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     

On the Chaotic Dynamics of a Spherical Pendulum with a Harmonically Vibrating Suspension

The equations of motion for a lightly damped spherical pendulum are considered. The suspension point is harmonically excited in both vertical and horizontal directions. The equations are approximated in the neighborhood of resonance by including the third order terms in the amplitude. The stability of equilibrium points of the modulation equations in a four-dimensional space is studied. The periodic orbits of the spherical pendulum without base excitations are revisited via the Jacobian elliptic integral to highlight the role played by homoclinic orbits. The homoclinic intersections of the stable and unstable manifolds of the perturbed spherical pendulum are investigated. The physical parameters leading to chaotic solutions in terms of the spherical angles are derived from the vanishing Melnikov–Holmes–Marsden (MHM) integral. The existence of real zeros of the MHM integral implies the possible chaotic motion of the harmonically forced spherical pendulum as a result from the transverse intersection between the stable and unstable manifolds of the weakly disturbed spherical pendulum within the regions of investigated parameters. The chaotic motion of the modulation equations is simulated via the 4th-order Runge–Kutta algorithms for certain cases to verify the analysis.     

Computational dynamics of a 3D elastic string pendulum attached to a rigid body and an inertially fixed reel mechanism

A high fidelity model is developed for an elastic string pendulum, one end of which is attached to a rigid body while the other end is attached to an inertially fixed reel mechanism which allows the unstretched length of the string to be dynamically varied. The string is assumed to have distributed mass and elasticity that permits axial deformations. The rigid body is attached to the string at an arbitrary point, and the resulting string pendulum system exhibits nontrivial coupling between the elastic wave propagation in the string and the rigid body dynamics. Variational methods are used to develop coupled ordinary and partial differential equations of motion. Computational methods, referred to as Lie group variational integrators, are then developed, based on a finite element approximation and the use of variational methods in a discrete-time setting to obtain discrete-time equations of motion. This approach preserves the geometry of the configurations, and leads to accurate and efficient algorithms that have guaranteed accuracy properties that make them suitable for many dynamic simulations, especially over long simulation times. Numerical results are presented for typical examples involving a constant length string, string deployment, and string retrieval. These demonstrate the complicated dynamics that arise in a string pendulum from the interaction of the rigid body motion, elastic wave dynamics in the string, and the disturbances introduced by the reeling mechanism. Such interactions are dynamically important in many engineering problems, but tend be obscured in lower fidelity models.     

Experimental Investigation of Dynamics and Bifurcations of an Impacting Spherical Pendulum

Since Newton first considered the motion of a spherical pendulum over 200 years ago, many researchers have studied its dynamic response under a variety of conditions. The characteristic of the problem that has invited so much investigation was that a spherical pendulum paradigms much more complex phenomena. Understanding the response of a paradigm gives an almost multiplicative effect in the understanding of other phenomena that can be modeled as a variant of the paradigm. The spherical pendulum has been used to damp irregular motion in helicopters and on space stations as well as for many other applications. In this study an inverted impacting spherical pendulum with large deflection was investigated. The model was designed to approximate an ideal pendulum, with the pendulum bob contributing the vast majority of the mass moment of inertia of the system. Two types of bearing mechanisms and tracking devices were designed for the system, one of which had low damping coefficient and the other with a relatively high damping coefficient. An experimental investigation was performed to determine the dynamics of an inverted, impacting spherical pendulum with large deflection and vertical parametric forcing. The pendulum system was studied with nine different bobs and two different base configurations. During the experiments, the frequency of the excitation remained between 24.6 and 24.9 Hz. It was found that sustained conical motions did not naturally occur. The spherical pendulum system was analyzed to determine under what conditions the onset of Type I response (a repetitive motion in which the pendulum bob does not traverse through the apex. The bob strikes the same general area of the restraint without striking the opposite side of the restraint.), sustainable Type II response (this is the repetitive motion in which the pendulum bob traverses through the apex. The bob strikes opposite sides of the restraint.), and mixed mode response (motion in which the pendulum bob randomly strikes either the same area of the restrain or the opposite side of the restraint) occurred.     

Oscillations of a body filled with a viscous fluid

The oscillations of a physical pendulum containing a spherical cavity filled with an incompressible viscous liquid were discussed in [1]. In this paper we consider the mote general problem of the motion of an axially symmetric solid with a spherical cavity filled with an incompressible viscous fluid and moving about a fixed point. It is assumed that the center of the cavity and the fixed point lie on the axis of symmetry of the body.     

Modeling of system dynamics of a slewing flexible beam with moving payload pendulum

When a tower crane is handling payload via rotation and moving the carriage simultaneously the jib structure and the payload can be modeled as a system consisting of a slewing flexible clamed-free beam with the spherical payload pendulum that moves along the beam. The present work completes the dynamic modeling of the system mentioned above. The clamed-free beam attached to a rotating hub is modeled by Euler–Bernoulli beam theory. The payload is modeled as a sphere pendulum of point mass attached to via massless inextensible cable the carriage moving on the rotating beam. Non-linear coupled equations of motion of the in- and out-of-plane of the beam and the payload pendulum are derived by means of the Hamilton principle. Some remarks are made on the equations of motion.     

Dynamic response of tower crane induced by the pendulum motion of the payload

Dynamic response of tower cranes coupled with the pendulum motions of the payload is studied in this paper. A simple perturbation scheme and the assumption of small pendulum angle are applied to simplify the governing equation. The tower crane is modeled by the finite element method, while the pendulum motion is represented as rigid-body kinetics. Integrated governing equations for the coupled dynamics problem are derived based on Lagrange’s equations including the dissipation function. Dynamics of a real luffing crane model with the spherical and planar pendulum motions is analyzed using the proposed formulations and computational method. It is found that the dynamic responses of the tower crane are dominated by both the first few natural frequencies of crane structure and the pendulum motion of the payload. The dynamic amplification factors generally increase with the increase of the initial pendulum angle and the changes are just slightly nonlinear for the planar pendulum motion.     

The behaviour of a simple pendulum with uniformly shortening string length

Whilst watching the loading of a ship by crane it is often observed that the cargo shifts abruptly during the lifting stage. In order to try to understand this kind of behaviour the motion of a simple undamped pendulum, which is being shortened at a constant speed, is determined. It is found that the tension in the string always increases initially from its static value and that this increase in tension is very large even when the initial angular speed is small. Finally, it is proved that the linearised approximation is most unsatisfactory even when the initial amplitude of the swing and the shortening rate are small.     

求解大柔度梁/绳索的ANCF单元及隐式迭代格式

为模拟大柔度梁/绳索结构的变形和大范围运动,基于绝对节点坐标方法ANCF(Absolute nodal coordi-nate formulation)和HHT(Hilber-Hughes-Taylor)积分方法,建立了ANCF单元的隐式动力学迭代格式.得到了简洁的节点等效力向量,且进一步导出了切线刚度矩阵的全部公式,...     

Cases of Integrability of Three-Dimensional Dynamic Equations for a Solid

A dynamic model of the interaction of a rigid body with a jet flow of a resistant medium is considered. This model allows us to obtain three-dimensional analogs of plane dynamic solutions for a solid interacting with the medium and to reveal new cases where the equations are Jacobi integrable. In such cases, the integrals are expressed in terms of elementary functions. The classical problems of a spherical pendulum in a flow and three-dimensional motion of a body with a servoconstraint are shown to be integrable. Mechanical and topological analogs of these problems are found     

Oscillatory–ballistic motion regularities of a gravitational pendulum

We simulate the motion of a gravitational pendulum that has initial angular amplitude larger than 90\(^{\circ }\) and smaller than 180\(^{\circ }\), and loses energy at each change from ballistic to oscillatory motion when the string is suddenly tensed (we name this event collision). Simulation is based on a velocity Verlet algorithm that is implemented in a numerical code. The numerical simulation of motion as function of time is checked against an analytical code that describes the trajectory. The string tension expression that respects the velocity Verlet algorithm requirements is identified and a criterion for collision occurrence is introduced. An interesting band-like structure of the number of collisions as function of the initial amplitude and damping modelling is obtained.     

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.

     

Controlled pendulum on a movable base

A plane motion of a multilink pendulum hinged to a movable base (a wheel or a carriage) is considered. The control torque applied between the base and the first link of the pendulum is independent of the base position and velocity and is bounded in absolute value. The coordinate determining the base position is cyclic. The mathematical model of the system permits one to single out the equations describing the pendulum motion alone, which differ from the well-known equations of motion of a pendulum with a fixed suspension point both in the structure and in the parameters occurring in these equations. The phase portrait of motions of a control-free one-link pendulum suspended on a wheel or a carriage is obtained. A feedback control ensuring global stabilization of the unstable upper equilibrium of the pendulum is constructed. Time-optimal control synthesis is outlined.     

Various analogies for the equilibrium shapes of an elastic thread on two-dimensional surfaces

In accordance with the Kirchhoff analogy, the equilibrium equations of an elastic thread on a plane are equivalent to the equations of motion of a simple pendulum. This analogy is generalized to the case when the thread is situated on a smooth curved surface. The equilibrium equations for the threads in the general case and in the particular cases of flat, cylindrical, and spherical surfaces are derived. For these surfaces the Kirchhoff analogy is generalized to the case of a simple pendulum in an additional force field. There are also considered the electromagnetic and nonholonomic analogies for the equilibrium equations of an elastic thread.     

Seismic behaviour of isolated multi-span continuous bridge to nonstationary random seismic excitation

This paper concentrates on the results of responses of a multi-span continuous bridge isolated with double concave friction pendulum bearings subjected to non-stationary random seismic excitation characterized by the incoherence, the wave-passage, and the site-response effects. The earthquake excitation is modelled as a non-stationary random process as uniformly modulated broad-band excitation. To perform the seismic isolation procedure, the double concave friction pendulum bearings which are sliding devices that utilize two spherical concave surfaces are placed at each of the six support points of the deck. The non-stationary response of the isolated bridge is compared with the corresponding stationary response in order to study the effects of non-stationary characteristics of the earthquake input motion. Solutions obtained from the stationary and non-stationary stochastic analyses for the isolated bridge to spatially varying earthquake ground motions are compared for the special cases of the earthquake ground motion model. The spatially varying earthquake ground motions are described stochastically based on an empirical coherency loss function and a filtered power spectral density function. The site effect is considered by a transfer function derived from one dimensional wave propagation theory. It is observed that the stationary assumption is reasonable for the considered ground motion duration.     

Identification of Friction Coefficients in the Hinge of a Controlled Physical Pendulum Using the Amplitudes of Steady-State Oscillations

A dynamic model of a controlled physical pendulum is considered. The Pontryagin method of searching for the periodic solutions to near-Hamiltonian systems is used to formulate a programmed law of pendulum oscillations such that the test modes of oscillations become steady and orbitally stable. An approach to identify the friction parameters in the hinge of the pendulum is proposed for the case of the active motor mode. This approach is based on the data available about the integral characteristics of motion. The motion of the system under consideration is numerically simulated.     

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.     

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.     

Plane Problem on a Heavy Ball Rolling in a Spherical Recess of an Inverted Pendulum

The Appell equations are used to formulate a plane problem on a heavy homogeneous ball moving without slippage in a spherical recess of an inverted pendulum. It is shown that this mechanical system falls into the class of Chaplygin's nonholonomic systems