Stability Of Periodic Response Research Articles

Stability of Periodic Orbits and Bifurcation Analysis of Ship Roll Oscillations in Regular Sea Waves

Response, stability, and bifurcation of roll oscillations of a biased ship under regular sea waves are investigated. The primary and subharmonic response branches are traced in the frequency domain employing the Incremental Harmonic Balance (IHB) method with a pseudo-arc-length continuation approach. The stability of periodic responses and bifurcation points are determined by monitoring the eigenvalues of the Floquet transition matrix. The primary and higher-order subharmonic responses experience a cascade of period-doubling bifurcations, eventually culminating in chaotic responses detected by numerical integration (NI) of the equation of motion. Bifurcation diagrams are obtained through the period-doubling route to chaos. Solutions are aided with phase portrait, Poincaré map, time history and Fourier spectrum for better clarity as and when required. Finally, the same ship model is investigated under variable excitation moments that may result from different wave heights in regular seas. The biased ship roll model exhibits primary and subharmonic responses, jump phenomena, coexistence of multiple responses, and chaotically modulated motion. The stable, periodic, and steady-state roll responses obtained by the IHB method are validated by the NI method. Results obtained by both methods are found to agree very well.

Theoretical and Numerical Analysis of Coupled Axial-Torsional Nonlinear Vibration of Drill Strings

Based on a lumped parameter model with two degrees of freedom, the periodic response of the coupled axial-torsional nonlinear vibration of drill strings is studied by HB-AFT (harmonic balance and alternating frequency/time domain) method and numerical simulations. The amplitude-frequency characteristic curves of axial relative displacement and torsional relative angular velocity are given to reveal the mechanism of bit bounce and stick-slip motion. The stability of periodic response is analyzed by Floquet theory, and the boundary conditions of bifurcation of periodic response are given when parameters such as nominal drilling pressure, angular velocity of turntable, and formation stiffness are varied. The results show that the amplitude of the periodic response of the system precipitates a spontaneous jump and Hopf bifurcation may occur when the angular velocity of the turntable is varied. The variation of parameters may lead to the complex dynamic behavior of the system, such as period-doubling motion, quasiperiodic motion, and chaos. Bit bounce and stick-slip phenomenon can be effectively suppressed by varying the angular velocity of turntable and nominal drilling pressure.

Dynamics and nonlinear effects of a compact near-zero frequency vibration isolator with HSLD stiffness and fluid damping enhancement

Aiming to isolate disturbance vibration for heavy payloads with low frequency, a novel hydro-pneumatic near-zero frequency vibration isolator is proposed, which possesses high-static and low-dynamic (HSLD) stiffness. And different from most isolators existing previously, a nonlinear damping strategy realized by fluid damping mechanism is implemented into the device in order to enhance vibration isolation performance. Due to that the natural frequency of isolation system is close to zero and loading capacity can be adjusted by the gas pressure, the proposed device is termed as pneumatic near-zero frequency (NZF) vibration isolator. To obtain the primary resonance response for the harmonically forced nonlinear system, the averaging method is employed first and verified numerically. A direct current (d. c.) term is added in the analytical trial solution, which reflects the asymmetry existed in dynamic response. As a result, multiple steady solutions are observed and the induced amplitude jump appears. Then the stability of periodic response is carried out by using the Jacobin matrix approach. It is concluded that increases of all types damping including linear, square and friction damping are able to shrink the unstable region and further reduce the possibility of occurrence of amplitude sudden-jump in primary resonance. More accurately, the critical border which can separate the jump and none jump zones is found analytically and numerically, and design criterion for avoidance of sudden-jump is developed. After that, equivalent damping analysis gives a new insight into how nonlinearity effects of each damping dominate vibration isolation transmissibility in resonance and effective isolation frequency band. It is shown that the square type damping based on fluid mechanism can suppress the resonance response and transmissibility and does not decrease the effectiveness in isolation band, which will bring the favorable advantage for NZF isolator in practical application.

Periodic solution and its stability of spring pendulum with horizontal base motion

In this paper, the nonlinear dynamic response of spring pendulum with horizontal base motion is studied. The dynamical equations of the system are established by using Lagrange equation. The discrete Fourier transform, harmonic balance method and homotopy continuation method are combined to solve the periodic response of the system, which avoids the limitation of the small amplitude caused by the Taylor expansion in the traditional analytical method. The comparison with the numerical results shows that the proposed method in this paper can not only be used to solve the large amplitude vibration of spring pendulum, but also has a high accuracy. The stability of periodic response is studied by using Floquet theory. The effects of amplitude and frequency of base motion on the periodic response of the system are given, and the bifurcation characteristics of the periodic solution are analyzed. The results show that the influence curve of the base frequency on the periodic response has two peaks, and with the increase of the amplitude of the base motion, the two peaks will shift to the different sides respectively. When the base amplitude is large, the periodic response amplitude changes with the frequency of the foundation motion, and there will be two jumps. The amplitude of the periodic solution increases with the base amplitude. For some base frequencies, the amplitude of the periodic solution will jump with the change of the base amplitude. When the amplitude and frequency of the system are large, the periodic response of the system may be unstable. After the instability, the spring pendulum enters the continuous rotation state, and the amplitude in the breathing direction is great, the system will be destroyed. It is found that Hopf bifurcation may occur in the periodic response of the system corresponding to some base frequencies and amplitudes. The variation of the system response with the base frequency and amplitude after the Hopf bifurcation is studied numerically by the Runge-Kutta method. Complex dynamical behaviors such as periodic motion, almost periodic motion, torus doubling and chaos are found. It is shown that the main path of the system entering chaos is almost periodic torus rupture and paroxysmal. Finally, the influence analysis of the base frequency and amplitude is synthesized, and the transition of the response form on the plane of the basic motion parameters is given. The results of this paper provide a theoretical reference for the analysis and design of spring pendulum in engineering.

Nonlinear dynamic analysis of a rub-impact rotor supported by oil film bearings

A general model of a rub-impact rotor system is set up and supported by oil film journal bearings. The Jacobian matrix of the system response is used to calculate the Floquet multipliers, and the stability of periodic response is determined via the Floquet theory. The nonlinear dynamic characteristics of the system are investigated when the rotating speed and damping ratio is used as control parameter. The analysis methods are inclusive of bifurcation diagrams, Poincare maps, phase plane portraits, power spectrums, and vibration responses of the rotor center and bearing center. The analysis reveals a complex dynamic behavior comprising periodic, multi-periodic, chaotic, and quasi-periodic response. The modeling results thus obtained by using the proposed method will contribute to understanding and controlling of the nonlinear dynamic behaviors of the rotor-bearing system.

A method for determining the periodic solution and its stability of non-linear bearing-rotor system based on observed states of the system

In this article, stability and bifurcation of a hydrodynamic bearing-rotor system are analysed. A numerical method is presented to find the periodic responses of the hydrodynamic bearing-rotor system with the change of the system parameter. The observed states of the system are used to solve inversely the Jacobian matrix and the obtained Jacobian matrix is used to calculate the Floquet multiplier; then the stability of periodic response can be determined by the Floquet theory. The periodic responses and their stability with the change of the system parameter can be calculated by the proposed method when steady-state and transient-state information are observed online. The proposed method is applied to a rotor system with elliptical bearing supports to determine non-linear periodic responses and their stability. The combination of the predictor-corrector mechanism and the Poincaré-Newton-Floquet method is also applied to the system. Comparison of the two methods proves the proposed method to be effective. Taking rotating speed as the bifurcation parameter, the periodic, quasi-periodic, coexistent, jump, and chaotic solutions of the system are computed. The numerical results reveal the rich and complex non-linear behaviours of the system.

Numerical and Perturbation Analysis of Dynamic Post-buckling of Rectangular Laminated Plates

A new phenomenon for laminated composite plates undergoing dynamic post-buckling is presented. The buckled composite plate subjected to in-plane dynamic load not only gains more non-linearity by the quadratic term, which arises from static buckling, but also changes its dynamic behavior from pure parametric vibration to combined parametric and forcing traverse vibrations. The nonlinear behavior of a simply supported laminated composite plate undergoing dynamic post-buckling is investigated. The range of bifurcation parameter (forcing frequency) for chaotic motion is obtained, and the characteristics of the dynamic behavior of the laminated composite plate in post-buckling are unveiled. Hamilton's Principle, Galerkin's Method, and Lagrange's Equation are utilized to obtain the equations of motion with higher-order shear deformation. The differential equation has quadratic and cubic non-linearities as well as parametric and harmonic excitation terms. The method of multiple scales is used to determine the equations that describe the second order modulation of the amplitude and phase. Floquet theory is used to analyze the stability of periodic responses.