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Some simple models of a child swinging on a playground swing are presented. These are analyzed using techniques from Lagrangian mechanics with a twist: the child changes the configuration of the system by sudden movements of their body at key moments in the oscillation. This can lead to jumps in the generalized coordinates describing the system and/or their velocities. Jump conditions can be determined by integrating the Euler--Lagrange equations over a short time interval and then taking the limit as this time interval goes to zero. These models give insights into strategies used by swingers, and answer such vexed questions such as whether it is possible for the swing to go through a full 360 ° turn over its pivot. A model of an instability at the pivot observed by Colin Furze in a rigid swing constructed to rotate through 360 ° is also described. This uses a novel double pendulum configuration in which the two components of the pendulums are constrained to move in orthogonal planes. [ABSTRACT FROM AUTHOR]

In recent years, energy harvesters using pendulum systems have often been applied in ultra-low-frequency environments, such as ocean waves, human motion, and structural vibration. To illustrate the research progress in pendulum-type energy harvesting, a comprehensive review is provided in the present study. Specifically, single- and double-pendulum energy harvesters based on different energy-conversion mechanisms are separately grouped. In addition, different improvement techniques and design schemes used in studies on pendulum energy harvesters are summarized. Theoretical studies have explored the dynamic characteristics of single and double pendulums. Various key aspects, including the fundamental mechanisms, optimization methods, core structures, and applications, to improve the performance of single- and double-pendulum energy harvesters are discussed. Finally, several potential research directions and applications are proposed. [ABSTRACT FROM AUTHOR]

In recent decades there has been an increasing interest in installing heavy and complex underwater equipment, like subsea manifolds and templates. More specifically, oil and gas companies frequently carry out this kind of operation, which involves various effects related to the dynamics of the vessel-cable-payload system under environmental loads. Many studies simplify these systems to unidimensional models, considering only the vertical dynamics. However, neglecting the horizontal and rolling motions of the equipment sometimes may result in misleading analysis. The present work shows through numerical and experimental tests that a vertically excited cable-payload system may exhibit parametric resonance, which culminates in a double pendulum motion of the subsea equipment during the installation process. The kinematics formulas are obtained using the Euler-Lagrange equations. After that, a linear stability analysis is carried out using the Floquet method. Finally, the numerical and experimental results are compared. The time-domain numerical results show good agreement when compared with the experimental tests, capturing both phenomena of cable's slacking and the equipment's parametric rolling motion. Furthermore, the stability analysis using the Floquet method was an effective approach to identifying the parametric resonance frequencies. Hence, this work shows the importance of considering even more realistic models when analyzing subsea equipment installations. • A nonlinear model that considers cable slacking and equipment's rolling. • The use of stability analysis to identify parametric resonance frequencies. • The system, despite oscillatory rolling, is stable per Floquet theory. • Unstable cases and bifurcation types are presented. [ABSTRACT FROM AUTHOR]

This article presents an energy-based method for examining the dynamics of an autoparametric double pendulum with quadratic coupling. The method is based on decomposing the total energy (Hamiltonian) into partial terms and studying these terms in the time domain. The article then demonstrates using this method to analyse internal resonances in various regimes, including regular periodic, quasiperiodic, and chaotic motion. This study also explores higher-order resonances linked to the geometric tuning parameter and initial energy, which enrich the complexity of the system. The work focuses on identifying conditions that lead to the development of internal resonances and analysing energy exchange mechanisms at various initial energy levels. This analysis uses Fast Lyapunov Indicators and Poincaré maps for these higher energy levels to identify chaotic regimes. Within these regimes, the assessment of coupling energy contributes to a deeper understanding of the studied system. Moreover, the research demonstrates how energy-based methods can be used to investigate auto-parametric systems. • Energy-based approach for autoparametric pendulum with quadratic coupling. • In-depth exploration of internal resonances across motion types. • Chaos detection using Fast Lyapunov Indicators. • Higher-order resonances linked to system parameters revealed. • Method applicable for various autoparametric systems. [ABSTRACT FROM AUTHOR]

The parametric pendulum and others different pendular mechanisms were extensively analyzed in current literature because of the chaotic motions. Recently, practical reasons, like energy harvesting, motivated the study of some pendular systems. The resonance and the chaos may occur in multiple and submultiple frequencies based on the natural frequency of the pendulum. In the present paper, a double pendulum is analyzed with two subharmonic frequencies and in the main resonance zone. The results found demonstrate high similarities of the double pendulum with the parametric pendulum when compared the bifurcation diagrams based on varying the amplitude. The conditions in the numerical simulation include a variation in the viscous friction in the joints and different amplitudes and frequency of excitation. The differential equations used dimensionless parameters representing time, frequency, amplitude, friction, position and velocities. Concerning the results, all tests performed have pointed that the same type of motion takes place in the two pendulums after the transient time have already elapsed. [ABSTRACT FROM AUTHOR]

This is a common way to improve the uniformity of microwave heating by using a mode stirrer. However, the conventional mode stirrer is a reciprocating periodic motion, which in some cases does not effectively improve the uniformity of microwave heating. In this study, a mode stirrer with double pendulum structure was creatively proposed. Due to the chaotic characteristics of the double pendulum motion process, the mode stirrer process is no longer a traditional periodic motion, and this chaotic stirrer process will affect the uniformity of heating. Based on the finite element method combined with the implicit function and level set method, the microwave heating process with double pendulum mode stirrer was calculated, and the multiphysics coupling simulation of structure‐electromagnetic‐heat transfer was realized. The heating performance under different double pendulum mode stirrers was studied and compared with the heating results of the single pendulum mode stirrer. The results show that the double pendulum mode stirrer with a specific initial position can effectively improve the heating uniformity without reducing the heating efficiency. [ABSTRACT FROM AUTHOR]

Forced linear oscillations of dissipative mechanical systems with two degrees of freedom under the action of time-periodic forces are considered. The Lagrange equations are expressed in terms of three positive-definite quadratic forms: kinetic energy, dissipative function, and potential energy. The necessary and sufficient condition for simultaneous reducibility to diagonal forms of symmetric matrices of three real quadratic forms of two variables is formulated and proved. The condition was reduced to the equality of the third-order determinant of the coefficients of quadratic forms to zero. In this case, by linear transformation, the quadratic forms are reduced to the sum of squares and the equations are split into two independent second-order equations. The solution of the system is in a general analytical form. The effectiveness of the method is demonstrated by analyzing the forced oscillations of a double pendulum. [ABSTRACT FROM AUTHOR]

Forced linear oscillations of dissipative mechanical systems with two degrees of freedom under the action of time-periodic forces are considered. The Lagrange equations are expressed in terms of three positive-definite quadratic forms: kinetic energy, dissipative function, and potential energy. The necessary and sufficient condition for simultaneous reducibility to diagonal forms of symmetric matrices of three real quadratic forms of two variables is formulated and proved. The condition was reduced to the equality of the third-order determinant of the coefficients of quadratic forms to zero. In this case, by linear transformation, the quadratic forms are reduced to the sum of squares and the equations are split into two independent second-order equations. The solution of the system is in a general analytical form. The effectiveness of the method is demonstrated by analyzing the forced oscillations of a double pendulum. [ABSTRACT FROM AUTHOR]

Mathematics is most beautiful thing in our daily life but when it comes to equations and formulae people seems to be unpleasant. This equations and formulae exhibits interactive properties and relation in our daily life. It is sure that the almost all beautiful thing in our nature even Universe unveils mathematical properties. These properties produce many patterns and shape and is directly related to geometric algebra as we can see in nature for example, fluorescent green beetle and colourful galaxies. These gorgeous properties of nature can be transformed into equations and logics. Therefore, the main aim of this paper is to generate creative graphics using mathematical equations such as number theory, position vectors and trigonometry functions. The sectional organization of this paper is as follows: section 1 gives the introduction of interactive graphic design and related work. Mathematical approach for graphic design has been briefed in section 2. In section 2 adopted algorithms and Graphical visualization of design is given. Graphic designs are applied to Indian ethnic wear and T-shirts. Last Section gives the conclusions and further scope of this work. [ABSTRACT FROM AUTHOR]

Highlights • Vibration based energy harvesting exploiting chaotic behavior of a base excited double pendulum. • Simple design that can harvest energy in linear as well as nonlinear regimes. • Derived equations of motion that consider the coupling effects due to magnetic field. • Optimal system parameters have been established that enhances energy harvesting. • Numerical predictions have been validated using experimental observations. Abstract This study focusses on exploiting the dynamics of a base excited double pendulum to generate electricity in small scale. The system consists of two pendulum attached in series with rigid massless links. The mass of the double pendulum is assumed to be concentrated at the bobs. A magnet is rigidly attached to the tip of second (lower) pendulum. A series of equally spaced coils are placed near the arc of the trajectory of the magnet. Small amplitude oscillations result in fluctuating magnetic field, generating electricity in the coils. The harvested power is significantly increased when the magnet undergoes chaotic motion. Parametric studies are carried out to optimize the system and excitation parameters that lead to enhancing the harvested energy. The proof-of-concept has been demonstrated through experiments. This involved recording the motion of the bobs, subsequent video processing, followed by time series analyses to establish the chaotic dynamics and correlating the dynamical behavior with the harvested energy. The experimental observations reveal good qualitative match with the numerical simulations. [ABSTRACT FROM AUTHOR]

Highlights • Double pendulum is used to improve the efficiency of piezoelectric energy cantilever. • The numerical analysis conducted and the prototype is designed. • The experimental results show a significant increase in the amount of average power. Abstract Harvesting electrical energy from various human motions using piezoelectric energy harvesters (PEH) is gaining research attention in recent years. The energy harvested could potentially power hand held electronic devices and medical devices without the need of external power source for recharging batteries. In this study, an attempt is made to improve the efficiency of PEH to harvest energy from human motions by adopting a double pendulum system coupled with magnetic force interactions. For the purpose of comparison, three configurations of PEH which includes the conventional PEH with cantilever beam (PEHCB), the PEH with single pendulum system (PEHSP) and the PEH with double pendulum system (PEHDP) are experimentally studied. Excitations by both mechanical shaker and major human body parts during walking and jogging motions are investigated. The performance of each configuration, in terms of voltage and power produced as well as the idle time between each cycle, are analysed, compared and discussed. ANSYS© software is used to analyse the proposed model and MATLAB© software is used to calculate the output power. The results demonstrate that, with the use of the proposed double pendulum system, multiple impacts in each motion cycle is generated, thus producing higher voltage and power as compared to the conventional PEHCB. The idle time between each motion cycle is also effectively reduced. The efficiency of the PEH is thus significantly increased. [ABSTRACT FROM AUTHOR]

Abstract Overhead cranes, which have been extensively studied, are mostly simplified as single pendulums. However, in practice, the existence of lifting hook usually makes the crane present double-pendulum swing, i.e., hook swing with respect to the trolley and payload swing with respect to the hook. Therefore, a severe gap between theory and practice is generated. Due to this fact, lots of researchers are now working on the automation for double-pendulum overhead cranes (DPOC). However, with an extra unactuated degree of freedom, the control of DPOC is much more challenging than that of the simplified system due to its complicated dynamics, which is made even worse when considering payload hoisting/lowering and uncertain system parameters. To solve this problem, an enhanced-coupling adaptive controller is proposed for DPOC in this paper. Specifically, the payload hoisting/lowering motion is elaborately considered. Moreover, to improve the swing suppression performance, more swing information are incorporated into the construction of control inputs. Particularly, the uncertain payload mass is online estimated by a new adaption law ensuring precise identification, which further enhances the robustness of the proposed method. By utilizing Lyapunov techniques and LaSalle's Invariance Theorem, the closed-loop system is proven to be asymptotically stable around the desired equilibria. Finally, convincing hardware experimental results are presented to demonstrate the efficiency and superior control performance of the proposed method. [ABSTRACT FROM AUTHOR]

Chaotic and regular behavior of the system consisting of a double physical pendulum with two repulsive permanent magnets is studied. We are focused on mathematical modelling, numerical simulations, experimental measurements and in particular on a novel magnetic interaction modelling. System parameters are identified by matching the output signals from experiments and numerical solutions to the developed mathematical model governed by a strongly non-linear set of two second order ODEs including the friction and the magnetic interaction torques. Considered system shows chaotic and periodic dynamics. Few chaotic zones have been detected numerically and confirmed experimentally. Scenarios of transition from regular to chaotic motion and vice versa, as well as the bifurcation diagrams are illustrated and discussed. Good agreement between the numerical simulation and experimental measurement is achieved. [ABSTRACT FROM AUTHOR]

Graphical abstract Highlights • The paper describes an experimental determination of the coefficient of restitution of two particles. • Three options for evaluating experiments are compared, including their results. • The influence of the pendulum thread and the shape and size of the particles was investigated. • The value of the coefficient of restitution of corn, wood pellets and black coal was determined. Abstract The coefficient of restitution e is determined as the ratio of the final to the initial relative velocity between two moving objects after their collision. It is also one of the basic features and parameters for contact models of the discrete element method (DEM). The virtual simulation of industrial processes using DEM modelling is currently undergoing a great development both in the ways of application and in the complexity of the individual models. Therefore, input values need to be constantly refined to ensure the quality outputs of these numerical models. This article describes an experimental method of measuring the coefficient of restitution of two particles by means of a double pendulum, which presents a number of problems especially for non-spherical particles. At the same time, this article presents three options for evaluating the proposed experiment, including their comparison and description of their pros and cons. A series of experiments with particles of different sizes, shapes and materials was performed and their coefficients of restitution were determined. The presented method allows to be used on a broad scale especially for spherical particles of >5 mm, the coefficient of restitution can be determined very accurately. To some degree, this applies to non-spherical particles of mass in tens of grams and of different materials as well. Their initial coefficient of restitution can be determined with a high degree of precision, which is important for using in contact models definition. The accuracy of the contact model is dependent on accurate specification of restitution coefficient. In this study, values of the coefficients of restitution of various shapes such as wood particles, particles of maize, spruce cylindrical pellets and coal were experimentally determined. [ABSTRACT FROM AUTHOR]

In this paper, we study the Jacobi stability on the nonlinear double pendulum by the Kosambi-Cartan-Chern (KCC) theory. We assume that the mass and length of rods of two kinds of pendulums are equal, respectively. Moreover, we consider the case that initial angles of the double pendulum are equal. Under these conditions, we obtain the boundary between Jacobi stable and unstable trajectories for initial angles. It is shown that the condition of Jacobi stable or unstable depends only on deflection angles of the nonlinear double pendulum. Then, we discuss relationships between Jacobi stability, physical parameters and other concepts of stability such as Lyapunov stability and chaos. We suggest that the ratio of length of rods and the mass ratio of pendulums of the double pendulum do not affect the Jacobi stability. It is suggested that the equilibrium points in the Jacobi stable region and in the Jacobi unstable region are Lyapunov stable and Lyapunov unstable, respectively, and that the motions in the Jacobi unstable region are related to the onset of chaotic behavior. [ABSTRACT FROM AUTHOR]

The externally excited and damped vibration of the double pendulum in the vertical plane are considered. The pendulum can collide many times during the motion with a motionless obstacle having the rough surface. The double pendulum colliding with this object has been modeled as a piecewise smooth system constrained by the unilateral constraint. In the relatively long time between the collisions, the differential equations govern the motion of the pendulum. When the contact with the obstacle appears, the pendulum exhibits a discontinuous behavior. An important element of the solving algorithm is aimed on the continuous tracking of the position of the pendulum in order to detect the collision with the unilateral constraints and to determine the state vector of the pendulum at the impact time instant. A single collision is described by the Euler's laws of motion in the integral form. The equations are supplemented by the Poisson's hypothesis and Coulomb's law of friction. The friction law is formulated for the instantaneous values of the reaction forces. The values of their impulses depend on the existence of a slip between the contacting bodies. Therefore, the Coulomb law cannot be generalized for the linear impulses of the forces in a simple way. We have applied the Routh method in order to solve the problem. The method has a simple geometrical interpretation in the impulse space. The angular velocities of both pendulum parts as well as the reaction forces at the joints of the system, which change in stepwise manner, have been presented in the paper. [ABSTRACT FROM AUTHOR]

Modern competitive shooting is a strenuous test of the human perceptual and motor systems. Like driving a race car or piloting a high performance aircraft, speed shooting matches require precise, rapid movements coordinated using a careful mixture of planning and reaction. In some disciplines of competitive shooting, this mixture is further complicated by complex moving targets. This paper develops a set of state space equations for a moving, reactive steel target used in professional 3-Gun competition. The equations for target motion are nonlinear, time-varying, and chaotic in certain regions of the state space. Once derived, the equations for target motion are validated against motion extracted from video. Then, the calibrated equations are implemented in real-time on a portable, laser-activated simulator. Applications of the simulation environment to marksmanship training and human motor control studies are also discussed. [ABSTRACT FROM AUTHOR]

In this paper energy harvesting from vibrating surfaces through electromagnetic induction is addressed. A double pendulum subject to base excitations generates electrical energy through energy harvesting coils mounted on the pendulum masses. Optimum double pendulum generates energy at a rate of 9 mW for excitation parameters characteristic of bridge vibration, frequency of 2 Hz and amplitude of 1 mm. [ABSTRACT FROM AUTHOR]

In this paper the dynamical interactions of a double pendulum arm and an electromechanical shaker is investigated. The double pendulum is a three degree of freedom system coupled to an RLC circuit based nonlinear shaker through a magnetic field, and the capacitor voltage is a nonlinear function of the instantaneous electric charge. Numerical simulations show the existence of chaotic behavior for some regions in the parameter space and this behaviour is characterized by power spectral density and Lyapunov exponents. The bifurcation diagram is constructed to explore the qualitative behaviour of the system. This kind of electromechanical system is frequently found in robotic systems, and in order to suppress the chaotic motion, the State-Dependent Riccati Equation (SDRE) control and the Nonlinear Saturation control (NSC) techniques are analyzed. The robustness of these two controllers is tested by a sensitivity analysis to parametric uncertainties. [ABSTRACT FROM AUTHOR]

Nonlinear dynamics of a double pendulum rotating at a constant speed about a vertical axis passing through the top hinge is investigated. Transitions of oscillations from chaotic to quasiperiodic and back to chaotic again are observed with increasing speed of rotation. With increasing speed, a pair of new stable equilibrium states, different from the normal vertical one, appear and the quasiperiodic oscillations occur. These oscillations are first centered around the origin, but with increasing rotation speed they cover the origin and the new fixed points. At a still higher speed, more than one pair of fixed points appear and the oscillation again turns chaotic. The onset of chaos is explained in terms of internal resonance. Analytical and numerical results confirm the critical values of the speed parameter at various transitions. [ABSTRACT FROM AUTHOR]

We consider two special types of double pendula, with the motion of masses restricted to various surfaces. In order to get quick insight into the dynamics of the considered systems the Poincaré cross sections as well as bifurcation diagrams have been used. The numerical computations show that both models are chaotic which suggests that they are not integrable. We give an analytic proof of this fact checking the properties of the differential Galois group of the system's variational equations along a particular non-equilibrium solution. [ABSTRACT FROM AUTHOR]

This paper presents a novel self-tuning anti-sway control system for shipboard cranes that provides full, six degree-of-freedom (DOF) motion compensation with respect to both the world (ocean) coordinate frame and the ship deck coordinate frame. The system can either manually or automatically switch between compensation modes during operation, and is capable of providing anti-sway action while tracking a time-varying operator input. Rather than requiring the operator to provide individual joint control, the entire system can track a simple three-axes Cartesian input from the operator, while simultaneously providing full anti-sway compensation in either coordinate frame. High-fidelity simulation case studies were performed with a nine-DOF shipboard knuckle boom crane, which included double pendulum dynamics between the hook and payload, in the presence of full six-DOF ship motion at sea-state six. The self-tuning anti-sway system showed at least a 92.70% reduction in payload tracking error when the corresponding compensation mode (either deck or world-frame) was activated, a significant reduction in undesired payload motion. Wind disturbances were added to the simulation using the Dryden wind model, where the system showed a maximum decrease in performance of only 0.73%; additional tests were run with self-tuning disabled, which resulted in a decrease in performance of up to 101%, indicating self-tuning is highly effective at minimizing the effect of wind disturbances. • A self-tuning anti-sway control system for a 9-DOF knuckle boom crane is developed. • The anti-sway system can smoothly switch between deck and world-frame compensation. • Compensation can be provided while tracking a time-varying input from an operator. • The system is tested in a high-fidelity simulation with wind and a double pendulum. • Self-tuning control reduces the effect of wind disturbances. [ABSTRACT FROM AUTHOR]

Optimal control problems for mechanical systems often arise in technical applications. To find solutions with minimal control effort, the system’s natural, uncontrolled dynamics can be used. Promising candidates to be considered for energy-efficient trajectories are highly dynamic, but uncontrolled motions on (un)stable manifolds of equilibria. In this contribution, we propose a control strategy for mechanical systems which sequences uncontrolled trajectories on (un)stable manifolds with short control manoeuvres to design a feedforward control. In particular, we present optimal swing-up solutions for a double pendulum which are based on trajectories on the stable manifold of the pendulum’s up–up equilibrium. To demonstrate the advantages of our approach compared to a black-box optimisation, we perform a post-optimisation with the optimal control sequence as an initial guess. The numerical results are evaluated in a simulation environment for the double pendulum on a cart and applied to a real test rig. [ABSTRACT FROM AUTHOR]

Born and Karman had pointed out a long time ago that a composite anisotropic solid must be characterized by multiple Debye temperatures (DTs). Since all non-elemental superconductors (SCs) are similar materials, we deal here with the problem of resolving the DT of a composite SC– usually given as single value --- into the DTs of its constituents. These DTs are a primary requirement for the application of the generalized BCS equations which have been shown to shed light on several properties of not only the high- T c SCs, but also a variety of others such as SrTiO 3 and the so-called exotic Fe-based heavy-fermion SCs which have been a serious concern for theoreticians for a long time. The SCs dealt with here are MgB 2 , YBCO, Tl-2212 and Bi-2212. [ABSTRACT FROM AUTHOR]

Abstract: This research provided a simple vibration control method for fast crane operation using an open-loop control approach. With the equations of double pendulum motion, we used two related acceleration and deceleration to propose a three-stage method consisting of (1) piecewise acceleration, (2) constant speed, and (3) piecewise deceleration. Our method is to reduce the sway angle and to reach high speed operation. The proposed method maintains a small sway angle in fast speed while the other open-loop methods do not. Our method is studied both numerically and experimentally. We found out that in numerical simulations the operational time was approximately proportional to the square root of the operation distance; a fourfold increase in distance only required a 200% increase in operational time. The method is stable with respect to the 10% variations of system parameters. In comparison to a common method in 100m operation distance, the operation time is 25% faster when the maximum sway angle is the same, and the sway angle is 44% smaller when the operation time is the same. We conducted a KUKA Robot KR 16 experiment to validate the proposed method. The KUKA experiments suggest a four times smaller vibration magnitude in the proposed method. Moreover, the numerical simulations of the proposed method compare favorably with the KUKA experiments with 95% confidence interval. With the scaled parameters, the KUKA experiment corresponds to 30s operation time and 180m operation distance in the real crane situation. The proposed method is safe and fast, and without the addition of any sensors the operator was able to control the sway of the crane cable. [Copyright &y& Elsevier]

Many activities present in the offshore technology are affected by sea waves and their safety strongly depends on the sea state. The fundamental question in operation risk analysis is the limit sea state which enables the safe operation controlled in all phases: from the sea surface to the sea bottom. The safe limit sea state is especially important when some lifting operation is performed in sea waves. This paper presents a dynamical model of a floating crane in sea waves with a hanging load at a high altitude above sea level. The model is defined as a linearized coupled double pendulum where the usual 6 DOF of the floating crane oscillatory motions in waves are extended by additional 3 DOF of the swinging load. The numerical model of the floating crane is calibrated and tuned according to the measured model in a towing tank. The significant values of the crane boom top accelerations were estimated for the given design sea state with different vertical load positions. The problem is illustrated by the example of a floating crane consisting of a simple single pontoon with an ordinary land crane mounted on the deck. The reliability of the proposed method is evaluated on the basis of correlation between the calculated and measured data. [ABSTRACT FROM AUTHOR]

This article focusses on the implementation of recently developed nonstationary linear-parameter varying (NSLPV) control algorithms for the regulation of nonlinear systems about pre-specified trajectories. The trajectories of interest eventually settle into periodic orbits, and hence are duly called eventually periodic trajectories. Parameterising the nonlinear system equations about such trajectories results in eventually periodic NSLPV models, and then NSLPV controllers are designed for these models to ensure accurate trajectory tracking despite various disturbances and uncertainties. These control algorithms will be applied to control a double pendulum where a vessel containing fluid is rigidly attached to the end of the second link of the pendulum. The mass of the fluid varies in time and, together with its rate of variation, is available for measurement during plant operation. [ABSTRACT FROM AUTHOR]

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 [ABSTRACT FROM AUTHOR]

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 [ABSTRACT FROM AUTHOR]

The single-inverted pendulum (SIP) model is still the paradigm describing dynamics and control of quiet human stance in the sagittal plane. We used two methods to verify this paradigm. First, in an experimental approach we acquired kinematic data of both legs of ten subjects at high spatial resolution while quietly standing on two force platforms. We calculated all leg joint angles, the belonging joint torques using inverse dynamics and estimates of joint stiffnesses. Some linear correlations and regressions of both local (joint) and global (COM, COP: centre of mass respectively pressure) variables predicted by the SIP model were investigated. All three verification criteria applied to mean values extracted from experimental data revealed that the SIP is not a valid model for quiet human stance. As a second method, we used computer synthesis to demonstrate that a double-inverted pendulum (DIP) model enters a stable attractor when just the 'hip' joint torque is regulated, whereas no torque is applied to the 'ankle' joint. Here, angle and torque fluctuations are necessary because such a DIP strategy is of inevitable dynamic character. The two predicted eigenfrequencies of this regulated DIP model approximate the upper and lower limits of the main part of measured power spectra of quiet human stance. We suggest this dynamic necessity to be representative of the biological constraints under which a mechanically unstable inverted multi-segment chain must be stabilised. [ABSTRACT FROM AUTHOR]

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

Abstract: Digital filters, frequently used in active control of mechanical systems, enable one to improve the signal-to-noise ratio and the control performance, but introduce group delays into the control loops simultaneously. In order to gain an insight into the effects of a digital filter on a controlled mechanical system, this paper presents the stability switches and the corresponding Hopf bifurcations of a double pendulum system with the linear quadratic control having a digital filter via theoretical analysis, numerical simulations and experiments. In this study, the digital filters are used to remove the undesired noise of high frequency, which is embedded in the control signal, and are modeled as the components of pure time delay during the theoretical analysis and numerical simulations. The study shows that a digital filter with moderate specifications can not only improve the vibration reduction effectively, but also save the energy consumption of the servo-motor remarkably. However, over demanding specifications will make the group delay of the filter exceed a critical value and cause either a divergent motion or a self-excited vibration through a Hopf bifurcation, the occurrence of which depends on both the stability and the size of the basin of attraction of the bifurcating periodic motion. The experimental results well coincide with the theoretical and numerical ones, and strongly support the simplification of the digital filters as the components of pure time delay. Finally, some suggestions are made to avoid the group delay induced instability. [Copyright &y& Elsevier]

This paper examines numerically the complex classical trajectories of the kicked rotor and the double pendulum. Both of these systems exhibit a transition to chaos, and this feature is studied in complex phase space. Additionally, it is shown that the short-time and long-time behaviours of these two PT -symmetric dynamical models in complex phase space exhibit strong qualitative similarities. [ABSTRACT FROM AUTHOR]

• Double pendulum crane with distributed mass beams (DMB) is considered. • Nonlinear sliding model control methods utilizing the velocity control are proposed. • State observer that adapts to the length variation of the suspension rope is designed. • Simulation and experiment verify the effectiveness of the method. Most of the existing anti-swing control methods treat the crane as a single pendulum system with fixed rope length. In this paper, considering the varying rope length, the dynamic model of the double pendulum crane system with distributed mass beams (DMB) is established. Four versions of sliding mode control are proposed by utilizing the equivalent control strategy. The system responses of the hierarchical linear sliding mode control method based on velocity control and displacement control are analyzed. Meanwhile, based on the pole placement method, a state observer that adapts to the length variation of the suspension rope is proposed. On this basis, the nonlinear sliding model control methods utilizing the velocity control, including the NTSMC and FTSMC, are presented. Simulations and experiments are conducted. Results indicate the effectiveness of the proposed methods. The control method can ensure the asymptotic stability of the system in a finite time, maintain robustness in disturbances and adapt the varying rope length. [ABSTRACT FROM AUTHOR]

The boundary of the phase domain of periodic solutions of a double pendulum is constructed and shown to be closed [ABSTRACT FROM AUTHOR]

The rotation of a double pendulum with a dc motor installed between the first and second pendulums was investigated using both experiments and numerical computations. The first pendulum can rotate freely around one end, while the second pendulum is attached to the other end of the first pendulum; the dc motor attached to the first pendulum causes the second pendulum to rotate with respect to the first pendulum. We clarify that there is a range of rotation speed in which the first pendulum rotates at the same, one-half, or one-third of the rotation speed of the second pendulum, and the stable rotation of the double pendulum is realized without torque control of the dc motor and report the characteristics of the stable rotation of the double pendulum. [ABSTRACT FROM AUTHOR]

Abstract: A neural oscillator model is applied to a giant swing control of an underactuated double pendulum. The oscillator receives the angle signal of the upper link and provides a relative angle of the lower link. The oscillator tunes itself to the natural dynamics of the pendulum increasing the swing amplitude, and finally the pendulum system enters the phase of giant swing. A most remarkable feature is that the transition from simple swing to giant swing can be made without changing the values of the parameters of the neural oscillator. [Copyright &y& Elsevier]

This paper describes an experimental apparatus that demonstrates some examples of self-excited vibrations in fluid mechanics. Such vibrations become unstable without an external perturbation but are characterized by an initially stable system with steady flow. The system extracts from the flow and exhibits increasing amplitude of vibration. Two experiments are described, one dependent on area change in ducted flow and the other dependent on the change in flow direction in a double pendulum. Mathematical analyses are developed which establish the stability criteria for the self-excitation observed. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]

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 [ABSTRACT FROM AUTHOR]

For the engineering of mechanical systems with a complex interplay of regular and chaotic behavior it is important to know the forces involved. It is shown how they can be computed and their time development evaluated. Characteristic features of periodic, quasiperiodic, and chaotic motion are identified. Classical methods such as Fourier transform and various statistics are used and compared to a redundant version of wavelet analysis. The latter is proposed as the most informative coherent representation of the distribution of times, frequencies, and amplitudes. [ABSTRACT FROM AUTHOR]

• The high frequency oscillation mechanism of the DMB is analyzed. • A new controller based on low-pass filter is proposed. • The higher DMB oscillation reduction with low jerk and smoothness is achieved. • Simulation and experiment verify the effectiveness of the method. Specific payload shape and hoisting mechanism will transform the crane to a complex double pendulum system with a distributed mass beams (DMB), which makes its control challenging. In this paper, the dynamic analysis of the double pendulum crane with DMB is conducted. Based on the low-pass filter, the system response law of the double pendulum system hoisting DMB is analyzed. On this basis, the previously proposed time optimal anti-swing control method is improved by utilizing low-pass filter to achieve the minimum maximum residual angle, as well as to smooth the velocity trajectory so that the jerk can be avoided. To evaluate the effectiveness of the proposed approach, simulations and experiments are performed. Results demonstrate that the proposed method can suppress and eliminate the DMB oscillation. The superiority of the proposed controller compared with the recently proposed controller is that it can achieve higher hook oscillation and DMB oscillation reduction. Besides, the advantage of the improved method is also illustrated in terms of low jerk and smoothness. [ABSTRACT FROM AUTHOR]

The Acrobot is a simple underactuated system consisting of a double pendulum with an actuator at the second joint only. We derive a set of exact trajectories of the nonlinear equations of motion, which involve inverted periodic motions. The trajectories can be made arbitrarily fast by an appropriate choice of the Acrobot mass and length parameters. Next, we present a nonlinear control law and show how it can be applied to the Acrobot to track these trajectories. In simulations we compare tracking results for our controller and one based on pseudo-linearization. The pseudo-linearizing controller produces significant error for a 1 Hz trajectory, while ours produces none. Finally, we present experimental results which demonstrate that the assumptions of the theory were not overly restrictive. In particular, peak-to-peak oscillations of joints as large as 850 were obtained, despite real-world effects, such as joint friction, inexact parameter values, and noisy and delayed joint velocity data. [ABSTRACT FROM PUBLISHER]

• Double pendulum crane with distributed mass beams(DMB) is considered. • Quasi-periodicity calculation method of DMB is designed. • Time optimal anti-swing control method is established. • Simulation, experiment and field test are conducted. The existing anti-swing control of crane is mostly used for the single pendulum system, but in practical application, specific payload shape and hoisting mechanism will induce to complex double pendulum dynamics. In this paper, the double pendulum crane with distributed mass beams(DMB) is considered. Based on the dynamic analysis, the DMB oscillations composed of the quasi-periodic single pendulum around the drum and the second-mode pendulum around the hook are obtained. And their period calculation equations can be acquired. On this basis, the time optimal control trajectory is planed to eliminate the oscillation of the quasi-periodic single pendulum by using the minimum principle. And the oscillation suppression strategy of the second-mode pendulum is designed with the step response law of the oscillation system. Finally, combining the time optimal control trajectory and the oscillation suppression strategy, the time optimal anti-swing control method is established. Simulation and experimental results demonstrate that the proposed method can suppress the DMB oscillation in short time. The adaptability of the control method to the parameters of suspension length and DMB quality is verified. Besides, the accelerated/decelerated time and distance traveled by the proposed controller is less than that of input shaping and command smoothing. Finally, the presented method is applied to the bridge crane in the rolling mills. [ABSTRACT FROM AUTHOR]

The aim of manifold learning is to extract low-dimensional manifolds from high-dimensional data. Manifold alignment is a variant of manifold learning that uses two or more datasets that are assumed to represent different high-dimensional representations of the same underlying manifold. Manifold alignment can be successful in detecting latent manifolds in cases where one version of the data alone is not sufficient to extract and establish a stable low-dimensional representation. The present study proposes a parallel deep autoencoder neural network architecture for manifold alignment and conducts a series of experiments using a protein-folding benchmark dataset and a suite of new datasets generated by simulating double-pendulum dynamics with underlying manifolds of dimensions 2, 3 and 4. The dimensionality and topological complexity of these latent manifolds are above those occurring in most previous studies. Our experimental results demonstrate that the parallel deep autoencoder performs in most cases better than the tested traditional methods of semi-supervised manifold alignment. We also show that the parallel deep autoencoder can process datasets of different input domains by aligning the manifolds extracted from kinematics parameters with those obtained from corresponding image data. [ABSTRACT FROM AUTHOR]