Simple Periodic Motion Research Articles

Tangle: A metric for quantifying complexity and erratic behavior in short time series.

Temporal complexity refers to qualities of a time series that are emergent, erratic, or not easily described by linear processes. Quantifying temporal complexity within a system is key to understanding the time based dynamics of said system. However, many current methods of complexity quantification are not widely used in psychological research because of their technical difficulty, computational intensity, or large number of required data samples. These requirements impede the study of complexity in many areas of psychological science. A method is presented, tangle, which overcomes these difficulties and allows for complexity quantification in relatively short time series, such as those typically obtained from psychological studies. Tangle is a measure of how dissimilar a given process is from simple periodic motion. Tangle relies on the use of a three-dimensional time delay embedding of a one-dimensional time series. This embedding is then iteratively scaled and premultiplied by a modified upshift matrix until a convergence criterion is reached. The efficacy of tangle is shown on five mathematical time series and using emotional stability, anxiety time series data obtained from 65 socially anxious participants over a 5-week period, and positive affect time series derived from a single participant who experienced a major depression episode during measurement. Simulation results show tangle is able to distinguish between different complex temporal systems in time series with as few as 50 samples. Tangle shows promise as a reliable quantification of irregular behavior of a time series. Unlike many other complexity quantification metrics, tangle is technically simple to implement and is able to uncover meaningful information about time series derived from psychological research studies. (PsycInfo Database Record (c) 2022 APA, all rights reserved).

Transition to turbulence in a five-mode Galerkin truncation of two-dimensional magnetohydrodynamics.

The chaotic dynamics of a low-order Galerkin truncation of the two-dimensional magnetohydrodynamic system, which reproduces the dynamics of fluctuations described by nearly incompressible magnetohydrodynamic in the plane perpendicular to a background magnetic field, is investigated by increasing the external forcing terms. Although this is the case closest to two-dimensional hydrodynamics, which shares some aspects with the classical Feigenbaum scenario of transition to chaos, the presence of magnetic fluctuations yields a very complex interesting route to chaos, characterized by the splitting into multiharmonic structures of the field amplitudes, and a mixing of phase-locking and free phase precession acting intermittently. When the background magnetic field lies in the plane, the system supports the presence of Alfvén waves thus lowering the nonlinear interactions. Interestingly enough, the dynamics critically depends on the angle between the direction of the magnetic field and the reference system of the wave vectors. Above a certain critical angle, independently from the external forcing, a breakdown of the phase locking appears, accompanied with a suppression of the chaotic dynamics, replaced by a simple periodic motion.

Flutter of variable stiffness composite laminates in supersonic flow with temperature effects

The nonlinear thermal flutter behavior of variable stiffness composite laminates (VSCL) with curvilinear fibers in high supersonic flow is investigated. The first order shear deformation theory (FSDT) combining von Karman large-deflection strain-displacement relations, quasi-steady first-order piston theory aerodynamics and quasi-steady thermal stress theory are used to formulate the nonlinear panel flutter finite element equations of motion. The fiber orientation within a layer is assumed to vary linearly from [Formula: see text] at the center to [Formula: see text] at the vertical edges of the rectangular lamina. The flutter characteristics of variable stiffness composite laminates with different temperature distributions are then studied. The results show that the critical dynamic pressure decreases as [Formula: see text] or [Formula: see text] increases, whereas the limit cycle amplitude increases as [Formula: see text] or [Formula: see text] increases for the same dynamic pressure. The critical dynamic pressure and limit cycle amplitude both increase when the temperature gradient along panel thickness increases. Simple harmonic motions, unharmonic but periodic motions, and chaotic motions can be observed on VSCL under different temperatures. It also turns out that temperature distribution has similar influence on both the critical dynamic pressure and limit cycle amplitude of VSCL.

Self-propelled Brownian spinning top: Dynamics of a biaxial swimmer at low Reynolds numbers

Recently the Brownian dynamics of self-propelled (active) rodlike particles was explored to model the motion of colloidal microswimmers, catalytically driven nanorods, and bacteria. Here we generalize this description to biaxial particles with arbitrary shape and derive the corresponding Langevin equation for a self-propelled Brownian spinning top. The biaxial swimmer is exposed to a hydrodynamic Stokes friction force at low Reynolds numbers, to fluctuating random forces and torques as well as to an external and an internal (effective) force and torque. The latter quantities control its self-propulsion. Due to biaxiality and hydrodynamic translational-rotational coupling, the Langevin equation can only be solved numerically. In the special case of an orthotropic particle in the absence of external forces and torques, the noise-free (zero-temperature) trajectory is analytically found to be a circular helix. This trajectory is confirmed numerically to be more complex in the general case of an arbitrarily shaped particle under the influence of arbitrary forces and torques involving a transient irregular motion before ending up in a simple periodic motion. By contrast, if the external force vanishes, no transient regime is found, and the particle moves on a superhelical trajectory. For orthotropic particles, the noise-averaged trajectory is a generalized concho-spiral. We furthermore study the reduction of the model to two spatial dimensions and classify the noise-free trajectories completely finding circles, straight lines with and without transients, as well as cycloids and arbitrary periodic trajectories.

Periodic motion of four spheres in a “kite” configuration

Cluster settling plays an important role in the sedimentation of dilute suspensions. Studies of isolated clusters indicate that they show considerable stability. In particular, spheres initially arranged in a horizontal isosceles triangle or rhombus exhibit simple periodic motion as they descend. Moving one sphere outward along the long diagonal of a rhombus produces a kite configuration that settles with a complicated periodic motion relative to its center of mass, which exhibits a back-and-forth movement as it descends. This makes the kite configuration the second member of a class between small clusters with simple periodic motions and isolated clusters that merely remain intact for a considerable time. If the displacement along the diagonal is too great, one sphere is left behind. Higher-order terms, which have little effect on widely separated spheres in a rhombus, are important in maintaining the periodicity of the kite configuration and in slowing its breakup.

Problem of stability of two-link trajectories in a multidimensional Birkhoff billiard

A linearized problem of stability of simple periodic motions with elastic reflections is considered: a particle moves along a straight-line segment that is orthogonal to the boundary of a billiard at its endpoints. In this problem issues from mechanics (variational principles), linear algebra (spectral properties of products of symmetric operators), and geometry (focal points, caustics, etc.) are naturally intertwined. Multidimensional variants of Hill’s formula, which relates the dynamic and geometric properties of a periodic trajectory, are discussed. Stability conditions are expressed in terms of the geometric properties of the boundary of a billiard. In particular, it turns out that a nondegenerate two-link trajectory of maximum length is always unstable. The degree of instability (the number of multipliers outside the unit disk) is estimated. The estimates are expressed in terms of the geometry of the caustic and the Morse indices of the length function of this trajectory.

DEVELOPMENT OF MOTION CAPTURE SYSTEM USING DUAL VIDEO CAMERAS FOR THE GAIT DESIGN OF A BIPED ROBOT

This paper, describes the development of a motion capture system with novel features for biped robots. In general, motion capture is effectively utilized in the field of computer animation. In the field of humanoid robotics, the number of studies attempting to design human-like gaits by using expensive optical motion capture systems is increasing. The optical motion capture systems used in these studies have involved a large number of cameras because such systems use small-sized ball markers; hence the position accuracy of the markers and the system calibration are very significant. However, since the human walking gait is a simple periodic motion rather than a complex motion, we have developed a specialized motion capture system for this study using dual video cameras and large band-type markers without high-level system calibration in order to capture the human walking gait. In addition to its lower complexity, the proposed capture method requires only a low-cost system and has high space efficiency. An image processing algorithm is also proposed for deriving the human gait data. Finally, we verify the reliability and accuracy of our system by comparing a zero moment point (ZMP) trajectory calculated by the motion captured data with a ZMP trajectory measured by foot force sensors.

Dynamics of a model of two delay-coupled relaxation oscillators

This paper investigates the dynamics of a new model of two coupled relaxation oscillators. The model replaces the usual DDE (differential-delay equation) formulation with a discrete-time approach with jumps. Existence, bifurcation and stability of in-phase periodic motions is studied. Simple periodic motions, which involve exactly two jumps per period, are found to have large plateaus in parameter space. These plateaus are separated by regions of complicated dynamics, reminiscent of the Devil’s Staircase. Stability of motions in the in-phase manifold are contrasted with stability of motions in the full phase space.

Numerical and asymptotic analysis of a localized heat source undergoing periodic motion

A highly localized heat source moves with simple periodic motion along a one-dimensional medium with reactive-diffusive properties. It is known that the system will experience a blow-up in finite time. Numerical results suggest that this blow-up, though unavoidable, can be delayed by increasing either the amplitude or the frequency of the motion. These numerical results are compared to known analytical results. Numerical and analytical results are also compared for the related case in which the heat source moves at a constant speed in one direction. The asymptotic behavior of the temperature of the material at the location of the heat source near the time of blow-up is also analyzed. It can be of interest to consider the asymptotic behavior of the solution in the context of the numerical solution in order to gain confidence in the numerical results.

Alteration of the equal settling ratio of fine particles by vertically-oscillating water

An equal settling ratio is an important factor in estimating particle separation accuracy. However, this factor is often calculated using the settling velocity in stationary water, there are no examples of calculation of the equal settling ratio in an actual separator. This is difficult because particle movement in a separator is very complicated, and even simple periodic motions, such as the oscillation field used with many separators, are ignored in many cases. The authors have previously reported on the relation between the equal settling ratio and the oscillation frequency by analysis of particle movement in vertically oscillating water, using spherical particles of glass (average size 435 μm) and zirconia (202 μm) which have the same settling velocity in stationary water. In this study, the influence of particle diameter on the change in the settling velocity in oscillating water was experimentally investigated for three pairs of glass and zirconia particles having different sizes under 0.5 mm, which have the same settling velocity in stationary water. The settling velocities of different-sized particles decreased at different rates in oscillating water, indicating that the equal settling ratio is reduced by water oscillation. We conclude that water oscillation improves the accuracy of size separation for glass particles over 300 μm and zirconia particles over 150 μm when glass and zirconia particles are separated from each other with the difference of these settling velocities.

Evolution and stability of the theoretically predicted families of periodic orbits in theN-body ring problem

We theoretically investigate the existence of families of periodic orbits in the planar N-body ring problem and we give a qualitative picture of the motion of a small particle. This study yields four families of periodic orbits which we also found numerically: two families of periodic orbits around the central body and two families around all the peripherals. These results are valid for different values of N and β. Also we investigate the evolution of simple periodic motions as well as their stability. We found stable and unstable orbits around the central body, around all the peripherals and around one or more peripherals which form rings of stability. Some families present other types of bifurcations, such as bifurcations of families of non-symmetric periodic orbits of the same period and period-doubling bifurcations.

The existence of periodic solutions to certain impulsive differential equations

For autonomous ordinary differential equations, it is shown that there always exist impulsive forcing patterns which permit the existence of periodic solutions. However, when the differential equation is nonautonomous, proving the existence of impulsive periodic solutions appears more complex. The authors study here an impulsive differential system which appears to have no periodic solutions. Continuous manifolds of initial vectors are found for which it is proven that the trajectory emanating from each initial vector is unstable, hence, nonperiodic. It is also established that a particular type of impulsive periodic solution, the so-called simple periodic motion, cannot exist. By changing the type of impulse applied, some periodic solutions are obtained, one of which, though not physically meaningful, is perhaps mathematically significant.

The Mercer online interactive chaotic pendulum

At Mercer University, we have developed an online interactive chaotic pendulum. This pendulum is unique in that it is not a computer simulation but an actual device that can be remotely controlled through the Internet. Our objective is threefold. We are attempting to provide an educational resource for chaos, demonstrate a new technology, and experiment with a new method of laboratory education. As students vary the driving frequency, they can watch the pendulum's motion switch from simple periodic motion to chaotic motion. The control software incorporates several mathematical analysis functions so that students can view the pendulum data in different ways. Students can conduct experiments using the pendulum, download their data, and conduct their own mathematical analysis.

Morphology development in polymer blends produced by chaotic mixing at various compositions

Whereas blending devices commonly entail complex flow fields and internal geometries, chaotic mixing can be instilled by simple periodic motion of bounding surfaces in simple devices. Breakup and coalescence of spatially expansive structures in components can give blends with a wide variety of morphologies. In this study, polystyrene and low density polyethylene were used as model components to study the effect of composition and processing time on gradual morphology development in immiscible binary blends. Inspections of samples disclosed attainable morphologies and also how transitions between morphologies occurred. Novel findings included blends with encapsulated fibers, abundant platelets, and two distinct morphologies having single phase continuity. Additionally, interpenetrating blends formed over a broad compositional range. Results suggest that chaotic mixing is a useful tool for studying relationships among processing conditions, morphology development, and blend properties and may serve as a means to more deliberately obtain target morphologies.

Blow-up in a reactive-diffusive medium with a moving heat source

A concentrated source of heat is allowed to move in one-dimension through a reactive-diffusive medium. For rather general motion, it is shown that if the speed remains sufficiently high then a blow-up can be avoided; whereas, if the speed remains sufficiently low then a blowup must eventually occur. Other criteria involving the displacement and velocity of the heat source are found to guarantee a blow-up. Special examples of (i) rapid deceleration from one velocity level to a lower level and (ii) a simple periodic motion are examined.

AN EXPERIMENTAL STUDY OF THE IMPULSE RESPONSE OF A VIBRO-IMPACTING CANTILEVER BEAM

The dynamics of a vibro-impacting cantilever beam experiment using an impact load cell is considered. The signal recorded from the cell produces spike train -type data. The issues related to the analysis of such data are discussed; particularly the sampling rate and threshold values. For vibro-impact motion of the beam, the duration of impacts is investigated by using a time of contact measure. The implications are discussed for vibro-impact systems mathematically modelled by using instantaneous impact assumptions (coefficient of restitution). Using the load cell to measure impact forces for the beam system is also considered. Then a delay reconstruction of the dynamics of the system by using interspike intervals is considered. It is demonstrated how this process is effected by the influence of noise and the data-acquision process using numerical simulations of the experimental data. It is shown how simple periodic motions can be identified by using a probability density approach and possible future research is highlighted.

The force acting on a moving current loop above a conducting plate

This paper describes a new mathematical procedure for the determination of the force acting on a direct current in a horizontal loop moving in a vertical direction, with simple periodic motion, above a conductive plate of finite thickness and conductivity. The density of eddy currents induced within the plate due to the harmonic motion and the electromagnetic field outside the plate are utilized in the calculation procedure of the force. A general expression for the force is given in the form of Fourier integral series. A graph of the force versus periodic motion frequency is presented.

Chaos in the classroom: Exposing gifted elementary school children to chaos and fractals

A unit of study for gifted 4th and 5th graders is described on the subject of mathematical periodicity and chaos and the underlying physical processes which produce these phenomena. A variety of hands-on experiments and the use of various data analysis tools and computer aids provide students with powerful raw material for their analysis, interpretation, and understanding. The concepts of simple periodic motion (e.g., a pendulum), complex superposition of motions (e.g., the vibrations in musical instruments), and chaotic sequences (e.g., stock prices) are covered, with numerous practical examples. Opportunities to involve related activities emphasizing language arts, history, and graphic art are included. The student response to the material is documented.

Functional equations, iterated mappings and clock problems

The current concept of time is reviewed with the aim of underlining its relation with the evolution of dynamical systems. It is shown that the index ordering the values of a variable in a sequence generated by a nonlinear mapping gives a different, system-dependent, time measure than the phase of a simple periodic motion. Some ergodicity problems in the construction of invariant distributions for iterated mappings are indicated.

The Dynamic Response of a System With Preloaded Compliance

We consider the dynamic response of a single degree of freedom system with preloaded, or “setup,” springs. This is a simple model for systems where preload is used to suppress vibrations. The springs are taken to be linear and harmonic excitation is applied; damping is assumed to be of linear viscous type. Using the piecewise linear features of the model equations we determine the amplitude and stability of the periodic responses and carry out a bifurcation analysis for these motions. Some parameter regions which contain no simple stable periodic motions are shown to possess chaotic motions.