Normal modes of a double pendulum at low energy levels

Nonlinear Dynamics - This study is concerned with a double pendulum and its regular behaviour associated with low energy levels and the influence of the associated initial conditions on the...     

一种提高冷镦件模具寿命的设计方法

 针对采用整模结构的冷镦模具疲劳寿命极低的现状，提出以三层圆筒组合模具结构 替代整体模具, 改变模具受力方式, 使冷镦件模具达到较高寿命的设计方法. 以螺栓 圆头的冷镦试验结果为依据，以力学的应力分析方法为基础，应用力学理论，导出了生产实 践中实用的组合模具设计原理与经验公式，并举例说明了该方法的应用.     

Bifurcations in the dynamics of an orthogonal double pendulum

The weakly nonlinear resonant response of an orthogonal double pendulum to planar harmonic motions of the point of suspension is investigated. The two pendulums in the double pendulum are confined to two orthogonal planes. For nearly equal length of the two pendulums, the system exhibits 1:1 internal resonance. The method of averaging is used to derive a set of four first order autonomous differential equations in the amplitude and phase variables. Constant solutions of the amplitude and phase equations are studied as a function of physical parameters of interest using the local bifurcation theory. It is shown that, for excitation restricted in either plane, there may be as many as six pitchfork bifurcation points at which the nonplanar solutions bifurcate from the planar solutions. These nonplanar motions can become unstable by a saddle-node or a Hopf bifurcation, giving rise to a new branch of constant solutions or limit cycle solutions, respectively. The dynamics of the amplitude equations in parameter regions of the Hopf bifurcations is then explored using direct numerical integration. The results indicate a complicated amplitude dynamics including multiple limit cycle solutions, period-doubling route to chaos, and sudden disappearance of chaotic attractors.     

Parametric resonances in a base-excited double pendulum

Two parametrically-induced phenomena are addressed in the context of a double pendulum subject to a vertical base excitation. First, the parametric resonances that cause the stable downward vertical equilibrium to bifurcate into large-amplitude periodic solutions are investigated extensively. Then the stabilization of the unstable upward equilibrium states through the parametric action of the high-frequency base motion is documented in the experiments and in the simulations. It is shown that there is a region in the plane of the excitation frequency and amplitude where all four unstable equilibrium states can be stabilized simultaneously in the double pendulum. The parametric resonances of the two modes of the base-excited double pendulum are studied both theoretically and experimentally. The transition curves (i.e., boundaries of the dynamic instability regions) are constructed asymptotically via the method of multiple scales including higher-order effects. The bifurcations characterizing the transitions from the trivial equilibrium to the periodic solutions are computed by either continuation methods and or by time integration and compared with the theoretical and experimental results.     

Estimating the chaos boundaries of a double pendulum

Loss of the orbital stability of a double pendulum is considered in terms of Lyapunov exponents. The boundaries of the domain of stochastic motion caused by bifurcational and chaotic processes are estimated     

Effect of initial conditions on decaying grid turbulence at low R λ

The transport equations for the second-order velocity structure functions 〈(δu)²〉 and 〈(δq)²〉 are used as a scale-by-scale budget to quantify the effect of initial conditions at low Reynolds numbers, typical of grid turbulence. The validity of these equations is first investigated via hot-wire measurements of velocity and transverse vorticity fluctuations. The transport equation for 〈(δq)²〉 is shown to be balanced at all scales, while anisotropy of the large scales leads to a significant imbalance in the equation for 〈(δu)²〉. The effect of using similarity to evaluate the transport equation is rigorously tested. This approach has the desirable benefit of requiring less extensive measurements to calculate the inhomogeneous term of the transport equation. The similarity form of the 〈(δq)²〉 equation produces nearly identical results as those obtained without the similarity assumption. In the case of the 〈(δu)²〉 equation, the similarity method forces a balance at large separation, although the imbalance due to large scale anisotropy remains. The initial conditions of the turbulence at constant R _M ≃ 10,400 (28≤ R _λ≤ 55) are changed by using three grids of different geometries. Initial conditions affect the shape and magnitude of the second- and third-order structure functions, as well as the anisotropy of the large scales. The effect of initial conditions on the scale-by-scale budget is restricted to the inhomogeneous term of the transport equations, while the dissipation term remains unaffected despite the low R _λ. Scales as small as λ are affected by the changes in initial conditions.     

Global stabilization of a double inverted pendulum with control at the hinge between the links

We study the plane motion of a double pendulum with fixed suspension point. The pendulum is controlled by a single moment applied to the internal hinge between the links. The moment is assumed to be bounded in absolute value. We construct a feedback control law bringing the pendulum from the position in which both links hang vertically downwards into the unstable upper position in which both links are inverted. The same feedback ensures the asymptotic stability of the pendulum in the upper equilibrium position. Since the pendulum can be brought to the lower equilibrium position from any initial states, it follows that the constructed control law ensures the global stability of the inverted pendulum.     

Influence of initial conditions on compressible vorticity dynamics

Prompted by the lack of a unique choice of pressure (P) and density () fields for a compressible free vortex and by the observed dependence of turbulence dynamics on initial P and in compressible simulations, we address the effects of initial conditions on the evolution of a single vortex, on the prototypical phenomenon of vortex reconnection, and on two-dimensional turbulence. Two previous choices of initial conditions used for numerical simulations of compressible turbulence have been: (i) both P and uniform (constant initial conditions, CIC), and (ii) uniform with P determined from the Poisson equation (constant density initial conditions, CDIC). We find these initial conditions to be inappropriate for compressible vorticity dynamics studies. Specifically, in compressible reconnection, the effects of baroclinic vorticity generation and shocklet formation cancel each other during early evolution for CDIC, thus leading to almost incompressible behavior. Although CIC captures compressibility effects, it incorrectly changes the initial vorticity distribution by introducing strong acoustic transients, thereby significantly altering the evolving dynamics.Here, a new initial condition, called polytropic initial condition (PIC), is proposed, for which the Poisson equation is solved for initially polytropically related P and fields. PIC provides P and distributions within vortices which are consistent with those observed in shock-wedge interaction experiment and also leads to compressible solutions with no acoustic transients. At low Mach number (M), we show that the effects of all these three initial conditions can be predicted by low-M asymptotic theories of the Navier-Stokes equations. At high M, it is shown here that inappropriate initial conditions may alter the evolutionary dynamics and, hence, lead to wrong conclusions regarding compressibility effects. We argue that PIC is a more appropriate choice.D. Virk acknowledges financial support of the Advanced Study Program at NCAR, Boulder, for 2 years during graduate studies. Part of this work and the writing of the paper have been supported by NSF Grant CTS-9214818.     

Limit cycles of a double pendulum with nonlinear springs

The limit cycles of a double pendulum with hard, soft, or linear springs subject to a follower force are drawn using computer simulation     

Ultimate energy of a double pendulum undergoing quasiperiodic oscillations

The boundary of the phase domain of periodic solutions of a double pendulum is constructed and shown to be closed __________ Translated from Prikladnaya Mekhanika, Vol. 43, No. 9, pp. 106–114, September 2007.     

Parametric identification of a chaotic base-excited double pendulum experiment

The parametric identification of a chaotic system was investigated for a double pendulum. From recorded experimental response data, the unstable periodic orbits (UPOs) were extracted and then used in a harmonic balance identification process. By applying digital filtering, digital differentiation and linear regression techniques for optimization, the results were improved. Verification of the related simulation system and linearized system also corroborated the success of the identification algorithm.     

Coupled flutter and divergence bifurcation of a double pendulum

The loss of stability of the equilibrium position of a double pendulum with follower force loading and elastic end support is studied. At a special parameter combination the linearized system is characterized by a zero root and a pure imaginary pair of eigenvalues. Therefore, the stability problem is a complicated critical case in the sense of Liapunov and requires a non-linear analysis. A complete post-bifurcation investigation of the coupled divergence and flutter motions is given by means of centre manifold theory, and bifurcation diagrams. Among the different types of motions even the appearance of chaotic behavior is shown.     

Bifurcation analysis of a double pendulum with internal resonance

ＩｎｔｒｏｄｕｃｔｉｏｎＡｎｏｎｌｉｎｅａｒｄｙｎａｍｉｃａｌｓｙｓｔｅｍｍａｙｅｘｈｉｂｉｔｃｏｍｐｌｅｘｄｙｎａｍｉｃｂｅｈａｖｉｏｒｉｎｔｈｅｖｉｃｉｎｉｔｙｏｆａｃｏｍｐｏｕｎｄｃｒｉｔｉｃａｌｐｏｉｎｔ[1].ＡｃｃｏｒｄｉｎｇｔｏｔｈｅｓｔｒｕｃｔｕｒｅｏｆｔｈｅＪａｃｏｂｉａｎｅｖａｌｕａｔｅｄａｔｔｈｅｃｒｉｔｉｃａｌｐｏｉｎｔ,ｔｈｅｓｙｓｔｅｍｓｍａｙｂｅｃｌａｓｓｉｆｉｅｄ,ｉｎｇｅｎｅｒａｌ,ａｓｃｏ＿ｄｉｍｅｎｓｉｏｎｏｎｅ,ｃｏ＿ｄｉｍｅｎｓｉｏｎｔｗｏ,ｅｔｃ.[2].Ｗｈｅｎｏ…     

On the motions of a double pendulum with vibrating suspension point

We consider the motions of a system consisting of two pivotally connected physical pendulums rotating about horizontal axes. We assume that the system suspension point, which coincides with the suspension point of one of the pendulums, performs harmonic vibrations of high frequency and small amplitude along the vertical. We also assume that the system has four relative equilibrium positions in which the suspension points and the pendulum centers of mass lie on one vertical line. We study the stability of these relative equilibria. For arbitrary physical pendulums, we obtain stability conditions in the linear approximation. For a system consisting of two identical rods, we solve the stability problem the in nonlinear setting. For the same system, we study the existence, bifurcations, and stability of high-frequency periodic motions of small amplitude other than the relative equilibria on the vertical line. The studies of dynamic stability augmentation in mechanical systems under the action of high-frequency perturbations was initiated in the paper [1], where it was shown that the unstable inverted equilibrium of a pendulum may become stable if the suspension point vibrates rapidly. This idea was developed in [2–10] and other papers, where several aspects of motion of a mathematical pendulum in the case of rapid small-amplitude vibrations of the suspension point were studied in the linear setting and also (without full mathematical rigor) in the nonlinear setting. The motions of the suspension point along an arbitrary oblique straight line [2, 4, 7, 8], along the vertical [3, 5, 6], along the horizontal [9], and in the case of damping [8] were considered. The monograph [10] deals with the stabilization of a pendulum or a system of pendulums under periodic and conditionally periodic vibrations of the suspension point along the vertical, along an oblique straight line, and along an ellipse. A rigorous nonlinear analysis of the existence and stability of periodic motions of the mathematical pendulum under horizontal and oblique vibrations of the suspension point at arbitrary frequencies and amplitudes can be found in [11, 12]. For the case of vertical vibrations of the suspension point at an arbitrary frequency and amplitude, a rigorous stability analysis of the relative equilibria of the pendulum on the vertical was carried out in [13].     

Stability of the stationary motion of a double pendulum interacting with a string

The equation of in-plane vertical motion of a double pendulum suspended at some point of a horizontal elastic string is derived using a hybrid model of this mechanical system. The conditions for the asymptotic stability of the stationary motion of the pendulum interacting with the string are established     

Limit cycles of a double pendulum subject to a follower force

It is shown that there is a magnitude of the follower force at which two limit cycles, stable and unstable, are born in the phase space of a double simple pendulum     

On the motion of a reversible double simple pendulum with tracking force

Stability domains of a pendulum in the presence of tracking force and viscoelastic elements are constructed. It is shown that the boundary of the stability domains consists of sections of two hyperbolas. The effect of the pendulum parameters on the configuration of the stability domains is considered. Translated from Prikladnaya Mekhanika, Vol. 35, No. 7, pp. 108–112, July, 1999.     

Projectile dynamics at low barrel pressures

A mathematical model for a projectile shot at low pressures in the space behind the projectile space is developed. The pressure rise is limited because of the nonsimultaneity of propellant ignition and combustion and the discharge of the propellant combustion products through the gap between the projectile and the walls of the gun barrel. The kinetic characteristics of flame propagation over the propellant particles are determined. A comparison of calculation and experimental data is performed. The calculation results are used in designing 2A85 self-propelled launchers and upgrading 2A30 self-propelled launchers. __________ Translated from Prikladnaya Mekhanika i Tekhnicheskaya Fizika, Vol. 48, No. 6, pp. 44–49, November–December, 2007.     

Suspension point vibration parameters for a given equilibrium of a double mathematical pendulum

The motion of a double mathematical pendulum under the action of the gravity force and a vibration force whose frequency substantially exceeds the system natural frequencies is considered. An oblique vibration stabilizing the pendulum in an arbitrarily given position is sought. The domain of existence of the pendulum equilibrium points and the vibration parameters corresponding to a given equilibrium of the pendulumare obtained analytically. In the domain of existence of equilibrium points, the subdomain of their stability is distinguished.     

The dynamics of an omnidirectional pendulum harvester

The pendulum applied to the field of mechanical energy harvesting has been studied extensively in the past. However, systems examined to date have largely comprised simple pendulums limited to planar motion and to correspondingly limited degrees of excitational freedom. In order to remove these limitations and thus cover a broader range of use, this paper examines the dynamics of a spherical pendulum with translational support excitation in three directions that operate under generic forcing conditions. This system can be modelled by two generalised coordinates. The main aim of this work is to propose an optimisation procedure to select the ideal parameters of the pendulum for an experimental programme intended to lead to an optimised pre-prototype. In addition, an investigation of the power take-off and its effect on the dynamics of the pendulum is presented with the help of Bifurcation diagrams and Poincaré sections.