On the stability of the dynamical system ‘rigid body + inviscid fluid’

1999 ◽  
Vol 386 ◽  
pp. 43-75 ◽  
Author(s):  
V. A. VLADIMIROV ◽  
K. I. ILIN
Keyword(s):  
Dynamical System ◽  
Rigid Body ◽  
Translational Motion ◽  
Sufficient Conditions ◽  
The Body ◽  
Inviscid Fluid ◽  
Rigid Boundary ◽  
First Case ◽  
Fixed Axis ◽  
The Stability

In this paper we study a dynamical system consisting of a rigid body and an inviscid incompressible fluid. Two general configurations of the system are considered: (a) a rigid body with a cavity completely filled with a fluid and (b) a rigid body surrounded by a fluid. In the first case the fluid is confined to an interior (for the body) domain and in the second case it occupies an exterior domain, which may, in turn, be bounded by some fixed rigid boundary or may extend to infinity. The aim of the paper is twofold: (i) to develop Arnold's technique for the system ‘body + fluid’ and (ii) to obtain sufficient conditions for the stability of steady states of the system. We first establish an energy-type variational principle for an arbitrary steady state of the system. Then we generalize this principle for states that are steady either in translationally moving in some fixed direction or rotating around some fixed axis coordinate system. The second variations of the corresponding functionals are calculated. The general results are applied to a number of particular stability problems. The first is the stability of a steady translational motion of a two-dimensional body in an irrotational flow. Here we have found that (for a quite wide class of bodies) the presence of non-zero circulation about the body does not affect its stability – a result that seems to be new. The second problem concerns the stability of a steady rotation of a force-free rigid body with a cavity containing an ideal fluid. Here we rediscover the stability criterion of Rumyantsev (see Moiseev & Rumyantsev 1965). The complementary problem – when a body is surrounded by a fluid and both body and fluid rotate with constant angular velocity around a fixed axis passing through the centre of mass of the body – is also considered and the corresponding sufficient conditions for stability are obtained.

Stability of rotating non-smooth complex fluids

2012 ◽  
Vol 708 ◽  
pp. 71-99 ◽  
Author(s):  
Ishan Sharma
Keyword(s):  
Angular Momentum ◽  
Complex Fluids ◽  
Energy Criterion ◽  
The Body ◽  
Stability Test ◽  
Future Application ◽  
Inviscid Fluid ◽  
Smooth Complex ◽  
Classical Energy ◽  
The Stability

AbstractWe extend the classical energy criterion for stability, the Lagrange–Dirichlet theorem, to rotating non-smooth complex fluids. The stability test so developed is very general and may be applied to most rotating non-smooth systems where the spectral method is inapplicable. In the process, we rigourously define an appropriate coordinate system in which to investigate stability – this happens to be the well-known Tisserand mean axis of the body – as well as systematically distinguish perturbations that introduce angular momentum and/or jumps in the stress state from those that do not. With a view to future application to planetary objects, we specialize the stability test to freely rotating self-gravitating ellipsoids. This is then employed to investigate the stability to homogeneous perturbations of rotating inviscid fluid ellipsoids. We recover results consistent with earlier predictions, and, in the process, also reconcile some contradictory conclusions about the stability of Maclaurin spheroids. Finally, we consider the equilibrium and stability of freely rotating self-gravitating Bingham fluid ellipsoids. We find that the equilibrium shapes of most such ellipsoids are secularly stable to homogeneous perturbations that preserve angular momentum, but not otherwise. We also touch upon the effect of shear thinning on stability.

scholarly journals Global Stability of Fractional Order Coupled Systems with Impulses via a Graphic Approach

Mathematics ◽  
2019 ◽  
Vol 7 (8) ◽  
pp. 744 ◽  
Author(s):  
Bei Zhang ◽  
Yonghui Xia ◽  
Lijuan Zhu ◽  
Haidong Liu ◽  
Longfei Gu
Keyword(s):  
Dynamical System ◽  
Graph Theory ◽  
Global Stability ◽  
Fractional Order ◽  
Trivial Solution ◽  
Sufficient Conditions ◽  
Fractional Order System ◽  
Coupled Systems ◽  
Order System ◽  
The Stability

Based on the graph theory and stability theory of dynamical system, this paper studies the stability of the trivial solution of a coupled fractional-order system. Some sufficient conditions are obtained to guarantee the global stability of the trivial solution. Finally, a comparison between fractional-order system and integer-order system ends the paper.

On Dynamically Equivalent Force Systems and Their Application to the Balancing of a Broom or the Stability of a Shoe Box

2004 ◽  
Author(s):  
Just L. Herder ◽  
Arend L. Schwab
Keyword(s):  
Rigid Body ◽  
Center Of Mass ◽  
The Body ◽  
Resultant Force ◽  
Dynamic Equivalence ◽  
Force Systems ◽  
The Stability ◽  
Metacentric Height ◽  
The Given

The stability of a rigid body on which two forces are in equilibrium can be assessed intuitively. In more complex cases this is no longer true. This paper presents a general method to assess the stability of complex force systems, based on the notion of dynamic equivalence. A resultant force is considered dynamically equivalent to a given system of forces acting on a rigid body if the contributions to the stability of the body of both force systems are equal. It is shown that the dynamically equivalent resultant force of two given constant forces applies at the intersection of its line of action and the circle put up by the application points of the given forces and the intersection of their lines of action. The determination of the combined center of mass can be considered as a special case of this theorem. Two examples are provided that illustrate the significance of the proposed method. The first example considers the suspension of a body, by springs only, that is statically balanced for rotation about a virtual stationary point. The second example treats the roll stability of a ship, where the metacentric height is determined in a natural way.

scholarly journals Stability of rigid body motion through an extended intermediate axis theorem: application to rockfall simulation

2021 ◽  
Author(s):  
Remco I. Leine ◽  
Giuseppe Capobianco ◽  
Perry Bartelt ◽  
Marc Christen ◽  
Andrin Caviezel
Keyword(s):  
Rigid Body ◽  
Direct Method ◽  
Lateral Spreading ◽  
The Body ◽  
Body Motion ◽  
Numerical Schemes ◽  
Stability Properties ◽  
Principal Axes ◽  
Rockfall Simulation ◽  
The Stability

AbstractThe stability properties of a freely rotating rigid body are governed by the intermediate axis theorem, i.e., rotation around the major and minor principal axes is stable whereas rotation around the intermediate axis is unstable. The stability of the principal axes is of importance for the prediction of rockfall. Current numerical schemes for 3D rockfall simulation, however, are not able to correctly represent these stability properties. In this paper an extended intermediate axis theorem is presented, which not only involves the angular momentum equations but also the orientation of the body, and we prove the theorem using Lyapunov’s direct method. Based on the stability proof, we present a novel scheme which respects the stability properties of a freely rotating body and which can be incorporated in numerical schemes for the simulation of rigid bodies with frictional unilateral constraints. In particular, we show how this scheme is incorporated in an existing 3D rockfall simulation code. Simulations results reveal that the stability properties of rotating rocks play an essential role in the run-out length and lateral spreading of rocks.

Effects of Flexibility on the Stability of Flying Aircraft

2004 ◽  
Vol 127 (1) ◽  
pp. 41-49 ◽  
Author(s):  
Ilhan Tuzcu ◽  
Leonard Meirovitch
Keyword(s):  
Feedback Control ◽  
Eigenvalue Problem ◽  
Rigid Body ◽  
Flight Dynamics ◽  
Flight Path ◽  
The Other ◽  
Flexible Aircraft ◽  
First Case ◽  
Rigid Body Motions ◽  
The Stability

Traditionally, flying aircraft have been treated within the confines of flight dynamics, which is concerned, for the most part, with rigid aircraft. On the other hand, flexible aircraft fall in the domain of aeroelasticity. In reality all aircraft possess some measure of flexibility and carry out rigid body maneuvers, so that the question arises as to whether rigid body motions and flexibility in combination can affect adversely the stability of flying aircraft. This paper addresses this question by solving the eigenvalue problem for the following three cases: (i) the flight dynamics of a flexible aircraft regarded as rigid and whose perturbations about the flight path are controlled by feedback control, (ii) the aeroelasticity of a corresponding flexible aircraft prevented from undergoing rigid body translations and rotations, and (iii) the control of the actual flexible aircraft using the control gains derived in the first case by regarding the aircraft as rigid. This investigation demonstrates that it is not always safe to treat separately rigid body and flexibility effects in a flying flexible aircraft.

Investigation of the influence of extreme floods on the stability of protective dams (using the example of Hanoi city)

2019 ◽  
Vol 12 (11-12) ◽  
pp. 26-34 ◽  
Author(s):  
T. H. Dinh ◽  
I. K. Fomenko ◽  
O. E. Vazkova ◽  
O. N. Sirotkina
Keyword(s):  
Mathematical Modeling ◽  
Groundwater Level ◽  
The Body ◽  
Dam Failure ◽  
Loss Of Stability ◽  
Important Indicator ◽  
First Case ◽  
Extreme Flood ◽  
The Stability ◽  
The City

According to the territorial development plans of the city of Hanoi, there is a need to build a new dam system to ensure the normal functioning of the riverside areas. Therefore, the solution of the problem of analyzing the causes of the loss of stability of the of dam slopes protecting the city from floods is of particular relevance. When assessing the possibility of dam failure as a result of a landslide process, the most important indicator is the appearance of cracks in the body of the dam. The possible mechanisms of loss of stability of the dam slopes as a result of changes in the hydrological situation are considered, their rationale based on the results of mathematical modeling is given. The studies used a “geotechnical” approach to estimating pore pressure values, which is new in Russian practice. This approach, without excluding the need to perform hydrogeological modeling, makes it possible to assess the stability of the dam slopes when the level of flood waters changes. The generalized engineering-geological model of the lithotechnical system (LTS) of the dam can be represented as the following scheme: at the base of the LTS lies an aquifer, represented by sands of various grain sizes and flowing sandy loams, which is overlapped by low-permeable loamy and clayey soils. Two scenarios for the violation of the stability of the dam slope are considered: when the groundwater level rises during the period of extreme flood; due to the rapid drawdown of the level of flood waters. By the methods of mathematical modeling it is proved that the decrease in the stability coefficient occurs in both cases. Thus, it is shown that the collapse of the dam slopes is possible both when the groundwater level rises during the flood period and during a sharp decrease after the end of the flood. At the same time, in the first case there will be a slide of the slopes falling in the direction “from the river”, and in the second — “to the river”. This fact must be taken into account when developing measures to strengthen the slopes of existing dams, as well as when designing a new dam system in Hanoicity.

On Nonholonomic Parametrization

1970 ◽  
Vol 37 (1) ◽  
pp. 128-132
Author(s):  
B. Vujanovic
Keyword(s):  
Dynamical System ◽  
Differential Equation ◽  
Fixed Point ◽  
Rigid Body ◽  
Force Field ◽  
First Integral ◽  
Central Force Field ◽  
The Stability ◽  
Asymmetric Rigid Body ◽  
System Time

This paper is concerned with the problem of obtaining the differential equation of the trajectory of dynamical system. Time is eliminated by means of some first integral linear with respect to the velocities. As an example, the stability of rotation of a heavy asymmetric rigid body with a fixed point under the action of the Newtonian central force field is considered.

scholarly journals Displacement: translation and rotation. Differences and Similarities in the Discrete and Continuous Models

2019 ◽  
Vol 4 (1) ◽  
pp. 104-124
Author(s):  
Géza Lámer
Keyword(s):  
Euclidean Space ◽  
Rigid Body ◽  
Solid Body ◽  
Discrete System ◽  
The Body ◽  
Closed Domain ◽  
Discrete Elements ◽  
Deformable Solid ◽  
First Case ◽  
Deformable Solid Body

The motion (displacement) of the Euclidean space can be decomposed into translation and rotation. The two kinds of motion of the Euclidean space based on two structures of the Euclidean space: The first one is the topological structure, the second one is the idea of distance. The motion is such a (topological) map, that the distance of any two points remains the same. The bounded and closed domain of the Euclidean space is taken as a model of the rigid body. The bounded and closed domain of the Euclidean space is also taken as a model of the deformable solid body. The map – i.e. the displacement field – of the deformable solid body is continuous, but is not (necessarily) motion; the size and the shape of body can change. The material has atomic-molecular structure. In compliance with it, the material can be comprehended as a discrete system. In this case the elements of the material, as an atom, molecule, grain, can be comprehended as either material point, or rigid body. In the first case the kinematical freedom is the translation, in the latter case the translation and the rotation. In the paper we analyse how the kinematical behaviour of the discrete and continuous mechanical system can be characterise by translation and rotation. In the discrete system the two motions are independent variable. At the same time they characterise the movement of the body different way. For instance homogeneous local translation gives the global translation, but the homogeneous local rotation does not give the global rotation. To realise global rotation in a discrete system on one hand global rotation of the position of the discrete elements, on the other hand homogeneous local rotations of the discrete elements in harmony with global rotation are required. In the continuous system the two kinds of movement cannot be interpreted: a point cannot rotate, a rotation of surrounding of a point or direction can be interpreted. The kinematical characteristics, as the displacement (practically this is equal to translation) of (neighbourhood of) point, the rotation of surrounding of that point and the rotation of a direction went through that point are not independent variables: the translation of a point determines the rotation of the surrounding of that point as well as the rotation of a direction went through that point. With accordance this statement the displacement (practically translation) (field) as the only kinematical variable can be interpreted in the continuous medium.

scholarly journals The Motion of a Rigid Body with Irrational Natural Frequency

2020 ◽  
Vol 2020 ◽  
pp. 1-8
Author(s):  
A. I. Ismail
Keyword(s):  
Rigid Body ◽  
Equations Of Motion ◽  
Geometric Interpretation ◽  
Minor Axis ◽  
The Body ◽  
Large Parameter ◽  
Small Component ◽  
Runge Kutta Method ◽  
The Stability ◽  
Euler’S Angles

In this paper, we consider the problem of the rotational motion of a rigid body with an irrational value of the frequency ω . The equations of motion are derived and reduced to a quasilinear autonomous system. Such system is reduced to a generating one. We assume a large parameter μ proportional inversely with a sufficiently small component r o of the angular velocity which is assumed around the major or the minor axis of the ellipsoid of inertia. Then, the large parameter technique is used to construct the periodic solutions for such cases. The geometric interpretation of the motion is obtained to describe the orientation of the body in terms of Euler’s angles. Using the digital fourth-order Runge-Kutta method, we determine the digital solutions of the obtained system. The phase diagram procedure is applied to study the stability of the attained solutions. A comparison between the considered numerical and analytical solutions is introduced to show the validity of the presented techniques and solutions.

scholarly journals The Stability Conditions for a Heavy Solid Motion

2020 ◽  
Vol 2020 ◽  
pp. 1-4
Author(s):  
A. I. Ismail
Keyword(s):  
Equations Of Motion ◽  
Center Of Mass ◽  
Sufficient Conditions ◽  
The Body ◽  
Stability Conditions ◽  
Moments Of Inertia ◽  
Principal Moments Of Inertia ◽  
The Stability ◽  
Nonlinear Equations Of Motion ◽  
Necessary And Sufficient

In this paper, the stability conditions for the rotary motion of a heavy solid about its fixed point are considered. The center of mass of the body is assumed to lie on the moving z-axis which is assumed to be the minor axis of the ellipsoid of inertia. The nonlinear equations of motion and their three first integrals are obtained when the principal moments of inertia are distributed as I 1 < I 2 < I 3 . We construct a Lyapunov function L to investigate the stability conditions for this motion. We give a numerical example to illustrate the necessary and sufficient conditions for the stability of the body at certain moments of inertia. This problem has many important applications in different sciences.

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