The motion of a particle sliding on a frictionless surface of revolution z(r) subject to a uniform gravitational field can be reduced to a one-dimensional radial motion governed by an “effective analogous potential energy function” (EAPEF), similar to the effective potential energy function used in two-dimensional problems with radial symmetry. The EAPEF depends on the angular momentum and total energy of the particle, and thus on the initial conditions of the motion. The general expression for the EAPEF is derived and the EAPEF is used to find the radial equation of motion, the condition for circular orbits on the surface, and the period and stability of those orbits. For stable circular orbits, the EAPEF is used to find the period of small radial oscillations in a perturbed but stable circular orbit. By comparing the period of the circular orbit with the period of small radial oscillations, the conditions for which the perturbed orbit will be periodic are derived. The period of more general radial oscillations, as well as the average orbital period, is given in terms of integrals involving the EAPEF, which can be computed numerically and used to find periodic orbits with large radial oscillations. We show results obtained by applying this technique to a variety of surfaces, as well as experimental results for a spherical pendulum that shows reasonable agreement with theoretical predictions. The EAPEF is a useful tool for analyzing motion on surfaces of revolution, and can be used as the basis for interesting student projects.

Recently, several researchers investigated mobile robots. However, when transporting objects in a complex and confined environment, robots that cannot move in any direction and cannot perform a super pivot turn have a limited range of motion. Therefore, omnidirectional mobile robots have been extensively studied. Mobile robots for transporting objects are often used in factories, public facilities, and restaurants. To shorten transportation time and avoid obstacles, mobile robots sometimes rapidly accelerate, decelerate, or traverse steps, which may cause vibrations and damage to transport objects. Transporting a person may give the person a sense of unease or discomfort. Therefore, vibration control is necessary. This study proposes the manual operation of an omnidirectional mobile robot to perform vibration control of a spherical pendulum, which is a tentative transport object. A notch filter and an optimal servo system were used for vibration control of the pendulum. The vibration of the pendulum generated during operation was reduced using a notch filter, and the vibration of the pendulum excited by the disturbances was suppressed by feedback control using the optimal servo system. The control gain of the optimal servo system was reasonably determined using a genetic algorithm based on a quantitative evaluation of the operability and damping performance. The effectiveness of the control system was verified through simulations and experiments.

For the study and experimentation of physical systems, it is essential to measure the physical variables, which implies choosing the most convenient method that does not affect the natural behavior of the system. This work presents the modeling and sensing of the spherical pendulum, integrating a novel non-invasive measurement scheme based on infrared sensors arranged in a quadrature configuration. The proposed method enables the estimation of angles around two axes, leveraging light reflection on a perpendicular plane aligned with the pendulum bar. A mathematical model was developed to create simulations, and a prototype was constructed to perform experiments and validate the detection method. The values recorded by the sensors enable the reproduction of the pendulum's trajectory, allowing for the correlation of real results with those of the simulations. The similarity of behavior between the simulations and the experimentation facilitates the validation of the proposal.

This paper considers a nonlinear spherical pendulum whose suspension point performs high-frequency spatial vibrations. The dynamics of this pendulum can be described by averaging its Hamiltonian over phases of vibrations. Rotationally symmetric conditions on vibrations are assumed in the averaged Hamiltonian. Under these conditions, a bifurcation diagram for the phase portraits of the averaged system is presented. Numerical simulations of different examples of vibrations are performed. The case of proper degeneration in KAM theory guarantees the coherence of dynamical characteristics between the averaged and exact systems.

Abstract This research explores the dynamics of an innovative three-degrees-of-freedom (3-DOF) spherical pendulum (SP) system integrated with a piezoelectric device to improve energy harvesting (EH) efficiency. The proposed model features a damped Duffing oscillator as the central component, coupled with a piezoelectric energy harvester and an attached SP. Lagrange’s equations are utilized to derive the system’s nonlinear differential governing equations. Analytical solutions (AS) are obtained using the multiple scales method (MSM) to achieve higher-order approximations. These solutions are then compared with numerical results to validate their accuracy and enhance the overall clarity of the analysis. Moreover, solvability criteria and characteristic exponents under resonant conditions are obtained. The stability of the steady-state solutions is analyzed using the Routh–Hurwitz criteria (RHC) and frequency response curves, providing deeper insights into the system’s behavior. Moreover, the basins of attraction have been simulated to analyze the behavior of the system’s nonlinear dynamics and its sensitivity to initial conditions. Additionally, the nonlinear stability analysis reveals both stable and unstable regimes, with resonance curves and time histories constructed for various parameter values. Furthermore, the phase portraits, graphs of bifurcation, and maps of Poincaré present a thorough view of the system’s dynamics that capture quasi-periodic and chaotic phenomena. This research has broad practical usages, evidenced by real-world applications of EH, including the power watch, power pucks, self-powered switches, Boeing wireless sensor nodes, and electrochemistry EH. The diversity and potential impact represented by these examples show the versatility of the proposed system in advancing EH technologies.

Pendulum oscillators study harmonic motion, energy conservation, and nonlinear dynamics, providing insights into mechanical vibrations, wave phenomena, weather patterns, and quantum mechanics, with real-world applications in engineering, seismology, and clock mechanisms. The present study addresses three distinct issues related to SPs; a charged magnetic spherical simple pendulum (SP), and a SP composed of heavy cylinders that roll freely in a horizontal plane, and a nonlinear model depicting the motion of a damped SP in a fluid flow. The SPs are analyzed via an innovative technology known as the non-perturbative approach (NPA), which is based on He’s frequency formula (HFF). This advanced approach linearizes a nonlinear ordinary differential equation (ODE), enabling more straightforward analysis and solution. As-well known, implementing the NPA has several advantages, chief among them the removal of the constraints associated with managing Taylor expansions. Consequently, there have been no augmentations to the current restorative forces. Secondly, the novel method enables us to assess the stability criteria of the system away from the traditional perturbation techniques. The numerical comparison of nonlinear ODEs into linear ones using Mathematica Software (MS) is conducted to validate this innovative method. An analysis of the two responses demonstrates a strong concordance, underscoring the necessity of precision of the methodology. Furthermore, to demonstrate the influence of the components on motion behavior, the time history of the calculated solution and the corresponding phase plane plots are accumulated. The use of multiple phase portraits aims to explore stability and instability near equilibrium points by examining the interaction between expanded and cyclotron frequencies, modulated by the magnetic field, for varying azimuthal angular velocities.

We study the precession of a Foucault pendulum using a new approach. We characterize the support anisotropy by the difference between the maximum and minimum periods of the pendulum along the principal axes of the support. Then, we compute the total precession rate, taking into account both the Airy precession of a spherical pendulum and the Coriolis precession due to the Earth's rotation. To study the resulting motion, we developed a calculation loop, period after period, that describes the movement of the oscillatory trajectory of the bob. To test our model, we mounted a test pendulum of 480.3 cm length and measured its periods and precession. The rate of precession is sensitive to the dimensions of the pendulum, the anisotropy of the support, and the initial conditions. We find that for certain amplitudes, the precession can stop entirely while the pendulum continues to oscillate. It is also possible to obtain continuous precession at lower oscillation amplitudes. We give an upper bound for this critical oscillation amplitude. We close with a discussion of the implications of our findings for the design of Foucault pendulums used in demonstrations and lab experiments.

This study focuses a trajectory control problem of a quadrotor, created by combining the dynamics of a spherical pendulum with the dynamics of a quadrotor. A spherical pendulum is a physical system comprising a mass suspended by a string or rod that has the freedom to swing in any direction. During the operation of a quadrotor, the payload connected to it behaves similar to the dynamics of a spherical pendulum. Consequently, it is essential to design a trajectory control problem by considering both quadrotor and spherical pendulum dynamics together, as this constitutes a significant aspect. Due to the varying masses of the payload that are attached to the quadrotor during each take off, the adaptability of the total mass in model-based control design poses a significant challenge. In addition, it is crucial to consider mass adaptation when the payload attached to the quadrotor is released during flight. The proposed Lyapunov based adaptive backstepping controller algorithm can predict the total mass of the quadrotor without prior knowledge of the actual payload. The ability to estimate varying payload masses is crucial for enhancing the quadrotor’s operational flexibility and reliability, particularly in real-world applications where payloads may change dynamically. This innovation paves the way for more versatile, real-world deployment of quadrotor systems in scenarios requiring precise and adaptive control.

Existing studies on the symmetric spherical pendulum are limited to small- and moderate-amplitude vibrations. This study was conducted to obtain accurate solutions for analysis of the large-amplitude vibration of a symmetric magnetic spherical pendulum using the continuous piecewise linearization method (CPLM). The stability conditions and bifurcation of the pendulum were derived based on the critical points, while the CPLM was used to estimate the frequency response and vibration histories to less than 0.1% and 1.0% relative error respectively when compared to numerical solutions. The CPLM was found to be significantly more accurate than the Laplace transform homotopy perturbation method and predicted the large-amplitude bi-stable vibrations accurately. The stability analysis that was conducted enabled the characterization of all bounded symmetric vibrations based on the relationship between the cyclotron frequency and azimuthal velocity, whereas the bifurcation analysis confirmed that the symmetric vibrations can undergo pitchfork bifurcation that results in transition from single-well to double-well (or bi-stable) vibrations and vice versa. Finally, a parametric analysis was conducted to study the effect of the cyclotron frequency and uniform azimuthal velocity on the frequency-amplitude response and vibration histories The parametric analysis showed that the frequency-amplitude response has a strong dependence on the cyclotron frequency and azimuthal velocity for all amplitudes. On the other hand, the oscillation profile only depends on the cyclotron frequency and azimuthal velocity for some amplitudes. The results of this study can be applied in the design of energy harvesters and elliptic tanks for liquid transport.

This paper explores the application of first integrals in constructing Lyapunov functions for stability analysis of dynamical systems in stochastic domains. A key advantage of using first integrals is their ability to embed system-specific structural and physical information, distinguishing the resulting Lyapunov functions from generic positive definite functions with no intrinsic connection to the system. However, since first integrals do not inherently satisfy Lyapunov conditions, additional constraints—often with direct physical interpretations—must be introduced to ensure positive definiteness and suitable monotonic behavior. The method is demonstrated on three mechanical systems subjected to parametric noise: a nonlinear aeroelastic single-degree-of-freedom oscillator, a spherical pendulum with two first integrals, and a gyroscope with three first integrals.

The movement of a nonlinear spherical pendulum in the noninertial Earth’s system of reference is considered. The article consists of three parts. The first part is devoted to the classical problem of the movement of the nonlinear spherical pendulum in inertial system of reference. We get some new results concerning the estimates of the apsidal angle. We give the critical analysis of previous publications (books and articles) on this problem. The second one is devoted to the classical problem of the Foucault pendulum. We study the problem in nonlinear case and get some new results concerning the precession of the trajectory of the nonlinear pendulum with initial conditions that were used in Foucault pendulum’s experiment. The third one (Application) is devoted to discussing and comparing this article’s results with previous results and explanations of the Foucault pendulum’ effects.

By complexifying a Hamiltonian system, one obtains dynamics on a holomorphic symplectic manifold. To invert this construction, we present a theory of real forms which not only recovers the original system but also yields different real Hamiltonian systems which share the same complexification. This provides a notion of real forms for holomorphic Hamiltonian systems analogous to that of real forms for complex Lie algebras. Our main result is that the complexification of any analytic mechanical system on a Grassmannian admits a real form on a compact symplectic manifold. This produces a 'unitary trick' for Hamiltonian systems which curiously requires an essential use of hyperkähler geometry. We demonstrate this result by finding compact real forms for the simple pendulum, the spherical pendulum, and the rigid body.

A new hybrid method for the construction of control actions of a linear control system with constant coefficients is considered in this paper. It is assumed in this paper that a part of the discussed system meets some conditions. Some states of the main system are considered to be control actions for a subsystem for which and LQR stabilizer is acquired. Then, those control actions of the subsystem are used to construct the control actions for the main system. In the problem of controlling the motion of a complex linear system of an inverted spherical pendulum on a moving base, a new approach to the construction of control actions (hybrid action method) was used. It is assumed that a component of the complex system under discussion satisfies certain conditions. The inertial forces at the center of mass of the base of the composite system are considered to be the controlling influences on the inversion of the pendulum, for which the LQR stabilizer was purchased. The determined internal control actions on the inverted pendulum are then used to construct external control actions on the base of the composite system. In the end, a numerical analysis was carried out.

Abstract. The orientation of ice crystals affects their microphysical behaviour, growth, and precipitation. Orientation also affects interaction with electromagnetic radiation, and through this it influences remote sensing signals, in situ observations, and optical effects. Fall behaviours of a variety of 3D-printed plate-like ice crystal analogues in a tank of water–glycerine mixture are observed with multi-view cameras and digitally reconstructed to simulate the falling of ice crystals in the atmosphere. Four main falling regimes were observed: stable, zigzag, transitional, and spiralling. Stable motion is characterised by no resolvable fluctuations in velocity or orientation, with the maximum dimension oriented horizontally. The zigzagging regime is characterised by a back-and-forth swing in a constant vertical plane, corresponding to a time series of inclination angle approximated by a rectified sine wave. In the spiralling regime, analogues consistently incline at an angle between 7 and 28°, depending on particle shape. Transitional behaviour exhibits motion in between spiral and zigzag, similar to that of a falling spherical pendulum. The inclination angles that unstable planar ice crystals make with the horizontal plane are found to have a non-zero mode. This observed behaviour does not fit the commonly used Gaussian model of inclination angle. The typical Reynolds number when oscillations start is strongly dependent on shape: solid hexagonal plates begin to oscillate at Re =237, whereas several dendritic shapes remain stable throughout all experiments, even at Re > 1000. These results should be considered within remote sensing applications wherein the orientation characteristics of ice crystals are used to retrieve their properties.

Buckling disappearance via merging/divergence in a nonlinear three-d.o.f. system with linear constitutive law

Adaptive Anti-Swing Control for 7-DOF Overhead Crane With Double Spherical Pendulum and Varying Cable Length

Real-time anomaly detection of the stochastically excited systems on spherical ([formula omitted]) manifold

Abstract Gait generation of a humanoid robot on a deformable terrain is a complex problem as the foot and terrain interaction and terrain deformation have to be included in the dynamics. To simplify the dynamics of walk on deformable terrain, we used a spherical inverted pendulum (SIP) to represent the single support phase, in which the effect of terrain deformation is represented by a spring and damper contact model. The impact model for leg transition is derived from angular momentum conservation. In order to minimize the energy loss due to impact, the double support phase is modeled as a suspended pendulum. Based on the motion of the SIP model, the hip and leg trajectories of a 10-degree-of-freedom (DOF) humanoid robot are generated. The joint trajectories of the robot are obtained from inverse kinematics. The motion of the center of mass is analyzed by inverse dynamics of a floating-base robot. The proposed gait generation method has been experimentally validated using a Kondo KHR-3HV humanoid robot on deformable terrain. The results show that the humanoid can effectively track the trajectories of the SIP model.

Hamiltonian Monodromy is the simplest topological obstruction to the existence of global action-angle coordinates in a completely integrable system. We show that this property can be studied in a neighborhood of a focus-focus singularity by a spectral Lax pair approach. From the Lax pair, we derive a Riemann surface which allows us to compute in a straightforward way the corresponding Monodromy matrix. The general results are applied to the Jaynes–Cummings model and the spherical pendulum.

Abstract This paper presents the problem of transporting a suspended load by a quadrotor. A full model considering the quadrotor and the dynamics of the suspended load, in a three‐dimensional space, is proposed considering as control inputs the torques and the thrust force due to the motors. The solution proposed consists of simple control strategies based on the tangent linearization of the model, which is controllable and therefore flat. A sliding mode control technique is developed for the thrust force and torques associated with the Euler angles in order to track a desired trajectory of the vehicle in a three‐dimensional space with minimum oscillation of the suspended load. The control strategy results to be relatively simple to implement and achieves local stability of the tracking errors, as well as robustness to internal nonlinearities, which are neglected in the linearization process and other external disturbances. The attractiveness of the sliding surfaces and the stability of the tracking errors are formally studied using Lyapunov stability theory. Simulation results are given to show the performance of the proposed control strategy.