Periodic Bouncing Research Articles

Modeling of Human-Induced Dynamic Bouncing Force Using a Self-Sustained Nonlinear Oscillator

This paper proposed a single-degree-of-freedom self-sustained nonlinear oscillator capable of precisely predicting the bouncing force induced by a person during bouncing activity on a flat and rigid surface. A bouncing person produces essential internal energy required to maintain its motion, so it can be modeled as a self-sustained oscillator that can generate (i) the stable limit cycle, (ii) the periodic bouncing force signal, and (iii) the self-sustained motion. A hybrid Van der Pol–Rayleigh oscillator added with two quadratic and one cubic nonlinear terms has been derived to yield desired softening and hardening effects as well as even and odd harmonics, as observed from the analysis of experimental bouncing force data. The force applied on the surface corresponds to its restoring force. The stability analysis of the oscillator has been performed using the energy balance and perturbation methods. The model parameters are extracted from the experimental bouncing force data resulting from a bouncing test on a group of seven subjects with shoe insoles at six different frequencies guided by a metronome. The bootstrapping method has been performed to examine the convergence of mean values of each model parameter by increasing the cardinality of the experimental set. The bouncing force signals produced by the proposed model and experimental results demonstrate an excellent agreement.

Periodic bouncing solutions of the Lazer–Solimini equation with weak repulsive singularity

We prove the existence and multiplicity of periodic solutions of bouncing type for a second-order differential equation with a weak repulsive singularity. Such solutions can be cataloged according to the minimal period and the number of elastic collisions with the singularity in each period. The proof relies on the Poincaré–Birkhoff Theorem.

Periodic bouncing of a plasmonic bubble in a binary liquid by competing solutal and thermal Marangoni forces

The physicochemical hydrodynamics of bubbles and droplets out of equilibrium, in particular with phase transitions, display surprisingly rich and often counterintuitive phenomena. Here we experimentally and theoretically study the nucleation and early evolution of plasmonic bubbles in a binary liquid consisting of water and ethanol. Remarkably, the submillimeter plasmonic bubble is found to be periodically attracted to and repelled from the nanoparticle-decorated substrate, with frequencies of around a few kilohertz. We identify the competition between solutal and thermal Marangoni forces as the origin of the periodic bouncing. The former arises due to the selective vaporization of ethanol at the substrate's side of the bubble, leading to a solutal Marangoni flow toward the hot substrate, which pushes the bubble away. The latter arises due to the temperature gradient across the bubble, leading to a thermal Marangoni flow away from the substrate, which sucks the bubble toward it. We study the dependence of the frequency of the bouncing phenomenon from the control parameters of the system, namely the ethanol fraction and the laser power for the plasmonic heating. Our findings can be generalized to boiling and electrolytically or catalytically generated bubbles in multicomponent liquids.

Periodic bouncing solutions for sublinear impact oscillator

The existence of periodic bouncing solutions for sublinear impact oscillator is proved by using Poincare-Birkhoff twist theorem. The approach of this paper is based on a well defined successor map and the phase-plane analysis of the spiral properties.

Periodic bouncing solutions for Hill's type sub-linear oscillators with obstacles

<p style='text-indent:20px;'>In this paper, we prove the existence and multiplicity of subharmonic bouncing motions for a Hill's type sublinear oscillator with an obstacle. Furthermore, we also consider the existence, multiplicity and dense distribution of symmetric periodic bouncing solutions when the weight function is even. Based on an appropriate coordinate transformation and the method of phase-plane analysis, we can study our main results via Poincar<inline-formula><tex-math id="M1">\begin{document}$ \acute{e} $\end{document}</tex-math></inline-formula> map by applying some suitable fixed point theorems.

Size-dependent spontaneous oscillations of Leidenfrost droplets

Abstract

Periodic bouncing modes for two uniformly magnetized spheres. II. Scaling.

A uniformly magnetized sphere moves without friction in a plane in response to the field of a second, identical, fixed sphere and makes elastic hard-sphere collisions with this sphere. Numerical simulations of the threshold energies and periods of periodic finite-amplitude nonlinear bouncing modes agree with small-amplitude closed-form mathematical results, which are used to identify scaling parameters that govern the entire amplitude range, including power-law scaling at large amplitudes. Scaling parameters are combinations of the bouncing number, the rocking number, the phase, and numerical factors. Discontinuities in the scaling functions are found when viewing the threshold energy and period as separate functions of the scaling parameters, for which large-amplitude scaling exponents are obtained from fits to the data. These discontinuities disappear when the threshold energy is viewed as a function of the threshold period, for which the large-amplitude scaling exponent is obtained analytically and for which scaling applies to both in-phase and out-of-phase modes.

Periodic bouncing modes for two uniformly magnetized spheres. I. Trajectories.

We consider a uniformly magnetized sphere that moves without friction in a plane in response to the field of a second, identical, fixed sphere, making elastic hard-sphere collisions with this sphere. We seek periodic solutions to the associated nonlinear equations of motion. We find closed-form mathematical solutions for small-amplitude modes and use these to characterize and validate our large-amplitude modes, which we find numerically. Our Runge-Kutta integration approach allows us to find 1243 distinct periodic modes with the free sphere located initially at its stable equilibrium position. Each of these modes bifurcates from the finite-amplitude radial bouncing mode with infinitesimal-amplitude angular motion and supports a family of states with increasing amounts of angular motion. These states offer a rich variety of behaviors and beautiful, symmetric trajectories, including states with up to 157 collisions and 580 angular oscillations per period.

Multiplicity of periodic bouncing solutions for generalized impact Hamiltonian systems

Applying the Generalized Nonsmooth Saddle Point Theorem, we obtain multiple nontrivial periodic bouncing solutions for systems ddot{x}=f(t,x) with new conditions. In particular, we generalize the collision axis from x=0 to the axis x=a, where a is an arbitrary constant.

Bouncing ball on a vibrating periodic surface.

We present an investigation of a partially elastic ball bouncing on a vertically vibrated sinusoidal surface. Following the work of McBennett and Harris [Chaos 26, 093105 (2016)], we begin by demonstrating that simple periodic vertical bouncing at a local minimum of the surface becomes unstable when the local curvature exceeds a critical value. The resulting instability gives rise to a period doubling cascade and results in persistent horizontal motion of the ball. Following this transition to horizontal motion, periodic "walking" states-where the ball bounces one wavelength over each vibration cycle-are possible and manifest for a range of parameters. Furthermore, we show that net horizontal motion in a preferred direction can be induced by breaking the left-right symmetry of the periodic topography.

Hořava-Lifshitz bouncing Bianchi IX universes: A dynamical system analysis

We examine the Hamiltonian dynamics of bouncing Bianchi IX cosmologies in Ho\v{r}ava-Lifshitz gravity. The $6$-dim phase space presents two critical points, one asymptotic de Sitter attractor at infinity and a $2$-dim invariant plane. We identified four distinct parameter domains $A$, $B$, $C$ and $D$ for which the pair of critical points engenders distinct features in the dynamics. In the domain $A$ the dynamics consists basically of periodic bouncing orbits, or oscillatory orbits with a finite number of bounces. The center with multiplicity two engenders in its neighborhood the topology of stable and unstable cylinders $R \times S^3$ of orbits. We show that the stable and unstable cylinders coalesce realizing a smooth homoclinic connection to the center manifold, a rare event of regular/non-chaotic dynamics in bouncing Bianchi IX cosmologies. The presence of a saddle of multiplicity two in the domain $B$ engenders a high instability in the dynamics so that the cylinders emerging from the center manifold towards the bounce have four distinct attractors: the center manifold itself, the de Sitter attractor at infinity and two further momentum-dominated attractors with infinite anisotropy. In the domain $C$ we examine the features of invariant manifolds of orbits about a saddle of multiplicity two. The presence of the saddle of multiplicity two engenders bifurcations of the invariant manifold as the energy $E_0$ of the system increases relative to the energy $E_{cr}$ providing structures that were not yet observed in the literature. The domain $D$ is not examined as most of its features are present already in the previous domains.

Generalized Nonsmooth Saddle Point Theorem and its applications on second order Hamiltonian systems

The Generalized Nonsmooth Saddle Point Theorem is proved, which generalizes the previous ones. As its application, we obtain the existence of nontrivial periodic bouncing solution for systems x¨=f(t,x) with new sublinear conditions, which has physical background.

Horizontal stability of a bouncing ball.

We present an investigation of a partially elastic ball bouncing on a vertically vibrated concave parabolic surface in two dimensions. In particular, we demonstrate that simple vertical motion, wherein the ball bounces periodically at the parabola's vertex, is unstable to horizontal perturbations when the parabolic coefficient defining the surface shape exceeds a critical value. The result is a new periodic solution where the ball bounces laterally over the vertex. As the parabola is further steepened, this new solution also becomes unstable which gives rise to other complex periodic and chaotic bouncing states, all characterized by persistent lateral motion.

Existence of periodic solutions of sublinear Hamiltonian systems

The variational method is used to obtain some existence theorems of periodic solutions of sublinear systems with or not with impacts under suitable growth conditions. Compared with normal systems, impact systems need additional conditions to ensure the existence of periodic bouncing solutions.

微通道内气泡扰流强化换热研究

为有效解决电子器件微集成化带来的高热流散热问题，本文提出用气泡扰动流思想来强化换热。具体做法是在带周期性锯齿结构的微通道内，以水与空气泡两相流为流体介质，惯性受限流动过锯齿微结构表面，产生周期性气泡弹跳运动扰动近壁区热边界层，从而实现强化换热的目的。本文将系统研究锯齿结构倾角、发泡频率以及主流流速对气泡弹跳运动与换热的影响，以揭示锯齿微结构与气泡弹跳运动的耦合流动换热特性，为多相流流固耦合高效换热研究提供新视角。 In order to effectively solve the problem of high-heat-flux heat transfer caused by the micro in-tegration of electronic devices, in this paper, we propose to strengthen the heat transfer by using the idea of bubble flow. Specific approach is to use water and air bubble as fluid medium in the microchannel with periodic saw tooth structure, inertial confined flow over the zigzag micro structure surface, resulting in periodic bubble bouncing motion disturbance near the wall thermal boundary layer, so as to achieve the purpose of strengthening heat transfer. In this paper, we will systematic study the dip angle of the zigzag structure, effects of foaming frequency and main flow velocity on the movement and heat transfer of air bubbles, and reveal the characteristics of coupled flow and heat transfer between zigzag micro structure and air bubble, providing a new perspective to study the multiphase fluid-solid coupling efficient heat transfer.

Influence of drilling mud rheology on the reduction of vertical vibrations in deep rotary drilling

Vertical vibrations in drilling (chatter) can be reduced by the viscous damping associated with the mud rheology. In previous studies, the effect of damping was modelled assuming that the drilling fluid is Newtonian. However, non-Newtonian shear-thinning muds are beneficial for the drilling process. In this study, we investigate the influence of the non-Newtonian mud rheology on the vertical drill bit vibrations using two different non-Newtonian rheology models: Herschel-Bulkley and Casson. In modelling the vibrations, we assume that the stiffness of the rock is much greater than the stiffness of the drill string, which corresponds to deep boreholes. This allows the use of the concept of impact oscillator, whereby the rock stiffness is assumed infinite, such that the vertical oscillations are associated with the drill bit detachment. The driving force for the oscillations is the periodic drill bit bouncing caused by impact of the roughness of the borehole floor. Parameters of the rheology models required for the efficient reduction of chatter have been determined. It has been established that with the muds used currently in the industry, a significant reduction of chatter cannot be achieved. Furthermore, the desirability of the shear-thinning behaviour in muds might limit the scope for a vibration-damping mud design.

Impact oscillators of Hill's type with indefinite weight: Periodic and chaotic dynamics

In this paper, we are concerned with superlinear impact oscillatorsof Hill's type with indefinite weight$$\left\{\begin{array}{lll} x''+f(x)x'+q(t)g(x)=0,~\text{for}~ x(t)>0;\\x(t)\geq0;\\x'(t_0+)=-x'(t_0-),~\text{if}~x(t_0)=0,\end{array}\right.$$ where the indefinite weight$q(t)$, defined in $(a,b)$ with $-\infty\leq a< b \leq+\infty,$has infinitely many zeros in $(a,b),$ $g$ is superlinear and $f$ isbounded. We prove the existence of globally defined bouncingsolutions with prescribed number of impacts in the intervalsof negativity and positivity of $q$. Furthermore, we showthat when $q$ is periodic, the equation under consideration exhibits an interesting phenomenonof chaotic-like dynamics. Finally, in case that $q$ is even and periodic,we prove the existence and multiplicity of the even and periodic bouncing solutions forthe Hill's type equation in case of $f\equiv0.$

Near-wall rising behaviour of a deformable bubble at high Reynolds number

The dynamics of a large deformable bubble ($\mathit{Re}\sim \mathit{O}(10^{3})$) rising near a vertical wall in quiescent water is experimentally investigated. The reference (without the wall) rising path of the considered bubble is a two-dimensional zigzag. For a range of wall configurations (i.e. initial wall distance and boundary condition), using high-speed shadowgraphy, various rising behaviours such as periodic bouncing, sliding, migrating away, and non-periodic oscillation without collisions are measured and analysed. Unlike low-Reynolds- and Weber-number bubbles, the contribution of the surface deformation to the transport between the energy components becomes significant during the bubble’s rise. In particular, across the bubble–wall collision, the excess surface energy compensates the deficit of kinetic energy. This enables a large deformable bubble to maintain a relatively constant bouncing kinematics, despite the obvious wall-induced energy dissipation. The wall effect, predominantly appearing as energy loss, is found to decrease as the initial distance from the bubble centre to the wall increases. Compared to the regular (no-slip) wall, a hydrophobic surface enhances or reduces the wall effect depending on the wall distance, whereas a porous surface reduces the energy loss due to the wall, regardless of the initial distance from the wall. Furthermore, the bubble–wall collision behaviour is assessed in terms of a restitution coefficient and modified impact Stokes number (ratio of the inertia to viscous forces), which shows a good correlation, the trends of which agree well with the variations in the energy components. The dependence of near-wall bubble motion on the wall distance and boundary condition may suggest a way of predicting or controlling the near-wall gas void-fraction distribution in gas–liquid flow systems.

Periodic Bouncing Solutions for Nonlinear Impact Oscillators

Abstract We prove the existence of a periodic solution to a nonlinear impact oscillator, whose restoring force has an asymptotically linear behavior. To this aim, after regularizing the problem, we use phase-plane analysis, and apply the Poincaré-Bohl fixed point Theorem to the associated Poincaré map, so to find a periodic solution of the regularized problem. Passing to the limit, we eventually find the “bouncing solution” we are looking for.

Discrete Electrostatic Charge Transfer by the Electrophoresis of a Charged Droplet in a Dielectric Liquid

We have experimentally investigated the electrostatic charging of a water droplet on an electrified electrode surface to explain the detailed inductive charging processes and use them for the detection of droplet position in a lab-on-a-chip system. The periodic bouncing motion of a droplet between two planar electrodes has been examined by using a high-resolution electrometer and an image analysis method. We have found that this charging process consists of three steps. The first step is inductive charge accumulation on the opposite electrode by the charge of a droplet. This induction process occurs while the droplet approaches the electrode, and it produces an induction current signal at the electrometer. The second step is the discharging of the droplet by the accumulated induced charge at the moment of contact. For this second step, there is no charge-transfer detection at the electrometer. The third step is the charging of the neutralized droplet to a certain charged state while the droplet is in contact with the electrode. The charge transfer of the third step is detected as the pulse-type signal of an electrometer. The second and third steps occur simultaneously and rapidly. We have found that the induction current by the movement of a charged droplet can be accurately used to measure the charge of the droplet and can also be used to monitor the position of a droplet under actuation. The implications of the current findings for understanding and measuring the charging process are discussed.