Variable Length Pendulum Research Articles

Variable-Length Pendulum-Based Mechatronic Systems for Energy Harvesting: A Review of Dynamic Models

The ability to power low-power devices and sensors has drawn a great deal of interest to energy harvesting from ambient vibrations. The application of variable-length pendulum systems in conjunction with piezoelectric or electromagnetic energy-harvesting devices is examined in this thorough analysis. Because of their changeable length, such pendulums may effectively convert mechanical vibrations into electrical energy. This study covers these energy-harvesting systems’ basic theories, design concerns, modeling methods, and performance optimization strategies. This article reviews several studies that look at dynamic models, the effects of damping coefficients, device designs, and excitation parameters on energy output. The advantages and disadvantages of piezoelectric and electromagnetic coupling techniques are demonstrated by comparative research. This review also looks at technical advances and future research prospects in variable-length, pendulum-based energy harvesting. An expanded model for an energy harvester based on a variable-length pendulum derived from the modified, swinging Atwood machine is more specifically presented. This model’s numerical simulations, estimated current and voltage outputs, and produced power from the electromagnetic and piezoelectric devices integrated at various points in a 4-DOF variable-length pendulum model all indicate encouraging results. This necessitates extra study, changes, and optimizations to improve the usefulness of the proposed model. Finally, important dynamic models on developing variable-length, pendulum-based energy harvesters for usage in a range of applications to create sustainable energy are summarized.

An approximate analytical solution of a 4-DOF variable-length pendulum model

An approximate analytical solution of a 4-DOF variable-length pendulum model

EMPLOYING ELECTROMAGNETIC ACTUATOR TO HARVEST ENERGY THROUGH A VARIABLE-LENGTH PENDULUM

This article explores the application of electromagnets in energy harvesting via a variable-length pendulum system. Harnessing the principles of electromagnetism, the study investigates the efficiency and feasibility of utilizing electromagnetic forces to extract energy from the motion of a pendulum with variable length, employing sinusoidal excitation as a means for vibrating machinery. This approach enables a more significant number of oscillations, consequently leading to a higher power output. The research investigates various aspects, including the design, implementation, and performance analysis of the electromagnet-based energy harvesting system. Through theoretical modeling, the article provides insights into the potential of electromagnets to generate sustainable energy from oscillatory motion. The findings show that up to 0.25W can be generated, providing power for small devices such as phone chargers and sensor units. The findings contribute to advancing renewable energy technologies and offer promising avenues for developing efficient and environmentally friendly energy harvesting mechanisms.

Dynamics of a heavy pendulum of variable length with a movable suspension point

Dynamics of a heavy pendulum of variable length with a movable suspension point

Characterization of out-of-plane libration of tethered satellite systems in elliptical orbits

This paper investigates the dynamic behavior of a tethered satellite system in an elliptical orbit as the tethered subsatellite is deployed from a mother satellite at an out-of-plane angle while maintaining the tether at a constant in-plane pitch angle by a predefined control force acting on the subsatellite. A variable-length pendulum model is employed to describe the purely out-of-plane tether deployment dynamics, from which an analytical tether deployment control law is derived. The stability condition of the analytical control law is examined by Floquet theory. The corresponding analytical deployment law is asymptotically stable, and the requirement for a positive tension control input is guaranteed to ensure the controllability of the tether system. Following deployment, the critical conditions for the out-of-plane libration of the tethered subsatellite to become chaotic during the station-keeping phase are examined. Finally, simulation results reveal that the proposed approach can effectively and rapidly identify periodic, quasiperiodic, and chaotic motions using bifurcation diagrams, which provides valuable insights for enhancing the stability and reliability of such tethered satellite systems.

A double variable-length pendulum with counterweight mass, kinematic excitation and electromagnetic forcing

This study introduces a novel double variable-length cable pendulum model and experimental setup featuring elastic suspension and counterweight mass. Our main goal is to investigate the complex dynamics resulting from variable length’s impact on vibration frequency and amplitude. Through numerical simulations and experiments, we explore the system’s response to different external forces. Utilizing methods like phase plots, bifurcation diagrams, and Lyapunov exponents, we delve into nonlinear dynamics. We also use vision-based techniques to assess friction damping-related vibrations and magnetic field interactions. The results reveal diverse behaviors, including chaotic and periodic oscillations, shedding light on control functions and parameter relationships. The developed cable system captures intricate nonlinear dynamics and attains stable vibration modes, as confirmed by vision-based measurements. This platform can analyze and control irregular dynamics in systems with elastically suspended weights driven by motors or mobile cranes. Its nature, encompassing kinematic excitation, electromagnetic interactions, and sliding friction, allows for exploring complex nonlinear dynamics. The system’s capacity to modulate vibration frequencies contributes to mitigating persistent vibrations.

Improving performance of a nonlinear absorber applied to a variable length pendulum using surrogate optimization

The paper investigates a nonlinear vibration mitigation strategy of a variable length pendulum subjected to a harmonic external excitation. A nonlinear absorber in a form of a tri-pendulum system is used to reduce the response of the primary pendulum. Thus, the paper investigates a non-stationary problem of nonlinear vibration mitigation of the primary pendulum using another nonlinear passive pendulum absorber. Due to genuine interest in capturing the nonlinear dynamic interaction, the paper numerically studies the performance of the primary mass and absorber, first, by constructing 2D maps in the unrestrained parametric space, which demonstrate the qualitative behavior of the system. Then, the surrogate optimization technique is used to tune the absorber’s parameters within a given bounded set of parameters’ values. The optimization is conducted based on a priori known reeling speed or acceleration/deceleration of the primary pendulum, thereby completely removing the need for acquiring a current system states essential for active feedback control. The obtained numerical results validate the proposed strategy and demonstrate high performance of the nonlinear passive absorber when it is properly tuned.

Dynamics analysis on the coordination of velocity and accuracy in baseball pitching.

Speed and accuracy are both critical to sports performance but remains poorly understood. In baseball pitching, both ball speed and accuracy must be constrained by human body and ball dynamics. This study examines the dynamics of pitching control ability that coordinates ball speed and control using the center of curvature of trajectories of ball and shoulder joint. Dynamics analysis based on a variable-length pendulum model with the center of curvature of the ball and shoulder joint trajectories reveals that the control of shoulder acceleration affects the coordination of ball speed and pitch control.

Controlling the variable length pendulum: Analysis and Lyapunov based design methods

The analysis and design of methods to damp the swing of the variable length pendulum by adjusting its length are presented here. To analyze the theoretical limits of such Coriolis force based damping, a comprehensive open-loop numerical analysis is performed for a two-dimensional model having the string length as the controlled input. Further, for a four dimensional model, having the force applied to the string as the controlled input, a smooth static state feedback controller is designed using backstepping. Results are verified both in simulations and through extensive laboratory experiments, and compared with previously published results achievable using an identical experimental setting.

Calculus of variations and optimal control for generalized functions

We present an extension of some results of higher order calculus of variations and optimal control to generalized functions. The framework is the category of generalized smooth functions, which includes Schwartz distributions, while sharing many nonlinear properties with ordinary smooth functions. We prove the higher order Euler–Lagrange equations, the D’Alembert principle in differential form, the du Bois–Reymond optimality condition and the Noether’s theorem. We start the theory of optimal control proving a weak form of the Pontryagin maximum principle and the Noether’s theorem for optimal control. We close with a study of a singularly variable length pendulum, oscillations damped by two media and the Pais–Uhlenbeck oscillator with singular frequencies.

Modeling, Simulation, and Analysis of a Variable-Length Pendulum Water Pump

Due to the long-term problem of electricity and potable water in most developing and undeveloped countries, predominantly rural areas, a novelty of the pendulum water pump, which uses a vertically excited parametric pendulum with variable-length using a sinusoidal excitation as a vibrating machine, is presented. With this, more oscillations can be achieved, reducing human effort further and having high output than the existing pendulum water pump with the conventional pendulum. The pendulum, lever, and piston assembly are modeled by a separate dynamical system and then joined into the many degrees-of-freedom dynamical systems. The present work includes friction while studying the system dynamics and then simulated to verify the system’s harmonic response. The study showed the effect of the pendulum length variability on the whole system’s performance. The vertically excited parametric pendulum with variable length in the system is established, giving faster and longer oscillations than the pendulum with constant length. Hence, more and richer dynamics are achieved. A quasi-periodicity behavior is noticed in the system even after 50 s of simulation time; this can be compensated when a regular external forcing is applied. Furthermore, the lever and piston oscillations show a transient behavior before it finally reaches a stable behavior.

Smooth second-order sliding mode controller for multivariable mechanical systems

This paper proposes a smooth second-order sliding mode controller for a class of multi-input multi-output mechanical systems with uncertain parameters and external disturbances. Since the control law is smooth, the chattering effect that can occur with non-smooth controllers is reduced. Lyapunov-based theorems are used to prove global and finite-time convergence of the sliding mode controller. Numerical simulations are presented to illustrate the performance of the proposed controller by applying it first to a variable-length pendulum and then to a two-link robotic manipulator. For the robotic manipulator, a detailed comparison is given of the finite-time convergence and chattering properties of the proposed controller, a super-twisting controller and a super-twisting like controller.Article HighlightsA new smooth second-order sliding mode controller is proposed for multivariablesystems with uncertain parameters and external disturbances.Global finite-time convergence of the controller is proved using Lyapunov theory.Numerical simulations are used to show the performance of the controller on thevariable-length pendulum and two-link robotic manipulator.

Damping Oscillation of Suspended Payload by Varying String Length

Abstract Many modern-day applications involve transport of objects suspended through cables such as in overhead cranes or landing of rovers on the Martian surface. Any undesired oscillation of the payload has the potential risk of instability, and the problem of damping such oscillation and stabilizing the payload at the desired length is the control objective of this article. The system is modeled as a variable length pendulum (VLP), which comprises a payload suspended via a string wrapped around a pulley. The length of the pendulum is varied using clockwise/counterclockwise rotation of the pulley through torque applied by a motor. For a known payload mass, a nonlinear control design is first presented that guarantees asymptotic stability of the desired equilibrium with limited state measurements. The design is then modified for it to handle significant uncertainty in payload mass. The effectiveness of both designs is validated in simulations.

The study of the motion of the truck vibration motors by the oscillating of a pendulum of variable length

The article provides a solution to the problem, according to the condition of which there is a body - a cart, which must be brought into directional motion by means of a swinging elastic pendulum of variable length, pivotally mounted on it. A variant of creating the initial momentum of the trolley movement due to the movement of the pendulum, made in the form of a spring-loaded load with the possibility of movement along a straight guide when it rotates relative to the pendulum suspension point, is proposed and justified. The mode of movement of the cart-pendulum system is studied when the individual components of the system are gradually released from the connections to provide the initial impulse of the cart movement. The full-turn shape of the trajectory of a spring-loaded pendulum pivotally mounted on a cart is shown. We use numerical analysis methods (numerical integration of systems of nonlinear differential equations) with modeling by means of the Mathcad package.

Amplitude and rotational speed control of variable length pendulum by periodic input

In this paper, a control law to stabilize the amplitude or rotational speed of a variable length pendulum to a desired value by periodically changing the position of the center of gravity is proposed. First, the motion of the pendulum oscillating around a lower equilibrium point is analyzed using the averaging method, and a first-order differential equation for the amplitude of the pendulum is derived. Subsequently, using the derived equation of motion, a control law is designed to control the amplitude of the pendulum to the desired value. Similarly, the motion of a pendulum rotating continuously around the rotation axis is analyzed, the first-order differential equation for the angular velocity of the pendulum is derived, and then a control law of the rotational speed is designed. The derived nonlinear feedback control law consists of the amplitude, angle, and angular velocity of the pendulum in the case of amplitude control, and in the case of rotational speed control, the rotational velocity and angular acceleration of the pendulum. Finally, by using the proposed control method, it is shown that the amplitude and rotational speed of the pendulum can be controlled to the desired values.

Design of a self-tunable, variable-length pendulum for harvesting energy from rotational motion

In this paper, a self-tunable energy harvester based on pendulum oscillations with a mechanical motion rectifier (MMR) system, which can convert vibration into electrical energy, is proposed. The harvester is composed of a pendulum excited by a slider-crank mechanism. The pendulum system is designed to automatically adjust its own natural frequency to match that of the imposed base excitation. Frequency adjustment in a proposed pendulum-type energy harvester is achieved by varying the length of the pendulum rod through changing the position of pendulum mass which mounted at its tip. The pendulum mass is driven by a ball screw through a stepper motor which controls the length of the pendulum automatically in accordance with the frequency of the external motion. The base motion frequency is detected by an infrared sensor. An ultrasonic distance sensor is used to detect the length of the pendulum rod and feeds this information to a microcontroller to obtain the corresponding natural frequency from a lookup table. The microcontroller calculates the frequency difference between natural frequency and excitation frequency and converts this value into a length difference through another lookup table. The microcontroller then gives instructions to drive a stepper motor through a sequence of steps to achieve the target length and keeps the device in resonance state to harvest maximum power during operation. Different time detection intervals were studied to investigate their effect on the tuning process. This study showed that the longer time intervals increase the detection accuracy for the calculation of low excitation frequency. The amount of energy consumed during the tuning process to adjust the pendulum length is presented. In this context, the consumed energy is only needed until the resonance of the device matches the excitation frequency. The harvester system was studied numerically and experimentally. Based on the findings of this work, the natural frequency of the harvester is successfully tuned below 0.7 Hz, when the length of pendulum rod is changed from 550 mm to 900 mm, generating power from 1.78 W to 4.1 W at an optimal load resistance value of 10 Ω and 3 Ω respectively at maximum excitation amplitude of 120 mm. Therefore, the proposed pendulum system can be used as an efficient harvester for producing power in low-frequency applications (< 1 Hz).

Design and Experimental Validation of a 3-DOF Underactuated Pendulum-Like Robot

This article presents the design and the experimental validation of a novel 3-degree-of-freedom (DOF) pendulum-like cable-driven robot capable of executing point-to-point motions by leveraging partial feedback linearization control and on-line trajectory planning based on adaptive frequency oscillators (AFOs). Unlike most cable-suspended parallel robots, which rely on at least n actuated cables to control n DOF, the proposed robot is capable of performing 3-DOF point-to-point motions, from a starting pose to a goal one within its dynamic workspace, by means of two actuators only. Feedback linearization allows the dynamics of the variable-length pendulum to be decoupled from the orientation of the end effector, enabling the device to use parametric excitation to control the oscillations of the variable-length pendulum, akin to a playground swing. A pool of AFOs is introduced to enable smooth, lag free on-line estimations of the current phase of the pendulum oscillation to inform the on-line planner and the parametric excitation controller. Experimental results demonstrate feasibility of the proposed design and control approach.

Variable-Gain Second-Order Sliding Mode Controller With Globally Fixed-Time Stability Guarantees

In this brief, the globally fixed-time control problem is investigated for a general class of uncertain nonlinear systems. The novelties of this brief are two folds. One novelty lies in the propose of a novel variable-gain second-order sliding mode (SOSM) controller. It overcomes the principal shortcomings of first-order sliding mode control and makes it possible to achieve fixed-time stability. The conventional SOSM algorithms can only achieve finite-time stability whose settling time will grow unboundedly when the initial conditions tend to infinity. Another novelty is the extension of constant upper bounds to time-varying upper functions. This enables us to propose a global result, while only local results can be obtained in most of the existing literatures. It is indicated that the proposed novel variable-gain SOSM controller can achieve the establishment of SOSMs in a fixed-time independent of initial conditions. In addition, Lyapunov analysis is given to show that the closed-loop system is globally fixed-time stable. An application to variable-length pendulum is given.

On the elastic pendulum, parametric resonance and ‘pumping’ swings

The ‘peculiar’ behavior of the spring/elastic pendulum, its parametric resonance, and ‘pumping’ swings are frequently studied phenomena and always exhibit interesting features, but the subject is not taught sufficiently in physics courses. Based on our calculations and physics teaching experience, we try to provide here some new results and judgments on these phenomena. Firstly, we discuss the coupling of the normal modes of the elastic pendulum oscillations. The nonapproximated equations of elastic pendulum motions are solved numerically. We then comment on experiments regarding exciting different normal modes. The ‘pumping’ of swings is treated here as a case of parametric resonance of a pendulum of variable length. A new reasonable model of ‘pumping’ swings is presented and the respective calculations show the oscillation amplitude increasing. In addition, a simple experimental setup for parametric resonance in the physics laboratory is presented.

Bifurcation study of a chaotic model variable-length pendulum on a vibrating base

It is known that in the dissipative system of an inverted pendulum of constant length on an oscillating base, a cascade of bifurcations arises, leading to chaos. In this paper, the appearance of chaotic behavior of a conservative variable-length pendulum on a vibrating base near the upper equilibrium position at high vibration frequencies and small amplitudes of harmonic oscillations of the length of the pendulum and the point of its suspension is discovered and investigated. As a mathematical model, a non-autonomous averaged second-order system with dissipation near resonance 1:2 between the oscillation frequencies of the length and oscillations of the suspension point is used. A numerically-analytical bifurcation study of an autonomous control system and a non-autonomous dissipative system is performed at a decrease in the dissipation coefficient to zero. Cascades of bifurcations of limit cycles in the neighborhood of the upper equilibrium position, leading to the formation of a chaotic attractor, are found. The presence of dynamic chaos is proved by graphs and maps of the largest Lyapunov exponent, by maps of dynamic regimes and bifurcation diagrams.