Influence of material and geometrical nonlinearities on the bifurcations of equilibrium states of a two-link pendulum

The equilibrium states of an inverted two-link simple pendulum with an asymmetric follower force are classified depending on the characteristics of the springs (hard, soft, or linear) at the upper end and at the hinges. Phase portraits are plotted. The bifurcation points on the equilibrium curves are identified. Emphasis is on fold and cusp catastrophes __________ Translated from Prikladnaya Mekhanika, Vol. 43, No. 8, pp. 115–128, August 2007.     

Evolution of the equilibrium states of an inverted pendulum

The influence of the pendulum parameters and the follower force on the evolution of equilibrium states is analyzed using a generalized mathematical model of inverted pendulum. Equilibrium curves are plotted using the parameter continuation method. It is shown that the pendulum with certain values of the angular eccentricity has one or three nonvertical equilibrium positions __________ Translated from Prikladnaya Mekhanika, Vol. 43, No. 3, pp. 122–131, March 2007.     

Equilibrium states of a pendulum with an asymmetric follower force

The effect of the linear eccentricity of the follower force on the equilibrium states of an inverted pendulum is examined. Bifurcation points and catastrophes associated with changes in pendulum parameters and type of springs are analyzed. Phase flows are plotted __________ Translated from Prikladnaya Mekhanika, Vol. 43, No. 4, pp. 121–129, April 2007.     

Influence of the Nonlinearity of the Elastic Elements on the Stability of a Double Pendulum with Follower Force in the Critical Case

A generalized mathematical theory of a double mathematical pendulum with follower force is used to analyze the stability of the vertical equilibrium position of the pendulum with both linear and nonlinear (hard and soft) elastic elements in the critical case of one zero root of the characteristic equation. The influence of the parameters of these elements on the safe and dangerous sections of the stability boundary is demonstrated__________Translated from Prikladnaya Mekhanika, Vol. 41, No. 4, pp. 133–142, April 2005.     

Evolution of Limit Cycles in the Stability Domain of a Double Pendulum under a Variable Follower Force

It is shown that stable and unstable limit cycles are born at a certain value of the follower force. As the absolute value of the force increases, the cycles move with different velocities in opposite directions. The unstable limit cycle moves toward the origin in the four-dimensional phase space. At a certain value of the follower force, this cycle gets at the origin, disturbing its stability as a singular point of the dynamic system. If the absolute value of the follower force decreases, then death of both cycles is possible. In this case, the attraction domain of the origin becomes infinite     

Theory of inverted pendulum with follower force revisited

The effect of the type of springs on the equilibrium states of an inverted pendulum is examined. The angular and linear eccentricities of the follower force are taken into account __________ Translated from Prikladnaya Mekhanika, Vol. 43, No. 6, pp. 126–137, June 2007.     

Bifurcation analysis of the motion of an asymmetric double pendulum subjected to a follower force: Codimension three problem

Local and global bifurcations in the motion of a double pendulum subjected to a follower force have been studied when the follower force and the springs at the joints have structural asymmetries. The bifurcations of the system are examined in the neighborhood of double zero eigenvalues. Applying the center manifold and the normal form theorem to a four-dimensional governing equation, we finally obtain a two-dimensional equation with three unfolding parameters. The local bifurcation boundaries can be obtained for the criteria for the pitchfork and the Hopf bifurcation. The Melnikov theorem is used to find the global bifurcation boundaries for appearance of a homoclinic orbit and coalescence of two limit cycles. Numerical simulation was performed using the original four-dimensional equation to confirm the analytical prediction.     

The equations of an inverted pendulum with an arbitrary number of links and an asymmetric follower force

This paper formulates a problem for and gives the differential equations of plane-parallel motion of an inverted n-link pendulum subject to an asymmetric follower force applied at the upper end via a spring. The physical nonlinearities of springs are taken into account. The possible mechanisms of energy dissipation are described __________ Translated from Prikladnaya Mekhanika, Vol. 43, No. 5, pp. 106–114, May 2007.     

Asymptotic stability boundaries for the equilibrium positions of a double pendulum with vibrating point of suspension

The mechanisms whereby a double pendulum with vibrating point of suspension loses stability in equilibrium positions are studied. Stability conditions for the equilibrium positions in critical cases are established __________ Translated from Prikladnaya Mekhanika, Vol. 44, No. 7, pp. 120–133, July 2008.     

Performance of ultra-low frequency passive vertical vibration isolation system with reverse pendulum

A novel long period passive vertical vibration isolator constructed by mounting reverse pendulums on two pairs of torsion springs is presented. By theoretical analysis and numerical calculation, it is shown that the isolator can achieve much longer resonant period due to gravitational positive feedback and is smaller in size than the current torsion spring isolators with the same geometric parameters. Project supported by the Post-doctoral Foundation of Huazhong University of Science and Technology.     

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.     

STABILITY FOR THE EQUILIBRIUM STATE OF CHAPLYGIN’S SYSTEMS

The stability for the equitibrium states of Chaplygin’s systems is considered. Theequations of motion of Chaplygin’s systems and the existence conditions of their equilibrium states are given. Some criteria of stability for the equilibrium.states of Chaplygin’s systems are obtained. Two examples are finally given.     

Thomas rotation and Foucault pendulum under a simple unifying geometrical point of view

In this paper we show that there is a complete parallelism between the Foucault pendulum and the Thomas rotation phenomena by using the concept of parallel transport in a surface. In the case of the Foucault pendulum the surface is the ordinary sphere corresponding to the Earth sphere, whereas in the case of the Thomas rotation the surface is the pseudosphere corresponding to the space of relativistic velocities. Moreover, in both cases we use a simple method that reduces the problem to the parallel transport in a conical surface, and so, to the plane.     

非保守力作用下FGM矩形板的稳定性分析

对受均布随从力作用的功能梯度材料(FGM)矩形板,引入应力函数,得到了以应力函数和挠度函数表示的耦合运动微分方程组。用Fourier级数法研究了四边简支FGM非保守矩形板的稳定性,给出了不同边长比和不同梯度指标下频率和发散载荷的变化曲线,以及梯度指标变化对频率和发散载荷的影响。     

Divergent Bifurcations of a Double Pendulum under the Action of an Asymmetric Follower Force

The paper studies the influence of the orientation parameter of the follower force on the behavior of the pendulum with a simple zero eigenvalue in the linearization matrix     

Experimental and numerical investigation of chaotic regions in the triple physical pendulum

The experimental and numerical analysis of triple physical pendulum is performed. The experimental setup of the triple pendulum with the first body externally excited by the square function and the widely used LabView measure-programming system, which is designed especially for measure data processing and acquisition, are described. The mathematical model of the system is then introduced. The parameters of the model are estimated by minimization of the sum of squares of deviations between the signal from the simulation and the signal from the experiment. A good agreement between results from experiment and from simulation is shown in few examples, including periodic as well as chaotic solutions.     

Steering performance of an inverted pendulum vehicle with pedals as a personal mobility vehicle

From the viewpoints of environmental protection, support for the aged and ensuring the right to mobility, there is a need to develop a new type of mobility vehicle that provides more effective transportation. The authors propose an inverted pendulum vehicle with pedals as one of the forms of personal mobility vehicles (PMVs). In this paper, the steering performance of the inverted pendulum vehicle with pedals is discussed based on experiments on a prototype. From the experimental results, it was confirmed that the errors from the five subjects for the target trajectory and the five-grade evaluation of the maneuverability were similar. Finally, we created an inverted pendulum vehicle with pedals to which was added a reaction actuator for the steering system. From the experimental results, it was found that setting appropriate feedback gains for the handle steering angle and its rate of rotation, which control the right and left wheel driving torques, resulted in greatly improved maneuverability.     

Stability of an equilibrium position of a pendulum with step parameters

A mathematical pendulum affected by parametric disturbance with potential energy being periodic step function is considered. Non-linear equation of the pendulum depends on two parameters characterizing the mean value in time of the parametric disturbance and range of its “ripple”. Values of the parameters can be set arbitrarily. The non-linear problem of stability for two particular solutions of the equation corresponding to a hanging and inverse pendulum is solved.     

Stability of the equilibrium state in a convection model with nonlinear temperature and pressure dependences of density

The convection of a heat-conducting viscous liquid is considered. It is assumed that the liquid density depends quadratically on the temperature and pressure. The instability of the equilibrium state of a free-boundary horizontal layer with respect to small perturbations is studied using a linearization method. It is found that the state of mechanical equilibrium is unstable. Neutral curves are constructed and the critical Rayleigh numbers are found. The results are compared with the well-known solution of the same problem for the limiting case where the density is a quadratic function of temperature and does not depend on pressure. __________ Translated from Prikladnaya Mekhanika i Tekhnicheskaya Fizika, Vol. 48, No. 2, pp. 66–74, March–April, 2007.