Model simulation of parametrically excited ship rolling

Two different models for simulating the ship motion in longitudinal or oblique seas are presented and studied in detail. Particular attention is devoted to the parametrically induced rolling which may be established by means of the nonlinear coupling between both heave-roll and/or pitch-roll motions. It is proved that the phenomenon is likely to occur with this mechanism when the roll frequency is subharmonic of the encounter wave frequency and when the vertical motions become resonant.     

Nonlinear Effects on Coexistence Phenomenon in Parametric Excitation

We investigate the effect of nonlinearites on a parametrically excited ordinary differential equation whose linearization exhibits the phenomenon of coexistence. The differential equation studied governs the stability of a mode of vibration in an unforced conservative two degree of freedom system used to model thefree vibrations of a thin elastica. Using perturbation methods, we show thatat parameter values corresponding to coexistence, nonlinear terms can cause the origin to become nonlinearly unstable,even though linear stability analysis predicts the origin to be stable.We also investigate the bifurcations associated with this instability.     

Simulation of parametric ship rolling: Effects of hull bending and torsional elasticity

An enhanced mechanical model for simulating ship body oscillations and both the induced fluxural and twisting vibrations of the hull in the case of longitudinal seas is presented. The onset of parametric rolling, which may result from nonlinearly coupled heave-pitch-roll motions, and the effects of bending and torsional elasticity of the hull are considered in detail. It is shown that in the above sea conditions the flexural and/or twisting vibrations are likely to occur under a mechanism similar to that of parametric rolling.     

Parametric Resonance of Hopf Bifurcation

We investigate the dynamics of a system consisting of a simple harmonic oscillator with small nonlinearity, small damping and small parametric forcing in the neighborhood of 2:1 resonance. We assume that the unforced system exhibits the birth of a stable limit cycle as the damping changes sign from positive to negative (a supercritical Hopf bifurcation). Using perturbation methods and numerical integration, we investigate the changes which occur in long-time behavior as the damping parameter is varied. We show that for large positive damping, the origin is stable, whereas for large negative damping a quasi-periodic behavior occurs. These two steady states are connected by a complicated series of bifurcations which occur as the damping is varied.     

Local bifurcation theory of nonlinear systems with parametric excitation

This paper summarizes the authors' research on local bifurcation theory of nonlinear systems with parametric excitation since 1986. The paper is divided into three parts. The first one is the local bifurcation problem of nonlinear systems with parametric excitation in cases of fundamental harmonic, subharmonic and superharmonic resonance. The second one is the experiment investigation of local bifurcation solutions in nonlinear systems with parametric excitation. The third one is the universal unfolding study of periodic bifurcation solutions in the nonlinear Hill system, where the influence of every physical parameter on the periodic bifurcation solution is discussed in detail and all the results may be applied to engineering.     

On unpredictable ship rolling in irregular seas

Indirectly excited large amplitude roll motions of container ships and roll-on/roll-off ferries represent a considerable threat to goods and life in modern navigation. Especially when ships are heading into rough seas, or sailing away from following seas, large roll responses have been reported to occur in an unpredictable, random way. We approach this topic from deterministic bifurcation theory and generalize the analysis by characterizing the seaway as narrow band stochastic processes. It is demonstrated how the intermittent behavior of zero and large amplitude motions in irregular seas is related to the deterministic bifurcation scenario. Further, we identify parameter regions of intermittent roll motions and show how the on-/off intermittency can be characterized by the probability density function of the response amplitude.     

随机干扰与随机参数激励联合作用下的Hopf分叉

本文研究van der Pol-Duffing型的非线性振子在随机干扰和随机参数联合作用下的Hopf分叉现象。本文所得结果证实了当系统处在于Hopf分叉点附近时,对系统的参数的变化具有敏感性。在研究过程中,我们利用Markov扩散过程逼近系统的随机响应,得到了沿稳定矩的概率1稳定和矩稳定的条件。对于非线性振子,我们得到了振幅过程的稳态概论密度函数。研究发现,确定性系统的Hopf分叉点在随机参数作用下具有漂移现象,这种漂移是由系统的性质所决定的,当分叉点为超临界的,分叉点向前漂移;而当分叉点为亚临界时,这种漂移是向后的。当系统处在外部随机干扰作用下时,系统出现非零响应。另外我们发现,稳态矩的分叉与其阶数无关。     

Electromagnetically driven parametric instability in an annular layer

Local measurements of the free surface amplitude of an annular layer of mercury submitted to a low frequency vertical magnetic filed are performed. The results confirm the linear analysis which predicts two types of classical transitions (harmonic and sub-harmonic) and a specific electromagnetic transition occurring with a combination of frequences.     

Simulation of parametric ship roll and hull twist oscillations

A new mechanical model for simulating both the ship oscillations and the induced twisting of the hull in the case of longitudinal seas is presented. Particular attention is given to the onset of parametric rolling, which may result from non-linearly coupled heave-pitch-roll motions. It is shown that in these sea conditions the phenomenon of twisting is likely to occur under a mechanism similar to that of parametric rolling.     

An experimental study of bifurcation,chaos, and dimensionality in a system forced through a bifurcation parameter

An experimental study of a system that is parametrically excited through a bifurcation parameter is presented. The system consits of a lightly-damped, flexible beam which is buckled and unbuckled magnetically: it is parametrically excited by driving an electromagnet with a low-frequency sine wave. For voltage amplitudes in excess of the static bifurcation value, the beam slowly switches between the one-and two-well configurations. Experimental static and dynamic bifurcation results are presented. Static bifurcatons for the system are shown to involve a butterfly catastrophe. The dynamic bifurcation diagram, obtained with an automated data acquisition system, shows several period-doubling sequences, jump phenomena, and a chaotic region. Poincaré sections of a chaotic steady-state are obtained for various values of the driving phase, and the correlation dimension of the chaotic attractor is estimated over a large scaling region. Singular system analysis is used to demonstrate the effect of delay time on the noise level in delay-reconstructions, and to provide an independent check on the dimension estimate by directly estimating the number of independent coordinates from time series data. The correlation dimension is also estimated using the delay-reconstructed data and shown to be in good agrement with the value obtained from the Poincaré sections. The bifurcation and dimension results are used together with physical sonsiderations to derive the general form of a single-degree-of-freedom model for the experimental system.     

参外联合激励下的簇发振荡及其机理

当周期激励频率远小于系统固有频率时,会存在快慢耦合效应,与单项激励不同,参外联合激励不仅会导致快子系统平衡曲线和分岔行为的复杂化,也会产生一些特殊的非线性现象,为此,本文以两耦合Hodgkin-Huxley细胞模型为例,引入周期参外联合激励,探讨在频域不同尺度耦合时该系统的簇发振荡的特点及其分岔机制．通过建立相应的快慢子系统,得到慢变参数变化下的快子系统的各种分岔模式以及相应的分岔行为,结合转换相图,揭示耦合系统随激励幅值变化时的动力学行为及其机理．研究表明,在激励幅值较小时,系统表现为概周期振荡,两频率分别近似于快子系统平衡曲线由Hopf分岔引起的两稳定极限环的振荡频率．概周期解随激励幅值的增加进入簇发振荡,导致这些簇发振荡的主要原因是在慢变参数变化的部分区间内,存在唯一稳定的平衡曲线,使得系统的轨迹逐渐趋向该平衡曲线,产生沉寂态,并随着慢变参数的变化,由分岔进入激发态．同时,快子系统中参与簇发振荡的稳定吸引子随激励幅值的变化也会不同,导致不同形式的簇发振荡．另外,与单项激励下的情形不同,联合激励时快子系统的部分稳定吸引子掩埋在其它稳定吸引子内,从而失去对簇发振荡的影响．     

Transition from Planar to Whirling Oscillations in a Certain Nonlinear System

A single-mass two-degrees-of-freedom system is considered, witha radially oriented nonlinear restoring force. The latter is smooth andbecomes infinite at a certain value of a radial displacement. Stabilityanalysis is made for planar oscillation, or motion along a givendirection. As long as this motion is periodic, the nonlinearity in therestoring force provides a periodic parametric excitation in thetransverse direction. The linearized stability analysis is reduced tostudy of the Mathieu equation for the (infinitesimal) motions in thetransverse direction. For the case of free oscillations in the givendirection an exact solution is obtained, since a specific analyticalform is used for the (strongly nonlinear) restoring force, which permitsexplicit integration of the equation of motion. Stability of the planarmotion in this case is shown to be very sensitive to even slightdeviations from polar symmetry in the restoring force (as well as to theamplitude of oscillations in the given direction). Numerical integrationof the original equations of motion shows the resulting motion to be awhirling type indeed in case of the transversal instability. For thecase of a sinusoidal forcing in the given direction solution for the(periodic) response is obtained by Krylov–Bogoliubov averaging. Thisresults in the transmitted Ince–Strutt chart – namely, stabilitychart for transverse direction on the amplitude-frequency plane of theexcitation in the original direction.     

Milling Bifurcations from Structural Asymmetry and Nonlinear Regeneration

This paper investigates multiple modeling choices for analyzing the rich and complex dynamics of high-speed milling processes. Various models are introduced to capture the effects of asymmetric structural modes and the influence of nonlinear regeneration in a discontinuous cutting force model. Stability is determined from the development of a dynamic map for the resulting variational system. The general case of asymmetric structural elements is investigated with a fixed frame and rotating frame model to show differences in the predicted unstable regions due to parametric excitation. Analytical and numerical investigations are confirmed through a series of experimental cutting tests. The principal results are additional unstable regions, hysteresis in the bifurcation diagrams, and the presence of coexisting periodic and quasiperiodic attractors which is confirmed through experimentation.     

参数激励与强迫激励联合作用下非线性振动系统的分叉

本文利用多尺度法研究了参数激励与强迫激励联合作用下非线性振动系统的分叉问题,给出了分叉集和八种分叉响应曲线。     

Periodic Response and Chaos in Nonlinear Systems with Parametric Excitation and Time Delay

In this paper, the periodic motions of a nonlinear system with quadratic,cubic, and parametrically excited stiffness terms and with time-delayterms are obtained by the incremental harmonic balance (IHB) method. Theelements of the Jacobian matrix and residue vector arising in the IHBformulation are derived in closed form. A mechanism model representingthe one-mode oscillation of beams and plates is considered as anexample. A path-following algorithm with an arc-length parametriccontinuation procedure is used to obtain the response diagrams. Thesystem also exhibits chaotic motion through a cascade of period-doublingbifurcations, which is characterized by phase planes, Poincaré sectionsand Lyapunov exponents. The interpolated cell mapping (ICM) procedure isused to obtain the initial condition map corresponding to multiplesteady-state solutions.     

1／2亚谐共振──主参数共振情况下非线性振动系统的余维2退化分叉

本文应用Normal Form理论和退化向量场的普适开折理论研究了参数激励与强迫激励联合作用下非线性振动系统的余维2退化分叉,用Melnikov方法讨论了全局分叉的存在性.     

The Study of a Parametrically Excited Nonlinear Mechanical System with the Continuation Method

The dynamic behavior of a one-degree-of-freedom, parametrically excited nonlinear system is investigated. The Galerkin method is applied to the principal and fundamental parameteric resonance of the system. The continuation method is used to study the change of harmonic oscillation with respect to the variation of excitation frequency. The numerical stability analysis of the trivial solution is carried out and the stable and unstable regions of the trivial solution are given. They are found to agree with the results obtained by the analytical method of Galerkin. Periodic solutions are traced and the coexistence of multi-periodic solutions is observed With the change of excitation frequency the large amplitude periodic-2 oscillation is found to be in the same closed branch with the small amplitude periodic-2 solution. In addition, the bifurcation pattern of the trivial solution is found to change from subcritical Hopf bifurcation into supercritical Hopf bifurcation with the increase of excitation amplitude. Combined with the conventional numerical integration method, new complex dynamic behavior is detected.     

Nonlinear dynamic modeling and periodic vibration of a cantilever beam subjected to axial movement of basement

Dynamic modeling of a cantilever beam under an axial movement of its basement is presented. The dynamic equation of motion for the cantilever beam is established by using Kane's equation first and then simplified through the Rayleigh-Ritz method. Compared with the older modeling method, which linearizes the generalized inertia forces and the generalized active forces, the present modeling takes the coupled cubic nonlinearities of geometrical and inertial types into consideration. The method of multiple scales is used to directly solve the nonlinear differential equations and to derive the nonlinear modulation equation for the principal parametric resonance. The results show that the nonlinear inertia terms produce a softening effect and play a significant role in the planar response of the second mode and the higher ones. On the other hand, the nonlinear geometric terms produce a hardening effect and dominate the planar response of the first mode. The validity of the present modeling is clarified through the comparisons of its coefficients with those experimentally verified in previous studies. Project supported by the Fundamental Fund of National Defense of China (No. 10172005).     

Application of Special Nonsmooth Temporal Transformations to Linear and Nonlinear Systems under Discontinuous and Impulsive Excitation

Linear and nonlinear mechanical systems under periodic impulsive excitation are considered. Solutions of the differential equations of motion are represented in a special form which contains a standard pair of nonsmooth periodic functions and possesses a convenient structure. This form is also suitable in the case of excitation with a periodic series of discontinuities of the first kind (a stepwise excitation). The transformations are illustrated in a series of examples. An explicit form of analytical solutions has been obtained for periodic regimes. In the case of parametric impulsive excitation, it is shown that a nonequidistant distribution of the impulses with dipole-like temporal shifts may significantly effect the qualitative characteristics of the response. For example, the sequence of instability zones loses its different subsequences depending on the parameter of the shifts. It is shown that the method's applicability can be extended for nonperiodic regimes by involving the idea of averaging.