dimensional ◽  
Three Dimensions ◽  
Rotational Flow ◽  
Conical Flow ◽  
Plane Flow ◽  
Flow Problems ◽  
The Method Of Characteristics

The method of characteristics for steady supersonic flow problems in three dimensions, due to Coburn & Dolph (1949), is extended so that flow with shocks and entropy changes may be treated. Equations of motion based on Coburn & Dolph's characteristic coordinate system are derived and a scheme is described for solving these by finite differences.A linearized method of characteristics is developed for calculating perturbations of a given three-dimensional field of flow. This is a generalization of the method evolved by Ferri (1952) for perturbations of plane flow and conical flow.

Solution of One-Dimensional Elastic Wave Problems by the Method of Characteristics

1967 ◽  
Vol 34 (3) ◽  
pp. 745-750 ◽  
Author(s):  
P. C. Chou ◽  
R. W. Mortimer
Keyword(s):  
Elastic Wave ◽  
Method Of Characteristics ◽  
Hyperbolic Partial Differential Equations ◽  
Unified Approach ◽  
The Method Of Characteristics ◽  
Dependent Variables ◽  
Generalized Displacements ◽  
The Media ◽  
Shell Equations ◽  
Wave Problems

A number of elastic wave problems which involve one space variable are treated, in a unified manner, by a system of second-order hyperbolic partial differential equations, with the generalized displacements as dependent variables. This system of n equations is analyzed by the method of characteristics yielding closed-form equations for the physical characteristics, the characteristic equations, and the propagation of discontinuities. Procedures for numerical integration along the characteristic curves are established. Among the elastic wave problems that may be represented by this unified approach are the Timoshenko beam, plates, bars, and sheets; in all cases, the media may be non-homogeneous. Various approximate shell equations also may be represented. Results of numerical calculations are in agreement with those obtained by other methods.

Microstructure Theory for a Composite Beam

1971 ◽  
Vol 38 (4) ◽  
pp. 947-954 ◽  
Author(s):  
C.-T. Sun
Keyword(s):  
Composite Beam ◽  
Continuum Model ◽  
Equations Of Motion ◽  
Beam Theory ◽  
Present Theory ◽  
Flexural Waves ◽  
Subsequent Application ◽  
External Forces ◽  
Work Done ◽  
Constituent Layer

A continuum model with microstructure is constructed for laminated beams. In deriving equations, each constituent layer is considered as a Timoshenko beam. With a certain kinematical assumption regarding the deformations of the composite beam the kinetic and strain energies, as well as the variation of the work done by external forces, are computed. The energy and work expressions are “smoothed out” by a smoothing operation, thus transforming the laminated structuring into microstructure of a macro-homogeneous beam. Subsequent application of Hamilton’s principle yields the equations of motion and the boundary conditions. The equations thus obtained are employed to investigate free flexural waves in a composite beam. It is found that the dispersion curves according to the present theory agree very well with the curves obtained according to an exact analysis. Results according to the effective modulus theory are presented for comparison. Two simplified versions of the microstructure beam theory are also developed and discussed.

scholarly journals FREE VIBRATION OF TWO-DIRECTIONAL FGM BEAMS USING A HIGHER-ORDER TIMOSHENKO BEAM ELEMENT

2018 ◽  
Vol 56 (3) ◽  
pp. 380 ◽  
Author(s):  
Tran Thi Thom ◽  
Nguyen Dinh Kien
Keyword(s):  
Free Vibration ◽  
Timoshenko Beam ◽  
Equations Of Motion ◽  
Beam Theory ◽  
Beam Element ◽  
Higher Order ◽  
Functionally Graded ◽  
The Finite Element Method ◽  
Power Law Distribution ◽  
Graded Material

Free vibration of two-directional functionally graded material (2-D FGM) beams is studied by the finite element method (FEM). The material properties are assumed to be graded in both the thickness and longitudinal directions by a power-law distribution. Equations of motion based on Timoshenko beam theory are derived from Hamilton's principle. A higher-order beam element using hierarchical functions to interpolate the displacements and rotation is formulated and employed in the analysis. In order to improve the efficiency of the element, the shear strain is constrained to constant. Validation of the derived element is confirmed by comparing the natural frequencies obtained in the present paper with the data available in the literature. Numerical investigations show that the proposed beam element is efficient, and it is capable to give accurate frequencies by a small number of elements. The effects of the material composition and aspect ratio on the vibration characteristics of the beams are examined in detail and highlighted.

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