Resonance Classification in a Cubic System

1971 ◽  
Vol 38 (3) ◽  
pp. 585-590 ◽  
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.

RESONANCE CONTROL FOR A FORCED SINGLE-DEGREE-OF-FREEDOM NONLINEAR SYSTEM

2004 ◽  
Vol 14 (04) ◽  
pp. 1423-1429 ◽  
Author(s):  
ANDREW Y. T. LEUNG ◽  
JIN CHEN JI ◽  
GUANRONG CHEN
Keyword(s):  
Feedback Control ◽  
Nonlinear System ◽  
Singularity Theory ◽  
Degree Of Freedom ◽  
Theory Approach ◽  
Nonlinear Feedback ◽  
Single Degree Of Freedom ◽  
Weakly Nonlinear ◽  
Single Degree ◽  
Resonance Control

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.

scholarly journals On the critical forcing amplitude of forced nonlinear oscillators

2013 ◽  
Vol 3 (4) ◽  
Author(s):  
Mariano Febbo ◽  
Jinchen Ji
Keyword(s):  
Primary Resonance ◽  
Nonlinear Oscillators ◽  
Analytical Form ◽  
Excitation Amplitude ◽  
Degree Of Freedom ◽  
Single Degree Of Freedom ◽  
Weakly Nonlinear ◽  
Single Degree ◽  
Saddle Node ◽  
Basis Method

AbstractThe steady-state response of forced single degree-of-freedom weakly nonlinear oscillators under primary resonance conditions can exhibit saddle-node bifurcations, jump and hysteresis phenomena, if the amplitude of the excitation exceeds a certain value. This critical value of excitation amplitude or critical forcing amplitude plays an important role in determining the occurrence of saddle-node bifurcations in the frequency-response curve. This work develops an alternative method to determine the critical forcing amplitude for single degree-of-freedom nonlinear oscillators. Based on Lagrange multipliers approach, the proposed method considers the calculation of the critical forcing amplitude as an optimization problem with constraints that are imposed by the existence of locations of vertical tangency. In comparison with the Gröbner basis method, the proposed approach is more straightforward and thus easy to apply for finding the critical forcing amplitude both analytically and numerically. Three examples are given to confirm the validity of the theoretical predictions. The first two present the analytical form for the critical forcing amplitude and the third one is an example of a numerically computed solution.

Fast Simulation of a Single Degree-of-Freedom System Consisting of An Iwan Element Using the Method of Averaging

2020 ◽  
Vol 142 (5) ◽  
Author(s):  
Drithi Shetty ◽  
Matthew Allen
Keyword(s):  
Equations Of Motion ◽  
Element Model ◽  
Degree Of Freedom ◽  
Single Degree Of Freedom ◽  
Weakly Nonlinear ◽  
Single Degree ◽  
Integration Schemes ◽  
Current Implementation ◽  
Modal Model ◽  
Stiffness And Damping

Abstract While Iwan elements have been used to effectively model the stiffness and energy dissipation in bolted joints, integrating the equations of motion of these elements is fairly expensive since implicit schemes, such as Newmark’s methods, need to be used. This paper presents a method of simulating dynamic systems containing nonlinear Iwan elements that significantly reduce the computation cost by using closed-form expressions for stiffness and damping in the microslip regime and an averaging method for regions of time in which no external force is applied. The proposed algorithm is demonstrated on a single degree-of-freedom (SDOF) system to evaluate the range over which it retains accuracy and the improvement in performance it offers. Although the current implementation is limited to SDOF systems, it can be used to simulate the response of each mode in structures exhibiting weak nonlinearity that can be modeled using the modal Iwan approach. To verify this, the dynamic response of a finite element model of a beam assembly, integrated using the Newmark-β method, has been compared with its equivalent modal model integrated using the proposed algorithm. The results show that the algorithm accurately predicts the response in a fraction of the time taken by implicit integration schemes, so long as the modes remain uncoupled and weakly nonlinear.

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

1986 ◽  
Vol 108 (2) ◽  
pp. 253-260 ◽  
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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