Gravity-Gradient Excitation of a Rotating Cable-Counterweight Space Station in Orbit

1963 ◽  
Vol 30 (4) ◽  
pp. 547-554 ◽  
Author(s):  
V. Chobotov
Keyword(s):  
Parametric Excitation ◽  
Equations Of Motion ◽  
Space Station ◽  
Type A ◽  
Gravity Gradient ◽  
Viscous Damping ◽  
Stability Criteria ◽  
Axial Vibrations ◽  
The Stability ◽  
Transverse Oscillations

The gravity-gradient excitation of a whirling cable-counterweight space station in orbit is investigated. The Lagrange’s equations of motion for transverse oscillations of the system are derived and shown to be of the Mathieu type. A few representative cases are investigated analytically and on an IBM 7090 computer. The stability criteria for the axial vibrations are also considered and shown to be of the same kind as those for the transver se vibrations of the cable. Viscous damping is included in the analyses and found to be effective and essential for prevention of parametric excitation instability of the system.

Dynamic analysis of a microelectromechanical systems resonant gyroscope excited using combination parametric resonance

2006 ◽  
Vol 220 (9) ◽  
pp. 1463-1479 ◽  
Author(s):  
B J Gallacher ◽  
J S Burdess
Keyword(s):  
Perturbation Analysis ◽  
Parametric Excitation ◽  
Microelectromechanical Systems ◽  
Equations Of Motion ◽  
Multiple Time ◽  
Excitation Scheme ◽  
Electrostatically Actuated ◽  
Modes Of Vibration ◽  
The Stability ◽  
Electrostatic Coupling

This paper investigates the application of parametric excitation to a resonant microelectromechanical systems (MEMS) gyroscope. The modal equations of motion of an electrostatically actuated ring are derived and shown to be coupled via the electrostatic stiffness. Such electrostatic coupling between in-plane modes of vibration permits parametric instabilities that may be exploited in a novel excitation scheme. A multiple time scale perturbation method is used to analyse the response of the ring gyroscope to the combination parametric excitations with the principal objective of separating the drive and response frequencies of the ring gyroscope. As pairs of flexural modes of the perfect ring are degenerate, the combination excitation between distinct modes demand the ring to be analysed as a four degree of freedom system. Slight mis-tuning between the otherwise degenerate modes is incorporated in the perturbation analysis. The results of the perturbation analysis are subsequently used to determine the stability boundaries for a typical ring gyroscope when excited using a sum combination resonance between the flexural modes of order 2 and 5. In this case, the ratio of the drive and response frequencies is approximately 10:1. Drive and sense configurations that enable effective parametric excitation of a desired mode are investigated. Simulation of the oscillator scheme is achieved using MATLAB Simulink and this validates the perturbation analysis. Agreement between the models within 10 per cent is demonstrated.

A Preliminary Investigation of the Dynamic Stability of Flexible Manipulators Performing Repetitive Tasks

1986 ◽  
Vol 108 (3) ◽  
pp. 206-214 ◽  
Author(s):  
D. A. Streit ◽  
C. M. Krousgrill ◽  
A. K. Bajaj
Keyword(s):  
Parametric Excitation ◽  
Floquet Theory ◽  
Equations Of Motion ◽  
Preliminary Investigation ◽  
Zero Solution ◽  
Flexible Manipulator ◽  
Governing Equations ◽  
Repetitive Tasks ◽  
The Stability ◽  
Two Degree Of Freedom

The governing equations of motion for the compliant coordinates describing a flexible manipulator performing repetitive tasks contain parametric excitation terms. The stability of the zero solution to these equations is investigated using Floquet theory. Analytical and numerical results are presented for a two-degree-of-freedom model of a manipulator with one prismatic joint and one revolute joint.

Parametric Study of Stability Criteria for Rotor Bearing Model With Viscoelastic Support

2017 ◽  
Author(s):  
S. Chandraker ◽  
J. K. Dutt ◽  
H. Roy
Keyword(s):  
Classical Method ◽  
Equations Of Motion ◽  
Weight Ratio ◽  
High Strength ◽  
Point Of View ◽  
Stability Criteria ◽  
Engineering Structures ◽  
Rotor Bearing ◽  
The Stability ◽  
Bearing System

In the last few decades, intensive research has been carried out on viscoelastic materials. Among them, most importantly polymers and composites thereof find extensive applications in engineering structures and rotors primarily due to quite high strength to weight ratio in comparison with metals. In dynamic modeling of rotor bearing system, incorporation of damping is very important as stationary (external) damping always helps in stability, however rotary damping (internal) promotes instability of rotors above a certain speed. Therefore for modeling point of view, it is very important to consider both internal or external damping effect. For this reason, the dissipation mechanism has been handled in such a way that it provides proper forces irrespective of its presence in a stationary or a rotary frame. Also in present work, both classical method and operator multiplier method are suggested to derive the equations of motion. The analysis also shows the stability zones of the rotor bearing system for various parametric values of different viscoelastic supports. It is found that choosing a right viscoelastic support can increase the stability criteria of the system to some extent.

On the Stability of the Damped Hill’s Equation With Arbitrary, Bounded Parametric Excitation

1993 ◽  
Vol 60 (2) ◽  
pp. 366-370 ◽  
Author(s):  
C. D. Rahn ◽  
C. D. Mote
Keyword(s):  
Asymptotic Stability ◽  
Parametric Excitation ◽  
Optimal Control Theory ◽  
Damping Ratio ◽  
Viscous Damping ◽  
Conservative Estimate ◽  
Hill’S Equation ◽  
Hill's Equation ◽  
Time Optimal ◽  
The Stability

The minimum damping for asymptotic stability is predicted for Hill’s equation with any bounded parametric excitation. It is shown that the response of Hill’s equation with bounded parametric excitation is exponentially bounded. The parametric excitation maximizing the bounding exponent is identified by time optimal control theory. This maximal bounding exponent is balanced by viscous damping to ensure asymptotic stability. The minimum damping ratio is calculated as a function of the excitation bound. A closed form, more conservative estimate of the minimum damping ratio is also predicted. Thus, if the general (e.g., unknown, aperiodic, or random) parametric excitation of Hill’s equation is bounded, a simple, conservative estimate of the damping required for asymptotic stability is given in this paper.

Stability Analysis of a Rotating System Due to the Effect of Ball Bearing Waviness

2002 ◽  
Vol 125 (1) ◽  
pp. 91-101 ◽  
Author(s):  
G. H. Jang ◽  
S. W. Jeong
Keyword(s):  
Parametric Excitation ◽  
Ball Bearing ◽  
Equations Of Motion ◽  
Contact Forces ◽  
Fourier Series Expansion ◽  
Time Varying ◽  
Algebraic Equations ◽  
Rotating System ◽  
Mathieu’S Equation ◽  
The Stability

This research presents an analytical model to investigate the stability due to the ball bearing waviness in a rotating system supported by two ball bearings. The stiffness of a ball bearing changes periodically due to the waviness in the rolling elements as the rotor rotates, and it can be calculated by differentiating the nonlinear contact forces. The linearized equations of motion can be represented as a parametrically excited system in the form of Mathieu’s equation, because the stiffness coefficients have time-varying components due to the waviness. Their solution can be assumed as a Fourier series expansion so that the equations of motion can be rewritten as the simultaneous algebraic equations with respect to the Fourier coefficients. Then, stability can be determined by solving Hill’s infinite determinant for these algebraic equations. The validity of this research is proven by comparing the stability chart with the time responses of the vibration model suggested by prior research. This research shows that the waviness in the ball bearing generates the time-varying component of the stiffness coefficient, whose frequency is called the frequency of the parametric excitation. It also shows that the instability takes place from the positions in which the ratio of the natural frequency to the frequency of the parametric excitation corresponds to i/2 i=1,2,3,….

The Equilibrium Stability of a System of Disk Dynamos

1960 ◽  
Vol 56 (2) ◽  
pp. 154-173 ◽  
Author(s):  
Norman R. Lebovitz
Keyword(s):  
Equilibrium State ◽  
Viscous Damping ◽  
Equilibrium Stability ◽  
Stability Criteria ◽  
Equilibrium Solutions ◽  
Current Case ◽  
Zero Current ◽  
Single Disk ◽  
The Stability ◽  
Disk Dynamo

ABSTRACTThe stability of the equilibrium solutions of the equations describing the behaviour of a system of coupled disk dynamos is investigated. In the absence of viscous damping, it is found that systems consisting of more than two dynamos are unstable. That a single disk dynamo is stable is known. The stability of the undamped two-dynamo system has not been ascertained. When viscous damping is present, there are two equilibrium solutions, one in which all the currents are zero, and one in which they are finite. In the zero-current case, a stability criterion is found. Stability criteria are also found in the finite-current case. Further, the existence of the finite-current equilibrium state excludes the stability of the zero-current equilibrium state.

Parametric Excitation of a Pendulum With Bilinear Hysteresis

1970 ◽  
Vol 37 (4) ◽  
pp. 1061-1068 ◽  
Author(s):  
W. K. Tso ◽  
K. G. Asmis
Keyword(s):  
Steady State ◽  
Parametric Resonance ◽  
Parametric Excitation ◽  
Viscous Damping ◽  
Sinusoidal Oscillation ◽  
Effective Mechanism ◽  
Steady State Responses ◽  
The Stability ◽  
State Responses ◽  
Steady State Solutions

The steady-state responses of a simple pendulum with a hinge exhibiting bilinear hysteretic moment-rotation characteristics and parametrically excited by a sinusoidal oscillation at the base is given. The stability of the steady-state solutions is discussed. It is shown that in contrast with viscous damping, the bilinear hysteresis is an effective mechanism to limit the growth of the response during parametric resonance.

scholarly journals A Study of the Stability of Externally Pressurized Gas Bearings

1960 ◽  
Vol 27 (2) ◽  
pp. 250-258 ◽  
Author(s):  
Lazar Licht ◽  
Harold Elrod
Keyword(s):  
Equations Of Motion ◽  
Stability Criteria ◽  
Small Deviations ◽  
Gas Bearings ◽  
Lumped Parameter ◽  
Nozzle Size ◽  
The Subject ◽  
The Stability ◽  
Annular Clearance ◽  
Limiting Values

The subject of this paper is the stability of externally pressurized gas bearings. The pertinent equations of motion are linearized and the stability criteria stated in terms of small deviations from the equilibrium operating point. The flow in the bearing clearance is treated on a distributed rather than on a lumped-parameter basis. Results obtained from present analysis when compared with those previously arrived at by means of simplified analyses [1, 3] show a marked divergence in the limiting values of parameters which influence the stability of the bearing. These results and divergences are discussed in terms of permissible compression volume, pressure ratios, supply-nozzle size, length of annular clearance, and bearing mass.

Impulsive Parametric Excitation: Theory

1972 ◽  
Vol 39 (2) ◽  
pp. 551-558 ◽  
Author(s):  
C. S. Hsu
Keyword(s):  
Stability Analysis ◽  
General Theory ◽  
Parametric Excitation ◽  
Periodic Systems ◽  
Stability Criteria ◽  
Dirac Delta ◽  
Special Cases ◽  
Delta Functions ◽  
Analytic Stability ◽  
The Stability

Given in this paper is the development of a theory for dynamical systems subjected to periodic impulsive parametric excitations. By periodic impulsive parametric excitation we mean those excitations representable by periodic coefficients which consist of sequences of Dirac delta functions. It turns out that for this class of periodic systems the stability analysis can be carried out in a remarkably simple and general manner without approximation. In the paper, after giving the general theory, many special cases are examined. In many instances simple and closed-form analytic stability criteria can be easily established.

scholarly journals On the Destabilizing Effect of Damping in Nonconservative Elastic Systems

1965 ◽  
Vol 32 (3) ◽  
pp. 592-597 ◽  
Author(s):  
G. Herrmann ◽  
Ing-Chang Jong
Keyword(s):  
Critical Load ◽  
Characteristic Equation ◽  
Elastic System ◽  
Viscous Damping ◽  
Stability Criteria ◽  
Elastic Systems ◽  
The Stability ◽  
Proper Interpretation ◽  
Degree Of Instability

The destabilizing effect of linear viscous damping in a nonconservative elastic system is investigated by studying the roots of the characteristic equation in addition to the stability criteria and by introducing the concept of degree of instability. A generic relationship between critical loadings for no damping and for slight damping as well as vanishing damping is established. It is found that while the presence of small damping may have a destabilizing effect, proper interpretation of the limiting process of vanishing damping leads to the same critical load as for no damping.

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