The Bifilar Pendulum: Numerical Solution to the Exact Equation of Motion

1985 ◽  
Vol 107 (2) ◽  
pp. 175-179 ◽  
Author(s):  
B. E. Karlin ◽  
C. J. Maday
Keyword(s):  
Numerical Solutions ◽  
Angular Displacement ◽  
Nonlinear Differential Equations ◽  
Equations Of Motion ◽  
Equation Of Motion ◽  
Radius Of Gyration ◽  
Exact Equation ◽  
Filament Length ◽  
Elementary Functions ◽  
Highly Nonlinear

The bifilar pendulum is often used for indirect measurements of mass moments of inertia of bodies that possess complex geometries. The exact equation of motion of the bifilar pendulum is highly nonlinear, and has not been solved in terms of elementary functions. Extensive use has been made, however, of the linearized approximation to the exact equation, and it has been assumed that the simple harmonic oscillator adequately describes the motion of the bifilar pendulum. It is shown here that such is generally not the case. Numerical solutions to the exact nonlinear differential equations of motion are obtained for a range of values of initial angular displacement, filament length, and radius of gyration. The filament length and the radius of gyration are normalized with respect to the half-spacing between the filaments. It is shown that the approximate solution gives good results only for small ranges of the system parameters.

Dynamic Characteristics of a Hydrostatic Gas Bearing Driven by Oscillating Exhaust Pressure

1984 ◽  
Vol 106 (4) ◽  
pp. 477-483 ◽  
Author(s):  
C. B. Watkins ◽  
H. D. Branch ◽  
I. E. Eronini
Keyword(s):  
Reynolds Equation ◽  
Numerical Solutions ◽  
Equations Of Motion ◽  
Equation Of Motion ◽  
Source Model ◽  
Regular Perturbation ◽  
Response Characteristics ◽  
Finite Difference Approximations ◽  
Bode Plots ◽  
Gas Bearing

Vibration of a statically loaded, inherently compensated hydrostatic journal bearing due to oscillating exhaust pressure is investigated. Both angular and radial vibration modes are analyzed. The time-dependent Reynolds equation governing the pressure distribution between the oscillating journal and sleeve is solved together with the journal equation of motion to obtain the response characteristics of the bearing. The Reynolds equation and the equation of motion are simplified by applying regular perturbation theory for small displacements. The numerical solutions of the perturbation equations are obtained by discretizing the pressure field using finite-difference approximations with a discrete, nonuniform line-source model which excludes effects due to feeding hole volume. An iterative scheme is used to simultaneously satisfy the equations of motion for the journal. The results presented include Bode plots of bearing-oscillation gain and phase for a particular bearing configuration for various combinations of parameters over a range of frequencies, including the resonant frequency.

Application of GDQ method in nonlinear analysis of a flexible manipulator undergoing large deformation

2013 ◽  
Vol 227 (12) ◽  
pp. 2671-2685 ◽  
Author(s):  
Mohammad R Fazel ◽  
Majid M Moghaddam ◽  
Javad Poshtan
Keyword(s):  
Differential Equations ◽  
Nonlinear Differential Equations ◽  
Equations Of Motion ◽  
Nonlinear Partial Differential Equation ◽  
Flexible Manipulator ◽  
Runge Kutta Method ◽  
Highly Nonlinear ◽  
Frechet Derivatives ◽  
Fréchet Derivatives ◽  
Generalized Differential

Analysis of a flexible manipulator as an initial value problem, due to its large deformations, involves nonlinear ordinary differential equations of motion. In the present work, these equations are solved through the general Frechet derivatives and the generalized differential quadrature (GDQ) method directly. The results so obtained are compared with those of the fourth-order Runge–Kutta method. It is seen that both the results match each other well. Further considering the same manipulator as a boundary value problem, its governing equation is a highly nonlinear partial differential equation. Again applying the general Frechet derivatives and the GDQ method, it is seen that the results are in good match with the linear theory. In both cases, the general Frechet derivatives are introduced and successfully used for linearization. The results of the present study indicate that the GDQ method combined with the general Frechet derivatives can be successfully used for the solution of nonlinear differential equations.

Double Dispersion Effects on MHD Squeezing Flow of UCM Fluid through a Porous Medium

2019 ◽  
Vol 392 ◽  
pp. 10-28
Author(s):  
N. Naresh Kumar ◽  
Pravin Kashyap Kambhatla ◽  
Odelu Ojjela
Keyword(s):  
Skin Friction ◽  
Wall Motion ◽  
Numerical Solutions ◽  
Nonlinear Differential Equations ◽  
Squeezing Flow ◽  
Ucm Fluid ◽  
Highly Nonlinear ◽  
Dimensional Parameters ◽  
Double Dispersion ◽  
The Impact

The objective of the current problem is to explore the impact of wall motion on flow, heat and species concentration of a UCM fluid in a magnetohydrodynamic Darcian channel. The flow is confined between two moving walls. The effects of the wall motion on the physical quantities for expanding and contracting cases are studied through non-dimensional numbers and variables. Numerical solutions for the highly nonlinear differential equations are obtained by reducing the governing PDE to ODE using well-established similarity variables. The variation of skin friction, Nusselt and Sherwood numbers has been investigated with the help of surface plots so that the influence of the squeezing number on the other non-dimensional parameters can be deeply understood. The results suggest that the squeezing channel intensifies the mass transfer and skin friction at the walls and it also increases the velocity, temperature and concentration of the fluid across the channel.

scholarly journals Analysis of the Oscillating Motion of a Solid Body on Vibrating Bearers

Machines ◽  
2019 ◽  
Vol 7 (3) ◽  
pp. 58 ◽  
Author(s):  
Bissembayev ◽  
Jomartov ◽  
Tuleshov ◽  
Dikambay
Keyword(s):  
Solid Body ◽  
Quantitative Methods ◽  
Nonlinear Differential Equations ◽  
Equations Of Motion ◽  
Horizontal Displacement ◽  
Oscillatory Process ◽  
Highly Nonlinear ◽  
Cumulative Curve ◽  
The Stability ◽  
Oscillating Motion

This article considers the oscillation of a solid body on kinematic foundations, the main elements of which are rolling bearers bounded by high-order surfaces of rotation at horizontal displacement of the foundation. Equations of motion of the vibro-protected body have been obtained. It is ascertained that the obtained equations of motion are highly nonlinear differential equations. Stationary and transitional modes of the oscillatory process of the system have been investigated. It is determined that several stationary regimes of the oscillatory process exist. Equations of motion have been investigated also by quantitative methods. In this paper the cumulative curves in the phase plane are plotted, a qualitative analysis for singular points and a study of them for stability are performed. In the Hayashi plane a cumulative curve of a body protected against vibration forms a closed path which does not tend to the stability of a singular point. This means that the vibration amplitude of a body protected against vibration does not remain constant in a steady state, but changes periodically.

On the Harmonic Oscillations for the Motion of a Dynamical System

2017 ◽  
Vol 13 (2) ◽  
pp. 4657-4670
Author(s):  
W. S. Amer
Keyword(s):  
Dynamical System ◽  
Numerical Solutions ◽  
Equation Of Motion ◽  
Parameter Method ◽  
Kutta Method ◽  
Small Parameter Method ◽  
Harmonic Oscillations ◽  
Pivot Point ◽  
Runge Kutta Method ◽  
High Consistency

This work touches two important cases for the motion of a pendulum called Sub and Ultra-harmonic cases. The small parameter method is used to obtain the approximate analytic periodic solutions of the equation of motion when the pivot point of the pendulum moves in an elliptic path. Moreover, the fourth order Runge-Kutta method is used to investigate the numerical solutions of the considered model. The comparison between both the analytical solution and the numerical ones shows high consistency between them.

scholarly journals Pole-skipping and hydrodynamic analysis in Lifshitz, AdS2 and Rindler geometries

2021 ◽  
Vol 2021 (6) ◽  
Author(s):  
Haiming Yuan ◽  
Xian-Hui Ge
Keyword(s):  
Boundary Condition ◽  
Green’S Function ◽  
Green's Function ◽  
Equations Of Motion ◽  
Equation Of Motion ◽  
Boundary Theory ◽  
Hydrodynamic Analysis ◽  
Dimensionless Variables ◽  
Pass Through ◽  
Special Points

Abstract The “pole-skipping” phenomenon reflects that the retarded Green’s function is not unique at a pole-skipping point in momentum space (ω, k). We explore the universality of pole-skipping in different geometries. In holography, near horizon analysis of the bulk equation of motion is a more straightforward way to derive a pole-skipping point. We use this method in Lifshitz, AdS2 and Rindler geometries. We also study the complex hydrodynamic analyses and find that the dispersion relations in terms of dimensionless variables $$ \frac{\omega }{2\pi T} $$ ω 2 πT and $$ \frac{\left|k\right|}{2\pi T} $$ k 2 πT pass through pole-skipping points $$ \left(\frac{\omega_n}{2\pi T},\frac{\left|{k}_n\right|}{2\pi T}\right) $$ ω n 2 πT k n 2 πT at small ω and k in the Lifshitz background. We verify that the position of the pole-skipping points does not depend on the standard quantization or alternative quantization of the boundary theory in AdS2× ℝd−1 geometry. In the Rindler geometry, we cannot find the corresponding Green’s function to calculate pole-skipping points because it is difficult to impose the boundary condition. However, we can still obtain “special points” near the horizon where bulk equations of motion have two incoming solutions. These “special points” correspond to the nonuniqueness of the Green’s function in physical meaning from the perspective of holography.

scholarly journals Single-particle motion under the influence of the perpendicular ponderomotive force

1987 ◽  
Vol 37 (3) ◽  
pp. 497-506
Author(s):  
M. C. Festeau-Barrioz ◽  
M. L. Sawley ◽  
J. Václavík
Keyword(s):  
Electromagnetic Field ◽  
Single Particle ◽  
Particle Motion ◽  
Ponderomotive Force ◽  
Equation Of Motion ◽  
Exact Equation ◽  
Particle Theory ◽  
Magnetostatic Field ◽  
Single Particle Motion

The motion of a single particle under the influence of the ponderomotive force directed perpendicular to the external magnetostatic field is analysed. By solving the exact equation of motion for a specific applied electromagnetic field, the resultant ponderomotive drift is compared with the prediction of a single-particle theory using the oscillation-centre approximation. The regime of validity of this theory is discussed. It is shown that, for certain values of the amplitude and frequency of the electromagnetic field, the particle motion is unstable and therefore the concept of a single-particle ponderomotive force is meaningless.

Finite Deflections of a Nonlinearly Elastic Bar

1970 ◽  
Vol 37 (1) ◽  
pp. 48-52 ◽  
Author(s):  
J. T. Oden ◽  
S. B. Childs
Keyword(s):  
Differential Equations ◽  
Axial Force ◽  
Classical Theory ◽  
Numerical Solutions ◽  
Nonlinear Differential Equations ◽  
Finite Deflections ◽  
Hyperbolic Tangent ◽  
Elastic Bar ◽  
Elastoplastic Materials ◽  
Nonlinearly Elastic

The problem of finite deflections of a nonlinearly elastic bar is investigated as an extension of the classical theory of the elastica to include material nonlinearities. A moment-curvature relation in the form of a hyperbolic tangent law is introduced to simulate that of a class of elastoplastic materials. The problem of finite deflections of a clamped-end bar subjected to an axial force is given special attention, and numerical solutions to the resulting system of nonlinear differential equations are obtained. Tables of results for various values of the parameters defining the material are provided and solutions are compared with those of the classical theory of the elastica.

scholarly journals Assessment of Linearization Approaches for Multibody Dynamics Formulations

2017 ◽  
Vol 12 (4) ◽  
Author(s):  
Francisco González ◽  
Pierangelo Masarati ◽  
Javier Cuadrado ◽  
Miguel A. Naya
Keyword(s):  
Mechanical System ◽  
Multibody Dynamics ◽  
Equations Of Motion ◽  
Differential Algebraic Equations ◽  
Algebraic Equations ◽  
Practical Applications ◽  
Linearization Methods ◽  
Highly Nonlinear ◽  
Wide Range ◽  
The Way

Formulating the dynamics equations of a mechanical system following a multibody dynamics approach often leads to a set of highly nonlinear differential-algebraic equations (DAEs). While this form of the equations of motion is suitable for a wide range of practical applications, in some cases it is necessary to have access to the linearized system dynamics. This is the case when stability and modal analyses are to be carried out; the definition of plant and system models for certain control algorithms and state estimators also requires a linear expression of the dynamics. A number of methods for the linearization of multibody dynamics can be found in the literature. They differ in both the approach that they follow to handle the equations of motion and the way in which they deliver their results, which in turn are determined by the selection of the generalized coordinates used to describe the mechanical system. This selection is closely related to the way in which the kinematic constraints of the system are treated. Three major approaches can be distinguished and used to categorize most of the linearization methods published so far. In this work, we demonstrate the properties of each approach in the linearization of systems in static equilibrium, illustrating them with the study of two representative examples.

NUMERICAL SOLUTIONS OF 2D AND 3D SLAMMING PROBLEMS

2021 ◽  
Vol 153 (A2) ◽  
Author(s):  
Q Yang ◽  
W Qiu
Keyword(s):  
Free Surface ◽  
Numerical Solutions ◽  
Stokes Equations ◽  
Type Equation ◽  
The Body ◽  
Navier Stokes ◽  
Time Step ◽  
Navier Stokes Equations ◽  
Highly Nonlinear ◽  
2D And 3D

Slamming forces on 2D and 3D bodies have been computed based on a CIP method. The highly nonlinear water entry problem governed by the Navier-Stokes equations was solved by a CIP based finite difference method on a fixed Cartesian grid. In the computation, a compact upwind scheme was employed for the advection calculations and a pressure-based algorithm was applied to treat the multiple phases. The free surface and the body boundaries were captured using density functions. For the pressure calculation, a Poisson-type equation was solved at each time step by the conjugate gradient iterative method. Validation studies were carried out for 2D wedges with various deadrise angles ranging from 0 to 60 degrees at constant vertical velocity. In the cases of wedges with small deadrise angles, the compressibility of air between the bottom of the wedge and the free surface was modelled. Studies were also extended to 3D bodies, such as a sphere, a cylinder and a catamaran, entering calm water. Computed pressures, free surface elevations and hydrodynamic forces were compared with experimental data and the numerical solutions by other methods.

Sign in / Sign up

Export Citation Format

Share Document