Periodic Motion Research Articles

About tautochronic movements

It is shown that a material point, under the influence of an attractive linear force and a repulsive force inversely proportional to the cube of the distance from the center of attraction, performs a periodic motion, the period of which does not depend on the initial data (tautochronic motion). The problem is reduced to a nonlinear autonomous second-order equation, the general solution of which is expressed in terms of elementary functions. It has also been proven that for other power laws of repulsive force, except for degrees 0, 1 and –3, the movement of a material point is not tautochronous.

Nonlinear Dynamic Analysis of Riemann–Liouville Fractional-Order Damping Giant Magnetostrictive Actuator

Abstract In view of the large errors in the integer-order prediction model of the current giant magnetostrictive actuator (GMA), existing studies have shown that the fractional-order theory can improve the classical integer-order error situation. To this end, the Riemann–Liouville (R–L) fractional-order calculus theory is applied to the damping part of the GMA system; based on the averaging method and the power series method, the analytical and numerical solutions of the system are obtained, respectively, the motion of the GMA system is obtained through simulation, the parameters affecting the main resonance response of the system are analyzed as well as the motion characteristics of the system under the parameters, and the bifurcation and chaotic characteristics of the system are analyzed qualitatively and quantitatively. It is shown that the fractional-order model can improve the prediction accuracy of the system, the fractional order has a significant effect on the motion of the system, and the interval of the periodical motion parameter is less than an integer when the order of the damping term is (0,1), and the system can be induced to shift to periodic motion by changing the parameters.

Simultaneous multi-slice cardiac real-time MRI at 0.55T.

To determine the feasibility of simultaneous multi-slice (SMS) real-time MRI (RT-MRI) at 0.55T for the evaluation of cardiac function. Cardiac CINE MRI is routinely used to evaluate left-ventricular (LV) function. The standard is sequential multi-slice balanced SSFP (bSSFP) over a stack of short-axis slices using electrocardiogram (ECG) gating and breath-holds. SMS has been used in CINE imaging to reduce the number of breath-holds by a factor of 2-4 at 1.5T, 3T, and recently at 0.55T. This work aims to determine if SMS is similarly effective in the RT-MRI evaluation of cardiac function. We used an SMS bSSFP pulse sequence with golden-angle spirals at 0.55T with an SMS factor of three. We cover the LV with three acquisitions for SMS, and nine for single-band (SB). Imaging was performed on 9 healthy volunteers and 1 patient with myocardial fibrosis and sternal wires. A spatio-temporal constrained reconstruction is used, with regularization parameters selected by a board-certified cardiologist. Images were quantitatively analyzed with a normalized contrast and an Edge Sharpness (ES) score. There was a statistically significant 2-fold difference in contrast between SMS and SB and no significant difference in ES score. The contrast for SMS and SB were 13.38/29.05 at mid-diastole and 10.79/22.26 at end-systole; the ES scores for SMS and SB were 1.77/1.83 at mid-diastole and 1.50/1.72 at end-systole. SMS cardiac RT-MRI at 0.55T is feasible and provides sufficient blood-myocardium contrast to evaluate LV function in three slices simultaneously without any gating or periodic motion assumptions.

Quantifying the Tip Leakage Vortex Wandering in a Low-Speed Axial Flow Fan via URANS Simulation

The tip leakage vortex is a dominant noise and aerodynamic loss source of low-speed axial flow fans. The tip leakage vortex often has an unsteady behavior, like the periodic motion of the vortex core: the so-called vortex wandering. This periodic vortex wandering results in additional noise, aerodynamic loss, and an oscillation in the moment acting on the blading, which causes an unfavorable vibration of the rotor. In the research campaigns focusing on vortex wandering, complicated measurement techniques (PIV) or high-fidelity but computationally costly simulations are commonly used. The present paper aims to introduce a relatively cost-effective unsteady Reynolds-averaged Navier-Stokes simulation-based investigation method of vortex wandering. The capabilities of the proposed technique are presented through a low-speed axial flow fan case study.

Two-parameter dynamics and multistability of a non-smooth railway wheelset system with dry friction damping.

A deep understanding of non-smooth dynamics of vehicle systems, particularly with dry friction damping offer valuable insights into the design and optimization of railway vehicle systems, ultimately enhancing the safety and reliability of railway operations. In this paper, the two-parameter dynamics of a non-smooth railway wheelset system incorporating dry friction damping are investigated. The effect of the crucial parameters on the complexity of the evolution process is comprehensively exposed by identifying different dynamic responses in the two-parameter plane. In addition, the multistability and the various routes transition to chaos for the system are also discussed. It is found that dry friction induces highly complex dynamics in the system, encompassing a range of behaviors such as periodic, quasi-periodic, and chaotic motions. These intricate dynamics are a direct result of the interplay between multiple parameters, such as speed and damping coefficients, which are critical in determining the system's stability and performance. The presence of multistability further complicates the system, resulting in unpredictable transitions between different motion states.

Orbits of a system of three point vortices and the associated chaotic mixing

We study the general periodic motion of a set of three point vortices in the plane, as well as the potentially chaotic motion of one or more tracer particles. While the motion of three vortices is simple in that it can only be periodic, the actual orbits can be surprisingly complex and varied. This rich behavior arises from the existence of both co-linear and equilateral relative equilibria (steady motion in a rotating frame of reference). Here, we start from a general (unsteady) co-linear array with arbitrary vortex circulations. The subsequent motion may take the vortices close to a distinct co-linear relative equilibrium or to an equilateral one. Both equilibrium states are necessarily unstable, as we demonstrate by a linear stability analysis. We go on to study mixing by examining Poincaré sections and finite-time Lyapunov exponents. Both indicate widespread chaotic motion in general, implying that the motion of three vortices efficiently mixes the nearby surrounding fluid outside of small regions surrounding each vortex.

Human short-latency reflexes show precise short-term gain adaptation after prior motion.

The central nervous system adapts the gain of short-latency reflex loops to changing conditions. Experiments on biomimetic robots showed that reflex modulation could substantially increase energy efficiency and stability of periodic motions if, unlike known mechanisms, the reflex modulation both acted precisely on the muscles involved and lasted after the motion. This study tests the presence of such a mechanism by having participants repeatedly rotate either their right elbow or shoulder joint, before perturbing either joint. The results demonstrate a mechanism that modulates short-latency reflex gains after prior motion with joint-specific precision. Enhanced gains were observed hundreds of milliseconds after movement cessation, a time scale well-suited to quickly adapt overall periodic motion cycles. A serotonin antagonist significantly decreased these post-movement gains diffusely across joints. But blocking serotonin did not affect the joint-specificity of the gain scaling more than a placebo, suggesting that serotonin sets the overall reflex gain across joints after movement by an effect that is modulated in a joint-specific manner by an unidentified neural circuit. These results confirm the existence of a new, joint-specific, fast, persistent adaptation of short-latency reflex loops induced by motion in human arms.

Chaos and attraction domain of fractional Φ6‐van der Pol with time delay velocity

This article investigates the chaotic analysis and attractive domain of a fractional‐order ‐van der Pol with time delay velocity under harmonic excitation. Firstly, eight different types of bifurcation states of the system under different parameters are calculated by using the undisturbed system. Secondly, the Melnikov method is used to explore the effect of time delay velocity on the threshold of chaos in the Smale horseshoe sense under the double‐well potential and three‐well potential of the system. Finally, through numerical analysis of the phase diagram, bifurcation diagram, and maximum Lyapunov exponent, the influence of time delay velocity on system chaos is studied. The results indicate that an increase in the delay velocity coefficient will lead to the system transitioning from a chaotic state to a periodic state, while an increase in the delay velocity term will lead to the system transitioning from a periodic state to a chaotic state. In the study of system bifurcation, it is found that the position of the equilibrium points of the system changes during periodic motion. Therefore, cell mapping is used to draw the attractive domain of the system is studying the influence of initial conditions on the equilibrium point of the system and the results show that there is a close relationship between the attraction domain and the process of chaos occurrence.

Global reorientation of a free-fall multibody system using periodical joint motions – theory and motion planning

Global reorientation of a free-fall multibody system using periodical joint motions – theory and motion planning

Utilization of the Resonance Behavior of a Tendon-Driven Continuum Joint for Periodic Natural Motions in Soft Robotics

Continuum joints use structural elastic deformations to enable joint motion, and their intrinsic compliance and inherent mechanical robustness are envisioned for applications in which the robot, the human, and the environment need to be safe during interaction. In particular, the intrinsic compliance makes continuum joints a competitor to soft articulated joints, which require additional integrated spring elements. For soft articulated joints incorporating rigid and soft parts, natural motions have been investigated in robotics research to exploit this energy-efficient motion property for cyclic motions, e.g., locomotion. To the best of the author’s knowledge, there is no robotic system to date that utilizes the natural motion of a continuum joint under periodic excitation. In this paper, the resonant behavior of a tendon-driven continuum joint under periodic excitation of the torsional axis is experimentally investigated in a functional sense. In the experiments, periodic inputs are introduced on the joint side of a tendon driven continuum joint with four tendons. By modulating the pretension of the tendons, both the resonant frequency and the gain can be shifted, from 3 to 4.3 Hz and 2.8 to 1.4, respectively, in the present experimental setup. An application would be the rotation of a humanoid torso, where gait frequencies are synchronized with the resonant frequency of the continuum joint.

Influence of the rigidity of the backbone and arms on the dynamical and conformational properties of the comb polymer in shear flow.

The dynamical and conformational properties of the comb polymer with various rigidities of the backbone and arms in steady shear flow are studied by using a hybrid mesoscale simulation approach that combines multiparticle collision dynamics with standard molecular dynamics. First, during the process of the comb polymer undergoing periodic tumbling motion, we find that the rigidity of the arms always promotes the tumbling motion of the comb polymer, but the rigidity of the backbone shifts from hindering to promoting it with increasing the rigidity of the arms. In addition, the comb polymer transitions from vorticity tumbling to gradient tumbling with the increase in shear rate. Second, the range of variation of the end-to-end distance of the backbone and the average end-to-end distance of the arms increases with the increase in the rigidity of the arms and backbone, respectively, and the range of both changes grows with the increase in shear rate. Furthermore, as the rigidity increases, the moldability of the comb polymer decreases and the orientation angle of the comb polymer increases.

Dynamical Casimir effect with screened scalar fields

Understanding the nature of dark energy and dark matter is one of modern physics' greatest open problems. Scalar-tensor theories with screened scalar fields like the chameleon model are among the most popular proposed solutions. In this article, we present the first analysis of the impact of a chameleon field on the dynamical Casimir effect, whose main feature is the particle production associated with a resonant condition of boundary periodic motion in cavities. For this, we employ a recently developed method to compute the evolution of confined quantum scalar fields in a globally hyperbolic spacetime by means of time-dependent Bogoliubov transformations. As a result, we show that particle production is reduced due to the presence of the chameleon field. In addition, our results for the Bogoliubov coefficients and the mean number of created particles agree with known results in the absence of a chameleon field. Our results initiate the discussion of the evolution of quantum fields on screened scalar field backgrounds.

Simulation method for dynamic characteristics of rotor-support system subjected to environmental loads

An improved finite element method was proposed to investigate the dynamic characteristics of rotor system with environmental loads (temperature and aerodynamic forces) in aeroengines. According to the axial temperature distribution within shaft elements, the polynomials were established for physical properties of material. The derivation for coefficient matrices under thermal loads, and the tensile stiffness matrix considering axial aerodynamic forces were completed, based on the Timoshenko beam theory. A high-pressure rotor was simulated with operating temperature 200∼600∘C, and the magnitude of aerodynamic force is 105N. Furthermore, a squeezed film damper (SFD) was adopted to this rotor-support system, its steady responses were solved by the equivalent coefficient method, and the Newmark-HHT method was utilized to solve the transient responses. The results indicate that the first two critical speeds are reduced by 1.85 % and 2.07 %, which is mainly caused by thermal loads. The rising temperature leads to lower elastic modulus, higher Poisson's ratio and expansion ratio. With the amount of a mass unbalance being 500 g·mm located at the 1st-stage compressor disk (CD1): the 1st resonance peaks at CD1 and turbine disk (TD) increase by 4.56 % and 4.98 %, respectively, and the 2nd resonance peaks at CD1 and TD rise 5.88 % and 5.00 %, separately. Owing to SFD damping, the first two resonance peaks at CD1 and TD decrease by 69 % and 51 %, respectively. Compared to 40 °C, the first two resonance peaks at CD1 increase by 88 % and 55 % at the oil temperature 100 °C. The oil-film damping deteriorates significantly due to the lower oil viscosity in higher temperature. When operating near the first-order critical speed, the transient responses of rotor perform as simple harmonics, circular precession trajectory, dominant rotational frequency, and periodic motions. Overall, the dynamic characteristics of rotor-SFD-support systems exposed to environmental loads can be predicted by the proposed simulation method.

Comparison of natural convection in liquid gallium under horizontal and vertical magnetic fields

The dynamics of two-dimensional natural convection in liquid metal under the influence of magnetic fields within a confined square enclosure are numerically investigated using a high-accuracy method. The study focuses on the flow transition process influenced by both horizontal and vertical magnetic fields, examining the impacts of temperature difference, magnetic field strength, and orientation on flow characteristics and heat transfer. The range of Rayleigh numbers (Ra) explored is from 104 to 106, while Hartmann numbers (Ha) vary from 0 to 100. Liquid gallium is used as the working fluid with a Prandtl number (Pr) of 0.025. Our investigation reveals rich variations in flow patterns, categorized into unsteady periodic, steady “two-in-one” vortex, and steady single vortex states. As Rayleigh numbers increase, a complex series of flow transitions is observed, leading to enhanced flow dynamics. The transition from a steady state to periodic motion is accompanied by a notable increase in heat transfer efficiency. However, increasing the strength of the magnetic field suppresses the flow. A sufficiently strong suppression transitions the flow from periodic motion back to a steady state, resulting in a gradual decrease in heat transfer efficiency. For the parameters considered, the suppression efficiency of horizontal magnetic fields is stronger than that of vertical ones, especially when Ra>6×105 and Ha=100, with a 14% greater suppression efficiency. A scaling law for heat transfer is also identified: Nu:Ra/Ha2γ, where 14≤γ≤12, independent of the magnetic field orientation.

Prediction of energy harvesting efficiency through a wake–foil interaction model for oscillating foil arrays

This research investigates the wake–foil interactions between two oscillating foils in a tandem configuration undergoing energy harvesting kinematics. Oscillating foils have been shown to extract hydrokinetic energy from free-stream flows through a combination of periodic heave and pitch motions, at relatively higher amplitudes and lower reduced frequency than thrust generating foils. When placed in tandem, the wake–foil interactions can govern the energy harvesting efficiency of the system due to a reduced relative flow velocity in combination with a structured and coherent wake of vortices shed from the high amplitude flapping of upstream foils. This work utilizes simulations of two tandem foils to parameterize and model the energy harvesting performance as a function of array configuration and foil kinematics. Once the wake of the leading foil has been fully parameterized, the placement, phase angle and kinematic stroke of the second foil is utilized to estimate the time-dependent power curve. The algorithm predicts the power of the second foil through the mean and unsteady wake characteristics, including the direct impingement of a vortex with the trailing foil.

On plunging using double-peak kinematics

This study investigates the thrust generation mechanism and performance of a plunging airfoil undergoing double-peak periodic motion, specifically utilizing superposition of the regular sinusoidal waveform with another less-weighted faster one. It is revealed that the proposed waveform produces 200% more thrust when compared to sinusoidal plunging with a small efficiency difference. Also, it produces two vortex pairs (2P) per cycle all over the plunging frequency-amplitude space, instead of the reverse von Karman vortices. Using this proposed waveform, the 2P wake is generated through interaction of three types of vortices shed during plunging, opposing the regular 2P pattern found in the literature.

Optical penetration depth and periodic motion of a photomechanical strip

Liquid crystal elastomers (LCEs) containing light-sensitive molecules exhibit large, reversible deformations when subjected to illumination. Here, we investigate the role of optical penetration depth on this photomechanical response. We present a model of the photomechanical behavior of photoactive LCE strips under illumination that goes beyond the common assumption of shallow penetration. This model reveals how the optical penetration depth and the consequent photomechanically induced deformation can depend on the concentration of photoactive molecules, their absorption cross-sections, and the intensity of illumination. Through a series of examples, we show that the penetration depth can quantitatively and qualitatively affect the photomechanical response of a strip. Shallow illumination leads to monotone curvature change while deep penetration can lead to non-monotone response with illumination duration. Further, the flapping behavior (a cyclic wave-like motion) of doubly clamped and buckled strips that has been proposed for locomotion can reverse direction with sufficiently large penetration depth. This opens the possibility of creating wireless light-driven photomechanical actuators and swimmers whose direction of motion can be controlled by light intensity and frequency.

Theoretical Analysis of Viscoelastic Friction System Characteristics of Robotic Arm Brake Based on Fractional Differential Theory

In engineering practice, the nonlinear vibration effect can easily lead to chaos in the system, which will not only reduce the performance of the system but also lead to premature fatigue of components, control failure, and increased safety risks. In view of the core position of the robotic arm in modern industry, this study relies on the robotic arm brake system to explore the theoretical basis of integrated viscoelastic materials as a vibration isolation layer. By analyzing the dynamic characteristics of the friction braking system with fractional differential terms, it aims to provide a new perspective for understanding and controlling the chaotic phenomena of a class of nonlinear friction systems. Firstly, we construct a model of a friction system and analyze its dynamic characteristics in detail. The self-excited vibration of the system under disturbance is studied. The relationship between amplitude and frequency is calculated by a nonlinear approximate analytical algorithm, and the accuracy of this relationship is verified by a numerical algorithm. Then, we compare the differences between non-fractional systems and fractional systems. It is found that with the increase in the fractional order term, the vibration amplitude of the system decreases significantly, which helps to reduce the nonlinear characteristics generated by the friction system and narrow the range of unstable solutions. Secondly, we also study the influence of parameter coefficients on the amplitude–frequency characteristics and analyze the local static bifurcation characteristics through singularity theory. Finally, we study the dynamic bifurcation behavior under different parameter perturbations and find that the change in system parameters will lead to the alternation of periodic motion and chaotic motion.

Tailoring of Self-Healable Polydimethylsiloxane Films for Mechanical Energy Harvesting.

Triboelectric nanogenerators (TENGs) have emerged as potential energy sources, as they are capable of harvesting energy from low-frequency mechanical actions such as biological movements, moving parts of machines, mild wind, rain droplets, and others. However, periodic mechanical motion can have a detrimental effect on the triboelectric materials that constitute a TENG device. This study introduces a self-healable triboelectric layer consisting of an Ecoflex-coated self-healable polydimethylsiloxane (SH-PDMS) polymer that can autonomously repair mechanical injury at room temperature and regain its functionality. Different compositions of bis(3-aminopropyl)-terminated PDMS and 1,3,5-triformylbenzene were used to synthesize SH-PDMS films to determine the optimum healing time. The SH-PDMS films contain reversible imine bonds that break when the material is damaged and are subsequently restored by an autonomous healing process. However, the inherent stickiness of the SH-PDMS surface itself renders the material unsuitable for application in TENGs despite its attractive self-healing capability. We show that spin-coating a thin layer (≈32 μm) of Ecoflex on top of the SH-PDMS eliminates the stickiness issue while retaining the functionality of a triboelectric material. TENGs based on Ecoflex/SH-PDMS and nylon 6 films show excellent output and fatigue performance. Even after incisions were introduced at several locations in the Ecoflex/SH-PDMS film, the TENG spontaneously attained its original output performance after a period of 24 h of healing. This study presents a viable approach to enhancing the longevity of TENGs to harvest energy from continuous mechanical actions, paving the way for durable, self-healable mechanical energy harvesters.

Diversity and transition of periodic motion of a periodically excited soft-impacting machinery

Dynamics of a periodically excited vibro-impact system with soft impacts is investigated. Essential features of period-one multi-impact motion group and correlated transition characteristics in low-frequency range are discussed in detail by the way of two-parameter bifurcation space providing qualitative domains for different periodic motions. The main focus is given to the effect of sensitive parameters including constraint stiffness k 0, clearance threshold b, and damping parameter ζ on the system response. The low-frequency characteristics in the finite-dimensional parameter space are particularly explored. It is found that the increase of k 0 induces multi-type bifurcation of period-one double-impact symmetrical motion, which induces a rich variety of periodic motions, and period-one multi-impact motion group orbit primarily exist in the small-clearance b and low-frequency ω zone. Based on the evolution irreversibility of adjacent period-one multi-impact orbit, the mechanism of singularies appearing in pairs and two different transition zones (hysteresis and liguliform zones) is studied, the result of which provides a theoretical reference value for the common low-frequency vibration instability phenomenon in the field of mechanical engineering. For small-damping coefficient ζ, period-one multi-impact motion has a large quantity, and the main bridge for the transition of adjacent period-one multi-impact motion is liguliform zone, which embraces period-one multi-impact asymmetrical motion and period-n multi-impact subharmonic motion and a certain chaotic zone. For large-damping coefficient ζ, the amount of period-one multi-impact motion group is reduced, and the main bridge for the transition of adjacent period-one multi-impact motion is hysteresis zone, where adjacent period-one multi-impact orbits can coexist according to initial conditions. As designing and renovating impact mechanical equipment, the reasonable matching law of dynamic parameters can be determined through two-parameter bifurcation space, which is conducive to making the system work in stable periodic motion and obtaining larger instantaneous impact velocity.