Double Pendulum

The article proposes a nonlinear optimal control method for the dynamic model of the parallel double inverted pendulum. Three different forms of this system are considered: (i) two poles mounted on the same cart resulting into a model with four state variables and one control input, (ii) two poles mounted on the same cart resulting into a model with six state variables and one control input, (iii) two different cart-pole systems connected through an elastic link, resulting into a model with eight state variables and two control inputs. To implement the proposed nonlinear optimal control method the dynamic model of the parallel double pendulum undergoes approximate linearization around a temporary operating point which is updated at each sampling instance. This operating point is defined within each sampling period by the present value of the state vector of the parallel double inverted pendulum and by the last sampled value of the control inputs vector. The linearization process is based on first-order Taylor series expansion and on the computation of the associated Jacobian matrices. For the approximately linearized model of the parallel double inverted pendulum a stabilizing H-infinity feedback controller is designed. To compute the feedback gains of this controller an algebraic Riccati equation is also being solved at each time-step of the control algorithm. The global stability properties of the control scheme are proven through Lyapunov analysis. First it is demonstrated that the H-infinity tracking performance criterion holds, which signifies robustness of the control method under model uncertainty and external perturbations. Moreover, under moderate conditions it is proven that the control loop of the parallel double inverted pendulum is globally asymptotically stable. To implement state estimation-based control the H-infinity Kalman Filter is used as a robust observer. The nonlinear optimal control scheme for the parallel double inverted pendulum achieves fast and accurate tracking of reference setpoints under moderate variations of the control inputs.

In the present paper, we introduce new models of pendulum motions for two cases: the first model consists of a pendulum with mass M moving at the end of a string with a suspended point moving on an ellipse and the second one consists of a pendulum with mass M moving at the end of a spring with a suspended point on an ellipse. In both models, we use the Lagrangian functions for deriving the equations of motions. The derived equations are reduced to a quasilinear system of the second order. We use a new mathematical technique named a large parameter method for solving both models’ systems. The analytical solutions are obtained in terms of the generalized coordinates. We use the numerical techniques represented by the fourth-order Runge–Kutta method to solve the autonomous system for both cases. The stabilities of the obtained solutions are studied using the phase diagram procedure. The obtained numerical solutions and analytical ones are compared to examine the accuracy of the mathema...

We introduce "Golden Resonance," a previously unrecognized stable state in classical mechanics, identified in a double pendulum system with lengths and masses proportioned by the golden ratio (ϕ ≈ 1.6180339887). In this configuration, π universally governs the periodicity of motion, as seen in the pendulum periods Ti = 2π Li/g, while ϕ emerges as a dynamic organizer of energy distribution, stabilizing the system against chaotic breakdown. Extensive numerical simulations, employing a symplectic integrator over 1000 seconds, reveal that the energy ratio between the pendulums (E1/E2) consistently converges to approximately 1.618 under diverse conditions-large initial angles, high energy inputs, and external perturbations. Poincaré sections display a robust toroidal attractor, and Lyapunov exponents remain near zero (∼ 0.001-0.005), contrasting sharply with the rapid chaotic divergence (Lyapunov ∼ 0.08-0.15) of a rational-ratio control system. This interplay of π's rhythmic foundation and ϕ's proportional stability suggests a new equilibrium in nonlinear dynamics, distinct from passive stability mechanisms like KAM theory. The phenomenon, tested to its limits, hints at a fundamental role for ϕ in mechanical systems, with potential applications in engineering stability, orbital dynamics, and natural systems exhibiting golden proportions.

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 this study, a computational intelligence technique is developed to find the solutions of nonlinear Mathieu system arising in parameter excitation, vertically derive pendulum and dusty plasma studies using the strength of artificial neural networks in the modeling of equation and effective optimization of the error function through bioinspired heuristics based on global search with genetic algorithm and rapid local convergence with interior-point algorithm. The proposed scheme is applied to number of scenarios for Mathieu system to analyze the its dynamics for parameter excitation, oscillatory pendulum and dusty plasma models. The comparison of the proposed solutions with numerical results shows a close match which establishes its correctness. The consistent accuracy of the proposed solver is verified through results of statistics in terms of different performance indices based on mean absolute error, root mean square error and Nash-Sutcliffe efficiency. These solutions greatly enrich stochastic numerical solver for the celebrated Mathieu systems.

A parametrically excited pendulum is a simple nonlinear dynamical system. The rotation of parametric pendulum [1] exhibits a conversion from the external vibration into its rotational motion. The converted motion is applicable to energy scavenging from vibration of external source. Since the periodic rotation coexists with low energy states and motions, the control is required to maintain the periodic rotation against irregularity, noise, and frequency variation of the vibration. We propose a control method for establishing the periodic rotation of the parametric pendulum based on the delayed feedback control . In the implementation of the control,

The nonlinear dynamic responses of moored crane vessels to regular waves are investigated experimentally and theoretically. The main subject of interest are nonlinear phenomena like bifurcations and the existence of multiple attractors. In the experimental part of the work, a moored model of a crane vessel has been excited by regular waves in a wave tank. A special mechanism has been developed to model the nonlinear behavior of real mooring systems. The theoretical part of the work concerns the mathematical modeling of the floating cranes. Two mathematical models of different levels of complexity are presented. Two different tools are used to systematically determine the responses of the systems to periodic forcing of waves. Firstly, the path-following techniques in combination with numerical integration of equations of motion applied to a full nonlinear model give insight into the dynamics in time domain. Secondly, the multiple scales method allows for an analytical investigation of simplified nonlinear models in frequency domain. Many results of computations for two crane vessels, barge and ship, are presented. Special attention is paid to oscillations near the frequencies of primary resonances and to subharmonic motions. An excellent agreement is found between the results of time-domain and frequency-domain analysis. The computational examples chosen correspond to the models used not only in the present experiments but in the experiments of others as well. The results presented in the work allow us to draw several important conclusions concerning the dynamic behavior of floating cranes during offshore operations. Both the developed models and the analytical tools can be used to identify the limits of the operating range of floating cranes.

The chattering oscillations for an inverted pendulum impacting between lateral walls, a prototype of a class of impact dampers, are analysed. Attention is focused on the periodic chattering appearing when the rest positions cease to be attracting, and the aim is that of computing the time τ required by the micro-oscillations to come to rest. An algorithm is proposed to compute τ , and it is shown how this time depends on the excitation amplitude. When τ becomes equal to the excitation period, the considered chattering is observed to lose attractivity. This occurs for a certain excitation amplitude threshold, which is computed and whose (very weak) dependence on the excitation frequency is illustrated. An asymptotic estimate of the chattering time with respect to a small parameter proportional to the excitation amplitude is given. This is obtained by an appropriate expansion in Taylor series of the relevant quantities involved in the analysis. It is shown that τ is asymptotically proportional to the square root of the excitation amplitude.

The chattering oscillations for an inverted pendulum impacting between lateral walls, a prototype of a class of impact dampers, are analysed. Attention is focused on the periodic chattering appearing when the rest positions cease to be attracting, and the aim is that of computing the time τ required by the micro-oscillations to come to rest. An algorithm is proposed to compute τ , and it is shown how this time depends on the excitation amplitude. When τ becomes equal to the excitation period, the considered chattering is observed to lose attractivity. This occurs for a certain excitation amplitude threshold, which is computed and whose (very weak) dependence on the excitation frequency is illustrated. An asymptotic estimate of the chattering time with respect to a small parameter proportional to the excitation amplitude is given. This is obtained by an appropriate expansion in Taylor series of the relevant quantities involved in the analysis. It is shown that τ is asymptotically proportional to the square root of the excitation amplitude.

The double concave friction pendulum system has been recognized as an efficient device for decreasing the seismic response of a structure during an earthquake excitation. Previous studies have focused mainly on the properties of the double concave friction pendulum system under constant vertical loading, and the width of the hysteretic loop changed by the vertical ground motion is less considered. In view of this, a theoretical study of the double concave friction pendulum system under variable vertical loading is conducted in this article. Meanwhile, the properties of the hysteretic loops of the double concave friction pendulum system with different friction coefficients between the articulated slider with the upper and lower sliding surfaces are investigated. The results show that the hysteretic loops of the double concave friction pendulum system will be affected by the variation of the vertical loading and the difference of the friction coefficients between the articulated slide...

In 1993, Scherpen generalized the balanced truncation method to the nonlinear setting. However, the Scherpen procedure is not easily computable and has not yet been applied in practice. We offer methods, tools, and algorithms for computing the energy functions and coordinate transformations involved in the Scherpen theory and procedure for nonlinear balancing. We then apply our approach to derive, for the first time, balanced representations of nonlinear state-space models.

This paper presents investigations into the development of input shaping techniques for swaying control of a double-pendulum-type overhead crane (DPTOC) system. A nonlinear DPTOC system is considered and the dynamic model of the system is derived using the Euler-Lagrange formulation. An unshaped bang-bang input force is used to determine the characteristic parameters of the system for design and evaluation of the input shaping control techniques. The positive and modified specified negative amplitude (SNA) input ...

Batik Pendulum is a new batik pattern created by Rumah Batik Komar using a single-string pendulum filled with wax. However, current production is still manual, so it is impossible to re-manufacture in large quantities. This research is part of a machine and software development project to produce Batik Pendulum, where this research focuses on software development. The designed software has a spherical pendulum trajectory planning feature through parameter changes. The spherical pendulum path was chosen because it has the same pattern as the currently produced Batik Pendulum. An algorithm that receives parameter values input has been designed to produce the spherical pendulum trajectory. These proposed parameters provide a variety of spherical pendulum patterns. After the user changes the parameters, the software requires 1-2 seconds to generate the trajectory. This duration is within the user's flow of thought when engaging with the software.

Vibrations of an autoparametric system, composed of a nonlinear mechanical oscillator with an attached damped pendulum, around the principal parametric resonance region, are investigated in this paper. The aim of the work is to show the chaotic motion in instability region. Two kinds of chaotic motion are detected: chaotic swings and chaotic motion composed of swings and rotation of pendulum. The results are confirmed experimentally on especially designed laboratory model. Additionally, the latest methods of chaos identification are applied to confirm chaotic dynamics experimentally.

A novel approach to solve multi-objective optimization problems of complex mechanical systems is proposed based on evolutionary algorithm. Discrete mechanics derives structure preserving constraint equations and objective functions. Standard non-linear optimization techniques used to obtain optimal solution to these equations fails to find global optimum solution and also requires system satisfying initial guess. Multi-objective optimization technique like non-dominated sorting genetic algorithm-II (NSGA-II) finds global optimal solution without giving any initial guess for multiple conflicting objectives. This method is numerically illustrated by optimizing an underactuated mechanical system called 2D SpiderCrane system. In SpiderCrane, fast and precise payload positioning is to be achieved while keeping payload swing minimum along the trajectory. Minimizing the time of operation requires greater amount of force which may lead to unacceptable payload sway, while decreasing forces increases the time of operation. Proposed control law to optimize this conflicting multi-objectives is validated with simulation results.

Single and double inverted pendulum systems subjected to delayed state feedback are analyzed in terms of stabilizability. The maximum (critical) delay that allows a stable closed-loop system is determined via the multiplicity-induced-dominancy property of the characteristic roots, that is the dominant (rightmost) roots are associated with higher multiplicity under certain conditions of the system parameters. Other methods such as tracking the changes of the D-curves with increasing delay and the Walton–Marshall method are also demonstrated for the example of the single pendulum. For the double inverted pendulum subjected to full state feedback, the number of control gains is four, and application of numerical methods requires therefore high computational effort (i.e. optimization in a four-dimensional space). It is shown that, with the multiplicity-induced-dominancy–based approach, the critical delay and the associated control gains can be determined directly using the characteristi...

This paper presents two control strategies for a parametrically excited pendulum with chaotic behavior. One of them considers active control obtained by nonlinear saturation control (NSC) and the other a passive rotational magnetorheological (MR) damper. Firstly, the active control problem was formulated in order to design the external torque for the pendulum, considering the NSC. Numerical simulations were carried out in order to show the effectiveness of this method for the active control of the pendulum oscillation. The ability of the control of the proposed NSC in suppression of the chaotic behavior, considering the proposed parameters, was tested by a sensitivity analysis to parametric uncertainties. In the case of the passive rotational MR damper, firstly the influence of the introduction of the MR in a pendulum was performed considering the 0-1 test. Different electric currents are applied to suppress the chaotic behavior of the system. The numerical results showed that the simple introduction of a passive rotational MR damper without electric current did not change the chaotic behavior of the system. However, it is possible to keep the pendulum oscillating with periodic behavior using the rotational MR damper with energizing discontinuity.

This paper presents a more accurate model of a Furuta double pendulum for efficient planning of optimal trajectories using the new approach of discrete mechanics and optimal control (DMOC). Based on the variation of discrete mechanics a direct discretization of the Lagrange-d'Alembert principle enables a novel formulation of the optimization problem. This leads to less optimization parameters and thus more computational efficiency. To shed the light on the advantages of the proposed method the swing-up maneuver of the Furuta double pendulum is used to demonstrate the point-to-point trajectory planning. For this purpose, a model of Furuta double pendulum based on Denavit-Hartenberg convention with higher accuracy is presented additionally.

A geometric form of Euler-Lagrange equations is developed for a chain pendulum, a serial connection of n rigid links connected by spherical joints, that is attached to a rigid cart. The cart can translate in a horizontal plane acted on by a horizontal control force while the chain pendulum can undergo complex motion in 3D due to gravity. The configuration of the system is in (S 2) n ×R 2. We examine the rich structure of the uncontrolled system dynamics: the equilibria of the system correspond to any one of 2 n different chain pendulum configurations and any cart location. A linearization about each equilibrium, and the corresponding controllability criterion is provided. We also show that any equilibrium can be asymptotically stabilized by using a proportional-derivative type controller, and we provide a few numerical examples.

A special combined gyro-pendulum stabilizer (a gyroscope with coupling to a pendulum) mounted on a vibrating mass is considered for investigation of the vibration responses. This paper mainly focuses on the derivation of the frequency equations and on finding the required angular momentum for vibration control of the system. Besides, there is also an ANSYS simulation model of gyro-pendulum, which was built to verify the mathematical model. The dynamic responses of both that obtained from ANSYS simulation and that obtained from numerical solving of a Lagrangian mathematical model are analyzed comparatively. The angular momentum ($Omega I_p$), in relation to the natural frequency ($omega_n$) of the primary mass, shows that this vibration control device is more adaptable than other conventional ones by producing unidirectional thrust along the forcing excitation axis whilst the gyroscope is spinning.

In this paper, the Parallel Distributed Compensation (PDC) method is used to stabilize a cart-mounted inverted pendulum and Overhead Crane model. One of the significant issues of using PDC approach for systems with nonlinear terms is finding a linear sectors. Since equations of motion (EoM) of an inverted pendulum and overhead crane include some complicated nonlinear terms, finding sectors is often impossible. Therefore, the traditional PDC has some difficulties from the practical point of view. In order to overcome such a problem, here, two different approaches have been proposed. In the first approach, prior to utilizing a PDC, complex equations have been simplified using feedback linearization method. PDC method is then applied to the obtained closed loop system. In the second strategy, named as the approximated PDC, after eliminating the small terms in the EoM, PDC method is applied. The results of simulations pertinent to PDC controller and traditional LQR method have been compared. Finally, for verification of the presented approximated approach, practical implementation on the experimental crane setup has been done and results reported and compared with simulation.

Experimental and numerical investigations are carried out on an autoparametric system consisting of a composite pendulum attached to a harmonically base excited mass-spring subsystem. The dynamic behavior of such a mechanical system is governed by a set of coupled nonlinear equations with periodic parameters. Particular attention is paid to the dynamic behavior of the pendulum. The periodic doubling bifurcation of the pendulum is determined from the semi-trivial solution of the linearized equations using two methods: a trigonometric approximation of the solution and a symbolic computation of the Floquet transition matrix based on Chebyshev polynominal expansions. The set of nonlinear differential equations is also integrated with respect to time using a finite difference scheme and the motion of the pendulum is analyzed via phase-plane portraits and Poincare maps. The predicted results are experimentally validated through an experimental set-up equipped with an optoelectronic set sensor that is used to measure the angular displacement of the pendulum. Period doubling and chaotic motions are observed.

This paper considers the nonlinear dynamics of an electromechanical device with a pendulum arm and a Nonlinear Energy Sink (NES) put on the point of the pendulum suspension. It is shown that the (NES) is capable of absorbing energy from the system. The numerical results are shown in a bifurcation diagram, phase plane, Poincaré map and Lyapunov exponents.

Dynamics of the nonlinear spring pendulum is analysed using two asymptotic approaches. The multiple scale method is commonly applied with using two time scales. The purpose of the research is to justify the introduction of an additional third scale. Results of the analysis clearly show that introducing the third scale improve correctness of the approximate analytical solution. The obtained results allow for qualitative and quantitative analysis of the behavior of the studied system with a high accuracy. Calculations are made both in the neighbourhood of the resonance and also far from it.

The dynamic response of a harmonically and kinematically excited spring pendulum is studied. This system is a multi-degree-of-freedom system and is considered as a good example for several engineering applications. The multiple-scale (MS) method allows us to analytically solve the equations of motion and recognize resonances. Also stability of the steadystate solutions can be verified. The transfer of energy from one to another mode of vibrations is illustrated.

This paper presents investigations into the development of input shaping techniques for swaying control of a doublependulum-type overhead crane (DPTOC) system. A nonlinear DPTOC system is considered and the dynamic model of the system is derived using the Euler-Lagrange formulation. An unshaped bang-bang input force is used to determine the characteristic parameters of the system for design and evaluation of the input shaping control techniques. The positive and modified specified negative amplitude (SNA) input shapers with the derivative effects respectively are designed based on the properties of the system. Simulation results of the response of the DPTOC system to the shaped inputs are presented in time and frequency domains. Performances of the control schemes are examined in terms of sway angle reduction and time response specifications. Finally, a comparative assessment of the proposed control techniques is presented and discussed.

It is well-known that a suitably designed u n p owered mechanical biped robot can \walk" down an inclined plane with a steady gait. The characteristics of the gait (e.g., velocity, time period, step length) depend on the geometry and the inertial properties of the robot and the slope of the plane. A p assive motion has the distinction of being \natural" and is likely to enjoy energy optimality. Investigation of such motions may potentially lead us to strategies useful for controlling active walking machines. In this paper we demonstrate that the nonlinear dynamics of a simple passive \compass gait" biped r obot can exhibit periodic and stable limit cycle. Kinematically the robot is identical to a double pendulum (or its variations such as the Acrobot and the Pendubot). Simulation results also reveal the existence of a stable gait with unequal step lengths. We also present an active control scheme which enlarges the basin of attraction of the passive limit cycle.

It is well-known that a suitably designed u n p owered mechanical biped robot can \walk" down an inclined plane with a steady gait. The characteristics of the gait (e.g., velocity, time period, step length) depend on the geometry and the inertial properties of the robot and the slope of the plane. A p assive motion has the distinction of being \natural" and is likely to enjoy energy optimality. Investigation of such motions may potentially lead us to strategies useful for controlling active walking machines. In this paper we demonstrate that the nonlinear dynamics of a simple passive \compass gait" biped r obot can exhibit periodic and stable limit cycle. Kinematically the robot is identical to a double pendulum (or its variations such as the Acrobot and the Pendubot). Simulation results also reveal the existence of a stable gait with unequal step lengths. We also present an active control scheme which enlarges the basin of attraction of the passive limit cycle.

We provide sufficient conditions for the existence of periodic solutions of the planar perturbed double pendulum with small oscillations having equations of motion θ 1 = −2aθ 1 + aθ 2 + εF 1 (t, θ 1 ,θ 1 , θ 2 ,θ 2), θ 2 = 2aθ 1 − 2aθ 2 + εF 2 (t, θ 1 ,θ 1 , θ 2 ,θ 2), where a and ε are real parameters. The two masses of the unperturbed double pendulum are equal, and its two stems have the same length l. In fact a = g/l where g is the acceleration of the gravity. Here the parameter ε is small and the smooth functions F 1 and F 2 define the perturbation which are periodic functions in t and in resonance p:q with some of the periodic solutions of the unperturbed double pendulum, being p and q positive integers relatively prime.

Chaos control techniques exploit the sensitivity of chaos to initial conditions by applying feedback perturbations to an accessible system parameter. Most methods apply only one perturbation per period and are thus susceptible to control failure when applied to highly unstable systems. Here we extend a recently developed model-independent, quasicontinuous chaos control technique to stabilize a high-dimensional chaotic system: the driven double pendulum.

The coupled slosh-vehicle dynamics of a rigid body in planar atmospheric flight carrying a sloshing liquid is considered as a multibody system with the sloshing motion modelled as a simple pendulum. The coupled, non-linear equations for the four-degree-of-freedom multibody system are derived using the method of Lagrangian dynamics. Careful non-dimensionalization reveals two crucial parameters that determine the extent of coupling between the rigid body and slosh modes, and also two important frequency parameters. Using a two-parameter continuation method, critical combinations of these four parameters for which the coupled slosh-vehicle dynamics can become unstable are computed. Results are displayed in the form of neutral stability curves (stability boundaries) in parameter space, and an analytical expression incorporating the four parameters that represents the neutral stability curves is obtained. Reduced-order linearized models and key transfer functions are derived in an effort to understand the instability phenomenon. Physically, the sloshing motion is seen to induce a static instability, sometimes called tumbling, in the vehicle pitch dynamics, depending on the slosh mass fraction and the location of the slosh pendulum hinge point above the rigid vehicle center of mass.

The basic aim of the present work was to swing up a real pendulum from the pending position and to balance stably the pendulum at the upright position and further move the pendulum cart to a specified position on the pendulum rail in the shortest time. Different control strategies are compared and tested in simulations and in real-time experiments, where maximum acceleration of the pendulum pivot and length of the pendulum rail are limited. A comparison of fuzzy swinging algorithm with energy-based swinging strategies shows advantages of using fuzzy control theory in nonlinear real-time applications. An adaptive state controller was developed for a stabile, and in the same time optimal balancing of an inverted pendulum and a switching mechanism between swinging and balancing algorithm is proposed.

The basic aim of the present work was to swing up a real pendulum from the pending position and to balance stably the pendulum at the upright position and further move the pendulum cart to a specified position on the pendulum rail in the shortest time. Different control strategies are compared and tested in simulations and in real-time experiments, where maximum acceleration of the pendulum pivot and length of the pendulum rail are limited. A comparison of fuzzy swinging algorithm with energy-based swinging strategies shows advantages of using fuzzy control theory in nonlinear real-time applications. An adaptive state controller was developed for a stabile, and in the same time optimal balancing of an inverted pendulum and a switching mechanism between swinging and balancing algorithm is proposed.

The paper considers a model of a vertical double pendulum with one suspension centre moving in a vertical plane. For the proposed system of pendulums, differential equations of motion and conditions for the collision of balls are obtained. When modelling the movement of pendulums, the central impact of the balls was considered for various variants of the movement of the suspension point: the suspension point oscillates in the vertical direction; the suspension point makes rotational movements in the vertical plane. In this case, various conditions of the central impact between the balls were considered: absolutely inelastic impact; absolutely elastic impact; impact with the transformation of impact energy (elastic impact). Comparison of the results of the numerical simulation and the results of experiments with the Kapitza pendulum in published sources confirmed the possibility of modelling an elastic impact between balls in a double pendulum and between balls in an autobalancer with a horizontal axis of rotation of the rotor.

Dynamical stabilization of an inverted pendulum through vertical movement of the pivot is a well-known counter intuitive phenomenon in classical mechanics. This system is also known as Kapitza pendulum and the stability can be explained with the aid of effective potential. We explore the effect of many body interaction for such a system. Our numerical analysis shows that interaction between pendula generally degrades the dynamical stability of each pendulum. This effect is more pronounced in nearest neighbour coupling than all-to-all coupling and stability improves with the increase of the system size. We report development of beats and clustering in network of coupled pendula.

This paper presents the design of a Linear Matrix Inequality (LMI) based state feedback controller for position tracking, hook and payload oscillations of a double pendulum crane. In this work, a linearised model of the crane was firstly obtained. The idea of formulating the stability condition in the form of LMIs using the linearised model was proposed. Using this approach, the stabilisation conditions constraints the closed-loop poles of the system to be in an LMI region to guarantee the system stability and gives satisfied transient performance. Based on the stabilization conditions formulated, an LMI based state feedback tracking controller for trolley position tracking and swing angles control of the double pendulum crane is proposed.  Mostly, two or more controllers were used for the control of double pendulum crane to control the trolley, hook and payload. However, in this work a single LMI based controller is used to achieve similar performance. This reduces the number of co...

This paper shows, for the first time, that the mathematical pendulum and generalized mathematical pendulum initial and boundary value problems may be computed from the explicit and exact general solution to the corresponding differential equation in a straightforward fashion by a direct method.

The objective in this paper is to show that the generalized Duffing-van der Pol and modified Emden type equations consist of limiting cases of the exactly integrable Monsia et al.[2] nonlinear oscillator equation by expanding the exponential-type damping and restoring forces in a Taylor series.

A non-linear pendulum model is developed to represent the motion of a sloshing fluid in real time. The forces imposed by the sloshing fluid are identified using multiphase RANS CFD simulations and subsequently included in the pendulum sloshing model. The pendulum sloshing ...

In this paper a full state feedback control of a double inverted pendulum on a cart (DIPC) are designed and compared. Modeling is based on Euler-Lagrange equations derived by specifying a Lagrangian, difference between kinetic and potential energy of the DIPC system. A full state feedback control with H infinity and H 2 is addressed. Two approaches are tested: open loop impulse response and a double inverted pendulum on a cart with full state feedback H infinity and H 2 controllers. Simulations reveal superior performance of the double inverted pendulum on a cart with full state feedback H infinity controller.

The paper considers a model of a vertical double pendulum with one suspension centre moving in a vertical plane. For the proposed system of pendulums, differential equations of motion and conditions for the collision of balls are obtained. When modelling the movement of pendulums, the central impact of the balls was considered for various variants of the movement of the suspension point: the suspension point oscillates in the vertical direction; the suspension point makes rotational movements in the vertical plane. In this case, various conditions of the central impact between the balls were considered: absolutely inelastic impact; absolutely elastic impact; impact with the transformation of impact energy (elastic impact). Comparison of the results of the numerical simulation and the results of experiments with the Kapitza pendulum in published sources confirmed the possibility of modelling an elastic impact between balls in a double pendulum and between balls in an autobalancer with a horizontal axis of rotation of the rotor.

In this paper we derive a modified energy based swing-up controller using Lyapunov functions. During the derivation, all effort has been made to use a more complex dynamical model for the single inverted pendulum (SIP) system than the simplified model that is most commonly used. We consider the electrodynamics of the DC motor that drives the cart, and incorporate viscous damping friction as seen at the motor pinion. Furthermore, we use a new method to account for the limitation of having a cart-pendulum system with a finite track length. Two modifications to the controller are also discussed to make the method more appropriate for real-time implementation. One of the modifications improves robustness using a modified Lyapunov function for the derivation, while the other one incorporates viscous damping as seen at the pendulum axis. We present both simulation and real-time experimental results implemented in MATLAB Simulink.

This study attempts to model the human arm as a dynamical triple pendulum system. The equation of motion of te human arm was obtained using Euler-Lagrange equation. The resulted second order differential equation was solved analytically. Simulated results were presented with the aid of a computer software-Maple. It was observed that the angular displacement values of the three segments are directly proportional to their respective angular acceleration, which is in the modelling and analysis of human arm motion as a multiple pendulum system. Generally, the longer the segments of the human arm the longer it takes to swing back and forth, and the fewer back-and-forth swings there are in a second.

Many researchers have worked on dynamical systems in recent times because of its relevance in Engineering and Applied Physics. This paper attempts to model a damped driven simple pendulum as dynamical system and solve the model using Undetermined Coefficients Method. The model is a second order differential equation. The results obtained show that the motion of the pendulum is affected, significantly, by the driving force and the torque of the pendulum. Specifically, as driving force increases the response maximum amplitude of the pendulum decreases.

This paper proposes a Model Reference Command Shaping (MRCS) approach for an effective vibration and oscillation control of multimode flexible systems. The proposed MRCS is designed based on a reference model and avoids the need for measurement or estimation of several modes of frequency and damping ratio as in the case of other input shaping and command shaping approaches. To test the effectiveness and robustness, the designed MRCS is implemented for oscillation control of a double-pendulum overhead crane. Simulations on a nonlinear crane model and experiments using a laboratory overhead crane are carried out under two cases, without and with payload hoisting. Without a prior knowledge of the system frequency and damping ratio, the MRCS is shown to provide the highest reductions in the overall hook and payload oscillations when compared to the multimode Zero Vibration and Zero Vibration Derivative shapers designed based on the Average Travel Length approach. In addition, the MRCS is more robust towards changes in the frequency during payload hoisting and changes in the payload mass. It is envisaged that the proposed method can be useful in designing effective vibration control of multimode flexible systems.

A Laboratory device is designed to emulate the Segway motion, modifying the Furuta pendulum experiment. To copy the person on the Segway transportation unit to the Furuta pendulum, a second pendulum is added to the main one. Using LMI theory, the control objective consists to manipulate the base of the Furuta pendulum according to the inclination of the added pendulum whereas the coupled pendulums are maintained near to theirs upright positions. According to the inclination of the second pendulum, the base will rotate as long as the second pendulum remains inclined, emulating the Segway behavior. The base rotation will depend on the side of the inclination too. Experimental results are offered too (video link: http://youtu.be/SHQdW3k3qiE).

We present new solutions for the dynamics of a pendulum energy converter which is vertically excited at its suspension point. Thereby, we deal with a random excitation by a non‐white Gaussian stochastic process. We formulate the pendulum energy converter as a weakly perturbed Hamiltonian system. The random process across the energy levels of the Hamiltonian system is then approximated by a Markov process, which is obtained by stochastic averaging. This procedure leads to analytical results for the energy of the pendulum motion, which are used for analyzing the required probability of reaching higher energy states of the pendulum energy converter in order to maximize the harvested energy.