On the Admissibility of Given Acceleration-Dependent Forces in Mechanics

2005 ◽  
Vol 74 (1) ◽  
pp. 107-110 ◽  
Author(s):  
Michael M. Zhechev
Keyword(s):  
Unique Solution ◽  
Superposition Principle ◽  
Equation Of Motion ◽  
Analytical Dynamics ◽  
Higher Derivatives

In his book “A Treatise on Analytical Dynamics,” Pars asserted that acceleration-dependent forces are inconsistent with one of the fundamental principles of mechanics, namely, with the superposition principle, thus spreading among mechanical scientists the idea that such forces are not admissible in mechanics. This article demonstrates that given forces that depend on acceleration or higher derivatives are admissible in mechanics and shows that this assertion in Pars’s book is fallacious and the only condition for the applicability of such forces is the equation of motion possessing a unique solution.

Hamel Paradox and Rosenberg Conjecture in Analytical Dynamics

2013 ◽  
Vol 80 (4) ◽  
Author(s):  
Y. H. Chen
Keyword(s):  
Kinetic Energy ◽  
Fundamental Equation ◽  
Equation Of Motion ◽  
Analytical Dynamics ◽  
Constrained Motion ◽  
Embedding Method ◽  
Constrained Equation ◽  
Unconstrained Motion

Hamel proposed a seemingly intuitive, simple, straightforward, but incorrect, method of formulating the constrained equation of motion. The method has to do with the direct embedding of the constraint into the kinetic energy of the unconstrained motion. His intention was to caution against its possible adoption. Rosenberg echoed Hamel's warning and followed up to explore more insight of this method. He proposed a conjecture that the Hamel's embedding method would work if the constraint was holonomic. It would not work if the constraint was nonholonomic. We investigate the Hamel paradox and Rosenberg conjecture via the use of the Fundamental Equation of Constrained Motion.

Viscoacoustic wave propagation simulation in the earth

Geophysics ◽  
1988 ◽  
Vol 53 (6) ◽  
pp. 769-777 ◽  
Author(s):  
Jose M. Carcione ◽  
Dan Kosloff ◽  
Ronnie Kosloff
Keyword(s):  
Wave Propagation ◽  
Seismic Waves ◽  
Superposition Principle ◽  
Time Integration ◽  
Equation Of Motion ◽  
Strain History ◽  
Porous Rocks ◽  
Arrival Times ◽  
Realistic Description ◽  
Geometrical Effects

Anelasticity of earth materials produces significant changes in the amplitude and phase spectra of seismic waves. The anelastic properties of real materials, particularly of porous rocks, are described using the theory of linear viscoelasticity based on Boltzmann’s superposition principle. Wave‐propagation simulation with this model requires implementing the convolutional relation in the equation of motion. The choice of a viscoacoustic constitutive relation based on a spectrum of relaxation mechanisms allows a realistic description of the anelastic effects, and the introduction of memory variables obviates storing the entire strain history required by the time convolution. A pseudospectral time‐integration technique is used to solve the equation of motion. Applications of viscoacoustic modeling suggest the need for considering the correct attenuation‐dispersion effects for various fundamental seismic problems in anelastic earth models. Comparison of acoustic and viscoacoustic synthetic seismograms shows differences in the amplitudes and arrival times of the wave fields which are enhanced for particular combinations of anelastic and geometrical effects.

scholarly journals Nonuniqueness of Attribution in Piecewise Potential Vorticity Inversion

2018 ◽  
Vol 75 (3) ◽  
pp. 875-883 ◽  
Author(s):  
Joseph Egger ◽  
Thomas Spengler
Keyword(s):  
Unique Solution ◽  
Vertical Velocity ◽  
Potential Vorticity ◽  
Superposition Principle ◽  
Balance Condition ◽  
Potential Vorticity Inversion ◽  
Inversion Procedure ◽  
Flow Domain ◽  
Ageostrophic Circulation ◽  
The Impact

Abstract Piecewise potential vorticity inversion (PPVI) seeks to determine the impact of observed potential vorticity (PV) anomalies on the surrounding flow. This widely used technique is based on dividing a flow domain D into subdomains D1 and D2 = D − D1. The influence of PV in D1 on the flow in D2 is assessed by removing all PV anomalies in D2 and then inverting the modified PV in D. The resulting flow with streamfunction ψ1 is attributed to the PV anomalies in D1. The relation of PV in D1 to ψ1 in D2 is not unique, because there are many PV distributions in D1 that induce the same ψ1. There is, however, a unique solution if the ageostrophic circulation is included in the inversion procedure. The superposition principle requires that the sum of inverted flows with PV = 0 in D2 and the complementary ones with PV = 0 in D1 equal the inverted flow for the complete observed PV in D. It is demonstrated, using two isolated PV balls as a paradigmatic example, that the superposition principle is violated if the ageostrophic circulation is included in PPVI, because the ageostrophic circulation cannot be associated with only one of the anomalies. Inversions of Ertel’s PV are carried out using Charney’s balance condition. PPVI is not unique in that case, because many different PV fields can be specified in D1, which all lead to the same inverted flow in D2. The balance condition assumes vanishing vertical velocity w so that uniqueness cannot be established by including w in the inversion, as was possible in the quasigeostrophic case.

Stability Properties of a Viscoelastic Column Under a Periodic Force

1992 ◽  
Vol 59 (1) ◽  
pp. 16-19 ◽  
Author(s):  
G. Cederbaum ◽  
M. Mond
Keyword(s):  
Superposition Principle ◽  
Viscoelastic Behavior ◽  
Equation Of Motion ◽  
Stable System ◽  
Elastic Model ◽  
Periodic Force ◽  
Stability Properties ◽  
Viscoelastic Column ◽  
The Stability ◽  
Boltzmann Superposition Principle

The dynamic stability of a viscoelastic column subjected to a periodic longitudinal load is investigated. The viscoelastic behavior is given in terms of the Boltzmann superposition principle which yields an integro-differential equation of motion. The stability boundaries of this equation are determined analytically by using the multiplescales method. It is shown that due to the viscoelasticity the stability regions are expanded, relative to the elastic ones, and the time for which a stable system becomes unstable is given. In addition, the stability properties of the viscoelastic column are time dependent and an initially stable system can turn unstable after a finite time, unlike columns that are described by the elastic model.

Particle motion with air resistance

2012 ◽  
Vol 8 (1) ◽  
pp. 1-15
Author(s):  
Gy. Sitkei
Keyword(s):  
Basic Equation ◽  
Initial Conditions ◽  
Equation Of Motion ◽  
Terminal Velocity ◽  
Dimensionless Form ◽  
Wood Industry ◽  
Acceptable Error ◽  
General Validity ◽  
Practical Applications ◽  
Air Resistance

Motion of particles with air resistance (e.g. horizontal and inclined throwing) plays an important role in many technological processes in agriculture, wood industry and several other fields. Although, the basic equation of motion of this problem is well known, however, the solutions for practical applications are not sufficient. In this article working diagrams were developed for quick estimation of the throwing distance and the terminal velocity. Approximate solution procedures are presented in closed form with acceptable error. The working diagrams provide with arbitrary initial conditions in dimensionless form of general validity.

Lifetime Prediction of Tires with Regard to Oxidative Aging5

2008 ◽  
Vol 36 (1) ◽  
pp. 63-79 ◽  
Author(s):  
L. Nasdala ◽  
Y. Wei ◽  
H. Rothert ◽  
M. Kaliske
Keyword(s):  
Finite Element Analysis ◽  
Finite Element ◽  
Temperature Distribution ◽  
Superposition Principle ◽  
Accelerated Aging ◽  
Material Model ◽  
Aging Behavior ◽  
Element Analysis ◽  
Material Parameters ◽  
Reaction Processes

Abstract It is a challenging task in the design of automobile tires to predict lifetime and performance on the basis of numerical simulations. Several factors have to be taken into account to correctly estimate the aging behavior. This paper focuses on oxygen reaction processes which, apart from mechanical and thermal aspects, effect the tire durability. The material parameters needed to describe the temperature-dependent oxygen diffusion and reaction processes are derived by means of the time–temperature–superposition principle from modulus profiling tests. These experiments are designed to examine the diffusion-limited oxidation (DLO) effect which occurs when accelerated aging tests are performed. For the cord-reinforced rubber composites, homogenization techniques are adopted to obtain effective material parameters (diffusivities and reaction constants). The selection and arrangement of rubber components influence the temperature distribution and the oxygen penetration depth which impact tire durability. The goal of this paper is to establish a finite element analysis based criterion to predict lifetime with respect to oxidative aging. The finite element analysis is carried out in three stages. First the heat generation rate distribution is calculated using a viscoelastic material model. Then the temperature distribution can be determined. In the third step we evaluate the oxygen distribution or rather the oxygen consumption rate, which is a measure for the tire lifetime. Thus, the aging behavior of different kinds of tires can be compared. Numerical examples show how diffusivities, reaction coefficients, and temperature influence the durability of different tire parts. It is found that due to the DLO effect, some interior parts may age slower even if the temperature is increased.

The Coarse Scale Velocity

2017 ◽  
Author(s):  
Philip Isett
Keyword(s):  
Velocity Field ◽  
Building Blocks ◽  
Coarse Scale ◽  
Divergence Equation ◽  
Higher Derivatives ◽  
Order Of Magnitude ◽  
Derivatives Of

This chapter deals with the coarse scale velocity. It begins the proof of Lemma (10.1) by choosing a double mollification for the velocity field. Here ∈ᵥ is taken to be as large as possible so that higher derivatives of velement are less costly, and each vsubscript Element has frequency smaller than λ‎ so elementv⁻¹ must be smaller than λ‎ in order of magnitude. Each derivative of vsubscript Element up to order L costs a factor of Ξ‎. The chapter proceeds by describing the basic building blocks of the construction, the choice of elementv and the parametrix expansion for the divergence equation.

Finite difference code for velocity and surface traction of a Fluid between Two Eccentric Spheres

2015 ◽  
Vol 11 (1) ◽  
pp. 2960-2971
Author(s):  
M.Abdel Wahab
Keyword(s):  
Velocity Field ◽  
Angular Velocity ◽  
Finite Difference ◽  
Numerical Study ◽  
Outer Sphere ◽  
Equation Of Motion ◽  
Annular Region ◽  
Surface Traction ◽  
Angular Velocities ◽  
Axis Passing

The Numerical study of the flow of a fluid in the annular region between two eccentric sphere susing PHP Code isinvestigated. This flow is created by considering the inner sphere to rotate with angular velocity 1  and the outer sphererotate with angular velocity 2  about the axis passing through their centers, the z-axis, using the three dimensionalBispherical coordinates (, ,) .The velocity field of fluid is determined by solving equation of motion using PHP Codeat different cases of angular velocities of inner and outer sphere. Also Finite difference code is used to calculate surfacetractions at outer sphere.

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