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To date, most control strategies have been designed for the two-dimensional and three-dimensional overhead crane systems with the single pendulum effect. In fact, three-dimensional overhead crane systems often exhibit the double-pendulum effect in practical applications, which is more controlled than the general single-pendulum crane, and also makes the controller design more challenging. This paper proposes a new nonlinear control strategy that enhances the coupling between the trolley speed and the swing angle components. Specifically, we devise a new energy-like function by introducing two generalized signals that can reflect the trolley speed and the hook and load swing angle components. This strategy greatly improves the effectiveness of the transient control process. Meanwhile, the asymptotic stability of the closed-loop system is not only guaranteed, but also the desired position of the trolley and swing angle suppression are achieved using the control strategy.

SUMMARY Double inverted pendulum model, stationary or on a cart, is computationally the simplest out of the range of reasonable models used for anthropomorphic robots motion synthesis. However, it is still not sufficient for describing more complex situations. The novel concept of variable double inverted pendulum (VDIP) for static postures and VDIP on cart (VDIPC) for dynamic cases is proposed. It provides a simplified but a sufficiently accurate tool for planning the human-like static and dynamic robot postures. Its variable parameters enable the description of both human static postures and motion dynamics. The variable length of the lower link is essential for the representation of postures attained by bending legs. The studies of a set of static and dynamic postures were used for deducing and verifying the locations of lower and upper joint of a double pendulum and the point masses. To justify the concept, human body and pendulum behaviors are compared taking into account a typical model of the human body. Static analysis was conducted by considering static human postures. Dynamic conditions were analyzed using the data acquired from human motion and thus the VDIPC definition was established. The zero moment point trajectories of the human and of VDIPC were compared, validating the correctness of VDIPC in dynamic situations. The formal description of VDIPC is provided together with the torques equilibrium condition needed for evaluating the dynamic postural stability, with the VDPIC representing the robot configuration. The VDPIC state equations are formulated in a form required by the predictive control method. The paper contributes to the motion synthesis methods of anthropomorphic robots taking into account postural control.

Conventional neural networks are universal function approximators, but they may need impractically many training data to approximate nonlinear dynamics. Recently introduced Hamiltonian neural networks can efficiently learn and forecast dynamical systems that conserve energy, but they require special inputs called canonical coordinates, which may be hard to infer from data. Here, we prepend a conventional neural network to a Hamiltonian neural network and show that the combination accurately forecasts Hamiltonian dynamics from generalised noncanonical coordinates. Examples include a predator–prey competition model where the canonical coordinates are nonlinear functions of the predator and prey populations, an elastic pendulum characterised by nontrivial coupling of radial and angular motion, a double pendulum each of whose canonical momenta are intricate nonlinear combinations of angular positions and velocities, and real-world video of a compound pendulum clock.

We propose new symplectic networks (SympNets) for identifying Hamiltonian systems from data based on a composition of linear, activation and gradient modules. In particular, we define two classes of SympNets: the LA-SympNets composed of linear and activation modules, and the G-SympNets composed of gradient modules. Correspondingly, we prove two new universal approximation theorems that demonstrate that SympNets can approximate arbitrary symplectic maps based on appropriate activation functions. We then perform several experiments including the pendulum, double pendulum and three-body problems to investigate the expressivity and the generalization ability of SympNets. The simulation results show that even very small size SympNets can generalize well, and are able to handle both separable and non-separable Hamiltonian systems with data points resulting from short or long time steps. In all the test cases, SympNets outperform the baseline models, and are much faster in training and prediction. We also develop an extended version of SympNets to learn the dynamics from irregularly sampled data. This extended version of SympNets can be thought of as a universal model representing the solution to an arbitrary Hamiltonian system.

In this paper, the H-infinity Sliding Mode Control (HSMC) is designed to produce a new dynamic output feedback controller for trajectory tracking of the nonlinear human swing leg system. The human swing leg system represents the support of human leg or the humanoid robot leg which is usually modeled as a double pendulum. The thigh and shank of a human leg is represented by two pendulum links and the hip joint will connect the upper body to the thigh and the knee joint will connect the thigh to the shank. The external torques (servo motors) are applied at the hip and knee joints to move the muscles of thigh and shank. The results show that the HSMC can robustly stabilize the system and achieve a desirable time response specification better than if only H-infinity or SMC is used. This controller achieves the following specifications: sec, for hip joint and sec, for knee joint.

The interesting features of chaos theory are utilized now a day’s in information security. The simplest chaotic dynamical system is the double pendulum. Here in this article, two double pendulums are used to enhance the chaotic behavior of a dynamical system. This system is sensitive to initial conditions and bears complex and chaotic trajectory. Moreover, being multi dimensional system it endures grander solution space for the generation of large number of S-boxes. Furthermore, a permutation comprising on only two cycles of symmetric group of order 256 is applied to generate integer values for the construction of desired substitution box. The algebraic analysis of suggested S-box emphasis on its application, thereafter, an image is encrypted with the help of this S-box, whose statistical analysis validates its efficacy.

Cart-double pendulum is a typical nonlinear under-actuated mechanical device, and highly sensitive to modeling error and external disturbance. Motion control of double pendulum is a very complex and difficult task, especially when it turns to real-time situation because a minor difference between kinetic model and actual device could lead to the necessity of significant adjustments according to control laws comparing to the usage of simulation controller. The use of accurate kinetic model is the key factor for fast transition from simulation to real-time operation. An improved genetic algorithm (IGA), which includes orthogonal experiment design, feedback mutation, pairing operation based on Hamming distance, and variable precision crossover operation, is introduced to identify the physical parameters of a double pendulum. The improved genetic algorithm (IGA) can overcome the shortcomings of the traditional genetic algorithm (TGA), for instance, premature convergence, and get results closer to the global optimal solution. The results of experiments demonstrate the effectiveness of the proposed method.

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.

The article considers the calculation and visualization of small oscillations of a double plane simplependulum. It contains the brief derivation of the motion equation and its solutions; the mathematical model of themotion in the form of the system of nonlinear differential equations. Experiments with the double pendulum areperformed at various ratios of bodies’ masses and initial angles, in particular for three values of /   m m 1 2 : μ1= 0.1,μ2= 0.2, μ3= 0.3 when the lengths of the pendulum are l = l1 = l2 = 0.25m and g = 9.8 m/s2. The angles of pendulumdeviation are given in radians. The graphs of small oscillations of the pendulum show that beats occur in the systemduring which energy cyclically passes from one pendulum to another. When one pendulum almost stops, the otherpendulum is at its maximum amplitude. After a while pendulums "exchange their states" and so on. Oscillations witha bigger frequency ω1 are modulated by lower frequency oscillations with a frequency ω2.

The topic of coupled oscillations is rich in physical content which is both interesting and complex. In this research work we aimed to study the kinematics behaviour of an important physical system known as coupled pendulum, in which two identical pendulums were coupled together by a spring. The Lagrangian of the system was first constructed, then the equations of motion have been derived. Analytical solutions were obtained for these equations, and in addition we apply a reliable algorithm based on an adaptation of the standard differential transformation method is presented, which is the multi-step differential transformation method to obtain numerical solutions for the considered system for some selected initial conditions. The solutions of the equations of motion were obtained by the multi-step differential transform method. Of course, this method is needed if it predicts the motion of the considered system. For this reason figurative comparisons between the multi-step differential transformation method, the standard differential transformation method and the classical fourth-order Runge–Kutta method are given. The results reveal that the proposed technique is a promising tool to predict the behaviours of the considered system in long time interval. Moreover, the motion of curves represents the initial conditions that sensitive depend and how much oscillatory motion. In conclusion, the proposed technique is a reliable method to solve the considered double pendulum problem in long time interval. We believe that the system considered in this work along with the method used is interesting for physicists, mathematicians and engineers.

This paper considers the dynamics of a double mathematical pendulum whose cylindrical joints axes are not collinear to each other and form a certain acute angle between them. Such system is the simplest scheme of spatial two-link manipulator and can also be a component of various multi-link robotic devices. Basic geometric and kinematic characteristics of a double pendulum are given, which are necessary for deriving the equations of its motion. Small oscillations of the system near the lower equilibrium position are studied. As a result, formulas for the natural frequencies and modes of small oscillations are obtained. The dependence of frequencies and modes on the angle between the pendulum joints is revealed using a qualitative and graphical analysis of found expressions. The obtained results are both of theoretical interest and important practical value.

This article presents a method to solve the impact of a kinematic chain in terms of a non-linear contact force. The nonlinear contact force has different expressions for elastic compression, elasto-plastic compression, and elastic restitution. Lagrange equations of motion are used to obtain the non-linear equations of motion with friction for the collision period. The kinetic energy during the impact is compared with the pre-impact kinetic energy. During the impact of a double pendulum the kinetic energy of the non-impacting link is increasing and the total kinetic energy of the impacting link is decreasing.

The dynamics of Long Reach Manipulators (LRM) produce undesirable vibrations creating disturbances in the mounting base. Such disturbances hamper proper mission tasks and hence controlling them has been and is an active area of research. This paper presents a numerical study incorporating eddy current (EC) damping for controlling the base disturbance of LRM applications. The LRM system has been idealized as a Double Pendulum. A short link augmented on the rigid link swings as a secondary pendulum so that the magnet, housed in it, continuously intercepts the coils mounted on the rigid link causing eddy currents and the consequent damping effect. The disturbances—in terms of the damping of oscillations, and angular displacement of the flexibly mounted base link, and damping of the angular displacement of the rigid link—are compared considering the absence and presence of EC damping. A parametric analysis involving the ratio of the lengths of the secondary link to the rigid link is also investigated. The results indicate the suitability of the damping arrangement in controlling the disturbances. Based on these results, mechanisms can be suitably developed for actual applications.

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The paper investigates the dynamics of a novel wave energy converter based on the parametrically excited pendulum. The herein developed concept of the parametric pendulum allows reducing the influence of the gravity force thereby significantly improving the device performance at a regular sea state, which could not be achieved in the earlier proposed original point-absorber design. The suggested design of a wave energy converter achieves a dominant rotational motion without any additional mechanisms, like a gearbox, or any active control involvement. Presented numerical results of deterministic and stochastic modeling clearly reflect the advantage of the proposed design. A set of experimental results confirms the numerical findings and validates the new design of a parametric pendulum based wave energy converter. Power harvesting potential of the novel device is also presented.

A two-wheeled vehicle balancing a passive inverted pendulum is analyzed based on an experimental device. The corresponding mechanical model is a wheeled double pendulum, where only the position of the lower pendulum is measured. The sampling effect of the digital control is modeled as a zero-order hold. It is shown that the stabilization of the upright position is possible by proper choice of the control parameters as function of the sampling period of the controller. The model can be applied to analyze the behavior of wheeled vehicles with passive human subjects standing on it. The results are demonstrated on small-scale experimental realization of the system.

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.

The paper is devoted to theoretical and experimental investigations of an electromechanical system consisting of DC motor, a physical pendulum with the repulsive magnets. The work consists of modeling, simulation and experimental measurements to validate the analytical predictions and the numerical simulation of the earlier introduced mathematical model. The parameters of the model are estimated using the experimental data. The analyzed system shows several types of non-linear effects, including hysteresis, jump phenomena, chaos and periodic dynamics. Good agreement between real and simulated behavior of the system is obtained.

In this paper, we consider a class of oscillatory mechanical systems described by nonlinear second-order differential equations that contain parameters with variable dissipation and possess the property of dynamic symmetry. We study properties of mathematical models of such systems depending on parameters. Special attention is paid to bifurcations and reconstructions of phase portraits and the appearance and disappearance of cycles that envelope the phase cylinder or lie entirely on its covering. We present a complete parametric analysis of system with position-viscous friction and four parameters, where the force action linearly depends on the speed. The space of parameters is split into domains in which the topological behavior of the system is preserved. For some classes of pendulum systems whose right-hand sides depend on smooth functions, a qualitative analysis is performed, i.e., the space of systems considered is split into domains of different behavior of trajectories on the phase cylinder of quasi-velocities. We also perform a systematic analysis of the problem on an aerodynamic pendulum and detect periodic modes unknown earlier; these modes correspond to cycles in the mathematical model.

This paper concerns the attitude control problem of a 3D rigid pendulum based on dynamic T-S fuzzy neural model. A generalized 3D rigid pendulum subjected to a constant gravitational force consists of a rigid body supported by a fixed and frictionless pivot with three degrees of freedom. There exist two equilibriums of the 3D rigid pendulum characterized with unstable manifolds near the inverted equilibrium and stable manifolds near the hanging equilibrium. For the attitude control problem at the equilibrium position, T-S fuzzy neural model is adopted to approximate the nonlinear part of the 3D rigid pendulum. The method of parallel distributed compensation is used to form the angular velocity feedback controller to cope with the nonlinearity. Moreover, an attitude feedback controller is designed for the linear part so as to guarantee the overall system’s stability. Numerical simulations are performed to illustrate the validity of the proposed method.

The article applies nonlinear optimal (H-infinity) control to underactuated robotic systems using as a case study Furuta’s pendulum. The pendulum’s dynamic model is first transformed to an equivalent form after applying partial feedback linearization. The later description of the pendulum’s dynamics undergoes approximate linearization which takes place round a temporary operating point (equilibrium) recomputed at each iteration of the control algorithm. The linearization makes use of Taylor series expansion of the state-space model of the system and needs computation of the associated Jacobian matrices. For the approximately linearized model of the pendulum an H-infinity feedback controller is developed. The controller’s gain is computed through the repetitive solution of an algebraic Riccati equation which is also performed at each step of the control method. The stability analysis is based on Lyapunov’s method. First it is shown that the control loop satisfies the H-infinity tracking performance condition. Next, under moderate conditions it is also shown that the global asymptotic stability of the control loop can be assured.

Purpose The purpose of this paper is to present an analytical approximate solution of the nonlinear mathematical model of the bifilar pendulum. Design/methodology/approach First, the equation of motion derived based on the classical dynamics law by only an angular oscillation assumption and vertical oscillation is neglected. The energy balance method is applied to solve an established model and an analytical formulation has been obtained for the nonlinear frequency of the bifilar pendulum. Findings A comparison of results with those obtained by a numerical solution of the exact model (without any simplifications) shows the precise accuracy even for a large amplitude of oscillation. Originality/value The proposed model and solution are relatively simple and can be applied instead to a linear model for achieving accurate results.

The problem of event-triggered sampled-data nonlinear control of Hamiltonian system is considered by the example of controlling the pendulum’s energy. A feedback control law based on the speed gradient method is chosen. The main contribution consists in precisely characterizing energy deviation bounds depending on event-trigger switching parameter.

This paper presents the control design problem for the real-time swing-up and stabilization of a double pendulum on a cart from a large set of non-stationary initial conditions. The proposed algorithm generates a tree of feedback stabilized trajectories leading to the upper equilibrium of the pendulum. For each trajectory in the tree, the algorithm determines the set of states near the nominal trajectory from which the system can be transfered to the upper equilibrium without violating the state and input constraints using the feedback controller of the trajectory. This set of states is defined as the funnel of the stabilized trajectory. The design of a control strategy for a large set of initial states is equivalent to the coverage of the desired region in state space using funnels. Several modifications of the original LQR-trees algorithm are introduced in this contribution, which includes a discretization based approximation of the funnel, the exploration of symmetric property of the double pendulum and the coverage of a design set which has a lower dimension than the state space. We show both simulation results and experimental validation of the proposed method.

In this paper we address the robust tracking control problem, against unknown but bounded matched disturbances, for an inertia wheel pendulum. The interested periodic motion of the pendulum will be at the upright position which corresponds to an unstable equilibrium point of the unforced system. A two-relay controller based reference model was developed for generating the reference trajectories where a second order sliding modes controller enforces the unactuated link to follow such prescribed trajectory. The desired amplitude and frequency were tuned by choosing the two-relay control gains properly. A variable structure observer was developed to estimate the velocity of the pendulum and the wheel. Lyapunov stability analysis was made to demonstrate robustness of the closed-loop system. Performance issues of the constructed controller-observer were illustrated in a numerical study.

This paper focuses on a novel rotating pendulum system which is derived from the coupling rotational model of simple pendulum and mass-spring oscillator. The remarkable feature of the proposed system is a cylindrical dynamical system with strongly irrational nonlinearity exhibiting smooth and discontinuous characteristics due to the geometry configuration. The equilibrium stability and bifurcations of the unperturbed system are explored showing the subcritical pitchfork bifurcation, supercritical pitchfork bifurcation and saddle-node bifurcation. Furthermore, the complex transitions of multiple well dynamics and various kinds of singular orbits are demonstrated by using nonlinear dynamical technique in both smooth and discontinuous case. For the perturbed system, the primary resonance of small angle oscillation is investigated analytically by introducing the complete Jacobian elliptic integrals. Finally, complicated resonant structures of period, quasi-period and stochastic phenomena are presented for the system with unique harmonic perturbation. Bifurcation diagram, Poincare section and basin analysis are carried out to demonstrate the co-existing different types of periodic motion and the transitions of chaotic motion between smooth and discontinuous case.

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.

We consider a simple pendulum whose suspension point undergoes fast vibrations in the plane of motion of the pendulum. The averaged over the fast vibrations system is a Hamiltonian system with one degree of freedom depending on two parameters. We give a complete description of bifurcations of phase portraits of this averaged system.

Stabilization of a pinned pendulum about its upright position via a reaction wheel is considered, where the pendulum’s angular position is measured by a single accelerometer attached directly to the pendulum. The control policy is modeled as a simple PD controller and different feedback mechanisms are investigated. It is shown that depending on the modeling concepts, the governing equations can be a retarded functional differential equation or neutral functional differential equation or even advanced functional differential equation. These types of equations have radically different stability properties. In the retarded and the neutral case the system can be stabilized, but the advanced equations are always unstable with infinitely many unstable characteristic roots. It is shown that slight modeling differences lead to significant qualitative change in the behavior of the system, which is demonstrated by means of the stability diagrams for the different models. It is concluded that digital effects, such as sampling, stabilizes the system independently on the modeling details.

We developed a technique to determine the non-chaotic oscillations of loads in the conservative pendulum systems by using the graphic technology of projection focusing. In this case, phase trajectories of the differential equations of oscillation are considered as projections of integral curves from the phase space onto the phase plane. The effect exerted by the value of one of the system's parameters on the image of phase trajectories was examined (at stable values of other parameters). By using projection focusing, the element that defines the critical value of variable parameter is selected among a family of phase trajectories, which, in a combination with other parameters, allows us to describe a non-chaotic trajectory in the load motion. The need for such studies is predetermined by the absence, in practice, of an engineering method for computing the non-chaotic trajectory of the load motion for a certain pendulum system. We proposed the notion of a focus-line of the parametric family of curves and the technique of projection focusing, which is based on it. We constructed integral curves in the phase space based on the numerical solution of second order Lagrange differential equations. A procedure is presented to determine the critical value of pendulum oscillation parameter by using the graphic notion of projection focusing of phase trajectories in the solutions of second order Lagrange differential equations. The examples are presented of determining the parameters of certain pendulums, which would provide for the non-chaotic trajectory of the load oscillations. The developed computerized projection technique for the simulation of oscillations in the pendulum mechanical systems makes it possible to choose the required values of parameters and initial conditions for initiating the oscillations, which provide for the non-chaotic technological character of oscillation trajectory of their elements, which is important for the practical implementation in the designs of pendulum systems.

The stability of motion of Maxwell’s pendulum is investigated in a uniform gravity field. By means of several canonic transforms of the equations of pendulum motion and the method of the surfaces of Poincare sections, the problem is reduced to investigation of the immobile-point stability retaining the area of mapping of the plane into itself. In the space of dimensionless parameters, the stability and instability regions are singled out.

This work touches two important cases for the motion of a pendulum called Sub and Ultra-harmonic cases. The small parameter method is used to obtain the approximate analytic periodic solutions of the equation of motion when the pivot point of the pendulum moves in an elliptic path. Moreover, the fourth order Runge-Kutta method is used to investigate the numerical solutions of the considered model. The comparison between both the analytical solution and the numerical ones shows high consistency between them.

The double pendulum consists of two pendulums attached with the pivot of the second pendulum located at the end of the first pendulum by a rigid bar, massless and considers the effect of gravity. The double pendulum system has strange and unpredictable behaviour. Chaos in a double pendulum system is usually used as an example of chaos in mechanical systems. The chaotic system is characterized by a sensitive system to the initial condition. It means that giving the initial condition with a small difference will give a different solution. Lyapunov Exponent is used to calculating the range of two trajectories with a small difference of initial condition, where the positive exponent value shows two different solutions close at the beginning then more diverge away. In this paper, we determine parameters that will cause the double pendulum system to reach chaos easily. To facilitate this, the equations of the double pendulum motion are simplified by =gl1,β=gl2,δ=m2m2. In particular, we focus on determining the mass ratio and length ratio from the double pendulum system. In some simulations with simulation program that have been made in Maple, the program has resulted in the greater value of δ at the specific problem shows that the system needs the value of θ1,θ2,θ˙1,θ˙2, as first initial condition smaller to reaches chaos easily.

The level of competency of playing cricket defensive drive was measured by using double pendulum model. Three defensive drives by three right-handed batters were recorded in the bat and wrist mounted inertial sensors. The sensor recorded acceleration profiles were tried to match with those obtained by modeling the drives with double pendulum dynamics imposing the initial conditions at start of each drives. In the analysis, the movement of the bat was modeled as the lower pendulum and the wrist as the upper pendulum of the double pendulum model. Phase portrait derivative method was used to analyze each drives in the pendulum dynamics to quantify the competency of the action for the drives. The method showed that the better competency of two batters (error 2.5% and 7.4%) compared to third one (error 13.9%) can easily be identified observing the statistical features (i. e. standard deviation of the bat velocity, ratio of major to minor axis of the phase portrait curve, null hypothesis between sensor data and data from the double pendulum model) extracted from the sensor and double pendulum model. Competency test can be a useful tool for cricket coaching environment; furthermore, incorporation of mathematics in cricket batting might add a new dimension in batting research.

An experimental testing rig has been proposed to study the dynamic interaction between a liquid cargo and the carrying vehicle during turning maneuvers. The basic operational principle of the testing rig consists in simulating the lateral accelerations associated to turning maneuvers through the use of a tilt table. While the experimental capabilities of the testing rig include the analysis of the effects of the cars on the whole transport infrastructure, including sleepers and ballast, a first use of the rig considered the effect of sloshing cargo on the level of dynamic forces transmitted to the rails. Such test was used to validate a simplified theoretical approach consisting of a two degree-of-freedom double pendulum mechanical system, where a simple pendulum, representing the sloshing cargo, is articulated to the spring-supported vehicle chassis, which is modelled as an inverted torsional pendulum. While the theoretical results exhibit a high correlation with the experimental data, the main discrepancy between both outputs, relates with the frequency of the residual vibration, once the lateral acceleration input is ceased. Such difference in frequency, would imply an underestimation of the number of loading cycles to which the infrastructure is subjected. On the other hand, the peak values are within acceptable difference levels.

A metachronal motion produces a net flow in a low Reynolds regime and is thus important for the development of artificial cilia. In this study, we report that a thermally actuated double pendulum exhibits a metachronal motion using a self-propulsion mechanism caused by spontaneous asymmetrical heat transfer. Specifically, by using a multiple pendulum structure in which three beams of different lengths are hung in parallel in order of length on a U-shaped nichrome heater, we demonstrate that phase differences can be produced among beams. Our device can be applied to biomedical microfluidic systems or microrobots in water.

The oscillation frequencies of a double pendulum are derived by extending the usual calculation for a single pendulum. Experimental results are presented as supplementary videos.

The article investigates nonlinear oscillations of the double mathematical pendulum having identical parameters of its links and end loads. Approximate solutions in the nonlinear zone are constructed and analyzed on the basis of exact solution in the linear zone. These solutions clearly illustrate the drift of frequencies and modes of double pendulum with increasing oscillation amplitudes. The obtained analytical results are very useful for the synthesis of controlled movement modes of wide range of manipulators and various robotic structures, and they are also necessary for constructing optimal locomotion modes of walking machines and mechanisms. These solutions are also interesting as illustrative examples of nonlinear mechanics in the teaching and engineering practice.

Centrifugal Pendulum Vibration Absorbers (CPVAs) reduce the torsional fluctuations of rotating shafts at fixed vibration order by changing the natural frequency in accordance with the excitation frequency. Attenuation of multiple orders is necessary in the case of the rear wheel drive vehicle and cylinder deactivation applications for achieving refined vehicle performance. Single pendulum vibration absorbers attenuate only one vibration order at a time and the pendulum length must be reduced for higher orders in the order of (1/n2) times the distance between the center of rotation and pivot point to attenuate nth order vibration. Use of single pendulum absorbers for reducing higher orders has limitations due to manufacturing inaccuracies and occurrence of wear of joints over prolonged usage which reduces the effectiveness of absorber in reducing the torsional vibration. In this study, a centrifugal double pendulum vibration absorber (CDPVA) is proposed to make the design easy for higher orders by deploying two pendulums in series. The CDPVA is designed to attenuate the fundamental excitation order (n) and its second harmonic (2n) without significant reduction in the pendulum length hence making the design robust for prolonged usage. The mass and length values of the double pendulum are tuned to match the two natural frequencies correspond to the fundamental and second harmonic order of the torsional vibration excitation. Torsional springs are used at the pivot joints of the double pendulum to counteract the gravitational effects at low speeds and the influence of spring stiffness on the natural frequencies is analyzed. The nonlinear second order equations of motion of pendulum masses and rotor are solved analytically using method of multiple time scales and the stability characteristics of pendulum motion are analyzed at the excitation frequencies with different torque levels. Numerical simulation of complete nonlinear equations is carried out and the analytical results are compared with the simulation results. The designed CDPVA for 2nd and 4th order is tested on a 4-cylinder the rear wheel drive vehicle and considerable reduction is achieved in the torsional vibration of the drive shaft and also in the vibration levels at seating locations. The proposed methodology helps in designing the CDPVAs for attenuating the torsional vibration of multiple higher orders yet avoiding impacts in low speed operation.

In this paper, we propose a parametrically excited pendulum with irrational nonlinearity which comprises a simple pendulum linked by a linear spring under base excitation. This parametric vibration system exhibits bistable state and discontinuous characteristics due to the geometry configuration. For small oscillations, this system can be described by Mathieu equation coupled with SD (Smooth and Discontinuous) oscillator whose dynamic response is examined analytically by using the averaging method in both smooth and discontinuous case. Numerical simulations are carried out to demonstrate the complicated dynamic behavior of multiple periodic motions and different types of chaotic motions.

The article presents an analysis of the dynamics of a double mathematical pendulum with variable mass. In the analyzed system, with time, under the influence of gravity, the mass of the upper member of pendulum decreases and the mass of the lower member of pendulum increases. The total mass of the system doesn’t change. For the analysis introduced dimensionless time and dimensionless parameters, which allows the presentation of the equations of motion in dimensionless form. It has been shown that the change of mass in the system has a significant impact on the dynamics. The increase in mass of the lower member reduces the amplitude of vibration of the pendulum. The numerical calculations were performer in Mathematica package.

A basic classical example of simple harmonic motion is the simple pendulum, consisting of a small bob and a massless string. In a vacuum with zero air resistance, such a pendulum will continue to oscillate indefinitely with a constant amplitude. However, the amplitude of a simple pendulum oscillating in air continuously decreases as its mechanical energy is gradually lost due to air resistance. To this end, it is generally perceived that the main role in the dissipation of mechanical energy is played by the bob of the pendulum, and that the string’s contribution is negligible. The purpose of this research is to experimentally investigate the merit of this assumption. Thus, we experimentally investigate the damping of a simple pendulum as a function of its string diameter and compare that to the contribution from its bob. We find out that although in some cases the effect of the string might be small or even negligible, in general the string can play a significant role, and in some cases even a greater role on the damping of the pendulum than its bob.

In this paper transition curves and periodic solutions of a parametric pendulum system are calculated analytically by employing the energy method. In previous studies this problem usually was dealt with by using the asymptotic method which is limited by small parameter. In our research, the hypothesis of small number in the pendulum system is not necessary, some different conclusions are obtained on the impacts of nonlinearity in the pendulum system on the transition curves in the parametric plane. The results based on the asymptotic method suggested that nonlinearity in the pendulum system only significantly causes decrease of the area of the stable regions in the parametric plane when the angular displacement of the pendulum is not very small. However, our analysis according to the energy method shows that nonlinearity does not significantly change the area of the stable regions in the parametric plane, but notably alter positions of the stable regions. Furthermore, position of the stable regions to a large extent is related to the amplitude of periodic vibrations of the pendulum especially when the angular displacement of the pendulum is large enough. Our results are very different from that reported in previous studies, which have been verified by numerical simulations.

In this article, nonlinear dynamics of pendulum systems is studied. The system of governing differential equations is derived using the Euler–Lagrangian approach. The recurrence plot method has been used for the nonlinear dynamics analysis. The natural frequency of a lumped pendulum system is smaller than the natural frequency corresponding to the distributed system. It is also observed that the bottom pendulum is the most chaotic than the middle and the top pendulums. It is also shown that a triple pendulum system with distributed mass is more chaotic than corresponding to the lumped system.

Pendulum is a simple system that is usually related to great discoveries in science and technology. Nonlinear dynamics of pendulum systems is related to a variety of responses being the objective of studies of oscillations, bifurcations and chaos. Smart material nonlinear effects are employed in several applications due to their adaptive behavior presenting great advantages in control strategies. This work deals with the dynamics of an SMA-pendulum system that consists of a pendulum coupled with shape memory alloy springs. A numerical investigation is carried out exploiting the temperature-dependent behavior of the SMA. Complex responses are presented and the possibility of thermal control of the system is investigated.

In this paper, we deal with the control method for rotational movements of a pendulum using a separatrix. We design a controller that attains a homoclinic motion or a heteroclinic motion of the pendulum and the asymptotic stability of the cart by using a kind of forwarding control design. First, we derive a controller that converges to a homoclinic orbit via a Lyapunov function of the pendulum subsystem. Next, we give a nonlinear stabilizing controller via another Lyapunov function of the cart subsystem. Moreover, using the third Lyapunov function and adding a complementary control input, we guarantee that the pendulum converges to the homoclinic orbit and the cart is stabilized. Finally, the simulation and the experiment using the rapid controller prototyping system based on MATLAB/Simulink are performed to demonstrate the forward upward circling and the giant swing of the pendulum.

Some special cases of a larger class of smooth models of the resultant friction force occurring on a planar contact area are developed and presented. These models are constructed under assumptions of classical Coulomb’s law of friction on each element of contact and instantaneous transition between two modes: fully developed sliding and rigid stick state without local slips and deformations of the contact area. They are able to model cases of different values of static and kinetic friction factors. The considered models are very effective tools for fast and reliable numerical simulations of mechanical systems with frictional contacts. They are applied in modelling and numerical analysis of a special kind of mechanical system, i.e. a kind of a pendulum with special frictional driving. The rotational joint of the pendulum is elastically suspended in the motion plane. The pendulum is driven by circular frictional contact with a rotating disk. Examples of self-excited bifurcation and chaotic dynamics as well as stick–slip behaviour of the pendulum are presented.