Piecewise Constant Forces Research Articles

DYNAMICS OF A BODY UNDER THE IMPULSE FORCE AND WITH OSCILLATION LIMITER

The paper investigates the dynamics of a body under a piecewise constant periodic force with an arbitrary duty cycle and in the presence of an oscillation limiter. The mathematical model is a strongly nonlinear non-autonomous dynamical system with a truncated phase space along one of the phase coordinates. Using the point mapping method for strongly nonlinear (vibro-impact) dynamical systems, equations for two-dimensional Poincare map are obtained in the form of explicit analytical formulas. The relations obtained for Poincare map made it possible to effectively study the rearrangements of arbitrarily complex periodic motions with both a finite and an infinite number of fixed points, depending on the change in the parameters of the dynamical system. This allowed us to represent for the first time an analytical form of the equations that define in the parameter space the boundaries of the existence and stability domain of periodic motions with an infinite number of fixed points on the Poincare surface (corresponds to the infinitely impact mode of motion). The numerical experiments presented in the paper using a software product developed in the C++ language made it possible to present bifurcation diagrams that clearly demonstrate the ranges of parameter values with chaotic motion modes of the body. A scenario for the emergence of chaos has been established. The transition of body motions with a change in the overload parameter from periodic motion modes to chaos is carried out according to the period doubling. Comparison of numerical analysis with analytical results for different sets of parameters of the system under study (overload parameter) showed their good agreement.

Toward an artificial intelligence physicist for unsupervised learning.

We investigate opportunities and challenges for improving unsupervised machine learning using four common strategies with a long history in physics: divide and conquer, Occam's razor, unification, and lifelong learning. Instead of using one model to learn everything, we propose a paradigm centered around the learning and manipulation of theories, which parsimoniously predict both aspects of the future (from past observations) and the domain in which these predictions are accurate. Specifically, we propose a generalized mean loss to encourage each theory to specialize in its comparatively advantageous domain, and a differentiable description length objective to downweight bad data and "snap" learned theories into simple symbolic formulas. Theories are stored in a "theory hub," which continuously unifies learned theories and can propose theories when encountering new environments. We test our implementation, the toy "artificial intelligence physicist" learning agent, on a suite of increasingly complex physics environments. From unsupervised observation of trajectories through worlds involving random combinations of gravity, electromagnetism, harmonic motion, and elastic bounces, our agent typically learns faster and produces mean-squared prediction errors about a billion times smaller than a standard feedforward neural net of comparable complexity, typically recovering integer and rational theory parameters exactly. Our agent successfully identifies domains with different laws of motion also for a nonlinear chaotic double pendulum in a piecewise constant force field.

Optimal navigation strategies for active particles

The quest for the optimal navigation strategy in a complex environment is at the heart of microswimmer applications like cargo carriage or drug targeting to cancer cells. Here, we formulate a variational Fermat's principle for microswimmers determining the optimal path towards a given target regarding travelling time, energy dissipation or fuel consumption. For piecewise constant forces (or flow fields), the principle leads to Snell's law, showing that the optimal path is piecewise linear, as for light rays, but with a generalized refraction law. For complex environments, like general 1D, shear or vortex fields, we obtain exact analytical expressions for the optimal path, showing, for example, that microswimmers sometimes have to temporarily navigate away from their target to reach it fastest. Our results apply to idealized microswimmers which can instantaneously steer, are fast enough so that translational noise is unimportant and might be useful, e.g., to benchmark algorithmic schemes for optimal navigation.

A systematic review of varicella seroprevalence in European countries before universal childhood immunization: deriving incidence from seroprevalence data.

Surveillance systems for varicella in Europe are highly heterogeneous or completely absent. We estimated the varicella incidence based on seroprevalence data, as these data are largely available and not biased by under-reporting or underascertainment. We conducted a systematic literature search for varicella serological data in Europe prior to introduction of universal varicella immunization. Age-specific serological data were pooled by country and serological profiles estimated using the catalytic model with piecewise constant force of infection. From the estimated profiles, we derived the annual incidence of varicella infection (/100·000) for six age groups (<5, 5-9, 10-14, 15-19, 20-39 and 40-65 years). In total, 43 studies from 16 countries were identified. By the age of 15 years, over 90% of the population has been infected by varicella in all countries except for Greece (86·6%) and Italy (85·3%). Substantial variability across countries exists in the age-specific annual incidence of varicella primary infection among the <5 years old (from 7052 to 16 122 per 100 000) and 5-9 years old (from 3292 to 11 798 per 100 000). The apparent validity and robustness of our estimates highlight the importance of serological data for the characterization of varicella epidemiology, even in the absence of sampling or assay standardization.

Green's Function and Periodic Solutions of a Spring-Mass System in which the Forces are Functionally Dependent on Piecewise Constant Argument

In this paper, damped spring-mass systems with generalized piecewise constant argument and with functional dependence on generalized piecewise constant argument are considered. These spring-mass systems have piecewise constant forces of the forms $Ax(\gamma(t))$ and $Ax(\gamma(t))+h(t,x_{t},x_{\gamma(t)})$, respectively. These spring-mass systems are examined without reducing them into discrete equations. While doing this examination, we make use of the results which have been obtained for differential equations with functional dependence on generalized piecewise constant argument in \cite{2}. Sufficient conditions for the existence and uniqueness of solutions of the spring-mass system with functional dependence on generalized piecewise constant argument are given. The periodic solution of the spring-mass system which has functional force is created with the help of the Green's function, and its uniqueness is proved. The obtained theoretical results are illustrated by an example. This illustration shows that the damped spring-mass systems with functional dependence on generalized piecewise constant argument with proper parameters has a unique periodic solution which can be expressed by Green's function.

Existence of periodic solutions for a mechanical system with piecewise constant forces

In this study, we consider spring-mass systems subjected to piecewise constant forces. We investigate sufficient conditions for the existence of periodic solutions of homogeneous and nonhomogeneous damped spring-mass systems with the help of the Floquet theory. In addition to determining conditions for the existence of periodic solutions, stability analysis is performed for the solutions of the homogeneous system. The Floquet multipliers are taken into account for the stability analysis [3]. The results are stated in terms of the parameters of the systems. These results are illustrated and supported by simulations for different values of the parameters.

Valuing guaranteed equity-linked contracts under piecewise constant forces of mortality

The work of this paper is motivated by the study in Gerber et al. (2012) and some following papers, in which equity-linked death benefits embedded in various variable annuity products are valuated for any time-until-death random variables whose density function can be approximated by a linear combination of densities of exponential random variables. Their analysis is made for the case where the time-until-death is exponentially distributed, i.e., under the assumption of a constant force of mortality. The main purpose of our study is to show that the discounted density approach can also be used to obtain similar explicit results on life-contingent options under the assumption of piecewise constant forces of mortality. Moreover, we study a term insurance product with the payoff at the time of death being equity-linked and inflation-indexed, and investigate two types of annuity-immediate products whose annual payments are equity-indexed with a minimum guaranteed amount. We also illustrate approximations and numerical calculations for some results obtained in this paper, and analyze parameter sensitivities.

Approximate calculation of channel flows under force and energy actions

Gasdynamic channel flows under force and energy actions are considered. An approximate method is proposed for solving the gasdynamic equations that describe these flows. Themethod includes the separation of an “active” flow volume in which the integral electric force and applied power, whose densities are assumed to be uniform, are concentrated and the numerical integration of the system of hydrodynamic equations over the entire channel (in laminar and turbulent variants) with the piecewise constant force and energy sources obtained. The results of experimental investigation are presented for the flow that arises after two accessories mounted on the opposite walls of the vertical rectangular channel of constant cross-section, which create a dielectric barrier discharge (DBD actuators). This flow is numerically simulated using the method developed. On the basis of the method proposed the flow characteristics are determined for a model subsonic diffuser on whose lower wall, immediately in front of the separation zone, the DBD actuator is mounted. The efficiency of this accessory in reducing the gasdynamic losses is demonstrated.

Stability analysis of a two-degree-of-freedom mechanical system subject to proportional–derivative digital position control

In this paper we present an analysis of the stability of a two-degree-of-freedom system, modeling a robotic arm connected to the actuator through an elastic joint and subject to digital position control. The system consists of two lumped masses connected to each other through a spring and a damper. In the model there is only one actuator, so the system is underactuated in a certain sense; two cases are considered, referring to a collocated and a noncollocated configuration. Stability analysis is presented using both a continuous and a discrete time approach. The discrete time approach is related to the case of a digital controller, typical in real applications. This samples the position and the velocity signals at discrete time intervals and, therefore, it generates a piecewise constant control force, introducing a delay in the control system as well. The stability charts are presented in the parameter space of the sampling time and the control gains. Their differences highlight the role played by the resonances between the finite sampling frequency and the natural frequency of the system in achieving robust stability with respect to parameter variations.

Ratchet effect on a relativistic particle driven by external forces

We study the ratchet effect of a damped relativistic particle driven by both asymmetric temporal bi-harmonic and time-periodic piecewise constant forces. This system can be formally solved for any external force, providing the ratchet velocity as a nonlinear functional of the driving force. This allows us to explicitly illustrate the functional Taylor expansion formalism recently proposed for this kind of systems. The Taylor expansion reveals particularly useful to obtain the shape of the current when the force is periodic, piecewise constant. We also illustrate the somewhat counterintuitive effect that introducing damping may induce a ratchet effect. When the force is symmetric under time-reversal and the system is undamped, under symmetry principles no ratchet effect is possible. In this situation increasing damping generates a ratchet current which, upon increasing the damping coefficient eventually reaches a maximum and decreases toward zero. We argue that this effect is not specific of this example and should appear in any ratchet system with tunable damping driven by a time-reversible external force.

Noise and Synchronization of a Single Active Colloid

A two-state oscillator in a viscous liquid is composed of a micron-scale particle whose intrinsic dynamics is defined by linear potentials that undergo configuration-coupled transitions and is externally driven by a piecewise constant periodic force of varying amplitude and frequency. This elementary example of "active matter" has the minimal elements that allow us to study synchronization in the presence of thermal fluctuations. Experiments reveal the presence of synchronized states (and Arnol'd tongues), which we explain using analytical and numerical calculations. The system maintains synchronization by adjusting the phase between the bead and the clock. We discuss the relevance of this model to synchronization in real-world systems, including the role of thermal noise.

Anomalous biased diffusion in a randomly layered medium

We present analytical results for the biased diffusion of particles moving under a constant force in a randomly layered medium. The influence of this medium on the particle dynamics is modeled by a piecewise constant random force. The long-time behavior of the particle position is studied in the frame of a continuous-time random walk on a semi-infinite one-dimensional lattice. We formulate the conditions for anomalous diffusion, derive the diffusion laws, and analyze their dependence on the particle mass and the distribution of the random force.

Nonlinear oscillator with discontinuity by generalized harmonic balance method

A generalized harmonic balance method is used to calculate the periodic solutions of a nonlinear oscillator with discontinuities for which the elastic force term is proportional to sgn ( x ) . This method is a modification of the generalized harmonic balance method in which analytical approximate solutions have rational form. This approach gives us not only a truly periodic solution but also the frequency of the motion as a function of the amplitude of oscillation. We find that this method works very well for the whole range of amplitude of oscillation in the case of the antisymmetric, piecewise constant force oscillator and excellent agreement of the approximate frequencies with the exact one has been demonstrated and discussed. For the second-order approximation we have shown that the relative error in the analytical approximate frequency is 0.24%. We also compared the Fourier series expansions of the analytical approximate solution and the exact one. Comparison of the result obtained using this method with the exact ones reveals that this modified method is very effective and convenient.

A numerical method for solving the curved rod model

AbstractIn this work a numerical procedure for the solution of the one‐dimensional equilibrium model of linear elastic curved rods is derived. The middle curve of the curved rod is approximated by a piecewise linear curve while applied loads are approximated by a piecewise constant function. The curved rod model for the piecewise linear curve and piecewise constant force is solved analytically producing a numerical procedure. The difference in sup–norm and H1–norm of the solutions of the boundary value problem of curved rods for the smooth and piecewise linear curve is estimated giving the error estimate of the procedure. Results of the numerical tests agree with the analytical error estimate.

Modal Analysis to Accommodate Slap in Linear Structures

The generalized momentum balance (GMB) methods, explored chiefly by Shabana and his co-workers, treat slap or collision in linear structures as sequences of impulses, thereby maintaining the linearity of the structures throughout. Further, such linear analysis is facilitated by modal representation of the structures. These methods are discussed here and extended. Simulations on a simple two-rod problem demonstrate how this modal impulse approximation affects the system both directly after each impulse as well as over the entire collision. Furthermore, these simulations illustrate how the GMB results differ from the exact solution and how mitigation of these artifacts is achieved. Another modal method discussed in this paper is the idea of imposing piecewise constant forces over short, yet finite, time intervals during contact. The derivation of this method is substantially different than that of the GMB method, yet the numerical results show similar behavior, adding credence to both models. Finally, a novel method combining these two approaches is introduced. The new method produces physically reasonable results that are numerically very close to the exact solution of the collision of two rods. This approach avoids most of the nonphysical, numerical artifacts of interpenetration or chatter present in the first two methods.

Analytical and numerical approaches to characteristics of linear and nonlinear vibratory systems under piecewise discontinuous disturbances

The characteristics of the dynamic systems exerted by discontinuous piecewise constant forces are investigated. Compared to the systems under continuous exertions, diverse behaviors of the systems with discontinuous exertions are found. Responses of the systems to various piecewise constant exertions under different initial conditions are analyzed theoretically and numerically. Conditions of oscillation and stability are also presented for such piecewise systems.

Mean first-passage time for an overdamped particle in a disordered force field

We derive a rigorous expression for the mean first-passage time of an overdamped particle subject to a constant bias in a force field with quenched disorder. Depending on the statistics of the disorder, the disorder-averaged mean first-passage time can undergo a transition from an infinite value for small bias to a finite value for large bias. This corresponds to a depinning transition of the particle. We obtain exact values for the depinning threshold for Gaussian disorder and also for a class of piecewise constant random forces, which we call generalized kangaroo disorder. For Gaussian disorder, we investigate how the correlations of the random force field affect the average motion of the particle. For kangaroo disorder, we apply the general results for the depinning transition to two specific examples, viz., dichotomous disorder and random fractal disorder.

Deformation of sandwich panels cylindrically bent by point forces. 2. Development of procedure

Our proposed method [1] is extended for the cylindrical bending of an asymmetrical composite panel upon piecewise constant loading and point forces, with and without allowance for transversal stiffness (perfect compliance) in shear or tension/compression along the normal. A set of fundamental functions was obtained for all the cases examined. The properties of functions were studied taking account of the discontinuous nature of the surface loading. The set of functions was normalized for initial values of the variable coordinate. Integral relationships required for analysis were derived and an identical expression of the unit function was represented in terms of the fundamental function set. The boundary problem of a panel supported along the surface of its lower face layer with free ends is reduced to the Cauchy problem. The solution is greatly simplified for a panel symmetrical relative to its mean plane. Asymptotic formulas were obtained for the case of infinite panel length. Relationships are give for the stresses and layer deflections, which permit consideration of all the features of the stress state in addition to simplified calculations for actual panel design.

On Oscillatory Motion of Spring-Mass Systems Subjected to Piecewise Constant Forces

Oscillatory motion of a spring-mass system subjected to a piecewise constant force of the form f( x([ t])) or f([ t]) is studied. Solutions of a damped spring-mass system subjected to linear, sinusoidal or non-linear piecewise constant forces are derived. Results are graphically presented and compared with those of the corresponding systems subjected to continuous forces. Oscillatory properties of an undamped spring-mass system subjected to a piecewise constant force are studied in detail. When in a non-linear system the non-linear function is treated as a piecewise constant force in a small enough time interval, the resulting response agrees with the solution of the original non-linear equation. To this effect, examples of linear, non-linear and chaotic systems have been treated in the text.