Dynamic Stability Characteristics of Axially Accelerated Beams

2006 ◽  
Vol 321-323 ◽  
pp. 1654-1658 ◽  
Author(s):  
Hong Hee Yoo ◽  
Sung Jin Eun
Keyword(s):  
Dynamic Stability ◽  
Natural Frequencies ◽  
Equations Of Motion ◽  
Stability Diagram ◽  
Accelerated Motion ◽  
Stability Diagrams ◽  
Divergence Type ◽  
The Stability ◽  
Direct Numerical Integration ◽  
Free Beam

Dynamic stability of axially accelerated beams is investigated in this paper. The equations of motion of a fixed-free beam undergoing axially accelerated motion are derived. Unstable regions due to the acceleration are obtained by using the Floquet’s theory. Stability diagrams are presented to illustrate the influence of the acceleration characteristics. Large unstable regions of flutter type instability exist around the first, twice the first, and twice the second bending natural frequencies. Divergence type instability also occurs when the acceleration exceeds a certain value. The validity of the stability diagram is confirmed by direct numerical integration of the equations of motion.

In-Plane Parametric Instability of a Rigid Body With a Dual-Rotor System

2015 ◽  
Author(s):  
Allen Anilkumar ◽  
V. Kartik
Keyword(s):  
Rigid Body ◽  
Floquet Theory ◽  
Natural Frequencies ◽  
Equations Of Motion ◽  
Parametric Instability ◽  
Harmonic Balance Method ◽  
Frequency Spectra ◽  
Stability Diagrams ◽  
The Stability ◽  
Lumped Masses

Rotating machines can be modeled at a basic level using lumped masses that are rotating about and attached using springs to an axis. Even such seemingly simple system can exhibit rich dynamics in the presence of time-varying terms in the governing differential equations. This paper investigates the dynamics of a rigid body with two attached rotors that rotate in the same plane. The system is parametrically-excited and the equations of motion are periodic in both rotor frequencies. The frequency spectra of the time responses show distinct side-band structures centered about the unforced natural frequencies. In addition to classical resonances, the stability diagrams generated using Floquet theory reveal instabilities at unexpected combinations of the forcing and natural frequencies. The harmonic balance method is employed to verify the stability boundaries obtained using Floquet theory. The study reveals safe regimes of parameter combinations that can help prevent the onset of instability in such systems.

On the Dynamic Stability of Self-Aligning Journal Gas Bearings

1977 ◽  
Vol 99 (4) ◽  
pp. 434-440 ◽  
Author(s):  
M. J. Cohen
Keyword(s):  
Dynamic Stability ◽  
Journal Bearing ◽  
Equations Of Motion ◽  
Stability Boundary ◽  
Initial Condition ◽  
Small Motion ◽  
Gas Bearings ◽  
Loading Case ◽  
The Stability ◽  
Small Disturbances

The report presents an investigation of the dynamic stability behaviour of self-aligning journal gas bearings when subjected to arbitrary small disturbances from an initial condition of operational equilibrium. The method is based on an approach similar to the nonlinear-ph solution of the author for the quasi-static loading case but the equations of motion of the journal are the linearized forms for small motion in the two degrees (translational) of freedom of the journal center. The stability domains for the infinite journal bearing are presented for the whole of the eccentricity (ε) and rotational speed (Λ) ranges for any given bearing geometry, in the shape of stability boundaries in that domain. It is shown that a given bearing will be stable within a corridor in the (ε, Λ) parametral domain having as its lower bound the so called “half-speed” whirl stability boundary and as its upper bound another whirling instability at a higher characteristic (relative) frequency, the instability occurs generally at the higher eccentricities and lower rotational speeds.

Numerical Analysis on Mathieu Instability of a Top-Tensioned Riser in a Dry-Tree Semisubmersible

2019 ◽  
Vol 142 (2) ◽  
Author(s):  
Hooi-Siang Kang ◽  
Moo Hyun Kim ◽  
Shankar S. Bhat Aramanadka
Keyword(s):  
Dynamic Stability ◽  
Parametric Resonance ◽  
Structural Integrity ◽  
Mathieu Equation ◽  
Stability Diagram ◽  
Stability Assessment ◽  
Dynamic Tension ◽  
Hydrocarbon Production ◽  
Mathieu Instability ◽  
The Stability

Abstract The development of a dry-tree semisubmersible (DTS), a new type offshore hydrocarbon production system, is facing unconventional challenges in the issues of dynamic stability, structural integrity, and parametric resonance. The Mathieu equation is used to assess the dynamic stability of a top-tensioned riser (TTR) in order to prevent the parametric resonance which leads to detrimental effects on structural integrity. The objectives of this paper are to (i) study a Mathieu stability diagram and its coefficients for an assessment of the stability of a TTR, (ii) identify the effects of the dynamic tension variations in the Mathieu stability assessment, and (iii) analyze the stability of the TTR on a DTS, which is equipped with a long-stroke tensioner, by using numerical simulation. The dynamic tension variation in the DTS was identified to induce instability in the TTR. Hence, the Mathieu stability assessment is recommended to be included in an analysis of TTR behaviors in a dry-tree interface of semisubmersibles.

Dynamic Stability of a Spinning Timoshenko Beam Subjected to a Moving Mass-Spring-Damper Unit

2010 ◽  
Author(s):  
T. H. Young ◽  
M. S. Chen
Keyword(s):  
Dynamic Stability ◽  
Timoshenko Beam ◽  
Multiple Scales ◽  
Equations Of Motion ◽  
Axial Direction ◽  
Longitudinal Axis ◽  
Moving Mass ◽  
The Stability ◽  
The Galerkin Method ◽  
Mass Spring

This paper investigates the dynamic stability of a finite Timoshenko beam spinning along its longitudinal axis and subjected to a moving mass-spring-damper (MSD) unit traveling in the axial direction. The mass of the moving MSD unit makes contact with the beam all the time during traveling. Due to the moving MSD unit, the beam is acted upon by a periodic, parametric excitation. In this work, the equations of motion of the beam are first discretized by the Galerkin method. The discretized equations of motion are then partially uncoupled by the modal analysis procedure suitable for gyroscopic systems. Finally the method of multiple scales is used to obtain the stability boundaries of the beam. Numerical results show that if the displacement of the MSD unit is equal to only one of the two transverse displacements of the beam, very large unstable regions may appear at main resonances.

Effect of Aerodynamic Loads on Dynamic Stability of Slender Launch Vehicle Subjected to an End Rocket Thrust

2002 ◽  
Author(s):  
Tsukasa Ohshima ◽  
Yoshihiko Sugiyama
Keyword(s):  
Stability Analysis ◽  
Dynamic Stability ◽  
Longitudinal Axis ◽  
Launch Vehicle ◽  
Aerodynamic Load ◽  
Aerodynamic Loads ◽  
Eigen Values ◽  
Rocket Thrust ◽  
The Stability ◽  
Free Beam

This paper deals with dynamic stability of a slender launch vehicle subjected to aerodynamic loads and an end rocket thrust. The flight vehicle is simplified into a uniform free-free beam subjected to an end follower thrust. Two types of aerodynamic loads are assumed in the stability analysis. Firstly, it is assumed that two concentrated aerodynamic loads act on the flight body at its nose and tail. Secondly, to take account of effect of unsteady flow due to motion of a flexible flight body, aerodynamic load is estimated by the slender body approximation. Extended Hamilton’s principle is applied to the considered beam for deriving the equation of motion. Application of FEM yields standard eigen-value problem. Dynamic stability of the beam is determined by the sign of the real part of the complex eigen-values. If aerodynamic loads are concentrated loads that act on the flight body at its nose and tail, the flutter thrust decreases by about 10% in comparison with the flutter thrust of free-free beam subjected only to an end follower thrust. If aerodynamic loads are distributed along the longitudinal axis of the flight body, the flutter thrust decreases by about 70% in comparison with the flutter thrust of free-free beam under an end follower thrust. It is found that the flutter thrust is reduced considerably if the aerodynamic loads are taken into account in addition to an end rocket thrust in the stability analysis of slender rocket vehicle.

On the Stability of Accelerated Motion: Some Thoughts on Linear Differential Equations with Variable Coefficients

1957 ◽  
Vol 8 (4) ◽  
pp. 309-330 ◽  
Author(s):  
A. R. Collar
Keyword(s):  
Differential Equations ◽  
Equations Of Motion ◽  
Variable Coefficients ◽  
Order Equation ◽  
Linear Differential Equations ◽  
Constant Speed ◽  
Accelerated Motion ◽  
Constant Coefficients ◽  
Linear Differential ◽  
The Stability

Summary:In studies of the stability of aeronautical systems, equations of motion are derived which have coefficients dependent on flight speed. Conventional practice treats the speed as constant, when a set of linear differential equations with constant coefficients results. Actually, since the speed varies during flight, it may be regarded as a prescribed function of time; the set of linear differential equations then has variable coefficients.The treatment of the problem of stability then becomes much more complex in this case. A simple example is given to show that a system which is stable at any constant speed can become unstable during deceleration; the ordinary constant-speed criteria are, strictly, therefore inadequate. Some approaches to the discussion of stability during acceleration are suggested; a solution is given of the single second-order equation which enables the amplitude of oscillation of the solution to be studied. Inverse methods of approach are suggested, both for single and sets of equations, in which particular forms of acceleration corresponding to prescribed solutions are derived; and some tentative conclusions are drawn. As would be expected, the effects of acceleration depend on a dimensionless “acceleration number.”

scholarly journals Motion of a Rigid Body Supported at One Point by a Rotating Arm

1993 ◽  
Vol 1 (2) ◽  
pp. 121-134
Author(s):  
Jeffrey D. Stoen ◽  
Thomas R. Kane
Keyword(s):  
Energy Dissipation ◽  
Rigid Body ◽  
Exact Solutions ◽  
Equations Of Motion ◽  
Stability Diagrams ◽  
Mass Properties ◽  
State Of Rest ◽  
The Stability ◽  
Nonlinear Equations Of Motion ◽  
System Geometry

This article details a scheme for evaluating the stability of motions of a system consisting of a rigid body connected at one point to a rotating arm. The nonlinear equations of motion for the system are formulated, and a method for finding exact solutions representing motions that resemble a state of rest is presented. The equations are then linearized and roots of the eigensystem are classified and used to construct stability diagrams that facilitate the assessment of effects of varying the body's mass properties and system geometry, changing the position of the attachment joint, and adding energy dissipation in the joint.

scholarly journals ДИНАМІЧНА РЕАКЦІЯ ТА СТІЙКІСТЬ ПОШКОДЖЕНОЇ КОНСТРУКЦІЇ ТРАНСПОРТНОГО ЛІТАКА

2020 ◽  
pp. 173-180
Author(s):  
Є. Ю. Іленко ◽  
В. М. Онищенко
Keyword(s):  
Dynamic Stability ◽  
Equations Of Motion ◽  
Initial Perturbation ◽  
Aerodynamic Characteristics ◽  
Operating Conditions ◽  
Ultimate State ◽  
Main Method ◽  
Operating Load ◽  
The Stability

In the process of designing and operating the aircraft, it is important to determine the ultimate state of the structure, taking into account the dynamic load of the structure and its stability. The ultimate state of the structure is characterized by damage, in which the structure still retains the ability to withstand without catastrophic destruction of the maximum operating load. The main method of studying the stability of the structure is the dynamic method. It allows us to investigate the perturbed motion of a structure as a nonconservative system for some initial perturbation. The monotonic departure of the system from the equilibrium position or its oscillations with increasing amplitudes indicate the instability of the structure. The paper analyzes the effect of damage to the aircraft structure on its dynamic stability based on the determination of the dynamic response of the aircraft to some non-stationary perturbation, for example, on the action of a turbulent atmosphere. The method of computational analysis is used to study the dynamic stability of the structure. The basis of this method is mathematical modeling (MM) of the operation of the aircraft in the form of a system of equations of motion and deformation of the structure. The problem of dynamic aeroelasticity is considered - the behavior of the elastic damaged structure of the aircraft in the air flow to the initial perturbation. On the basis of computer simulation, the dynamic stability of the elastic structure, its oscillating or quasi-static (aperiodic) deformation motion within the flight range of the aircraft is estimated. On the basis of parametric researches the limits of instability of a design at the set damages for typical operating conditions are estimated. The relevance of the direction focused on the creation and advanced operation of MM aircraft - their mathematical backups in the process of design and operation of aircraft due to the complexity and limited capabilities of ground experimental installations and flight experiment. It is noted that the condition for the application of this method is the formed MM operation of the aircraft and the availability of information on the mass-inertial, stiffness and aerodynamic characteristics of the aircraft.

scholarly journals Inviscid Faraday waves in a brimful circular cylinder

2013 ◽  
Vol 724 ◽  
pp. 671-694 ◽  
Author(s):  
R. Kidambi
Keyword(s):  
Circular Cylinder ◽  
Contact Line ◽  
Stability Diagram ◽  
Bond Number ◽  
Faraday Waves ◽  
Almost Periodic ◽  
Contact Lines ◽  
Stability Diagrams ◽  
The Stability ◽  
Subharmonic Resonances

AbstractWe study inviscid Faraday waves in a brimful circular cylinder with pinned contact line. The pinning leads to a coupling of the Bessel modes and leads to an infinite system of coupled Mathieu equations. For large Bond numbers, even though the stability diagrams and the subharmonic and harmonic resonances for the free and pinned contact lines are similar, the free surface shapes can be quite different. With decreasing Bond number, not only are the harmonic and subharmonic resonances very different from the free contact line case but also interesting changes in the stability diagram occur with the appearance of combination resonance tongues. Points on these tongue boundaries correspond to almost-periodic states. These do not seem to have been reported in the literature.

Dynamic Stability of Disks With Periodically Varying Spin Rates Subjected to Stationary In-Plane Edge Loads

2004 ◽  
Vol 71 (4) ◽  
pp. 450-458 ◽  
Author(s):  
T. H. Young ◽  
M. Y. Wu
Keyword(s):  
Dynamic Stability ◽  
Multiple Scales ◽  
Equations Of Motion ◽  
Parametric Instability ◽  
Stress Distributions ◽  
The Finite Element Method ◽  
Nodal Diameter ◽  
The Stability ◽  
Small Periodic ◽  
Small Periodic Perturbation

This paper presents an analysis of dynamic stability of an annular plate with a periodically varying spin rate subjected to a stationary in-plane edge load. The spin rate of the plate is characterized as the sum of a constant speed and a small, periodic perturbation. Due to this periodically varying spin rate, the plate may bring about parametric instability. In this work, the initial stress distributions caused by the periodically varying spin rate and the in-plane edge load are analyzed first. The finite element method is applied then to yield the discretized equations of motion. Finally, the method of multiple scales is adopted to determine the stability boundaries of the system. Numerical results show that combination resonances take place only between modes of the same nodal diameter if the stationary in-plane edge load is absent. However, there are additional combination resonances between modes of different nodal diameters if the stationary in-plane edge load is present.

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