Motion Of Pendulum Research Articles

On the Orbital Stability of Pendulum Motions of a Rigid Body in the Hess Case

On the Orbital Stability of Pendulum Motions of a Rigid Body in the Hess Case

Dynamics of heavy subaqueous spherical pendulums

We present a systematic study to investigate the fluid–structure interaction (FSI) of subaqueous spherical pendulums with several solid-to-fluid mass ratios $m^*\in [1.14, 14.95]$ and corresponding Reynolds numbers of up to $\textit {Re}\sim 10^4$ . A digital object tracking (DOT) method was employed to track the oscillating pendulum spheres whereas the time-resolved 3-D particle tracking velocimetry (tr-3D-PTV) was used to measure the flow field around the spheres. The data obtained from the coupling of the two measuring techniques provide novel insights into the dynamics of pendulum sphere oscillations, instantaneous pressure fluctuations related to vortex shedding around the spheres and the way they are influenced by the vortex and wake interactions. Namely, we show that during the downward motion of the pendulum spheres, vortex rings are shed off the spheres which, in turn, induce short-lived propulsion and, subsequently, distinct deceleration. Further, we used the measured data to improve an existing basic model of pendulum motion, which has significant discrepancies for the period and peak amplitude predictions. We did this by incorporating a vortex-induced drag term and a wake interaction term into the equation. Finally, the improved equations are shown to be capable of predicting the subaqueous pendulum dynamics with high accuracy, for the investigated range of $m^*$ . The study thus extends the current understanding of basic fluid dynamic mechanisms such as added mass, nonlinear drag, vortex and pressure dynamics.

Hollow cathode effect in radio frequency hollow electrode discharge in argon

Radio frequency capacitively coupled plasma source (RF-CCP) with a hollow electrode can increase the electron density through the hollow cathode effect (HCE), which offers a method to modify the spatial profiles of the plasma density. In this work, the variations of the HCE in one RF period are investigated by using a two-dimensional particle-in-cell/Monte-Carlo collision (PIC/MCC) model. The results show that the sheath electric field, the sheath potential drop, the sheath thickness, the radial plasma bulk width, the electron energy distribution function (EEDF), and the average electron energy in the cavity vary in one RF period. During the hollow electrode sheath’s expansion phase, the secondary electron heating and sheath oscillation heating in the cavity are gradually enhanced, and the frequency of the electron pendular motion in the cavity gradually increases, hence the HCE is gradually enhanced. However, during the hollow electrode sheath’s collapse phase, the secondary electron heating is gradually attenuated. In addition, when interacting with the gradually collapsed hollow electrode sheaths, high-energy plasma bulk electrons in the cavity will lose some energy. Furthermore, the frequency of the electron pendular motion in the cavity gradually decreases. Therefore, during the hollow electrode sheath’s collapse phase, the HCE is gradually attenuated.

Dynamics of a double pendulum with viscous friction in the joints. II. Dissipative oscillation modes and optimization of damping parameters

This paper is a continuation of the article "Dynamics of a double pendulum with viscous friction in the joints. I. Mathematical model of motion and construction of the regime diagram", in which a linear mathematical model of the motion of a double mathematical pendulum with identical parameters of links and end loads in the presence of viscous friction in both articulations was given, and a diagram of dissipative regimes of its motion was also constructed. The question of a particular variant of proportional damping is considered, in which the oscillation modes of a dissipative system are not distorted by friction forces, and basic formulas are given that describe the dynamics of the system in this situation. For the general case of damping, through a rational combination of analytical and numerical research methods, all key quantities characterizing the motion of the system on each of the dissipative oscillation modes are identified and determined. In addition, several problems of optimal damping of system oscillations are considered, and the best dissipative parameters are selected based on the criterion of the maximum degree of stability. The obtained results are accompanied by a series of graphical illustrations, which make it possible to establish their dependence on the damping coefficients and note their main qualitative and quantitative features. The solutions found can be useful in practice when designing two-link manipulators and studying their dynamic behavior.

Assessment of the roughness of the machined surface during pendulum surface plastic deformation

The influence of pendular surface plastic deformation mode on the roughness parameters of the machined surface of a cylindrical workpiece is studied. The optimal values of the processing mode parameters are determined. Keywords surface plastic deformation, roughness parameters, experimental study, device for pendular motion

Taphonomic significance of Aristotle's 'Wheel Paradox': an example with the cerithid gastropod Cerithium atratum (Born, 1778) from Conceição Lagoon, Santa Catarina State, Southern Brazil

In the book Mechanical Problems, attributed to Aristotle, the author relates the differential degrees of wearing and rounding of pebbles and shells moved by waves on the seashore to the variable radii and diameters of these particles. This phenomenon, which became known as Aristotle’s ‘Wheel Paradox,’ seems to explain the pattern of abrasion observed in shells of gastropods Cerithium atratum (Born) found in Conceição Lagoon, on the Island of Santa Catarina, Brazil. In situ observations show that the elongated turriform shells of C. atratum deposited at shallow (< 30 cm) depths on a sandy bottom with pavements of disarticulated valves of Anomalocardia flexuosa Gmelin are subject to bidirectional wave-generated currents under fairweather conditions. The 47 specimens analyzed exhibit a pattern of abrasion characterized by the loss of the shell wall, apparently proceeding from the wider last whorl toward the narrow apex, culminating in the exposure of the columella. Experiments designed to test whether this pattern could be related to the ‘Wheel Paradox’ principle show that under bidirectional flow, the shells describe circles on the vertical plane around the longitudinal axis (rolling) and a pendulum movement (revolution) on the horizontal plane centered on the apex. The whorls located farther from the apex have larger diameters and describe broader arc segments. Thus, they have higher tangential velocities and travel longer paths, which could explain why they are abraded earlier than the narrower whorls. Recognizing similar abrasion features in fossil shells can provide information on the local paleo-currents, hydrodynamics, and relative time-averaging of death assemblages and allow distinguishing between natural or anthropogenic modification in archaeological specimens. Although the effects vary between shells with different shapes, Aristotle’s ‘Wheel Paradox’ principle explains the general abrasion pattern among gastropods under certain hydrodynamic conditions. Keywords: Actualistic Taphonomy, Aktuopaläontologie, shell abrasion, shell orientation, biostratinomy, hydrodynamics

Tracking the Acceleration Due to Gravity and Damping of a Pendulum: A Video Analysis

This study focused on using video analysis to track the acceleration due to gravity and damping of a pendulum. The video tracker analysis software was proven effective in collecting data from several physics experiments. This study offered new insights on using a tracker and combining it with MS Excel to generate a damping graph. A pendulum was set in motion and recorded with a high-speed camera from the iPhone 13. The footage was then analyzed using tracker video analysis, modeling software, and Microsoft Excel to determine the amplitude and frequency of the pendulum's motion. At the same time, the other variables were measured using traditional tools (e.g., the length of the yarn using a meter ruler). The measured values were then used to calculate the acceleration due to gravity and the damping constant of the system. The results showed that the acceleration due to gravity was consistent with the average accepted value, while the damping constant accurately represented what was seen from the generated graph. Overall, this study demonstrated the effectiveness of video analysis in tracking the behavior of physical systems and its potential applications in various fields.

Simple pendulum: period dependent on amplitude of oscillation

In this study, with a pedagogical aim suited for the undergraduate, the differential equations of motion that characterize and determine the motion of a simple pendulum were obtained considering small and large amplitudes of oscillation. These differential equations were solved through differential equation solution methods, numerical, expansion of functions and integrations. The solutions obtained using the different methods were compared. It was possible to verify, both experimentally and theoretically, that for the oscillatory movement of the simple pendulum, its oscillation period increases and its angular frequency decreases with the increase of the oscillation amplitude. The validity range of the approximation for small ranges of motion was also determined. It was verified that the theoretical and experimental results present a good agreement for angles smaller than 55°. The experimental measurements were made with “a low-cost home-built” equipment.

A New Method Investigation for Robotic Inverted Pendulum Movement and Control

A New Method Investigation for Robotic Inverted Pendulum Movement and Control

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.

Testing flax fibers using high-speed bending

The article presents a method of measuring bending rigidity of flax fiber during high-speed bending. This method is an important part of the effort to reduce the dependency on imported cotton. Unlike the standardized testing methods, the suggested approach closer imitates the conditions present during fiber processing and exploitation of fiber products. For evaluation of bending rigidity of flax fiber during high-speed bending there was used the scheme of loading (meeting the factors of Eule – Bernoulli theory) of a sample fixed in a cantilevered way with the following determination of the work done by bending and deformation by interaction with impact pendulum. The value of sample deformation was determined on the basis of the control of moments of the time of angle velocities deformation which took place by operating movement (during the tests) and free movement of the pendulum. The specific feature of determining the time of deformation were more precise calculations: of angle velocity of the pendulum considering the force of resistance to its free movement; of pendulum motion periods at different stages of its movement during its interaction with the sample; of the time of starting and finishing the interaction by sample bending using the method of dichotomy by approximation of angle velocity values; of angle velocity variation at different stages of its movement during its interaction with the sample. These more precise calculations has become the basis of generalized algorithm for determining the bending deformation by interaction with impact pendulum, done by electronic calculator. A hardware-software complex has been created that employs this algorithm to measure and visualize both sample deformation and pendulum energy loss, as well as angular velocity of the pendulum. A comparison of different kinds of natural bast fibers has been performed – specifically, flax, nettle and hemp. The bending rigidity of these samples has been determined to be 1.42, 2.53 and 6.45 newton‧mm2 accordingly. These differences match the well-known properties of those materials.

Dynamical analysis for the motion of a 2DOF spring pendulum on a Lissajous curve

This study examines the motion of a spring pendulum with two degrees-of-freedom (DOF) in a plane as a vibrating system, in which its pivot point is constrained to move along a Lissajous curve. In light of the system’s coordinates, the governing equations of motion (EOM) are obtained utilizing the equations of Lagrange’s. The novelty of this work is to use the approach of multiple scales (AMS), as a traditional method, to obtain novel approximate solutions (AS) of the EOM with a higher degree of approximation. These solutions have been compared with the numerical ones that have been obtained using the fourth-order Runge–Kutta algorithm (4RKA) to reveal the accuracy of the analytic solutions. According to the requirements of solvability, the emergent resonance cases are grouped and the modulation equations (ME) are established. Therefore, the solutions at the steady-state case are confirmed. The stability/instability regions are inspected using Routh–Hurwitz criteria (RHC), and examined in accordance with the steady-state solutions. The achieved outcomes, resonance responses, and stability areas are demonstrated and graphically displayed, to evaluate the positive effects of different values of the physical parameters on the behavior of the examined system. Investigating zones of stability/instability reveals that the system’s behavior is stable for a significant portion of its parameters. A better knowledge of the vibrational movements that are closely related to resonance is crucial in many engineering applications because it enables the avoidance of on-going exposure to potentially harmful occurrences.

Symmetry and Asymmetry of Chaotic Motion in a Crank Arm and Connecting Rod Due to the Movement of the Follower

The symmetry and asymmetry of chaotic motion in the planar mechanism is investigated for a crank arm and connecting rod due to the motion of a flat-faced follower. The level of chaos is investigated using the conception of the Lyapunov exponent parameter and phase-plane diagram at different cam speeds with and without the use of coefficients of restitution. Moreover, the fast Fourier transform (FFT) of power spectrum analysis technique is used based on SNR factor values at different cam speeds and different coefficients of restitution. The wave forms and histograms of nonlinear responses are analyzed using the SolidWorks program for the crank arm, connecting rod, and flat-faced follower. There is a clearance between the flat-faced follower and its guides while the oscillation motion of the crank arm and connecting rod is described as the motion of a double pendulum. The level of chaos is increased with increases in the cam speeds and coefficients of restitution.

Controlled motion modes of a gyroscopic pendulum in a gimbal suspension

The issues of controlling the motion of a gyroscopic pendulum fixed in a gimbal suspension and having three degrees of freedom are discussed in the paper. It is proposed to use physical analogies and correlate their action with the dissipative forces of external viscous friction, friction in gimbal joints and internal friction in the pendulum rod, as well as inertial forces and gyroscopic forces to form the control torques in gimbal joints. Six modes of controlled movement of the system are considered, which allow achieving various goals. It is shown that it is possible to completely suppress the motions of a gyroscopic pendulum, switch it to the mode of a spherical or physical pendulum, enter the mode of rotation around a fixed vertical axis, bring it to the mode of regular precession, and also strengthen or weaken the gyroscopic structure of the motion equations. The discussion of these modes of controlled motion is accompanied by the construction of motion equations both within the framework of the original nonlinear model and for simplified linear or weakly nonlinear models. The obtained results are theoretically interesting and may be useful for specific practical applications in the field of gyroscopic technology.

A solvable walking model for a two-legged robot

We present a solvable biped walking model based on an inverted pendulum with two massless articulated legs capable of walking on uneven floors and inclined planes. The stride of the two-legged robot results from the pendular motion of a standing leg and the articulated motion of a trailing leg. Gaiting is possible due to the pendular motion conservation of energy and the alternating role of the legs, the standing and the trailing leg. The motion on uneven surfaces and inclined planes is possible by imposing the same maximal opening angle between the two legs in the transition between strides and the adaptability of the time of each stride. This model is solvable in closed form and is reversible in time, modelling the different types of biped motion. Several optimisation results for the speed of gaiting as a function of the robot parameters have been derived.

Proof of the small angle approximation sinθ≈θ using the geometry and motion of a simple pendulum

The small angle approximation is central to all treatments of the simple pendulum as a harmonic oscillator and is typically asserted as a result that follows from calculus. Here, however, we show that the geometry of the pendulum itself offers a route to understanding the origin of the small angle approximation without recourse to calculus. Rather charmingly, our approach exploits the motion of the pendulum to visualise the process of taking an important limit and can be used to explore the meaning of mathematical approximation in physical systems.

Information-theory analysis of mouse string-pulling agrees with Fitts’s Law: Increasing task difficulty engages multiple sensorimotor modalities in a dual oscillator behavior

Mouse string pulling, in which a mouse reels in a string with hand-over-hand movements, can provide insights into skilled motor behavior, neurological status, and cognitive function. The task involves two oscillatory movements connected by a string. The snout oscillates to track the pendulum movement of the string produced by hand-over-hand oscillations of pulling, and so the snout guides the hands to grasp the string. The present study examines the allocation of time required to pull strings of varying diameter. Movement is also described with end-point measures, string-pulling topography with 2D markerless pose estimates based on transfer learning with deep neural networks, and Mat-lab image-segmentation and heuristic algorithms for object tracking. With reduced string diameter, mice took longer to pull 60 cm long strings. They also made more pulling cycles, misses, and mouth engagements, and displayed changes in the amplitude and frequency of pull cycles. The time measures agree with Fitts’s law in showing that increased task difficulty slows behavior and engages multiple compensatory sensorimotor modalities. The analysis reveals that time is a valuable resource in skilled motor behavior and information-theory can serve as a measure of its effective use.

A piezoelectric cantilever-asymmetric-conical-pendulum-based energy harvesting under multi-directional excitation

This paper introduces a novel approach to address the challenges faced by traditional unidirectional cantilever-based energy harvesters in adapting to multi-directional vibration environments. The proposed solution combines a flexible cantilever with an asymmetric-conical-pendulum structure which consists of two different concentrated end masses and a rigid thin rod By investigation of energy interchange relationship between the kinetic feature of the asymmetric pendulum and beam bending vibration under horizontal and vertical excitations respectively from base movements, utilizing the Lagrange's theorem and multiple-scale method, 1:2 internal resonance can be induced for enabling the delivery of multi-directional motion of pendulums to the unidirectional bending of cantilever. On this basis, the advantages of the proposed system are revealed by comparing with traditional piezoelectric cantilever and piezoelectric cantilever-single-pendulum systems. Furthermore, the performance of the voltage amplitude and operational bandwidth was potentially improved up to 128.5 % and 229.8 % under x-direction. In addition, it is confirmed that its maximum voltage RMS value is 57.9 % and 11.4 % higher than that of the piezoelectric cantilever-single-pendulum in the x- and z-directions, owing to its multi-peak energy harvesting over variable excitation amplitudes. Therefore, the feasibility and superiorities of the proposed configuration are demonstrated theoretically and numerically in this paper.

Stability and performance benefits of system tautochrones for vibration control

Centrifugal pendulum vibration absorbers have become a prominent means of controlling torsional oscillations that are typically generated by the combustion pulses within an internal combustion engine. The dynamic stability and performance of devices designed to smooth torsional oscillation are highly dependent upon the motion path defined for their pendulous masses. Generally speaking, dynamic responses are more difficult to manage when pendulum and system resonances shift as a function of the excitation levels due to nonlinear effects. As a result, paths for pendulums that are so-called tautochronic, meaning their resonance does not change as the pendulum amplitude varies, facilitate designing vibration absorbing systems that efficiently reduce vibration even as excitation levels increase. Approximate tautochronic paths for centrifugal pendulum vibration absorbers (CPVAs) are typically derived by making simplifying assumptions, which uncouple the pendulum dynamics from that of the rotor (base) system in which it acts upon. The resulting family of motion paths enable absorber designs that generate desirable and reliable vibration absorption, when slightly over-tuned with respect to the excitation frequency. In this paper, we begin the process of relaxing the assumptions that isolate a pendulum subsystem from its operating context by considering the complete system, which consists of a pendulum with inertial-coupling to the base system. For a simplified vibration system related to a CPVA, wherein a base mass rolls on a horizontal plane and a pendulous mass slides within a cut-out on the sliding base mass, we identify shapes of cut-outs that produce system brachistochrone curves. As in the case of a classical brachistochrone for a fixed base mass, we show the system brachistochrone is a system tautochrone, meaning for a range of input energy, system response that includes pendulum and base mass motions together have a constant period. We show that system tautochrones have distinct vibration control advantages, including superior dynamic stability and vibration absorbing performance. Our results provide a guide to how one can find solutions for the analogous CPVA system.

Acquiring high-resolution wind measurements by modifying radiosonde sounding procedures

Abstract. High-resolution wind measurements are crucial for the understanding of dynamic processes in the atmosphere. In the troposphere and lower stratosphere, radiosondes provide a good vertical resolution of a few metres, but the wind data are usually low-pass filtered by the manufacturer in order to suppress disturbances in the data caused by the spurious motions of the sonde. As an example, the filter within the standard processing of Vaisala radiosondes becomes effective at vertical scales below 300 m for an ascent rate of 5 m s−1. We describe a method for increasing the usable resolution of radiosonde wind measurements. The main ideas are to avoid self-induced motions of the balloon by keeping it in the subcritical Reynolds number range, to avoid typical pendulum motions of a 15 s period by using a shorter string, and to use data from a descending balloon in order to avoid disturbances from the wake of the balloon affecting temperature and humidity measurements due to the decreased string length. We demonstrate that our changes in hardware and software allow for artefact-free wind data down to scales of 50 m, while the remaining disturbances on even smaller scales are removed. Accordingly, the usable resolution of the wind data has been increased by a factor of 6, compared to the standard data output, at relatively low cost.