Pendulum Motion Damped by Speed-Independent Friction

2021 ◽  
Vol 03 (03) ◽  
pp. 2150008
Author(s):  
Carl E. Mungan
Keyword(s):  
Numerical Solution ◽  
Turning Point ◽  
Analytic Solution ◽  
Angular Position ◽  
Rolling Friction ◽  
Pendulum Motion ◽  
Circular Trajectory ◽  
Air Drag ◽  
Small Block

A pendulum without a supporting string or rod is obtained if a small block or marble is released at the rim of a spherical bowl or cylindrical half-pipe. This setup also applies to the familiar loop-the-loop demonstration. However, the bob will then experience sliding or rolling friction, which is speed independent in contrast to the linear or quadratic air drag which is more commonly used to model damping of oscillators. An analytic solution can be found for the speed of the bob as a function of its angular position around the vertical circular trajectory. A numerical solution for the time that the object takes to move from one turning point to the next shows that it is smaller than it would be in the absence of friction.

On the convective diffusion under partial slip at wall

1989 ◽  
Vol 54 (4) ◽  
pp. 967-980 ◽  
Author(s):  
Ondřej Wein ◽  
Petr Kučera
Keyword(s):  
Mass Transfer ◽  
Numerical Solution ◽  
Analytic Solution ◽  
Convective Diffusion ◽  
Linear Velocity ◽  
Partial Slip ◽  
Accurate Calculation ◽  
Transfer Coefficients ◽  
Empirical Formulas ◽  
Mass Transfer Coefficients

Extended Leveque problem is studied for linear velocity profiles, vx(z) = u + qz. The existing analytic solution is reconsidered and shown to be inapplicable for the accurate calculation of mean mass-transfer coefficients. A numerical solution is reported and its accuracy is checked in detail. Simple but fairly accurate empirical formulas are suggested for the calculating of local and mean mass-transfer coefficients.

The numerical solution for problems of turning point without resonance

1985 ◽  
Vol 6 (5) ◽  
pp. 439-446 ◽  
Author(s):  
Lin Peng-cheng ◽  
Yan Peng-xiang
Keyword(s):  
Numerical Solution ◽  
Turning Point

Propagation of obliquely incident water waves over a trench

1983 ◽  
Vol 133 ◽  
pp. 47-63 ◽  
Author(s):  
James T. Kirby ◽  
Robert A. Dalrymple
Keyword(s):  
Numerical Solution ◽  
Water Waves ◽  
Analytic Solution ◽  
Small Scale ◽  
Obliquely Incident ◽  
Long Wave ◽  
Propagating Waves ◽  
Tank Experiment ◽  
Wave Region ◽  
Wave Limit

The diffraction of obliquely incident surface waves by an asymmetric trench is investigated using linearized potential theory. A numerical solution is constructed by matching particular solutions for each subregion of constant depth along vertical boundaries; the resulting matrix equation is solved numerically. Several cases where the trench-parallel wavenumber component in the incident-wave region exceeds the wavenumber for freely propagating waves in the trench are investigated and are found to result in large reductions in wave transmission; however, reflection is not total owing to the finiteness of the obstacle.Results for one case are compared with data obtained from a small-scale wave-tank experiment. An approximate solution based on plane-wave modes is derived and compared with the numerical solution and, in the long-wave limit, with a previous analytic solution.

Numerical solution of a turning point problem in elasticity

CALCOLO ◽  
1989 ◽  
Vol 26 (1) ◽  
pp. 93-102
Author(s):  
A. Schiaffino ◽  
V. Valente
Keyword(s):  
Numerical Solution ◽  
Turning Point ◽  
Point Problem

Near-circular satellite orbits in an oblate, diurnally varying atmosphere

1983 ◽  
Vol 389 (1796) ◽  
pp. 153-170 ◽  
Keyword(s):  
Gravitational Field ◽  
Differential Equations ◽  
Diurnal Variation ◽  
Numerical Solution ◽  
Initial Conditions ◽  
Satellite Orbits ◽  
Atmospheric Parameters ◽  
Orbital Elements ◽  
Air Drag ◽  
Circular Satellite

The influence of air drag and the geopotential on near-circular satellite orbits, eccentricity e < 0.01, is considered. A model of the atmosphere is adopted that allows for oblateness, and in which the density behaviour approximates to the observed diurnal variation. Differential equations governing the variation in e and the argument of perigee ω are derived by combining the effects of air drag with those of the Earth’s gravitational field. These are solved numerically using initial conditions obtained from a series of computed orbits of the satellite 1963-27 A. The behaviour of the orbital elements predicted by the numerical solution is compared with the observed elements to test the developed theory, and to obtain values of atmospheric parameters at heights near 400 km.

scholarly journals Numerical Solution for Complex Systems of Fractional Order

2012 ◽  
Vol 2012 ◽  
pp. 1-11 ◽  
Author(s):  
Rabha W. Ibrahim
Keyword(s):  
Complex Systems ◽  
Numerical Solution ◽  
Fractional Order ◽  
Analytic Solution ◽  
Approximate Solutions ◽  
Fractional Operators ◽  
Homotopy Perturbation

By using a complex transform, we impose a system of fractional order in the sense of Riemann-Liouville fractional operators. The analytic solution for this system is discussed. Here, we introduce a method of homotopy perturbation to obtain the approximate solutions. Moreover, applications are illustrated.

Finite Analytic Numerical Solution Axisymmetric Navier-Stokes and Energy Equations

1983 ◽  
Vol 105 (3) ◽  
pp. 639-645 ◽  
Author(s):  
Ching-Jen Chen ◽  
Young Hwan Yoon
Keyword(s):  
Heat Transfer ◽  
Numerical Solution ◽  
Numerical Method ◽  
Stream Function ◽  
Analytic Solution ◽  
Numerical Solutions ◽  
Stokes Equations ◽  
Navier Stokes ◽  
Inlet Temperature ◽  
Local Analytic Solution

Connective heat transfer for steady-state laminar flow in axisymmetric coordinates is considered. Numerical solutions for flow pattern and temperature distribution are obtained by the finite analytic numerical method applied to the Navier-Stokes equations expressed in terms of vorticity and stream function, and the energy equation. The finite analytic numerical method differs from other numerical methods in that it utilizes a local analytic solution in an element of the problem to construct the total numerical solution. Finite analytic solutions of vorticity, stream function, temperature, and heat transfer coefficients for flow with Reynolds numbers of 5, 100, 1000, and 2000, and Prandtl numbers of 0.1, 1.0, and 10.0 with uniform grid sizes, are reported for an axisymmetric pipe with a sudden expansion and contraction. The wall temperature is considered to be isothermal and differs from the inlet temperature. It is shown that the finite analytic is stable, converges rapidly, and simulates the convection of fluid flow accurately, since the local analytic solution is capable of simulating automatically the influence of skewed convection through the element boundary on the interior nodal values, thereby minimizing the false numerical diffusion.

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