Length Of Pendulum Research Articles

Development of a dual-DOF vibration energy harvester using a foldable spring pendulum mechanism

Abstract The novelty of this design is that it changes the traditional idea of an unchangeable pendulum length to achieve folding and stretching through the shear fork structure. The concept of spring as the modulation mechanism of the vibration energy harvester is introduced to complete the dynamic change of the pendulum length and realize the dual degrees of freedom output of the roll and heave motions. The spring mechanism integrated into the design, including tension and torsion springs, not only realizes the flexible expansion of the structure but also significantly enhances the ability of the vibration energy harvester to capture external vibration energy through the unique energy storage and release mechanism. The vibration energy harvester has been designed, mathematically simulated, constructed, and experimentally tested with good results, characterized by dual mode state, low-frequency characteristics, and high-power output. Experimental results show that the prototype can achieve an average power output of 5.47 W at an excitation frequency of 0.8 Hz, corresponding to a normalized power output of 377.42 W g−2 and a normalized power density of 26.25 W g−2 kg−1.

Theoretical characteristics of a three-point Roberts linkage.

The Roberts linkage is recognized for enabling long-period pendulum motion in a compact format. Utilizing this characteristic, we are developing a three-point Roberts linkage for vibration isolation systems, with an eye toward its potential contribution to the development of next-generation interferometric gravitational wave antennas. In this article, we derived the equations to determine the essential parameters when using this linkage as a vibration isolation system, namely, the equivalent pendulum length and the relationship between translational motion of the center of mass and rigid body rotation, from size parameters. In addition, we analyzed the behavior in response to various errors.

ANALISIS KORELASI ANTARA SUDUT SIMPANGAN DAN WAKTU RATAAN PADA BANDUL FISIS DENGAN LATIN SQUARE DESIGN

This study aims to examine the relationship between the angle of deviation and the average time length of a physical pendulum. This study uses primary data, namely practicum report data in a physics laboratory. Data collection was conducted under closed conditions with control variables (shaft length, number of oscillations, and radius of gyration) and dependent variables (angle of deviation and average time). The analysis technique used is a statistical approach technique, namely Latin Square Design analysis. The findings of this study indicate that the variables of deviation angle and average time on the physical pendulum have a strong and unidirectional correlation relationship. This result is evident from the value of F0 > Fcritical and the value of Pr (> P) > 0, 05(significant level) where both of these prove that there is a strong reason to reject H0 which means that there is at least 1 difference in the effect of the angle of deviation on the average time. It shows that the angle of deviation affects the average time of the physical pendulum

Study on the influence of orthogonal double pendulum on the ice shedding of transmission lines

ABSTRACT Ice shedding and vibration of ice-covered transmission lines can pose a serious threat to the safe operation of power lines in heavy ice areas. However, the common preventative practice of installing anti-vibration hammers, spacers and other devices on the line has limited effects on the safe operation of transmission lines. This paper focused on the simulation and experimental study of the influence of orthogonal double pendulum damper, a new type of anti-vibration device for transmission lines, on the ice shedding and vibration reduction of transmission lines. The 3-DOF calculation model of the orthogonal double pendulum damper was established, and the influence of the parameters of orthogonal double pendulum damper on the ice shedding of the line was simulated and analysed. In addition, natural ice shedding tests of lines were carried out at Xuefeng Mountain National Field Station, and ice shedding and vibration displacement response of lines was measured after the installation of orthogonal double pendulum dampers. The results showed that The three main structural parameters of mass, pendulum length and arm length can affect the maximum amplitude of ice shedding and vibration of transmission lines. The mass of the hammerhead has an effect on the vibration in both vertical and horizontal directions; the pendulum length mainly affects the vibration of the line in the horizontal direction; the arm length mainly affects the vibration in the vertical direction.

Study on the variable length simple pendulum oscillation based on the relative mode transfer method.

In this study, we employed the principle of Relative Mode Transfer Method (RMTM) to establish a model for a single pendulum subjected to sudden changes in its length. An experimental platform for image processing was constructed to accurately track the position of a moving ball, enabling experimental verification of the pendulum's motion under specific operating conditions. The experimental data demonstrated excellent agreement with simulated numerical results, validating the effectiveness of the proposed methodology. Furthermore, we performed simulations of a double obstacle pendulum system, investigating the effects of different parameters, including obstacle pin positions, quantities, and initial release angles, on the pendulum's motion through numerical simulations. This research provides novel insights into addressing the challenges associated with abrupt and continuous changes in pendulum length.

A robust Gated-PINN to resolve local minima issues in solving differential algebraic equations

Physics-Informed Neural Network (PINN) has emerged as a promising tool for solving various physical problems with differential equations. However in practice, PINN often suffers from the local minima issue while solving problems with minimal initial conditions. In this paper, we propose a robust Gated-PINN to address this issue, which blends numerical methods and PINN. The Gated-PINN overcomes the local minima issue by controlling the flow of information of the neural network similar to a numerical method. We first investigate the use of conventional PINN to the Cartesian-coordinate simple pendulum problem with minimal initial conditions, as one of basic differential algebraic equation (DAE) problem. Our results show the advantage of PINN over existing numerical methods in that it does not require sophisticated mathematical techniques such as index reduction. But we also observe that conventional PINN can lead to inaccurate solutions; such solutions partially satisfy the differential equation requirement, but does not meet the given initial condition and fail to further improve. We demonstrate the effectiveness of the proposed Gated-PINN by showing that it yields accurate solutions, such that for a pendulum of length 1, the mean Euclidean error between the Gated-PINN model and the traditional numerical method model is less than 0.01 and pointwise maximum Euclidean error is less than 0.04. Moreover, Gated-PINN can operate without any complicated index reduction, and unlike conventional PINN, accurate solution can be obtained consistently without falling into a local minima. Overall, our study presents the potential of the Gated-PINN for solving DAE problems and provides a valuable insight into the challenges and limitations of using PINN for solving physical problems.

Nonlinear double-mass pendulum for vibration-based energy harvesting

To enhance the performance of a vibration-based energy harvester, typical approaches employ frequency-matching strategies by either using nonlinear broadband or frequency-tunable harvesters. This study systematically analyzes the nonlinear dynamics and energy harvesting performance of a recently emerging tunable low-frequency vibration-based energy harvester, namely, a double-mass pendulum (DMP) energy harvester. This energy harvester can, to some extent, eliminate frequency dependence on pendulum length but exhibit vibration-amplitude-dependent softening nonlinearity. The natural frequency of the DMP structure is theoretically derived, showing several unique characteristics compared with the typical simple pendulum. The DMP energy harvester exhibits alternate single-period, multiple-period, and chaotic vibration behaviors with increase in excitation amplitudes. The analysis of gross output power indicates that the rotating motion, regardless of chaotic or periodic rolling motions, improves the energy harvesting performance in terms of power leap and broader bandwidth. Based on the parameter space analysis, the rotating motions usually occur at the shift-left locations of frequency ratios 1 and 2; a smaller damping ratio corresponds to a lower on-demand excitation amplitude for the rotating-motion occurrence. Numerical results confirm that the DMP is suitable for low-frequency energy harvesting scenarios, suggesting the realization of rotating motion for improving energy harvesting performance. Moreover, a shake table test was performed, and the experimental results validated the accuracy and effectiveness of the DMP modeling analysis. Practical issues related to DMP energy harvesters under different types of excitations are finally discussed. Although the analysis is for the DMP, the corresponding conclusions may shed light on other pendulum-type energy harvesters.

The Variational Iteration Method for a Pendulum with a Combined Translational and Rotational System

Abstract The dynamic analysis of complex mechanical systems often requires the application of advanced mathematical techniques. In this study, we present a variation iteration-based solution for a pendulum system coupled with a rolling wheel, forming a combined translational and rotational system. Furthermore, the Lagrange multiplier is calculated using the Elzaki transform. The system under investigation consists of a pendulum attached to a wheel that rolls without slipping on a horizontal surface. The coupled motion of the pendulum and the rolling wheel creates a complex system with both translational and rotational degrees of freedom. To solve the governing equations of motion, we employ the variation iteration method, a powerful numerical technique that combines the advantages of both variational principles and iteration schemes. The Lagrange multiplier plays a crucial role in incorporating the constraints of the system into the equations of motion. In this study, we determine the Lagrange multiplier using the Elzaki transform, which provides an effective means to calculate Lagrange multipliers for constrained mechanical systems. The proposed solution technique is applied to analyse the dynamics of a pendulum with a rolling wheel system. The effects of various system parameters, such as the pendulum length, wheel radius and initial conditions, are investigated to understand their influence on the system dynamics. The results demonstrate the effectiveness of the variation iteration method combined with the Elzaki transform in capturing the complex behaviour of a combined translational and rotational system. The proposed approach serves as a valuable tool for analysing and understanding the dynamics of similar mechanical systems encountered in various engineering applications.

P-Bifurcation Analysis of a Quarter-Car Model With Inerter-Based Pendulum Vibration Absorber: A Wiener Path Integration Approach

Abstract In this theoretical study, a recently developed inerter-based pendulum vibration absorber (IPVA) coupled with energy harvesting capabilities is applied to the quarter car model with class C road conditions (ISO 8608). The impact of varying the pendulum length parameter on power harvesting, ride comfort (sprung mass acceleration), and road handling is investigated. It is discovered that P-bifurcation of the probability density function (PDF), can simultaneously occur with enhanced output power (40% improvements), low sprung mass acceleration (60% improvements), and better road handling (60% improvements) when compared with the linear benchmark system. To predict this bifurcation, a Wiener path integration (WPI) method coupled with curvature checking is developed for the PDF. An efficient bifurcation detection algorithm is developed which leads to the prediction of monomodal, bimodal, and rotation PDF regions in the noise intensity-electrical damping plane. Using Monte Carlo simulations (MCS), the performance metrics were then compared against the optimal linear benchmark for varying driving speed on a class F road while varying the electrical damping so that the system is at or near P-bifurcation. Energy transfer into the electrical domain and power harvested are shown to be up to 43% and 20% higher than for the optimized linear system, respectively. Electrical efficiency considerations show that generator selection is also a factor. Ride comfort and road handling still saw improvements of at least 59%. Finally, the new algorithm effectively reduces an exhaustive MCS for various parameter configurations when qualitative changes in the PDF are linked to performance.

Light-induced motion of three-dimensional pendulum with liquid crystal elastomeric fiber

Liquid crystal elastomers (LCEs) typically exhibit slow deformation in response to environmental stimuli. However, recent studies have demonstrated that when LCE is reduced to a fiber form, it can undergo rapid and reversible contractions. This unique property allows LCE fibers to convert light energy into fast mechanical motion, enabling them to act as actuators that can swiftly flip the wings or lift an object. In this paper, we present a theoretical investigation of a three-dimensional pendulum constructed by a LCE fiber, which differs from the conventional pendulum with a fixed string length. The string length of the LCE pendulum is time-dependent and responsive to light radiation. We develop a nonlinear dynamic model based on Lagrangian mechanics and the photodynamic theory of LCE to study the effects of initial conditions, light intensity, radiation region, damping factor, and gravity on the characteristics of the three-dimensional pendulum. Our findings reveal the existence of three distinct modes of motion: static, light-excited self-oscillation, and light-excited self-rotation. We elucidate the mechanisms underlying these modes by investigating the time-dependent length of the string, the forces acting on the string, and the work done by light radiation during each period. The results and dynamic model presented in this paper provide valuable insights into light-driven motion structures and offer references for the design of soft robotics, energy harvesting devices, and wing-flapping micro aircraft applications based on smart soft materials.

Theoretical and experimental research on damping performance of suspended multiple mass pendulums damper

Conventional suspended mass pendulum damper (SMPD) include suspended single mass pendulum damper (SSMPD), tandem suspended multiple mass pendulums damper without springs (TSMP), and parallel suspended multiple mass pendulums damper without springs (PSMP), these types of SMPD have the advantages such as clear mechanism, simple construction, and good damping effect, but they have a narrow tuning frequency band and the pendulum length needs to be set too long for low-frequency super tall structures and too short for high-frequency tall structures, limiting their application in practical engineering. Because of this, the paper proposes connecting the spring to the mass pendulum, forming a tandem suspended multiple mass pendulums damper with spring (TSMP-S) and a parallel suspended multiple mass pendulums damper with spring (PSMP-S), and establishing structural system dynamics equations and performing the dynamic characterization of the TSMP-S and PSMP-S, obtain their frequency resolved solutions to explain the damping mechanism. In order to study the vibration-damping performance of these two types of SMPD, through numerical simulations to compare the damping effect of the conventional SMPD, TSMP-S, and PSMP-S on the controlled structures, and a series of shaking table tests on controlled and uncontrolled structures to verify the effectiveness of SSMPD, TSMP, PSMP, TSMP-S and PSMP-S for damping control of controlled structures under conventional earthquake and earthquake at four types of sites. Numerical simulations and shaking table test results in both show that the TSMP-S has the best damping effect, with the advantages of increasing the tuning frequency band, achieving multi-order vibration control, and flexible adjustment of the pendulum length, which is suitable for actual engineering.

Pemanfaatan Tangki Riak Untuk Mengukur Kecepatan Rambat Gelombang Permukaan Air

The study intends to investigate the relationship between wavelength and frequency to calculate the speed of propagating waves on the water's surface. The researchers used a ripple tank to collect data by recording the pendulum's movement as well as the image of the ripples on the water's surface. The data is then analyzed with the Tracker application. The wavelength is determined by analyzing the recorded ripple shadows with the Tracker application. The Tracker application is used to analyze the recorded pendulum movement to obtain data on the period of vibration and calculate the frequency of the vibrations. The average surface wave velocity at a pendulum length of 10 cm is 3.38 10-2 m/s. In addition, for a 30 cm pendulum length, an average surface wave speed of 3.26 10-2 m/s is obtained.

Extreme rotational events in a forced-damped nonlinear pendulum.

Since Galileo's time, the pendulum has evolved into one of the most exciting physical objects in mathematical modeling due to its vast range of applications for studying various oscillatory dynamics, including bifurcations and chaos, under various interests. This well-deserved focus aids in comprehending various oscillatory physical phenomena that can be reduced to the equations of the pendulum. The present article focuses on the rotational dynamics of the two-dimensional forced-damped pendulum under the influence of the ac and dc torque. Interestingly, we are able to detect a range of the pendulum's length for which the angular velocity exhibits a few intermittent extreme rotational events that deviate significantly from a certain well-defined threshold. The statistics of the return intervals between these extreme rotational events are supported by our data to be spread exponentially at a specific pendulum's length beyond which the external dc and ac torque are no longer sufficient for a full rotation around the pivot. The numerical results show a sudden increase in the size of the chaotic attractor due to interior crisis, which is the source of instability that is responsible for triggering large amplitude events in our system. We also notice the occurrence of phase slips with the appearance of extreme rotational events when the phase difference between the instantaneous phase of the system and the externally applied ac torque is observed.

Piezoelectric energy extraction from a cylinder undergoing vortex-induced vibration using internal resonance

A novel concept of utilizing the kinetic energy from ocean currents/wind by means of internal resonance is proposed to address the increasing global energy demand by generating clean and sustainable power. In this work, a non-linear rotative gravity pendulum is employed to autoparametrically excite the elastically mounted cylinder for a wide range of flow velocities. This concept is adopted to increase the oscillation amplitude of the cylinder due to vortex-induced vibration (VIV) in the de-synchronized region for energy harvesting. In this regard, a VIV-based energy harvesting device is proposed that consists of a cylinder with an attached pendulum, and energy is harvested with bottom-mounted piezoelectric transducers. The cylinder undergoes VIV when it is subjected to fluid flow and this excites the coupled fluid-multibody cylinder-pendulum system autoparametrically. In the de-synchronized region, when the vortex shedding frequency becomes two times the natural frequency of the pendulum, an internal resonance occurs. This helps in achieving a higher oscillation amplitude of the cylinder which does not happen otherwise. This study is focused on the two degree-of-freedom (2-DoF) cylinder-pendulum system where the cylinder is free to exhibit cross-flow vortex-induced vibrations subjected to the fluid. The objective of this work is to numerically investigate the effect of a non-linear rotative gravity pendulum (NRGP) on the VIV characteristics and piezoelectric efficiency of the system. The numerical model is based on the wake-oscillator model coupled with the piezoelectric constitutive equation. The influence of the frequency ratio, mass ratio, torsional damping ratio, and ratio of cylinder diameter to pendulum length of the NRGP device on response characteristics due to VIV is also investigated. A detailed comparative analysis in terms of electric tension and efficiency is performed numerically for flows with a wide range of reduced velocities for the cylinder with and without NRGP. A comprehensive study on the implications of internal resonance between the pendulum and a cylinder undergoing VIV on generated electric tension is also reported.

Equivalent Dynamic of Liquid Sloshing Coupled with Mass Transfer During On-Orbit Refueling

During an on-orbit refueling mission, the liquid fuel partially filling in the tanks will slosh and produce coupled motion with the servicer-to-client transfer, which may bring dangers to the spacecraft and even lead to attitude divergence. To effectively simulate the motion and the inertia of liquid fuel in a cylindrical tank during refueling, equivalent dynamic modeling of liquid fuel sloshing coupled with mass transfer is developed in this paper. A spring–mass–damper model with time-varying elastic and damping coefficients is implemented in the axial direction to equal the motion of the liquid mass center, while a pendulum model with time-varying pendulum length is adopted in the radial direction. An equivalent liquid elliptical cylinder is developed to approximate the inertia of dispersed liquid while sloshing. A series of computational fluid dynamics simulations are carried out to observe the liquid behaviors in the fuel tanks and to verify the proposed equivalent dynamic model. Finally, three-dimensional sloshing is theoretically and numerically analyzed.

A multi-stable ultra-low frequency energy harvester using a nonlinear pendulum and piezoelectric transduction for self-powered sensing

This paper presents the design, theoretical modelling and experimental validation of a quad-stable energy harvester for harnessing ultra-low frequency random motions using a nonlinear pendulum and piezoelectric transduction. The multi-stable pendulum is created by the magnetic forces between magnets on the pendulum and a tip magnet on a piezoelectric cantilever beam. Two attractive and one repulsive magnetic forces in combination with the gravitational force of the pendulum create multiple stable positions for the pendulum. The multi-stable dynamics allow the pendulum to effectively convert low-frequency random kinetic motions from the host, e.g. human motion or wind turbine tower oscillation into the pendulum oscillation, enabling effective plucking of the piezoelectric beam with enhanced output power. A theoretical model, including the magnetic interaction, piezoelectric conversion and pendulum dynamics, is established to describe the electromechanical dynamics of the whole harvester. A prototype is fabricated and tested on a linear shaker at ultra-low frequencies (1–3 Hz) to showcase the capability of the harvester and to validate the theoretical results. Around 8 μW Root-Mean-Square output power was obtained at 2.5 Hz and 0.8 g of excitation. Using the experimentally validated theoretical model, a parametric study was carried out to examine the influence of different structural and operating parameters, such as pendulum mass and length, magnetic coupling strength, excitation frequency and amplitude, on the output power and operating frequency range of the energy harvester. The operation frequency range and output power can be effectively adjusted by changing the above-mentioned parameters. The self-powered sensing capability is then illustrated by integrating the harvester with an off-the-shelf power management circuit and a 22 μF storage capacitor. The capacitor was charged from 2.8 V to 4 V in 90 s, showing its capability of implementing battery-free wireless sensing for different Internet of Things (IoT) applications.

Multi-Objective Optimization and Tradespace Analysis of a Mechanical Clock Movement Design

Abstract Pendulum clocks were the prevalent time keeping standard for centuries to regulate commerce and public activities. These mechanical movements were the most accurate timekeepers globally until replaced by electric clocks. Although mainly used for decorative purposes today, the pendulum clock's working principles and mechanical behavior can serve to demonstrate fundamental science and engineering concepts. The tradeoff between a clock's quality factor, pendulum properties, and period can best be explored with multiple objective optimization and tradespace analysis methods. In this project, a Multi-Objective Genetic Algorithm (MOGA-II) and a Multi-Objective Simulated Annealing (MOSA) optimization approaches are applied to evaluate a Graham escapement street clock for pendulum mass and time accuracy with a range of the period. These clock designs vary the pendulum length, pendulum bob radius, and bob thickness. Horological concepts are used to calculate the overall performance and general utility. The numerical results show a 0.7% increase in the quality factor, and a 0.56% reduction in the mass, while maintaining the designed period by modifying the clock parameters. More importantly, these changes can provide material cost savings in a mass production scenario. Overall, the study highlights the tradeoff designer engineers have considered for decades which can now be visualized using computer tools for greater insight.

Inclined Large-angle Pendulum May Produce Endless Linear Motion of a Cart When Friction is Negligible

We present the mechanics for the oscillation of an inclined large-angle pendulum-drive attached to a cart which is allowed to perform translation in one direction only. Neglecting the overall friction, the application of Newton’s second law shows that the oscillation of the pendulum is continuously converted into oscillating linear motion thus achieving a travel of infinite length. It is also shown that the frequency depends on the usual data of any pendulum plus the mass of the cart on which it is attached. After the determination of a novel effective pendulum length, a closed-form accurate analytical expression is presented for the amplitude of the pendulum, whereas semi-analytical formulas are provided for the period as well as the time-variation of the large azimuthal-like angle. Moreover, a simple expression was found for the position of the cart in terms of the azimuthal angle of the pendulum and the elapsed time. The extraction of the analytical formulas was facilitated by a computer model programmed in MATLAB®.

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.

GRAVITATIONAL CONUNDRUM: CONFUSING CLOCK-RATE MEASUREMENTS ON THE 'FIRST FLEET’ FROM ENGLAND TO AUSTRALIA

Voyages of exploration often included astronomers among their crew to aid with maritime navigation. William Dawes, a British Marine who had been trained in practical astronomy, was assigned to the 'First Fleet’, a convoy of eleven ships that left England in May 1787 bound for Botany Bay (Sydney, Australia). Dawes was also expected to take measurements of the local gravitational acceleration, g, at any port of call by measuring the daily rate by which his Shelton pendulum clock differed from that at Greenwich, its calibration location. Although Dawes and Nevil Maskelyne, Britain’s fifth Astronomer Royal, had planned to obtain clock-rate measurements in the Canary Islands, San Sebastian (Rio de Janeiro) and Table Bay, Captain Arthur Phillip, Commander of the First Fleet, only allowed Dawes to disembark the clock in Rio de Janeiro. Therefore, we have just one set of clock-rate measurements from the voyage, in addition to land-based measurements obtained in New South Wales. If gravity was the dominant factor affecting the clock’s changing rate, Dawes’ measurement of –48.067 sec per (sidereal) day obtained in Rio de Janeiro implies a local gravitational acceleration, <italic>g</italic> = 9.7946 m sec^–2. On the other hand, if we adopt the modern value, <italic>g</italic> = 9.7878 m sec^–2, the implied daily decay rate is almost exactly 30 sec greater than Dawes’ clock-rate determination, a difference that is well in excess of the prevailing uncertainties. This suggests that the pendulum’s regulator nut may have been offset by a full turn, thus implying that our assumptions regarding the pendulum length may have to be revisited.