Asymptotic Integration Methods Applied to Rotating Beams

1980 ◽  
Vol 47 (4) ◽  
pp. 884-890 ◽  
Author(s):  
C. R. Steele ◽  
K. E. Barry
Keyword(s):  
General Theory ◽  
Equations Of Motion ◽  
Vector Form ◽  
Rotating Beams ◽  
Integration Methods ◽  
Computational Costs ◽  
Tensile Forces ◽  
Vibrational Characteristics ◽  
Steady Rotation ◽  
Analytical Complexity

The in-plane vibrational characteristics of an off-axis clamped beam subjected to either compressive or tensile forces arising from steady rotation are studied. The differential equations of motion are cast into state vector form and solved using asymptotic matrix integration methods. The general theory of these methods is described in this paper and their application to the analysis of rotating beams is made. The advantages inherent in these methods with regard to accuracy, reduction of analytical complexity, and savings in computational costs are discussed.

The mechanics of general relativity

1959 ◽  
Vol 251 (1264) ◽  
pp. 55-65 ◽  
Keyword(s):  
General Relativity ◽  
General Theory ◽  
Equations Of Motion ◽  
Stress Singularity ◽  
Theory Of Relativity ◽  
General Theory Of Relativity

It is shown how to obtain, within the general theory of relativity, equations of motion for two oscillating masses at the ends of a spring of given law of force. The method of Einstein, Infeld & Hoffmann is used, and the force in the spring is represented by a stress singularity. The detailed calculations are taken to the Newtonian order.

scholarly journals Bending–torsional-coupled vibrations and buckling characteristics of single and double composite Timoshenko beams

2019 ◽  
Vol 11 (3) ◽  
pp. 168781401983445
Author(s):  
Ma’en S Sari ◽  
Wael G Al-Kouz ◽  
Rafat Al-Waked
Keyword(s):  
Slenderness Ratio ◽  
Equations Of Motion ◽  
Mode Shapes ◽  
Timoshenko Beams ◽  
Engineering Structures ◽  
Axial Forces ◽  
Double Beam ◽  
Tensile Forces ◽  
Torsional Coupling ◽  
The Stability

The stability and free vibration analyses of single and double composite Timoshenko beams have been investigated. The closed-section beams are subjected to constant axially compressive or tensile forces. The double beams are assumed to be connected by a layer of elastic translational and rotational springs. The coupled governing partial differential equations of motion are discretized, and the resulted eigenvalue problem is solved numerically by applying the Chebyshev spectral collocation method. The effects of the elastic layer parameters, the axial forces, the slenderness ratio, the bending–torsional coupling, and the boundary conditions on the critical buckling loads, mode shapes, and natural transverse frequencies have been studied. A parametric study was performed, and the obtained results revealed different features, which hopefully can be useful for single- and double-beam-like engineering structures.

Vibrational Characteristics of Ball Bearings

1977 ◽  
Vol 99 (2) ◽  
pp. 284-287 ◽  
Author(s):  
P. K. Gupta ◽  
L. W. Winn ◽  
D. F. Wilcock
Keyword(s):  
Initial Conditions ◽  
Equations Of Motion ◽  
Vibrational Analysis ◽  
Dominant Frequency ◽  
Hertzian Contact ◽  
Analytical Simulation ◽  
Contact Frequency ◽  
Ball Contact ◽  
Vibrational Characteristics ◽  
General Motion

The classical differential equations of motion of the ball mass center in an angular contact thrust loaded ball bearing are integrated with prescribed initial conditions in order to simulate the natural high frequency vibrational characteristics of the general motion. Two distinct frequencies are identified in the analytical simulation and their existence is also confirmed experimentally. One of the frequencies is found to be associated with the Hertzian contact spring at the ball race contact and it is therefore defined as the “elastic contact frequency”, Ωe. The other dominant frequency corresponding to oscillatory motion of the ball in the raceway groove appears to be kinematic in nature and it is, therefore, termed as the “bearing kinematic frequency”, Ωk. It is shown that for a given bearing Ωe and Ωk, vary as, respectively, 1/6 and 1/2 powers of the ball contact load and, therefore, for a given load these frequencies correspond to the natural frequencies of the bearing as applied in any vibrational analysis or simulation.

scholarly journals Charged anti-de Sitter BTZ black holes in Maxwell-f(T) gravity

2018 ◽  
Vol 33 (13) ◽  
pp. 1850076 ◽  
Author(s):  
G. G. L. Nashed ◽  
S. Capozziello
Keyword(s):  
Black Holes ◽  
Special Form ◽  
Energy Content ◽  
Equations Of Motion ◽  
Main Task ◽  
Cauchy Horizon ◽  
De Sitter ◽  
Vector Form ◽  
Charged Case ◽  
General Vector

Inspired by the Bañados, Teitelboim and Zanelli (BTZ) formalism, we discuss the Maxwell-[Formula: see text] gravity in [Formula: see text] dimensions. The main task is to derive exact solutions for a special form of [Formula: see text], with [Formula: see text] being the torsion scalar of Weitzenböck geometry. To this end, a triad field is applied to the equations of motion of charged [Formula: see text] and sets of circularly symmetric noncharged and charged solutions have been derived. We show that, in the charged case, the monopole-like and the [Formula: see text] terms are linked by a correlative constant despite the known results in teleparallel geometry and its extensions.[Formula: see text] Furthermore, it is possible to show that the event horizon is not identical with the Cauchy horizon due to such a constant. The singularities and the horizons of these black holes are examined: they are new and have no analogue in the literature due to the fact that their curvature singularities are soft. We calculate the energy content of these solutions by using the general vector form of the energy–momentum within the framework of [Formula: see text] gravity. Finally, some thermodynamical quantities, like entropy and Hawking temperature, are derived.

An Emerging O(N) Model and Algorithm for Virtual Prototyping of Dynamics of Molecular Conformation

2008 ◽  
Author(s):  
Andrew Ries ◽  
Shanzhong Shawn Duan
Keyword(s):  
Degrees Of Freedom ◽  
Molecular Conformation ◽  
Molecular System ◽  
Equations Of Motion ◽  
Integration Step ◽  
Computational Costs ◽  
System A ◽  
Hard Constraints ◽  
Solution Of Equations ◽  
Computationally Intensive

Molecular dynamics is effective for nano-scale phenomenon analysis. There are two major computational steps associated with computer simulation of dynamics of molecular conformation and they are the calculation of the interatomic forces and the formation and solution of the equations of motion. Currently, these two computational steps are treated separately, but in this paper an O(N) (order N) procedure is presented for an integration between these computational steps. For computational costs associated with calculating the interatomic forces, an internal coordinate method (ICM) approach is used for determining potentials due to both the bonding and non-bonding interactions. Thus, the potential gradients can be expressed as a combination of the potential in absolute and relative coordinates. For computational costs associated with the formation and solution of the equations of motion for the system, a constraint method that is used in computational multibody dynamics is utilized. This frees some degrees of freedom so that Kane’s method can be applied for the recursive formation and solution of equations of motion for the atomistic molecular system. Because the inclusion of lightly excited high frequency degrees of freedom, such as inter-atomic oscillations and rotation about double bonds would force the use of very small integration step sizes, holonomic constraints are introduced to freeze these “uninteresting” degrees of freedom. By introducing these hard constraints the time scale can be appropriately sized for to provide a less computationally intensive dynamic simulation of molecular conformation. The algorithm developed improves computational speed significantly when compared with any traditional O(N3) procedure.

On sound generated aerodynamically I. General theory

1952 ◽  
Vol 211 (1107) ◽  
pp. 564-587 ◽  
Keyword(s):  
Fluid Flow ◽  
Mach Number ◽  
General Theory ◽  
Stress Tensor ◽  
High Power ◽  
Velocity Vector ◽  
Unit Volume ◽  
Equations Of Motion ◽  
Sound Field ◽  
Velocity Of Sound

A theory is initiated, based on the equations of motion of a gas, for the purpose of estimating the sound radiated from a fluid flow, with rigid boundaries, which as a result of instability contains regular fluctuations or turbulence. The sound field is that which would be produced by a static distribution of acoustic quadrupoles whose instantaneous strength per unit volume is ρv i v j + p ij - a 2 0 ρ δ ij , where ρ is the density, v i the velocity vector, p ij the compressive stress tensor, and a 0 the velocity of sound outside the flow. This quadrupole strength density may be approximated in many cases as ρ 0 v i v j . The radiation field is deduced by means of retarded potential solutions. In it, the intensity depends crucially on the frequency as well as on the strength of the quadrupoles, and as a result increases in proportion to a high power, near the eighth, of a typical velocity U in the flow. Physically, the mechanism of conversion of energy from kinetic to acoustic is based on fluctuations in the flow of momentum across fixed surfaces, and it is explained in § 2 how this accounts both for the relative inefficiency of the process and for the increase of efficiency with U . It is shown in § 7 how the efficiency is also increased, particularly for the sound emitted forwards, in the case of fluctuations convected at a not negligible Mach number.

Effects of Warping and Pretwist on Torsional Vibration of Rotating Beams

1984 ◽  
Vol 51 (4) ◽  
pp. 913-920 ◽  
Author(s):  
K. R. V. Kaza ◽  
R. E. Kielb
Keyword(s):  
Aspect Ratio ◽  
Torsional Vibration ◽  
Equations Of Motion ◽  
Turbine Blades ◽  
Mode Shapes ◽  
Compressor Blades ◽  
Rotating Beams ◽  
Special Cases ◽  
Torsional Frequency ◽  
Compressor And Turbine Blades

The effect of pretwist and warping on the torsional vibration of short-aspect-ratio rotating beams is examined for application to the modeling of turbofan, turboprop, and compressor blades. The equations of motion and the associated boundary conditions by using both Wagner’s hypothesis and Washizu’s theory are derived and a few minor limitations of the Wagner’s hypothesis, as applied to thick blades, are pointed out and discussed. The equations for several special cases are solved in a closed form. Results are presented indicating the effect of warping, pretwist, and rotation on torsional vibration of beams as aspect ratio is varied. The results show that the structural warping and pretwist terms have a significant effect on torsional frequency and mode shapes of short-aspect-ratio blades whereas the inertial warping terms have negligible effect. Since the torsional frequencies and mode shapes are very important in aeroelastic analyses by using modal methods, the structural warping terms should be included in modeling turbofan, turboprop, compressor, and turbine blades.

scholarly journals Semi-Implicit and Semi-Explicit Adams-Bashforth-Moulton Methods

Mathematics ◽  
2020 ◽  
Vol 8 (5) ◽  
pp. 780
Author(s):  
Aleksandra Tutueva ◽  
Timur Karimov ◽  
Denis Butusov
Keyword(s):  
Simulation Software ◽  
Van Der Pol Oscillator ◽  
Great Success ◽  
General Description ◽  
Implicit Integration ◽  
Variable Stepsize ◽  
Integration Methods ◽  
Computational Costs ◽  
Accuracy Order ◽  
Predictor Corrector

Multistep integration methods are widespread in the simulation of high-dimensional dynamical systems due to their low computational costs. However, the stability of these methods decreases with the increase of the accuracy order, so there is a known room for improvement. One of the possible ways to increase stability is implicit integration, but it consequently leads to sufficient growth in computational costs. Recently, the development of semi-implicit techniques achieved great success in the construction of highly efficient single-step ordinary differential equations (ODE) solvers. Thus, the development of multistep semi-implicit integration methods is of interest. In this paper, we propose the simple solution to increase the numerical efficiency of Adams-Bashforth-Moulton predictor-corrector methods using semi-implicit integration. We present a general description of the proposed methods and explicitly show the superiority of ODE solvers based on semi-implicit predictor-corrector methods over their explicit and implicit counterparts. To validate this, performance plots are given for simulation of the van der Pol oscillator and the Rossler chaotic system with fixed and variable stepsize. The obtained results can be applied in the development of advanced simulation software.

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