Nonlinear dynamic singularity analysis of two interconnected synchronous generator system with 1:3 internal resonance and parametric principal resonance

The bifurcation analysis of a simple electric power system involving two synchronous generators connected by a transmission network to an infinite-bus is carried out in this paper. In this system, the infinite-bus voltage are considered to maintain two fluctuations in the amplitude and phase angle. The case of 1：3 internal resonance between the two modes in the presence of parametric principal resonance is considered and examined. The method of multiple scales is used to obtain the bifurcation equations of this system. Then, by employing the singularity method, the transition sets determining different bifurcation patterns of the system are obtained and analyzed, which reveal the effects of the infinite-bus voltage amplitude and phase fluctuations on bifurcation patterns of this system. Finally, the bifurcation patterns are all examined by bifurcation diagrams. The results obtained in this paper will contribute to a better understanding of the complex nonlinear dynamic behaviors in a two-machine infinite-bus （TMIB） power system.

PRINCIPAL RESONANCE IN TRANSVERSE NONLINEAR PARAMETRIC VIBRATION OF AN AXIALLY ACCELERATING VISCOELASTIC STRING

To investigate the principal resonance in transverse nonlinear parametric vibration of an axially accelerating viscoelastic string,the method of multiple scales is applied directly to the nonlinear partial differential equation that governs the transverse vibration of the string.To derive the governing equation,Newton's second law,Lagrangean strain,and Kelvin's model are respectively used to account the dynamical relation,geometric nonlinearity and the viscoelasticity of the string material. Based on the solvability condition of eliminating the secular terms,closed form solutions are obtained for the amplitude and the existence conditions of nontrivial steady-state response of the principal parametric resonance.The Lyapunov linearized stability theory is employed to analyze the stability of the trivial and nontrivial solutions in the principal parametric resonance.Some numerical examples are presented to show the effects of the mean transport speed,the amplitude and the frequency of speed variation.

CHAOTIC BEHAVIOUR OF FORCED OSCILLATOR CONTAINING A SQUARE NONLINEAR TERM ON PRINCIPAL RESONANCE CURVES

In this paper based on [1]we go further into the study of chaotic behaviour of theforced oscillator containing a square nonlinear term by the methods of multiple scalesand numerical simulation. Relation between the chaotic domain and principal resonance curve is discussed. By analyzing the stability of principal resonance curve weinfer that chaotic motion would occur near the frequency at which the principalresonance curve has vertical tangent.Results of numerical simulation confirm thisinference.Thus we offer an effective way to seek the chaotic motion of the systems which are hard to he investigated by Melnikoy method.

PRINCIPAL RESPONSE OF VAN DER POL-DUFFING OSCILLATOR UNDER COMBINED DETERMINISTIC AND RANDOM PARAMETRIC EXCITATION

The principal resonance of Van der Pol-Duffing oscillator to combined deterministic and random parametric excitations is investigated. The method of multiple scales was used to determine the equations of modulation of amplitude and phase. The behavior, stability and bifurcation of steady state response were studied. Jumps were shown to occur under some conditions. The effects of damping, detuning, bandwidth, and magnitudes of deterministic and random excitations are analyzed. The theoretical analysis were verified by numerical results.

VISCO-ELASTIC SYSTEMS UNDER BOTH DETERMINISTIC AND BOUND RANDOM PARAMETRIC EXCITATION

The principal resonance of a visco_elastic systems under both deterministic and random parametric excitation was investigated. The method of multiple scales was used to determine the equations of modulation of amplitude and phase. The behavior, stability and bifurcation of steady state response were studied by means of qualitative analysis. The contributions from the visco_elastic force to both damping and stiffness can be taken into account. The effects of damping, detuning, bandwidth, and magnitudes of deterministic and random excitations were analyzed. The theoretical analysis is verified by numerical results.

RESPONSE OF NONLINEAR OSCILLATOR UNDER NARROW-BAND RANDOM EXCITATION

The principal resonance of Duffing oscillator to narrow_band random parametric excitation was investigated. The method of multiple scales was used to determine the equations of modulation of amplitude and phase. The behavior, stability and bifurcation of steady state response were studied by means of qualitative analyses. The effects of damping, detuning, bandwidth and magnitudes of deterministic and random excitations were analyzed. The theoretical analyses were verified by numerical results. Theoretical analyses and numerical simulations show that when the intensity of the random excitation increases, the nontrivial steady state solution may change from a limit cycle to a diffused limit cycle. Under some conditions the system may have two steady state solutions.

RESPONSE OF PARAMETRICALLY EXCITED DUFFING-VAN DER POL OSCILLATOR WITH DELAYED FEEDBACK

The dynamical behaviour of a parametrically excited Duffing-van der Pol oscillator under linear-plus-nonlinear state feedback control with a time delay is concerned. By means of the method of averaging together with truncation of Taylor expansions, two slow-flow equations on the amplitude and phase of response were derived for the case of principal parametric resonance, it is shown that the stability condition for the trivial solution is only associated with the linear terms in the original systems besides the amplitude and frequency of parametric excitation. And the trivial solution can be stabilized by appreciate choice of gains and time delay in feedback control. Different from the case of the trivial solution, the stability condition for nontrivial solutions is also associated with nonlinear terms besides linear terms in the original system. It is demonstrated that nontrivial steady state responses may lose their stability by saddle-node （SN） or Hopf bifurcation （HB） as parameters vary. The simulations, obtained by numerically integrating the original system, are in good agreement with the analytical results.

PARAMETRIC SHIMMY ANALYSIS

A more reliable parametric shimmy analysis model taking the elasticity of strut, the elasticity of damper linkage system and the influence of shimmy damping into consideration is set up in this paper. The Multiple Scales Method is utilized to obtain the expressions of stability condition and transition curve under the conditions of non resonant case, principal parametric resonance and combination resonance respectively. The parametric shimmy analysis for a real type of aircraft is carried out based on the perturbation analysis results, and the stability chart and critical taxiing velocities of two types of aircraft in the case of principal parametric resonance are given. It is indicated that there exist two critical taxiing velocities near which parametric shimmy is susceptible to occur, while the landing gear system is stable when taxiing velocity is far from them. This characteristic makes the parametric shimmy distinct from the shimmy in general case. Because few previous studies of nose gear shimmy have dealt with parametric shimmy problem, the results given in this paper provide useful reference for aircraft design and maintenance.