Added Fluid Mass and the Equations of Motion of a Parachute

1983 ◽  
Vol 34 (3) ◽  
pp. 226-242 ◽  
Author(s):  
John A. Eaton
Keyword(s):  
Equations Of Motion ◽  
Added Mass ◽  
Unsteady Forces ◽  
Fluid Mass ◽  
Mass Tensor ◽  
Body Shapes ◽  
The Stability ◽  
External Forces ◽  
Six Degree Of Freedom ◽  
Nonlinear Solutions

SummaryWhile it has long been known that added fluid mass may be important in the dynamics of parachutes, due to inadequate or incorrect derivation and/or implementation of the added mass tensor its full significance in the stability of parachutes has yet to be appreciated. The concept of added mass is outlined and some general conditions for its significance are presented. Its implementation in the parachute equations of motion is reviewed, and the equations used in previous treatments are shown to be erroneous. A general method for finding the equivalent external forces and moments due to added mass is given, and the correct, anisotropic forms of the added mass tensor are derived for the six degree-of-freedom motion in an ideal fluid of rigid body shapes with planar-, twofold- and axisymmetry, These derivations may also be useful in dynamic stability studies of other low relative density bodies such as airships, balloons, submarines and torpedoes. Full nonlinear solutions of the equations of motion for the axisymmetric parachute have been obtained, and results indicate that added mass effects are more significant than previously predicted. In particular, the component of added mass along the axis of symmetry has a strong influence on stability. Better data on unsteady forces and moments on parachutes are needed.

Calculation of Dynamic Tire Forces from Aircraft Instrumentation Data

2010 ◽  
Vol 38 (3) ◽  
pp. 182-193 ◽  
Author(s):  
Gary E. McKay
Keyword(s):  
System Performance ◽  
Equations Of Motion ◽  
Test Laboratory ◽  
Excellent Correlation ◽  
Friction Behavior ◽  
Aircraft Landing ◽  
Data Scatter ◽  
Tire Forces ◽  
Six Degree Of Freedom ◽  
Tire Friction

Abstract When evaluating aircraft brake control system performance, it is difficult to overstate the importance of understanding dynamic tire forces—especially those related to tire friction behavior. As important as they are, however, these dynamic tire forces cannot be easily or reliably measured. To fill this need, an analytical approach has been developed to determine instantaneous tire forces during aircraft landing, braking and taxi operations. The approach involves using aircraft instrumentation data to determine forces (other than tire forces), moments, and accelerations acting on the aircraft. Inserting these values into the aircraft’s six degree-of-freedom equations-of-motion allows solution for the tire forces. While there are significant challenges associated with this approach, results to date have exceeded expectations in terms of fidelity, consistency, and data scatter. The results show excellent correlation to tests conducted in a tire test laboratory. And, while the results generally follow accepted tire friction theories, there are noteworthy differences.

scholarly journals Numerical methods for General Relativistic particles

2018 ◽  
Vol 14 (S342) ◽  
pp. 19-23
Author(s):  
Fabio Bacchini ◽  
Bart Ripperda ◽  
Alexander Y. Chen ◽  
Lorenzo Sironi
Keyword(s):  
Numerical Methods ◽  
Gravitational Lensing ◽  
Complex Dynamics ◽  
Equations Of Motion ◽  
Numerical Algorithms ◽  
Relativistic Effects ◽  
Compact Objects ◽  
Light Rays ◽  
Massive Particles ◽  
External Forces

AbstractWe present recent developments on numerical algorithms for computing photon and particle trajectories in the surrounding of compact objects. Strong gravity around neutron stars or black holes causes relativistic effects on the motion of massive particles and distorts light rays due to gravitational lensing. Efficient numerical methods are required for solving the equations of motion and compute i) the black hole shadow obtained by tracing light rays from the object to a distant observer, and ii) obtain information on the dynamics of the plasma at the microscopic scale. Here, we present generalized algorithms capable of simulating ensembles of photons or massive particles in any spacetime, with the option of including external forces. The coupling of these tools with GRMHD simulations is the key point for obtaining insight on the complex dynamics of accretion disks and jets and for comparing simulations with upcoming observational results from the Event Horizon Telescope.

Interactions between steady and oscillatory convection in mushy layers

2010 ◽  
Vol 645 ◽  
pp. 411-434 ◽  
Author(s):  
PETER GUBA ◽  
M. GRAE WORSTER
Keyword(s):  
Standing Waves ◽  
Mushy Layer ◽  
Two Dimensional ◽  
Oscillatory Convection ◽  
Mushy Layers ◽  
Theoretical Predictions ◽  
Convection Patterns ◽  
The Stability ◽  
Nonlinear Solutions ◽  
Steady Convection

We study nonlinear, two-dimensional convection in a mushy layer during solidification of a binary mixture. We consider a particular limit in which the onset of oscillatory convection just precedes the onset of steady overturning convection, at a prescribed aspect ratio of convection patterns. This asymptotic limit allows us to determine nonlinear solutions analytically. The results provide a complete description of the stability of and transitions between steady and oscillatory convection as functions of the Rayleigh number and the compositional ratio. Of particular focus are the effects of the basic-state asymmetries and non-uniformity in the permeability of the mushy layer, which give rise to abrupt (hysteretic) transitions in the system. We find that the transition between travelling and standing waves, as well as that between standing waves and steady convection, can be hysteretic. The relevance of our theoretical predictions to recent experiments on directionally solidifying mushy layers is also discussed.

Efficient Dynamic Computer Simulation of Simple-Closed Spatial Linkages

1990 ◽  
Author(s):  
L. T. Wang
Keyword(s):  
Computer Simulation ◽  
Computational Algorithm ◽  
Equations Of Motion ◽  
Joint Space ◽  
Kinematic Chains ◽  
Spatial Linkages ◽  
Dynamic Computer Simulation ◽  
The Stability ◽  
Coordinate Partitioning ◽  
Generalized Coordinate

Abstract A new method of formulating the generalized equations of motion for simple-closed (single loop) spatial linkages is presented in this paper. This method is based on the generalized principle of D’Alembert and the use of the transformation Jacobian matrices. The number of the differential equations of motion is minimized by performing the method of generalized coordinate partitioning in the joint space. Based on this formulation, a computational algorithm for computer simulation the dynamic motions of the linkage is developed, this algorithm is not only numerically stable but also fully exploits the efficient recursive computational schemes developed earlier for open kinematic chains. Two numerical examples are presented to demonstrate the stability and efficiency of the algorithm.

Stability of a String in Axial Flow

1985 ◽  
Vol 107 (4) ◽  
pp. 421-425 ◽  
Author(s):  
G. S. Triantafyllou ◽  
C. Chryssostomidis
Keyword(s):  
Stability Analysis ◽  
Frictional Coefficient ◽  
Axial Flow ◽  
Added Mass ◽  
Equation Of Motion ◽  
Bending Stiffness ◽  
Diameter Ratio ◽  
Constant Speed ◽  
Hamilton’S Principle ◽  
The Stability

The equation of motion of a long slender beam submerged in an infinite fluid moving with constant speed is derived using Hamilton’s principle. The upstream end of the beam is pinned and the downstream end is free to move. The resulting equation of motion is then used to perform the stability analysis of a string, i.e., a beam with negligible bending stiffness. It is found that the string is stable if (a) the external tension at the free end exceeds the value of a U2, where a is the “added mass” of the string and U the fluid speed; or (b) the length-over-diameter ratio exceeds the value 2Cf/π, where Cf is the frictional coefficient of the string.

scholarly journals A Study of the Stability of a Plate-Like Load Towed beneath a Helicopter

1971 ◽  
Vol 13 (5) ◽  
pp. 330-343 ◽  
Author(s):  
D. F. Sheldon
Keyword(s):  
Equations Of Motion ◽  
Physical Parameters ◽  
Recent Experience ◽  
Wind Tunnel Testing ◽  
Stability Derivatives ◽  
Direct Indication ◽  
Metric Dimensions ◽  
Set Up ◽  
The Stability ◽  
Derivative Information

Recent experience has shown that a plate-like load suspended beneath a helicopter moving in horizontal forward flight has unstable characteristics at both low and high forward speeds. These findings have prompted a theoretical analysis to determine the longitudinal and lateral dynamic stability of a suspended pallet. Only the longitudinal stability is considered here. Although it is strictly a non-linear problem, the usual assumptions have been made to obtain linearized equations of motion. The aerodynamic derivative data required for these equations have been obtained, where possible, for the appropriate ranges of Reynolds and Strouhal number by means of static and dynamic wind tunnel testing. The resulting stability equations (with full aerodynamic derivative information) have been set up and solved, on a digital computer, to give direct indication of a stable or unstable system for a combination of physical parameters. These results have indicated a longitudinal unstable mode for all practical forward speeds. Simultaneously the important stability derivatives were found for this instability and modifications were made subsequently in the suspension system to eliminate the instabilities in the longitudinal sense. Throughout this paper, all metric dimensions are given approximately.

Stability Analysis of a Hydrodynamic Journal Bearing With Rotating Herringbone Grooves

2003 ◽  
Vol 125 (2) ◽  
pp. 291-300 ◽  
Author(s):  
G. H. Jang ◽  
J. W. Yoon
Keyword(s):  
Journal Bearing ◽  
Equations Of Motion ◽  
Fourier Series Expansion ◽  
Algebraic Equations ◽  
High Rotational Speed ◽  
Dynamic Coefficients ◽  
Stiffness Coefficients ◽  
Hydrodynamic Journal Bearing ◽  
The Stability ◽  
Herringbone Grooves

This paper presents an analytical method to investigate the stability of a hydrodynamic journal bearing with rotating herringbone grooves. The dynamic coefficients of the hydrodynamic journal bearing are calculated using the FEM and the perturbation method. The linear equations of motion can be represented as a parametrically excited system because the dynamic coefficients have time-varying components due to the rotating grooves, even in the steady state. Their solution can be assumed as a Fourier series expansion so that the equations of motion can be rewritten as simultaneous algebraic equations with respect to the Fourier coefficients. Then, stability can be determined by solving Hill’s infinite determinant of these algebraic equations. The validity of this research is proved by the comparison of the stability chart with the time response of the whirl radius obtained from the equations of motion. This research shows that the instability of the hydrodynamic journal bearing with rotating herringbone grooves increases with increasing eccentricity and with decreasing groove number, which play the major roles in increasing the average and variation of stiffness coefficients, respectively. It also shows that a high rotational speed is another source of instability by increasing the stiffness coefficients without changing the damping coefficients.

scholarly journals Fully localized post-buckling states of cylindrical shells under axial compression

2017 ◽  
Vol 473 (2205) ◽  
pp. 20170177 ◽  
Author(s):  
Tobias Kreilos ◽  
Tobias M. Schneider
Keyword(s):  
Axial Compression ◽  
Stability Boundary ◽  
Axial Direction ◽  
Edge State ◽  
Equilibrium Solutions ◽  
Thin Cylindrical Shell ◽  
Nonlinear Force ◽  
Post Buckling ◽  
The Stability ◽  
Nonlinear Solutions

We compute nonlinear force equilibrium solutions for a clamped thin cylindrical shell under axial compression. The equilibrium solutions are dynamically unstable and located on the stability boundary of the unbuckled state. A fully localized single dimple deformation is identified as the edge state —the attractor for the dynamics restricted to the stability boundary. Under variation of the axial load, the single dimple undergoes homoclinic snaking in the azimuthal direction, creating states with multiple dimples arranged around the central circumference. Once the circumference is completely filled with a ring of dimples, snaking in the axial direction leads to further growth of the dimple pattern. These fully nonlinear solutions embedded in the stability boundary of the unbuckled state constitute critical shape deformations. The solutions may thus be a step towards explaining when the buckling and subsequent collapse of an axially loaded cylinder shell is triggered.

A Sequential Integration Method

1988 ◽  
Vol 110 (4) ◽  
pp. 382-388
Author(s):  
Liang-Wey Chang ◽  
James F. Hamilton
Keyword(s):  
Integration Method ◽  
Equations Of Motion ◽  
Slow Motion ◽  
Explicit Integration ◽  
Time Step ◽  
Lumped Model ◽  
Guaranteed Stability ◽  
Fast Motion ◽  
The Stability ◽  
Secular Terms

This paper presents a method for simulating systems with two inertially coupled motions, i.e., a slow motion and a fast motion. The equations of motion are separated into two sets of coupled nonlinear ordinary differential equations. For each time step, the two sets of equations are integrated sequentially rather than simultaneously. Explicit integration methods are used for integrating the slow motion since the stability of the integration is not a problem and the explicit methods are very convenient for nonlinear equations. For the fast motion, the equations are linear and the implicit integrations can be used with guaranteed stability. The size of time step only needs to be chosen to provide accuracy of the solution for the modes that are excited. The interaction between the two types of motion must be treated such that secular terms do not appear due to the sequential integration method. A lumped model of a flexible pendulum will be presented in this paper to illustrate the application of the method. Numerical results for both simultaneous and sequential integration are presented for comparison.

scholarly journals Dark matter from non-relativistic embedding gravity

2021 ◽  
pp. 2150101
Author(s):  
S. A. Paston
Keyword(s):  
General Relativity ◽  
Dark Matter ◽  
Explicit Form ◽  
Equations Of Motion ◽  
Additional Contribution ◽  
Relativistic Limit ◽  
The Core ◽  
The Stability ◽  
Dimensional Surface

We study the possibility to explain the mystery of the dark matter (DM) through the transition from General Relativity to embedding gravity. This modification of gravity, which was proposed by Regge and Teitelboim, is based on a simple string-inspired geometrical principle: our spacetime is considered here as a four-dimensional surface in a flat bulk. We show that among the solutions of embedding gravity, there is a class of solutions equivalent to solutions of GR with an additional contribution of non-relativistic embedding matter, which can serve as cold DM. We prove the stability of such type of solutions and obtain an explicit form of the equations of motion of embedding matter in the non-relativistic limit. According to them, embedding matter turns out to have a certain self-interaction, which could be useful in the context of solving the core-cusp problem that appears in the [Formula: see text]CDM model.

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