On the Stability Properties of a Periodically Forced, Time-Varying Mass System

2009 ◽  
Author(s):  
W. T. van Horssen ◽  
O. V. Pischanskyy ◽  
J. L. A. Dubbeldam
Keyword(s):  
Periodic Solutions ◽  
Forced Vibrations ◽  
Time Varying ◽  
Mass System ◽  
Single Degree Of Freedom ◽  
Stability Properties ◽  
Single Degree ◽  
Existence Of Periodic Solutions ◽  
The Stability ◽  
Varying Mass

In this paper the forced vibrations of a linear, single degree of freedom oscillator (sdofo) with a time-varying mass will be studied. The forced vibrations are due to small masses which are periodically hitting and leaving the oscillator with different velocities. Since these small masses stay for some time on the oscillator surface the effective mass of the oscillator will periodically vary in time. Not only solutions of the oscillator equation will be constructed, but also the stability properties, and the existence of periodic solutions will be discussed.

On the Natural Modes and Their Stability in Nonlinear Two-Degree-of-Freedom Systems

1959 ◽  
Vol 26 (3) ◽  
pp. 377-385
Author(s):  
R. M. Rosenberg ◽  
C. P. Atkinson
Keyword(s):  
Free Vibrations ◽  
Degree Of Freedom ◽  
Single Degree Of Freedom ◽  
Single Degree ◽  
Natural Modes ◽  
Theoretical Predictions ◽  
Stability Chart ◽  
The Stability ◽  
Two Parameters ◽  
Two Degree Of Freedom

Abstract The natural modes of free vibrations of a symmetrical two-degree-of-freedom system are analyzed theoretically and experimentally. This system has two natural modes, one in-phase and the other out-of-phase. In contradistinction to the comparable single-degree-of-freedom system where the free vibrations are always orbitally stable, the natural modes of the symmetrical two-degree-of-freedom system are frequently unstable. The stability properties depend on two parameters and are easily deduced from a stability chart. For sufficiently small amplitudes both modes are, in general, stable. When the coupling spring is linear, both modes are always stable at all amplitudes. For other conditions, either mode may become unstable at certain amplitudes. In particular, if there is a single value of frequency and amplitude at which the system can vibrate in either mode, the out-of-phase mode experiences a change of stability. The experimental investigation has generally confirmed the theoretical predictions.

scholarly journals The Integrability and Existence of Periodic Solutions on a First-Order Nonlinear Differential Equation with a Polynomial Nonlinear Term

2012 ◽  
Vol 2012 ◽  
pp. 1-26
Author(s):  
Ni Hua ◽  
Tian Li-Xin
Keyword(s):  
Differential Equation ◽  
Periodic Solutions ◽  
Nonlinear Differential Equation ◽  
Nonlinear Term ◽  
Order Differential Equation ◽  
First Order ◽  
Order Nonlinear Differential Equation ◽  
Existence Of Periodic Solutions ◽  
The Stability

This paper deals with a first-order differential equation with a polynomial nonlinear term. The integrability and existence of periodic solutions of the equation are obtained, and the stability of periodic solutions of the equation is derived.

Stochastic Dynamics of a Variable Inertia System via Lyapunov Exponents

2008 ◽  
Author(s):  
S. F. Asokanthan ◽  
X. H. Wang ◽  
W. V. Wedig ◽  
S. T. Ariaratnam
Keyword(s):  
Lyapunov Exponent ◽  
Lyapunov Exponents ◽  
Stochastic Systems ◽  
Stochastic Dynamics ◽  
Excitation Frequency ◽  
Excitation Amplitude ◽  
Single Degree Of Freedom ◽  
Single Degree ◽  
Frequency Excitation ◽  
The Stability

Torsional instabilities in a single-degree-of-freedom system having variable inertia are investigated by means of Lyapunov exponents. Linearised analytical model is used for the purpose of stability analysis. Numerical schemes for simulating the top Lyapunov exponent for both deterministic and stochastic systems are established. Instabilities associated with the primary and the secondary sub-harmonic resonances have been identified by studying the sign of the top Lyapunov exponent. Predictions for the deterministic and the stochastic cases are compared. Instability conditions have been presented graphically in the excitation frequency-excitation amplitude-top Lyapunov exponent space. The effects of fluctuation density as well as that of damping on the stability behaviour of the system have been examined. Predicted instability conditions are adequate for the design of a variable-inertia system so that a range of critical speeds of operation may be avoided.

scholarly journals The Spectrum Dip of Deck Mounted Systems

2010 ◽  
Vol 17 (1) ◽  
pp. 55-69
Author(s):  
R.J. Scavuzzo ◽  
G.D. Hill ◽  
P.W. Saxe
Keyword(s):  
Vertical Motion ◽  
Base Point ◽  
Degree Of Freedom ◽  
Mass System ◽  
Detailed Model ◽  
Single Degree Of Freedom ◽  
Single Degree ◽  
Fixed Base ◽  
Grid Points ◽  
Modal Mass

In this paper, a detailed model of a ship deck and attached dynamic systems was developed and subjected to dynamic studies using two different shock inputs: a triangular shaped velocity pulse and the vertical motion of the innerbottom of the standard Floating Shock Platform (FSP). Two studies were conducted, one considering four single degree-of-freedom systems attached at various deck locations and another considering a three-mass system attached at one location. The two shock inputs were used only for the multi-mass system study. The triangular pulse was used for the four single degree-of-freedom systems study. For the single degree-of-freedom systems study, shock spectra were first calculated at the four mounting locations assuming the oscillators were not present. Then the oscillator systems were added to these grid points to determine the change in the shock spectra. First, the oscillators were added one at a time, and then all the oscillators were added to the deck. The multi-mass system was analyzed using both shock inputs. First, the fixed-base modal masses and frequencies were determined. Then, the system as a whole was attached to the deck and the spectrum values at the base point were determined and compared to those for the free deck case. In the last step each mode of the multi-mass system, represented by a single degree-of-freedom system with the modal mass and appropriate spring stiffness, was considered individually to determine the spectrum responses. Results of the free deck, the entire system and individual modal responses are compared.

Predictions of the Periodic Response of a Single-Degree-of-Freedom Foam-Mass System by Using Incremental Harmonic Balance

2012 ◽  
Author(s):  
Yousof Azizi ◽  
Patricia Davies ◽  
Anil K. Bajaj
Keyword(s):  
Steady State ◽  
Harmonic Balance ◽  
Driving Frequency ◽  
Steady State Response ◽  
Base Excitation ◽  
Mass System ◽  
Single Degree Of Freedom ◽  
State Response ◽  
Single Degree ◽  
Incremental Harmonic Balance

Vehicle occupants are exposed to low frequency vibration that can cause fatigue, lower back pain, spine injuries. Therefore, understanding the behavior of a seat-occupant system is important in order to minimize these undesirable vibrations. The properties of seating foam affect the response of the occupant, so there is a need for good models of seat-occupant systems through which the effects of foam properties on the dynamic response can be directly evaluated. In order to understand the role of flexible polyurethane foam in characterizing the complex seat-occupant system behavior better, the response of a single-degree-of-freedom foam-mass system, which is the simplest model representing a seat-occupant system, is studied. The incremental harmonic balance method is used to determine the steady-state behavior of the foam-mass system subjected to sinusoidal base excitation. This method is used to reduce the time required to generate the steady-state response at the driving frequency and at harmonics of the driving frequency from that required when using direct time-integration of the governing equations to determine the steady state response. Using this method, the effects of different viscoelastic models, riding masses, base excitation levels and damping coefficients on the response are investigated.

On the forced vibrations of an oscillator with a periodically time-varying mass

2010 ◽  
Vol 329 (6) ◽  
pp. 721-732 ◽  
Author(s):  
W.T. van Horssen ◽  
O.V. Pischanskyy ◽  
J.L.A. Dubbeldam
Keyword(s):  
Forced Vibrations ◽  
Time Varying ◽  
Varying Mass

scholarly journals Stability of Sliding Frictional Surfaces with Varying Normal Force

1994 ◽  
Vol 116 (2) ◽  
pp. 237-242 ◽  
Author(s):  
P. E. Dupont ◽  
D. Bapna
Keyword(s):  
Normal Force ◽  
Elastic System ◽  
Functional Dependence ◽  
Low Velocity ◽  
Single Degree Of Freedom ◽  
Single Degree ◽  
State Dependent ◽  
Velocity Motion ◽  
The Stability ◽  
Frictional Surfaces

This paper presents the stability analysis of a single degree-of-freedom elastic system following a rate-and state-dependent friction law. Normal force is assumed to depend on the displacement, velocity and acceleration of the sliding interface. The history dependence of friction on normal force is included in the analysis. It is shown that to achieve steady sliding, system stiffness must exceed a critical value which depends on the expression for normal force. A system in which normal force depends on spring displacement is analyzed in detail. These results indicate that the functional dependence of normal force on system state can have a significant effect on the stability of low-velocity motion.

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