Stability Analysis of a Single-Degree-of Freedom Mechanical Model with Distinct Critical Points: I. Bifurcation Theory Approach

The main characteristic of a forced single-degree-of-freedom weakly nonlinear system is determined by its primary, super- and sub-harmonic resonances. A nonlinear parametric feedback control is proposed to modify the steady-state resonance responses, thus to reduce the amplitude of the response and to eliminate the saddle-node bifurcations that take place in the resonance responses. The nonlinear gain of the feedback control is determined by analyzing the bifurcation diagrams associated with the corresponding frequency-response equation, from the singularity theory approach. It is shown by illustrative examples that the proposed nonlinear feedback is effective for controlling three kinds of resonance responses.

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Static and dynamic stability of a single-degree-of-freedom autonomous system with distinct critical points

STRUCTURAL ENGINEERING AND MECHANICS ◽

10.12989/sem.1996.4.5.529 ◽

1996 ◽

Vol 4 (5) ◽

pp. 529-540 ◽

Cited By ~ 4

Author(s):

D.S. Sophianopoulos

Keyword(s):

Dynamic Stability ◽

Critical Points ◽

Autonomous System ◽

Degree Of Freedom ◽

Single Degree Of Freedom ◽

Single Degree ◽

Static And Dynamic Stability

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The Effect of Circumferential Aerodynamic Detuning on Coupled Bending-Torsion Unstalled Supersonic Flutter

Journal of Turbomachinery ◽

10.1115/1.3262045 ◽

1986 ◽

Vol 108 (2) ◽

pp. 253-260 ◽

Cited By ~ 10

Author(s):

D. Hoyniak ◽

S. Fleeter

Keyword(s):

Stability Analysis ◽

Natural Frequency ◽

Frequency Ratio ◽

Center Of Gravity ◽

Degree Of Freedom ◽

Mode Analysis ◽

Single Degree Of Freedom ◽

Single Degree ◽

Unsteady Aerodynamic ◽

Supersonic Flutter

A mathematical model is developed to predict the enhanced coupled bending-torsion unstalled supersonic flutter stability due to alternate circumferential spacing aerodynamic detuning of a turbomachine rotor. The translational and torsional unsteady aerodynamic coefficients are developed in terms of influence coefficients, with the coupled bending-torsion stability analysis developed by considering the coupled equations of motion together with the unsteady aerodynamic loading. The effect of this aerodynamic detuning on coupled bending-torsion unstalled supersonic flutter as well as the verification of the modeling are then demonstrated by considering an unstable twelve-bladed rotor, with Verdon’s uniformly spaced Cascade B flow geometry as a baseline. It was found that with the elastic axis and center of gravity at or forward of the airfoil midchord, 10 percent aerodynamic detuning results in a lower critical reduced frequency value as compared to the baseline rotor, thereby demonstrating the aerodynamic detuning stability enhancement. However, with the elastic axis and center of gravity at 60 percent of the chord, this type of aerodynamic detuning has a minimal effect on stability. For both uniform and nonuniform circumferentially spaced rotors, a single degree of freedom torsion mode analysis was shown to be appropriate for values of the bending-torsion natural frequency ratio lower than 0.6 and higher than 1.2. However, for values of this natural frequency ratio between 0.6 and 1.2, a coupled flutter stability analysis is required. When the elastic axis and center of gravity are not coincident, the effect of detuning on cascade stability was found to be very sensitive to the location of the center of gravity with respect to the elastic axis. In addition, it was determined that when the center of gravity was forward of an elastic axis located at midchord, a single degree of freedom torsion model did not accurately predict cascade stability.

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Resonance Classification in a Cubic System

Journal of Applied Mechanics ◽

10.1115/1.3408856 ◽

1971 ◽

Vol 38 (3) ◽

pp. 585-590 ◽

Cited By ~ 8

Author(s):

D. J. Ness

Keyword(s):

Stability Analysis ◽

Parametric Excitation ◽

Degree Of Freedom ◽

Time Varying ◽

Single Degree Of Freedom ◽

Weakly Nonlinear ◽

Single Degree ◽

Resonance Phenomena ◽

Cubic System

A weakly nonlinear, single-degree-of-freedom cubic system subject simultaneously to a time-varying force and parametric excitation is considered. The various types of resonance phenomena exhibited by the system are classified and a detailed stability analysis is presented for one case of particular interest.

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Stability Analysis of a SDOF Mechanical Model with Distinct Critical Points: II. Catastrophe Theory Approach

World Journal of Mechanics ◽

10.4236/wjm.2015.512025 ◽

2015 ◽

Vol 05 (12) ◽

pp. 266-273

Author(s):

Dimitrios S. Sophianopoulos ◽

Vasiliki S. Pantazi

Keyword(s):

Stability Analysis ◽

Critical Points ◽

Catastrophe Theory ◽

Mechanical Model ◽

Theory Approach

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Computing the Branches, Singularity Trace, and Critical Points of Single Degree-of-Freedom, Closed-Loop Linkages

Journal of Mechanisms and Robotics ◽

10.1115/1.4025752 ◽

2013 ◽

Vol 6 (1) ◽

Cited By ~ 9

Author(s):

David H. Myszka ◽

Andrew P. Murray ◽

Charles W. Wampler

Keyword(s):

Critical Points ◽

Design Parameter ◽

Closed Loop ◽

Turning Points ◽

Critical Curve ◽

Degree Of Freedom ◽

Single Degree Of Freedom ◽

Single Degree ◽

General Method ◽

Number Of Branches

This paper considers single degree-of-freedom (DOF), closed-loop linkages with a designated input angle and one design parameter. For a fixed value of the design parameter, a linkage has input singularities, that is, turning points with respect to the input angle, which break the motion curve into branches. Motion of the linkage along each branch can be driven monotonically from the input. As the design parameter changes, the number of branches and their connections, in short the topology of the motion curve, may change at certain critical points. Allowing the design parameter to vary, the singularities form a curve called the critical curve, whose projection is the singularity trace. Many critical points are the singularities of the critical curve with respect to the design parameter. The critical points have succinct geometric interpretations as transition linkages. This paper presents a general method to compute the singularity trace and its critical points. As an example, the method is used on a Stephenson III linkage, and a range of the design parameter is found where the input angle is able to rotate more than one revolution between singularities. This characteristic is associated with critical points that appear as cusps on the singularity trace.

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