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 Control of Near-Grazing Dynamics in the Two-Degree-of-Freedom Vibroimpact System with Symmetrical Constraints

Complexity ◽  
2020 ◽  
Vol 2020 ◽  
pp. 1-12
Author(s):  
Zihan Wang ◽  
Jieqiong Xu ◽  
Shuai Wu ◽  
Quan Yuan
Keyword(s):  
Control Strategy ◽  
Stability Criterion ◽  
Control Method ◽  
Degree Of Freedom ◽  
Local Analysis ◽  
Grazing Bifurcation ◽  
Vibroimpact System ◽  
The Stability ◽  
Two Parameters ◽  
Two Degree Of Freedom

The stability of grazing bifurcation is lost in three ways through the local analysis of the near-grazing dynamics using the classical concept of discontinuity mappings in the two-degree-of-freedom vibroimpact system with symmetrical constraints. For this instability problem, a control strategy for the stability of grazing bifurcation is presented by controlling the persistence of local attractors near the grazing trajectory in this vibroimpact system with symmetrical constraints. Discrete-in-time feedback controllers designed on two Poincare sections are employed to retain the existence of an attractor near the grazing trajectory. The implementation relies on the stability criterion under which a local attractor persists near a grazing trajectory. Based on the stability criterion, the control region of the two parameters is obtained and the control strategy for the persistence of near-grazing attractors is designed accordingly. Especially, the chaos near codimension-two grazing bifurcation points was controlled by the control strategy. In the end, the results of numerical simulation are used to verify the feasibility of the control method.

Chaotic Responses of a Two-Degree-of-Freedom Elastic-Plastic Beam Model to Short Pulse Loading

1992 ◽  
Vol 59 (4) ◽  
pp. 711-721 ◽  
Author(s):  
J.-Y. Lee ◽  
P. S. Symonds ◽  
G. Borino
Keyword(s):  
Short Pulse ◽  
Direct Method ◽  
Degree Of Freedom ◽  
Beam Model ◽  
Single Degree Of Freedom ◽  
Single Degree ◽  
Arch Action ◽  
Time Histories ◽  
Energy Plane ◽  
Two Degree Of Freedom

The paper discusses chaotic response behavior of a beam model whose ends are fixed, so that shallow arch action prevails after moderate plastic straining has occurred due to a short pulse of transverse loading. Examples of anomalous displacement-time histories of a uniform beam are first shown. These motivated the present study of a two-degree-of-freedom model of Shanley type. Calculations confirm these behaviors as symptoms of chaotic unpredictability. Evidence of chaos is seen in displacement-time histories, in phase plane and power spectral diagrams, and especially in extreme sensitivity to parameters. The exponential nature of the latter is confirmed by calculations of conventional Lyapunov exponents and also by a direct method. The two-degree-of-freedom model allows use of the energy approach found helpful for the single-degree-of-freedom model (Borino et al., 1989). The strain energy is plotted as a surface over the displacement coordinate plane, which depends on the plastic strains. Contrasting with the single-degree-of-freedom case, the energy diagram illuminates the possibility of chaotic vibrations in an initial phase, and the eventual transition to a smaller amplitude nonchaotic vibration which is finally damped out. Properties of the response are further illustrated by samples of solution trajectories in a fixed total energy plane and by related Poincare section plots.

Graphical Methods to Locate the Secondary Instant Centers of Single-Degree-of-Freedom Indeterminate Linkages

2005 ◽  
Vol 127 (2) ◽  
pp. 249-256 ◽  
Author(s):  
David E. Foster ◽  
Gordon R. Pennock
Keyword(s):  
Degree Of Freedom ◽  
Graphical Methods ◽  
Single Degree Of Freedom ◽  
Revolute Joint ◽  
Single Degree ◽  
Revolute Joints ◽  
Straight Line ◽  
Single Link ◽  
Prismatic Joints ◽  
Two Degree Of Freedom

This paper presents graphical techniques to locate the unknown instantaneous centers of zero velocity of planar, single-degree-of-freedom, linkages with kinematic indeterminacy. The approach is to convert a single-degree-of-freedom indeterminate linkage into a two-degree-of-freedom linkage. Two methods are presented to perform this conversion. The first method is to remove a binary link and the second method is to replace a single link with a pair of links connected by a revolute joint. First, the paper shows that a secondary instant center of a two-degree-of-freedom linkage must lie on a unique straight line. Then this property is used to locate a secondary instant center of the single-degree-of-freedom indeterminate linkage at the intersection of two lines. The two lines are obtained from a purely graphical procedure. The graphical techniques presented in this paper are illustrated by three examples of single-degree-of-freedom linkages with kinematic indeterminacy. The examples are a ten-bar linkage with only revolute joints, the single flier eight-bar linkage, and a ten-bar linkage with revolute and prismatic joints.

A Note on a New Stability Method for the Linear Modes of Nonlinear Two-Degree-of-Freedom Systems

1962 ◽  
Vol 29 (2) ◽  
pp. 258-262 ◽  
Author(s):  
Jack Porter ◽  
C. P. Atkinson
Keyword(s):  
Equations Of Motion ◽  
Analog Computer ◽  
Degree Of Freedom ◽  
Oscillatory Systems ◽  
Coupled Equations ◽  
Theoretical Predictions ◽  
Stability Method ◽  
The Stability ◽  
Two Degree Of Freedom

This paper presents a method for analyzing the stability of the linearly related modes of nonlinear two-degree-of-freedom oscillatory systems. For systems described by the coupled equations x¨1 = f(x1, x2) and x¨2 = g(x1, x2) there exist solutions related by the linear modal restraint x1 = cx2 where c is a constant. Such oscillations are not always stable. The method of this paper allows the prediction of the stability of the modes in terms of the amplitudes of the oscillations and the parameters of the equations of motion. Analog-computer results are presented which confirm the theoretical predictions.

Graphical Methods to Locate the Secondary Instant Centers of Single-Degree-of-Freedom Indeterminate Linkages

2004 ◽  
Author(s):  
David E. Foster ◽  
Gordon R. Pennock
Keyword(s):  
Degree Of Freedom ◽  
Graphical Methods ◽  
Single Degree Of Freedom ◽  
Single Degree ◽  
Revolute Joints ◽  
Straight Line ◽  
Single Link ◽  
Prismatic Joints ◽  
Instantaneous Center ◽  
Two Degree Of Freedom

This paper presents graphical techniques to locate the unknown instantaneous centers of zero velocity of planar, single-degree-of-freedom, linkages with kinematic indeterminacy. The approach is to convert a single-degree-of-freedom indeterminate linkage into a two-degree-of-freedom linkage. Two methods are presented to perform this conversion. The first method is to remove a binary link and the second method is to replace a single link with a pair of links connected by a revolute joint. First, the paper shows that a secondary instantaneous center of a two-degree-of-freedom linkage must lie on a unique straight line. Then this property is used to locate a secondary instant center of the single-degree-of-freedom linkage at the intersection of two lines. The two lines are obtained from a purely graphical procedure. The graphical techniques presented in this paper are illustrated by three examples of single-degree-of-freedom linkages with kinematic indeterminacy. The examples are a ten-bar linkage with only revolute joints, the single flier eight-bar linkage, and a ten-bar linkage with revolute and prismatic joints.

scholarly journals Higher order stochastic averaging method for mechanical systems having two degrees of freedom

2004 ◽  
Vol 26 (2) ◽  
pp. 103-110
Author(s):  
Nguyen Duc Tinh
Keyword(s):  
Degrees Of Freedom ◽  
Averaging Method ◽  
Stochastic Averaging ◽  
Stochastic Averaging Method ◽  
Higher Order ◽  
Degree Of Freedom ◽  
Approximation Solution ◽  
Single Degree Of Freedom ◽  
Single Degree ◽  
Two Degree Of Freedom

Higher order stochastic averaging method is widely used for investigating single-degree-of-freedom nonlinear systems subjected to white and coloured random noises.In this paper the method is further developed for two-degree-of-freedom systems. An application to a system with cubic damping is considered and the second approximation solution to the Fokker-Planck (FP) equation is obtained.

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