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The walking ability of the elderly is a major global health concern. Most of the elderly experience walking difficulties such as decreased walking speed and loss of balance as well as coordination. To address this issue, a mechanical orthosis called the hip-knee-ankle-foot orthosis (HKAFO) has been designed to assist elderly individuals and hospital patients in performing therapeutic treatments. However, the existing standard HKAFOs have several limitations including high energy consumption and overloading of the hip and lower limb joints during walking activities. Therefore, this study aims to enhance the standard HKAFO by developing an electromechanical HKAFO (EM-HKAFO). The study involves developing a two-degree-of-freedom mathematical model of a double pendulum using Lagrange formulation to represent the HKAFO structure and evaluate the kinematic motions of the lower limbs. The performance of the proposed walking aid is evaluated using an actual instrumented EM-HKAFO focusing on its effectiveness in achieving a desired walking angle of 25°. The result indicates that the difference between simulation and experimental data is within an acceptable error range of 14%. Overall, this study contributes to the advancement of mobility aids and enhances the quality of life for elderly individuals and patients in need. [ABSTRACT FROM AUTHOR]

In this paper, we apply the method of Lagrangian descriptors as an indicator to study the chaotic and regular behaviors of trajectories in the phase space of the classical double pendulum system. To successfully quantify the degree of chaos with this tool, we first derive Hamilton's equations of motion for the problem in nondimensional form and demonstrate how they can be expressed compactly using matrix algebra. Once the dynamical equations are obtained, we conduct a parametric study based on the system's total energy and other key parameters such as the lengths and masses of the pendulums, as well as gravity, to determine the extent of the chaotic and regular regions in the phase space. Our numerical results reveal that, for a given mass ratio, the maximum chaotic fraction of phase space trajectories is attained when the pendulums have equal lengths. Furthermore, we characterize how chaos grows and decays within the system in terms of the model parameters, and explore the hypothesis that the chaotic fraction follows an exponential law over different energy regimes. [ABSTRACT FROM AUTHOR]

In this paper, internal resonances appearing in mechanical systems with several degrees of freedom, when the ratio between the frequencies of their oscillations represent integers, are studied. As specific examples, such systems with two degrees of freedom like a double mathematical pendulum and a double physical pendulum are considered. The relationships for the oscillation frequencies of each of these systems are presented in dimensionless form depending on two dimensionless parameters characterizing the ratio between the weights of the end loads or weighty links, as well as the ratio between the lengths of the links. The conditions that should be imposed on these dimensionless parameters in order to provide for the appearance of internal resonances have been revealed. The obtained solutions are presented graphically in the form of curves on the plane of dimensionless parameters corresponding to internal resonances. In addition, a detailed investigation into the nature of these curves is presented, too. The results found upon analyzing the two problems are compared with each other. The obtained relationships have fundamental theoretical significance, which could be useful for the case of different technical applications. [ABSTRACT FROM AUTHOR]

For conservative mechanical systems, the method of normal coordinates is known, which uses the theorem on the reduction of two quadratic forms to the sum of squares. In this case, the system of differential equations is split into a system of independent oscillators. A linear dissipative mechanical system with a finite number of freedom degrees is defined by three quadratic forms: the kinetic energy of the system and potential energy of the system, and the dissipative Rayleigh function. We study the linear problem of forced oscillations of a double pendulum when the friction coefficients are proportional to the masses. Then all three quadratic forms are reduced to the sum of squares by a single transformation. In normal coordinates the system splits into two independent systems of second order. An analytical solution is constructed in the most general form for arbitrary rod lengths and point masses. A complete analysis of the oscillations in the non-resonant case and in the case of resonances is given. Formulas for the error of the analytical formulas if the proportionality of the friction coefficients and masses is violated are also obtained. [ABSTRACT FROM AUTHOR]

The interaction between the double pendulum and nonlinear sloshing in two cranes transporting a liquid tank corrupts the efficiency and safety of the cooperative transportation. Although significant work has been directed at the double-pendulum dynamics of industrial cranes and the liquid-sloshing dynamics in the moving tank, less effort has been focused on the interaction between the double pendulum and nonlinear sloshing. An analytical model of two cranes carrying a liquid tank has been derived in this article. The interaction between the double pendulum and nonlinear sloshing has been captured by the model. Furthermore, a new method has been developed to control coupled oscillations caused by the interaction. Simulated and experimental data obtained from a testbench of two cranes carrying a liquid tank validated the effectiveness of the dynamic modeling and vibration control methods. [ABSTRACT FROM AUTHOR]

We present analytical and numerical results on integrability and transition to chaotic motion for a generalized Ziegler pendulum, a double pendulum subject to an angular elastic potential and a follower force. Several variants of the original dynamical system, including the presence of gravity and friction, are considered, in order to analyze whether the integrable cases are preserved or not in presence of further external forces, both potential and non-potential. Particular attention is devoted to the presence of dissipative forces, that are analyzed in two different formulations. Furthermore, a study of the discrete version is performed. The analysis of periodic points, that is presented up to period 3, suggests that the discrete map associated to the dynamical system has not dense sets of periodic points, so that the map would not be chaotic in the sense of Devaney for a choice of the parameters that corresponds to a general case of chaotic motion for the original system. [ABSTRACT FROM AUTHOR]

In most material handling industries, input‐ and command‐shaping techniques are extensively used to transfer objects safely. In these techniques, most of the work assumes a single‐degree‐of‐freedom system with zero initial conditions, and some of these cases are not practical. In crane systems, a crane jib cannot be positioned exactly on top of the payload at the start of transfer motion, which forces the system to start with initial angles. Furthermore, when the hook is large and/or far from the payload, the system cannot be considered a single‐degree‐of‐freedom system. Not accounting for such conditions can create unwanted oscillations at the end of the motion. In this work, a closed‐form command shaping control of a double pendulum is suggested to eliminate the residual oscillations while considering nonzero initial angles. The equation of motion is derived and then solved to find the shaper constants analytically. This shaper has a selectable maneuvering time and ensures maximum cruising velocity. Several examples are implemented numerically and experimentally to evaluate the shaper's performance. Despite the nonzero initial conditions, the results showed that the shaper can effectively eliminate all induced vibrations at the end of the maneuver. [ABSTRACT FROM AUTHOR]

In order to quickly eliminate the swing angle of plate hoisting under wind load, an improved dynamic sliding mode control method is proposed in this paper. Firstly, the plate hoisting model under wind load is simplified into a double pendulum system, and the dynamic model of a double pendulum cable crane with wind disturbance is established. To enhance the coupling relationship between trolley displacement and double swing angle, a new coupling state vector for the double pendulum system is defined. On this basis, a dynamic sliding surface is constructed and then an integral function of the system's discontinuous term is established, essentially ensuring the continuity of the sliding surface and reducing the system's chattering characteristics. Finally, the stability of the system is rigorously proved by Lyapunov theorem and Barbalat lemma. The simulation and experimental results show that the controller proposed in this paper has a good effect on the anti-swing problem of plate hoisting under wind load. [ABSTRACT FROM AUTHOR]

In this paper, the behavior of the double pendulums has been studied by considering the effect of initial conditions (angular displacement of the outer pendulum for four cases 12, 30, 90, and 150 degrees) and the influence of system geometry (increasing rod length and mass of the second pendulum), also. After extracting motion equations using the Lagrangian method, in order to deal with the frequency spectrum, traces of bobs, and understanding the system behavior for every case, the Fast Fourier Transform (FFT) technique and Poincare sections have been applied. The obtained results show that the consequence of the rising angular displacement the outer pendulum is to increase the energy level of the system and the change of its behavior from quasi-periodic for angular displacement is less than 90 degrees to chaotic when it is 150 degrees. Therefore, the energy level, in this case, has increased more than twice compared to the first. In addition, it seems a quasi-periodic behavior is forming at the heart of chaotic behavior. On the other hand, the results indicate a very significant effect of geometrics on a system's behavior. According to the calculations, the consequence of increasing the length of the rod of the second pendulum has only led to a different behavior from its similar case (case number 3) completely. however, their energy level is the same. Increasing the mass of the outer by twice not only to lead decrease energy level of the system by 330J but also has shown chaotic behavior (in comparison to the case3).

In this paper, the dynamic behavior of a double pendulum under horizontal harmonic excitation is studied. The mathematical model is described by a four-dimensional non-autonomous system with smooth nonlinearities. Based on the sensitivity analysis of parameters, two representative parameters are selected and their influences on the system behavior are reported with a set of high-resolution stability diagrams. In addition to explore the classical dynamic behavior of the system, our study also investigates the issue of multistability arising from attractor self-reproducing. To enhance the reliability of our findings, simulations were conducted within a multi-body simulation environment, which yielded consistent and robust results. Furthermore, utilizing the experimental platform developed with Qualisys, we identified several pairs of attractors with specific offsets, a significant indication of attractor self-reproducing. This paper will contribute to understand the rich and intriguing behaviors of the double pendulum. [ABSTRACT FROM AUTHOR]

The paper is devoted to the investigation of the possibility of constructing a double pendulum fixed at the L1 libration point in the framework of the planar circular restricted three-body problem. Possible configurations of pendulum equilibrium positions depending on the ratios of masses and lengths of single pendulums composing the double pendulum are shown. The stability of two equilibrium positions is proved using Sylvester's criterion. In the first position, the pendulum is oriented toward a Moon, and it is oriented toward a Planet in the second position. Small motions near these stable equilibrium configurations are studied. The natural frequencies and mode ratios are obtained analytically, and their dependence on the mass and length ratios of the pendulums is analyzed. The conducted studies demonstrate the possibility of building a space elevator in the Mars–Phobos system from the L1 libration point to a Moon (distance from the L1 point to the surface of Phobos ~ 3.4 km) or to a Planet (distance from the L1 point to the surface of Mars ~ 7800 km). This also opens up the opportunity of building a two-part space elevator from Mars to Phobos. The obtained natural frequencies and mode ratios allow us to predict in advance the possible motions of a space elevator under small perturbations relative to the stable equilibrium position. [ABSTRACT FROM AUTHOR]

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]

The single, double, and triple pendulum has served as an illustrative experimental benchmark system for scientists to study dynamical behavior for more than four centuries. The pendulum system exhibits a wide range of interesting behaviors, from simple harmonic motion in the single pendulum to chaotic dynamics in multi-arm pendulums. Under forcing, even the single pendulum may exhibit chaos, providing a simple example of a damped-driven system. All multi-armed pendulums are characterized by the existence of index-one saddle points, which mediate the transport of trajectories in the system, providing a simple mechanical analog of various complex transport phenomena, from biolocomotion to transport within the solar system. Further, pendulum systems have long been used to design and test both linear and nonlinear control strategies, with the addition of more arms making the problem more challenging. In this work, we provide extensive designs for the construction and operation of a high-performance, multi-link pendulum on a cart system. Although many experimental setups have been built to study the behavior of pendulum systems, such an extensive documentation on the design, construction, and operation is missing from the literature. The resulting experimental system is highly flexible, enabling a wide range of benchmark problems in dynamical systems modeling, system identification and learning, and control. To promote reproducible research, we have made our entire system open-source, including 3D CAD drawings, basic tutorial code, and data. Moreover, we discuss the possibility of extending our system capability to be operated remotely, enabling researchers all around the world to use it, thus increasing access.

Pendulums are simple mechanical systems that have been studied for centuries and exhibit many aspects of modern dynamical systems theory. In particular, the double pendulum is a prototypical chaotic system that is frequently used to illustrate a variety of phenomena in nonlinear dynamics. In this work, we explore the existence and implications of codimension-1 invariant manifolds in the double pendulum, which originate from unstable periodic orbits around saddle equilibria and act as separatrices that mediate the global phase space transport. Motivated in part by similar studies on the three-body problem, we are able to draw a direct comparison between the dynamics of the double pendulum and transport in the solar system, which exist on vastly different scales. Thus, the double pendulum may be viewed as a table-top benchmark for chaotic, saddle-mediated transport, with direct relevance to energy-efficient space mission design. The analytical results of this work provide an existence result, concerning arbitrarily long itineraries in phase space, that is applicable to a wide class of two degree of freedom Hamiltonian systems, including the three-body problem and the double pendulum. This manuscript details a variety of periodic orbits corresponding to acrobatic motions of the double pendulum that can be identified and controlled in a laboratory setting. [ABSTRACT FROM AUTHOR]

We study a version of the two-degree-of-freedom double pendulum in which the two point masses are replaced by rigid bodies of irregular shape and nonconservative forces are permitted. We derive the equations of motion by analysing the forces involved in the framework of screw theory. This distinguishes the work from similar studies in the literature, which typically consider a double pendulum composed with rods and assume equations of motion without derivation. The equations of motion are solved numerically using the fourth-order Runge-Kutta method to show that decreasing the friction of the axles can cause the trajectory of one of the pendulums to become aperiodic. The stability of steady state solutions is also analysed.

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]

The geometry dynamical modeling method for a double pendulum is explored with the Lie group and a double spherical space method. Four types of Lagrange equations are built for relative and absolute motion with the above two geometry methods, which are then used to explore the influence of different expressions for motion on the dynamic modeling and computations. With Legendre transformation, the Lagrange equations are transformed to Hamilton ones which are dynamical models greatly reduced. The models are solved by the same numerical method. The simulation results show that they are better for the relative group than for the absolute one in long time simulation with the same numerical computations. The Lie group based result is better than the double spherical space one. [ABSTRACT FROM AUTHOR]

The aim of the experiment is to put online some practical work on the double pendulum by highlighting the chaotic aspect of this system. Then, we experimentally compare the ideal case with the real case (Effect of frictions) and we study the evolution of the system by changing the initial conditions. After having chosen the initial conditions, the learner launches the double pendulum, via the web. Thanks to a fast camera, she/he recovers the video showing the evolution of the system. Then, using a specific pointing software, the positions of the both masses of the double pendulum are detected over time, which allows to compare the real case with the theoretical case (without friction) and therefore to see the friction effect. The purpose of this paper is to carry out remote physics experiments for students to handle and control in real time. The experiment adopted for this work is the double pendulum. It consists of a pendulum attached to the end of another pendulum. The assembly is then set in motion from a configuration, determined by the initial conditions. To conduct this experiment remotely, we proceeded as follows: First of all, the mechanical part, in which we made the 3D design and the simulation using Solidworks in order to validate the design, then, we chose the electronic components that are compatible with the mechanical part. Finally, the IT part for the on-line setting. [ABSTRACT FROM AUTHOR]

Gantry cranes are mostly regarded as single pendulum models to research. However, gantry cranes will produce a double pendulum effect during the actual operation when the hook mass or cable length between the load and hook cannot be ignored. Aiming at the problems of working inefficiency, poor positioning accuracy and violent hook/load swing during the lifting process of gantry cranes, an enhanced-coupling-based tracking control method is proposed. By referring to a smooth tracking trajectory, the proposed method ensures that the trolley runs steadily. And by combining with the passivity analysis of the gantry crane system, a coupling signal, which enhances the relationship between system variables, is constructed. The system stability is proved strictly by Barbarat theorem and Lyapunov method. Experiments and simulations are performed to demonstrate the feasibility of the proposed method. The final results reflect that the proposed method, which can not only ensure the precise positioning of the trolley, but also restrain and eliminate the system swings promptly, has excellent control performance. When the system parameters are changed or external disturbances are added, the proposed method can also obtain outstanding control performance and has excellent robustness. Not only does the proposed method provide an anti-swing strategy for double pendulum underactuated gantry cranes, but also it provides a possibility for the research and development of the automatic driving of gantry cranes, which has great practical significance and application prospects. [ABSTRACT FROM AUTHOR]

Gantry cranes have attracted extensive attention that are mostly simplified as nonlinear single pendulum systems without load hoisting/lowering. However, due to the existence of the hook in practice, gantry cranes produce double pendulum effect. With an extra underactuated degree of freedom, the anti-swing control of double pendulum gantry cranes becomes more difficult than that of single pendulum gantry cranes. Moreover, double pendulum gantry cranes with load hoisting/lowering may cause large swings, which lead to inaccurate positioning and low transportation efficiency. In this paper, a novel nonlinear coupled tracking anti-swing controller is proposed to solve these problems. The proposed controller can ensure the stable startup and operation of the trolley by introducing a smooth expected trajectory. In addition, a composite signal is constructed to suppress and eliminate the swing angles of the gantry crane system. The system stability is analyzed by utilizing Lyapunov techniques and Barbalat's lemma. Theoretical derivation, simulation and experimental results indicate that the proposed controller suppresses and eliminates the hook/load swing angle effectively. Furthermore, it can achieve superior control effects and strong robustness against the changes of the load mass, trolley target displacement, initial rope lengths, initial system swing angles and external disturbances. [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.

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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]

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]

Implicit methods for the numerical solution of initial-value problems may admit multiple solutions at any given time step. Accordingly, their nonlinear solvers may converge to any of these solutions. Below a critical timestep, exactly one of the solutions (the consistent solution) occurs on a solution branch (the principal branch) that can be continuously and monotonically continued back to zero timestep. Standard step-size control can promote convergence to consistent solutions by adjusting the timestep to maintain an error estimate below a given tolerance. However, simulations for symplectic systems or large physical systems are often run with constant timesteps and are thus more susceptible to convergence to inconsistent solutions. Because simulations cannot be reliably continued from inconsistent solutions, the critical timestep is a theoretical upper bound for valid timesteps.

In this article, a control strategy approach is proposed for a system consisting of a quadrotor transporting a double pendulum. In our case, we attempt to achieve a swing free transportation of the pendulum, while the quadrotor closely follows a specific trajectory. This dynamic system is highly nonlinear, therefore, the fulfillment of this complex task represents a demanding challenge. Moreover, achieving dampening of the double pendulum oscillations while following a precise trajectory are conflicting goals. We apply a proportional derivative (PD) and a model predictive control (MPC) controllers for this task. Transportation of a multiple pendulum with an aerial robot is a step forward in the state of art towards the study of the transportation of loads with complex dynamics. We provide the modeling of the quadrotor and the double pendulum. For MPC we define the cost function that has to be minimized to achieve optimal control. We report encouraging positive results on a simulated environmentcomparing the performance of our MPC-PD control circuit against a PD-PD configuration, achieving a three fold reduction of the double pendulum maximum swinging angle. [ABSTRACT FROM AUTHOR]