A qualitative investigation of the oscillations of a pendulum with a periodically varying length and a mathematical model of a swing

The behaviour of the amplitude-frequency characteristics of families of periodic solutions, produced from the equilibrium position of a system, is established by a qualitative investigation of the equation of the oscillations of a pendulum, the length of which is an arbitrary periodic function of time. The non-local conditions for their stability and instability, expressed in terms of the amplitude and frequency of the oscillations, are obtained. The results are used when discussing the parametric and self-excited oscillatory model of a swing. In the parametric model the length of a swing is a specified periodic function of time, and in the self-excited oscillatory model it is a function of the phase coordinates of the system. For an appropriate choice of these functions, both systems have a common periodic solution. It is shown that the parametric model leads to an erroneous conclusion regarding the instability of the periodic mode, which is in fact realized in the oscillations of a swing, whereas the self-excited oscillatory model indicates its stability.     

Bifurcation of travelling wave solutions for the generalized ZK equations

By using the bifurcation theory of planar dynamical systems to the generalized ZK equations, the existence of smooth and non-smooth travelling wave solutions is proved. Under different regions of parametric spaces, various sufficient conditions to guarantee the existence of above solutions are given. Some exact explicit parametric representations of the above waves are determined.     

Bifurcation of travelling wave solutions for the generalized ZK-BBM equations

By using the bifurcation theory of planar dynamical systems to the generalized ZK-BBM equations, the existence of smooth and non-smooth travelling wave solutions is proved. Under different regions of parametric spaces, various sufficient conditions to guarantee the existence of above solutions are given. Some exact explicit parametric representations of the above waves are determined.     

Hyperbolic systems of conservation laws with a symmetry

Hyperbolic systems of conservaiton laws with a symmetry are studied. Some peculiar phenomena for such systems are shown. Admissibility criteria for solutions to such systems are discussed. propagation and cancellation of initial oscillations for the systems are classified. As a byproduct of this study, an existence theorem of global solutions for the Cauchy of the systems is established.     

Bifurcation of travelling wave solutions for the generalized KP-MEW equations

By using the theory of bifurcations of planar dynamical systems to the generalized KP-MEW equations, the existence of smooth and non-smooth travelling wave solutions is proved. Under different regions of parametric spaces, various sufficient conditions to guarantee the existence of the above solutions are given. Some exact explicit parametric representations of the above waves are obtained.     

Bifurcation of travelling wave solutions for the generalized Camassa–Holm–KP equations

By using the bifurcation theory of planar dynamical systems to the generalized Camassa–Holm–KP equations, the existence of smooth and non-smooth travelling wave solutions is proved. Under different regions of parametric spaces, various sufficient conditions to guarantee the existence of above solutions are given. Some exact explicit parametric representations of the above waves are determined.     

轴向变速运动弦线的非线性振动的稳态响应及其稳定性

研究具有几何非线性的轴向运动弦线的稳态横向振动及其稳定性．轴向运动速度为常平均速度与小简谐涨落的叠加．应用Hamilton原理导出了描述弦线横向振动的非线性偏微分方程．直接应用于多尺度方法求解该方程．建立了避免出现长期项的可解性条件．得到了近倍频共振时非平凡稳态响应及其存在条件．给出数值例子说明了平均轴向速度、轴向速度涨落的幅值和频率的影响．应用Liapunov线性化稳定性理论,导出倍频参数共振时平凡解和非平凡解的不稳定条件．给出数值算例说明相关参数对不稳定条件的影响．     

The effect of transverse crack upon parametric instability of a rotor-bearing system with an asymmetric disk

It is well known that either the asymmetric disk or transverse crack brings parametric inertia (or stiffness) excitation to the rotor-bearing system. When both of them appear in a rotor system, the parametric instability behaviors have not gained sufficient attentions. Thus, the effect of transverse crack upon parametric instability of a rotor-bearing system with an asymmetric disk is studied. First, the finite element equations of motion are established for the asymmetric rotor system. Both the open and breathing transverse cracks are taken into account in the model. Then, the discrete state transition matrix (DSTM) method is introduced for numerically acquiring the instability regions. Based upon these, some computations for a practical asymmetric rotor system with open or breathing transverse crack are conducted, respectively. Variations of the primary and combination instability regions induced by the asymmetric disk with the crack depth are observed, and the effect of the orientation angle between the crack and asymmetric disk on various instability regions are discussed in detail. It is shown that for the asymmetric angle around 0, the existence of transverse (either open or breathing) crack has attenuation effect upon the instability regions. Under certain crack depth, the instability regions could be vanished by the transverse crack. When the asymmetric angle is around π/2, increasing the crack depth would enhance the instability regions.     

The stability of the equilibrium of a pendulum of variable length

The frequencies and modes of parametric oscillations of a pendulum of variable length for values of the modulation index from the smallest to the limit admissible values are investigated. The limits of the resonance zones of the first four oscillation modes are constructed and investigated by analytical and numerical methods, and the fundamental qualitative properties of the higher modes are established. Complete degeneracy of the modes with even numbers, i.e., coincidence of the frequencies of the odd and even eigenmodes for admissible values of the modulation parameter, is proved. The global pattern of the limits of the regions of stability of the lower position of equilibrium is constructed and it is shown that it differs considerably from the Ince–Strutt diagrams. Specific properties of the eigenmodes are established.     

A frequency-parametric analysis of natural oscillations of non-uniform rods

A constructive numerical-analytical method of investigating free transverse oscillations of a highly non-uniform rod with the boundary conditions of elastic clamping is developed. Standard special cases of the boundary conditions are also considered. To solve the corresponding self adjoint boundary eigenvalue and eigenfunction problem, an effective computational procedure for determining the frequencies and shapes of the oscillations is set up, similar to the shooting method. Assertions equivalent to the Sturm comparison theorems and corollaries of them for second-order boundary-value problems are formulated. The algorithm is tested on model examples with known solutions. A parametric synthesis is carried out for a family of conical rods for different boundary conditions, which is important for applications. The results obtained are compared with the classical results of Kirchhoff, Timoshenko and Gould.     

Oscillation theorems for nonlinear hyperbolic systems with impulses

In this paper, we are mainly concerned with the oscillation problems of nonlinear hyperbolic differential systems with impulses. Some sufficient conditions for oscillations of the solutions of nonlinear impulsive hyperbolic systems with Robin boundary conditions are obtained and the criteria of oscillation of the systems are established.     

Features of the behavior of solutions to a nonlinear dynamical system in the case of two-frequency parametric resonance

Two-frequency parametric resonance in nonlinear dynamical systems is studied by analyzing a delay differential equation with the delay obeying a two-frequency law, which arises in the mathematical simulation of some physical processes. It is shown that the system can exhibit chaotic oscillations (strange attractors) when the parametric excitation frequencies are both close to the doubled eigenfrequency of the system (degenerate case). The formation mechanisms of chaotic attractors are discussed, and the Lyapunov exponents and the Lyapunov dimension are calculated for them. If only one of the parametric excitation frequencies is close to the double eigenfrequency, a two-frequency regime occurs in the system.     

变速度轴向运动粘弹性梁的动态稳定性

研究速度变化的轴向运动粘弹性梁在亚谐波共振及组合共振范围内的参数振动．通过平均法,在运动参数激励频率为2倍固有频率或为两阶固有频率之和附近时得到了自治的常微分方程组．在参数激励频率和激励振幅平面上,可以找到由于共振而产生的失稳区域,并应用数值方法验证了理论推导结果的正确性．分析了粘弹性阻尼,速度和预紧张力对失稳区域的影响．粘弹性阻尼使得共振失稳区域减小,而速度和预紧张力使共振失稳区域在频率-振幅平面上发生漂移．     

Explicit exact periodic wave solutions and their limit forms for a long waves-short waves model

A long waves-short waves model is studied by using the approach of dynamical systems. The sufficient conditions to guarantee the existence of solitary wave, kink and anti-kink waves, and periodic wave in different regions of the parametric space are given. All possible explicit exact parametric representations of above traveling waves are presented. When the energy of Hamiltonian system corresponding to this model varies, we also show the convergence of the periodic wave solutions, such as the periodic wave solutions converge to the solitary wave solutions, kink and anti-kink wave solutions, and periodic wave solutions, respectively.     

The Cauchy Problem for a Class of Coupled Systems Containing a Convolution Operator

Some results on the invariant regions ,existence and uniqueness of solutions to a class of integrodifferential systems are established. Applying these results to integrodifferential systems with a small parameter ε > 0, we obtain,in particular, some estimates of solutions uniform in ε > 0.     

The stability of the parametric oscillations of second-order non-linear systems

The parametric oscillations of strongly non-linear systems with one degree of freedom are considered using a more general definition of these oscillations than the generally accepted definition. Stability criteria, that are verifiable using the signs of the derivatives of the amplitude-frequency characteristics, are found for the two families of periodic solutions corresponding to the fundamental parametric resonance. A condition is indicated under which the latter are monotonic and, as a result, one of the families is stable and the other is unstable. It is shown that, in a system with a concave non-monotonic elastic characteristic, the stable family loses stability for fairly large amplitudes and this effect is not revealed by the well-known analytical methods of non-linear mechanics.     

BIFURCATIONS OF TRAVELLING WAVE SOLUTIONS FOR THE GENERALIZED DODD-BULLOUGH-MIKHAILOV EQUATION

In this paper, the generalized Dodd-Bullough-Mikhailov equation is studied. The existence of periodic wave and unbounded wave solutions is proved by using the method of bifurcation theory of dynamical systems. Under different parametric conditions, various sufficient conditions to guarantee the existence of the above solutions are given.Some exact explicit parametric representations of the above travelling solutions are obtained.     

BIFURCATIONS OF TRAVELLING WAVE SOLUTIONS TO A COUPLED NONLINEAR WAVE SYSTEM

Employ theory of bifurcations of dynamical systems to a system of coupled nonlin-ear equations, the existence of solitary wave solutions, kink wave solutions, anti-kink wave solutions and periodic wave solutions is obtained. Under different parametric conditions, various suffcient conditions to guarantee the existence of the above so-lutions are given. Some exact explicit parametric representations of travelling wave solutions are derived.     

Parametric vibrations of an orthotropic cylindrical shell with an elastic core. 2. Some numerical results

It is shown that the operator of the problem belongs to the class of oscillation operators. The first five regions of dynamic instability of an elastic orthotropic cylindrical shell with an elastic isotropic core are determined in the first approximation. The effect of the core and transverse shear in the shell on the width and location of these regions of dynamic instability of the system is determined. The effect of transverse shear on the natural frequencies of the empty shell is established.     

一类二次KdV类型水波方程的行波解

应用平面动力系统理论研究了一类非线性KdV方程的行波解的动力学行为．在参数空间的不同区域内,给出了系统存在孤立波解,周期波解,扭子和反扭子波解的充分条件,并计算出所有可能的精确行波解的参数表示．