Simple Pendulum Model Research Articles

Hierarchical Three-Body Problem at High Eccentricities = Simple Pendulum II: Octupole including Brown’s Hamiltonian

Abstract The very long-term evolution of the hierarchical restricted three-body problem with a massive perturber is analyzed analytically in the high eccentricity regime. Perturbations on the time scale of the outer orbit can accumulate over long timescales and be comparable to the effect of the octupole term. These perturbations are described by Brown’s Hamiltonian - having different forms in the literature. We show that at the high eccentricity regime - the effect of Brown’s Hamiltonian is an azimuthal precesssion of the eccentricity vector and can be solved analytically. In fact, the dynamics are equivalent to a simple pendulum model allowing an explicit flip criterion.

Hierarchical three-body problem at high eccentricities = simple pendulum I: octupole

ABSTRACT The gradual evolution of the restricted hierarchical three body problem is analysed analytically, focusing on conditions of Kozai–Lidov cycles that may lead to orbital flips from prograde to retrograde motion due to the octupole (third order) term which are associated with extremely high eccentricities. We revisit the approach described by Katz, Dong and Malhotra (2011) and show that for most initial conditions, to an excellent approximation, the analytical derivation can be greatly simplified and reduces to a simple pendulum model allowing an explicit flip criterion. The resulting flip criterion is much simpler than the previous one but the latter is still needed in a small fraction of phase space. We identify a logical error in the earlier derivation but clarify why it does not affect the final results.

Analysis of the dynamic characteristics of an eccentric cylinder in axial flow

The simple pendulum model is a fundamental and widely studied concept in physics, its movement follows a tendency to stabilize towards the side with lower gravitational potential energy. Consequently, the simple pendulum structures have been extensive applications in petroleum, aerospace machinery, and others, particularly in gravity induction and actuator systems. The utilization of this model offers versatile solutions for complex engineering challenges. The research object of this study is an eccentric cylindrical stabilized platform. A nonlinear dynamic model of the eccentric cylindrical stable platform in axial flow with variable damping is established. Due to the complexity of the analytical solution of the simple pendulum motion, we adopt the method of numerical analysis to building the simulation and analysis model. Combined with the fluid calculation and dynamics simulation methods, the motion characteristics of the eccentric cylindrical model under steady and unsteady flow fields are analyzed by adjusting the rotational speed, inclination angle, and initial pendulum angle. The experimental platform is built to analyze the motion law of the eccentric cylinder under the action of axial flow, and the experimental results of the motion law of the eccentric cylindrical platform are consistent with the numerical calculation results.

Does a physical pendulum ever act like a simple pendulum?

We show that for a physical pendulum comprising a massive sphere swinging from a massive string, there is, in general, a length of string for which its oscillatory period equals the period calculated by the simple pendulum model with a point-like mass swinging from a massless string whose model length equals the summed length of the real string and the sphere’s radius.

The Relationship between Air Resistance and the Vibrating Period of Simple Pendulum

The simple pendulum is a typical model of the compound pendulum. In this model, resistance and buoyancy are negligible. This paper researches how air resistance affects the vibration period in the simple pendulum model but with exact conditions. There are two simplified air resistance models that can apply in the models, one is the first power air resistance model, and the second is the second power air resistance model. But the first power air resistance model is quite complex, there is no equipment to measure its coefficient of it, so it only applies the second power air resistance. This paper studies the formula that revealed air resistance and related suitable cases. The results show that the time from the formula is longer than the experiment a little bit.

Controlling of payload swinging of an overhead crane

Abstract A new two-level hierarchical approach to control the trolley position and payload swinging of an overhead crane is proposed. At the first level, a simple mathematical pendulum model is investigated considering the time delay due to the use of a vision system. In the second level, a chain model is developed, extending the previous pendulum model considering the vibration of the suspending chain. The relative displacement of the payload is measured with a vision sensor, and the rest of the state-space variables are determined by a collocated observer. The gain parameters related to the state variables of the chain vibration are determined by the use of a pole placement method. The proposed controller is verified by numerical simulation and experimentally on a laboratory test bench.

The motion of trees in the wind: a data synthesis

Abstract. Interactions between wind and trees control energy exchanges between the atmosphere and forest canopies. This energy exchange can lead to the widespread damage of trees, and wind is a key disturbance agent in many of the world's forests. However, most research on this topic has focused on conifer plantations, where risk management is economically important, rather than broadleaf forests, which dominate the forest carbon cycle. This study brings together tree motion time-series data to systematically evaluate the factors influencing tree responses to wind loading, including data from both broadleaf and coniferous trees in forests and open environments. We found that the two most descriptive features of tree motion were (a) the fundamental frequency, which is a measure of the speed at which a tree sways and is strongly related to tree height, and (b) the slope of the power spectrum, which is related to the efficiency of energy transfer from wind to trees. Intriguingly, the slope of the power spectrum was found to remain constant from medium to high wind speeds for all trees in this study. This suggests that, contrary to some predictions, damping or amplification mechanisms do not change dramatically at high wind speeds, and therefore wind damage risk is related, relatively simply, to wind speed. Conifers from forests were distinct from broadleaves in terms of their response to wind loading. Specifically, the fundamental frequency of forest conifers was related to their size according to the cantilever beam model (i.e. vertically distributed mass), whereas broadleaves were better approximated by the simple pendulum model (i.e. dominated by the crown). Forest conifers also had a steeper slope of the power spectrum. We interpret these finding as being strongly related to tree architecture; i.e. conifers generally have a simple shape due to their apical dominance, whereas broadleaves exhibit a much wider range of architectures with more dominant crowns.

Synthetic nonlinear second-order oscillators on Riemannian manifolds and their numerical simulation

<p style='text-indent:20px;'>The present paper outlines a general second-order dynamical system on manifolds and Lie groups that leads to defining a number of abstract non-linear oscillators. In particular, a number of classical non-linear oscillators, such as the simple pendulum model, the van der Pol circuital model and the Duffing oscillator class are recalled from the dedicated literature and are extended to evolve on manifold-type state spaces. Also, this document outlines numerical techniques to implement these systems on a computing platform, derived from classical numerical schemes such as the Euler method and the Runke-Kutta class of methods, and illustrates their numerical behavior by a great deal of numerical examples and simulations.</p>

Modeling and Simulation of a Human Knee Exoskeleton's Assistive Strategies and Interaction.

Exoskeletons are increasingly used in rehabilitation and daily life in patients with motor disorders after neurological injuries. In this paper, a realistic human knee exoskeleton model based on a physical system was generated, a human–machine system was created in a musculoskeletal modeling software, and human–machine interactions based on different assistive strategies were simulated. The developed human–machine system makes it possible to compute torques, muscle impulse, contact forces, and interactive forces involved in simulated movements. Assistive strategies modeled as a rotational actuator, a simple pendulum model, and a damped pendulum model were applied to the knee exoskeleton during simulated normal and fast gait. We found that the rotational actuator–based assistive controller could reduce the user's required physiological knee extensor torque and muscle impulse by a small amount, which suggests that joint rotational direction should be considered when developing an assistive strategy. Compared to the simple pendulum model, the damped pendulum model based controller made little difference during swing, but further decreased the user's required knee flexor torque during late stance. The trade-off that we identified between interaction forces and physiological torque, of which muscle impulse is the main contributor, should be considered when designing controllers for a physical exoskeleton system. Detailed information at joint and muscle levels provided in this human–machine system can contribute to the controller design optimization of assistive exoskeletons for rehabilitation and movement assistance.

Subharmonic resonance bifurcation and chaos of simple pendulum system with vertical excitation and horizontal constraint

In order to improve the working performance and optimize the working parameters of the typical engineering pendulum of a typical system that it is abstracted as a physical simple pendulum model with vertical excitation and horizontal constraint. The dynamical equation of the system with vertical excitation and horizontal constraint is established by using Lagrange equation. The multiple-scale method is used to analyze the subharmonic response characteristics of the system. The amplitude-frequency response equation and the phase-frequency response equation are obtained through calculation. The effects of the system parameters on the amplitude resonance bandwidth and variability are clarified. According to the singularity theory and the universal unfolding theory, the bifurcation topology structure of the subharmonic resonance of the system is obtained. The Melnikov function is applied to the study of the critical conditions for the chaotic motion of the system. The parameter equation of homoclinic orbit motion is obtained through calculation. The threshold conditions of chaos in the sense of Smale are analyzed by solving the Melnikov function of the homoclinic motion orbit. The dynamic characteristics of the system, including single-parameter bifurcation, maximum Lyapunov exponent, bi-parameter bifurcation, and manifold transition in the attraction basin, are analyzed numerically. The results show that the main path of the system entering into the chaos is an almost period doubling bifurcation. Complex dynamical behaviors such as periodic motion, period doubling bifurcation and chaos are found. The bi-parameter matching areas of the subharmonic resonance bifurcation and chaos of the system are clarified. The results reveal the global characteristics of the system with vertical excitation and horizontal constraint, such as subharmonic resonance bifurcation, periodic attractor multiplication, and the coexistence of periodic and chaotic attractors. The results further clarify the mechanism of the influence of system parameters change on the movement form transformation, energy distribution and evolution law of the system. The mechanism of the influence of relevant parameters on the performance of the engineering system with vertical excitation and horizontal constraint is also obtained. The results of this research provide theoretical bases for adjusting the parameters of working performances of this typical physical system in engineering domain and the vibration reduction and suppression of the system in actual working conditions.

Dynamics for an electric pendulum

The Lagrangian formulation has been an extensive tool for the analysis of physical systems. In particular, we have applied the Lagrangian procedure to deduce the dynamics and stability for an electric pendulum system. We have considered two cases, a repulsive and attractive electric interactions as perturbations to the classical simple pendulum model. We study both cases, the repulsive and attractive electric interactions that can be considered as perturbations to the classical simple pendulum model. We have contrast both situations studying their restrictions, phase trajectories and stability points for this purpose.

Steady state obliquity of a rigid body in the spin–orbit resonant problem: application to Mercury

We investigate the stable Cassini state 1 in the p : q spin–orbit resonant problem. Our study includes the effect of the gravitational potential up to degree and order 4 and p : q spin–orbit resonances with \(p,q\le 8\) and \(p\ge q\). We derive new formulae that link the gravitational field coefficients with its secular orbital elements and its rotational parameters. The formulae can be used to predict the orientation of the spin axis and necessary angular momentum at exact resonance. We also develop a simple pendulum model to approximate the dynamics close to resonance and make use of it to predict the libration periods and widths of the oscillatory regime of motions in phase space. Our analytical results are based on averaging theory that we also confirm by means of numerical simulations of the exact dynamical equations. Our results are applied to a possible rotational history of Mercury.

Disturbance Rejection of Interval Type-2 Fuzzy Systems Based on Equivalence-Input-Disturbance Approach

This paper focuses on the problem of disturbance rejection for a class of interval type-2 (IT-2) fuzzy systems via equivalence-input-disturbance (EID)-based approach. The main objective of this work is to design a fuzzy state-feedback controller combined with a disturbance estimator such that the output of the fuzzy system perfectly tracks the given reference signal without steady-state error and produces an EID to eliminate the influence of the actual disturbances. By constructing a suitable Lyapunov function and using linear matrix inequality (LMI) technique, a new set of sufficient conditions is established in terms of linear matrix inequalities for the existence of fuzzy controller. Finally, a simple pendulum model is considered to illustrate the effectiveness and applicability of the proposed EID-based control design.

Simultaneous Longitudinal and Transverse Oscillations in an Active-Region Filament

We report on the co-existence of longitudinal and transverse oscillations in an active filament. On March 15$^{th}$ 2013, a M1.1 class flare was observed in the active region AR 11692. A CME was found to be associated with the flare. {The CME generated a shock wave that triggered the oscillations in a nearby filament}, situated at the south-west of the active region as observed from National Solar Observatory (NSO)\textit{Global Oscillation Network Group}(GONG) H$\alpha$ images. In this work we report the longitudinal oscillations in the two ends of the filament, co-existing with the transverse oscillations. We propose a scenario in which {an} incoming shock wave hits the filament obliquely and triggers both longitudinal and transverse oscillations. Using the observed parameters, we estimate the lower limit of the magnetic field strength. We use simple pendulum model with gravity as the restoring force to estimate the radius of curvature. We also calculate the mass accretion rate which causes the filament motions to damp quite fast.

Adjustable maneuvering time wave-form command shaping control with variable hoisting speeds

Classical input shaping is based on convolving a general input signal with a sequence of timed impulses. These impulses are chosen to match certain modal parameters of the system under control to eliminate residual vibrations in rest-to-rest maneuvers. This type of input shaping is strongly dependent on the system period. In this work, an adjustable maneuvering time wave form command shaper is presented. The equation of motion of a simple pendulum model of a crane is derived and solved in order to eliminate residual vibrations at the end of motion. Several cases are simulated numerically and validated experimentally on an experimental model of an overhead crane. Results show that the proposed command shaper is capable of eliminating residual vibrations effectively with a single continuous wave form command. The work is extended to include the effect of hoisting on the shaper performance. Several functions are used to simulate hoisting. To overcome the added complexity of hoisting on the system, an approximation technique is used to determine initial shaped command parameters, which are later used in a genetic algorithm optimization scheme. Numerical and experimental results prove that the proposed command shaper can effectively eliminate residual vibrations in rest-to-rest maneuvers.

3P1-N02 周期性に着目した人体の運動モデルによる異常検知(感覚・運動・計測(2))

For example, for the prevention of the fall down, new technologies are required to obtain information related to humans. Many methods are used for detection of human body movement. However, because the human body is complicated, the measurement of many points is necessary to express the movement of the human body. Many sensors are necessary, and therefore time and costs increase to measure many points. Therefore we made human walking simple as possible and tried to express it. The movement of an arm and foot are linked to the movement of a pendulum. The simple pendulum model was useful to express a rhythm of the movement of the human.

Sliding Mode Control for Slosh-Free Motion Using a Nonlinear Sliding Surface

A new nonlinear switching surface is proposed for the design of a sliding mode controller for a class of mismatched uncertain systems. A control system for slosh-free motion of a partially filled liquid container is sought. This is representative of a broader class of systems to which the proposed methodology is applicable. A fundamental mode of lateral slosh is considered. A simple pendulum model is used to represent the lateral slosh. Given the difficulty of measuring the slosh states directly, a sliding mode observer is used for implementation of the resulting sliding mode control. The effectiveness of the theoretical developments is demonstrated via both simulation and experimental results.

Sliding Mode Observer Based Sliding Mode Controller for Slosh-Free Motion Through PID Scheme

This paper presents a new approach to design a sliding mode controller for a class of mismatched uncertain systems. A method is proposed for the design of a switching surface in the presence of mismatched uncertainties. A design method for a sliding mode observer based on high gain is also proposed in this paper to reconstruct the states of the system for the implementation of sliding mode control. The design technique is simple and computationally efficient. A control problem for the slosh-free motion of a container is considered as the representative of a typical class of systems. A simple pendulum model is considered to represent the lateral slosh. The validity of the proposed scheme is demonstrated by simulation along with the experimental results.

Sliding mode control for slosh-free motion-A class of underactuated system

Underactuated systems are the representative of a large class of systems. This paper presents a new method for the design of sliding surface for a class of second order underactuated system in which a virtual input is considered in unactuated subsystem. A sliding mode controller is proposed to ensure sliding along the sliding surface. A problem of slosh free-motion of a container is considered as representative of a typical class of underactuated system. A simple pendulum model is considered to represent the lateral slosh. Extensive simulation studies are conducted with the controller to demonstrate the proposed approach.

Paraspinal Muscle Vibration Alters Dynamic Motion of the Trunk

Loss in dynamic stability of the low back has been identified as a potential factor in the etiology of low back injuries. A number of factors are important in the ability of a person to maintain an upright trunk posture including the preparatory stiffness of the trunk and the magnitude and timing of the neuromotor response. A neuromotor response requires appropriate sensing of joint motion. In this research, the role of this sensory ability in dynamic performance of the trunk was examined using a simple pendulum model of the trunk with neuromotor feedback. An increased sensory threshold was found to lead to increased torso flexion and increased delay in neuromotor response. This was confirmed experimentally using paraspinal muscle vibration which is known to alter proprioception of the muscle spindle organs. Before, during and after exposure to bilateral, paraspinal muscle vibration for 20 minutes, the dynamic response of subjects to an unexpected torso flexion load was examined. Subjects were found to have a 19.5% slower time to peak muscle activity and a 16.1% greater torso flexion during exposure to paraspinal muscle vibration. Torso flexion remained significantly increased after vibration exposure relative to before exposure. These results suggest that the neuromotor response plays an important role in trunk dynamics. Loss in sensitivity of the sensory system can have a detrimental effect on trunk dynamics, increasing delays in neuromotor response and increasing the motion of the trunk in response to an unexpected load.