Periodic Motion Research Articles

On the effects of rotor induced vibrational stability on helicopter flight dynamics

This article investigates vibrational stabilization effects in rotorcraft flight dynamics. This study is motivated by recent results in flapping-wing flight, which showed that the time-varying aerodynamic and inertial loads due to the insect wing periodic motion induce a vibrational stabilization mechanism in hover. The dynamics of flapping-wing flyers and rotary-wing vehicles are both described by time-periodic systems so vibrational stabilization mechanisms can also have an effect on stability characteristics of rotary-wing vehicles. The article extends the use of the harmonic decomposition method to vibrational stability analysis of rotorcraft. Two cases are considered: vibrational stability due blade imbalance at hover, and vibrational stability due to number-of-blades-per-rotor-revolution (Nb\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$N_b$$\\end{document}/rev) in high-speed forward flight. Results show that while vibrations induced by rotor blade imbalance do not stabilize the hovering dynamics of a helicopter, these vibrations can still have a significant effect on the hovering dynamics. Rotor blade imbalance results in a symmetric effect on the roll and pitch axes, in that it tends to decrease the frequency of the subsidence modes of the hovering cubic, while the unstable oscillatory modes tend to increase in frequency and decrease in damping (destabilizing effect). On the other hand, the yaw/heave dynamics are relatively unaffected compared to the lateral and longitudinal axes. Moreover, Nb\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$N_b$$\\end{document}/rev rotor loads in forward flight are shown to reduce the damping of the coupled roll/pitch oscillation mode.

Research on a novel compliant mechanism applied in the vehicle side door latch and stability analysis via Lyapunov exponents

The functional reliability of the vehicle latch is very important and is determined by the internal mechanism’s motion stability. In this paper, a novel compliant mechanism constructed with a novel compliant joint is applied to the vehicle latch, and multiple mode motions of the mechanism realize the newly added electric power release function. The repeatability of mechanism trajectories and phase diagrams of several different periodic motions is creatively regarded as the stability of the compliant mechanism. In this compliant mechanism, the endpoint of the groove-coupler is chosen as a characteristic point and the compliant joint’s spring stiffness kT is taken as the key parameter to obtain trajectories and phase diagram. It is found that all cycle overlapping phase diagrams possess intuitively good repeatability, except in two locally specific regions. Then, Wolf’s numerical calculation method of Lyapunov exponents is utilized to analyze the motion stability of the compliant mechanism. An appropriate range (1.0–1.05 N mm/°) of the joint stiffness kT is obtained, satisfying the stability criterion. Finally, characteristic point trajectory and functional experiments with kT = 1 N mm/° are conducted to verify the motion stability of the compliant mechanism. The method of mechanism motion stability analysis by means of Lyapunov exponents can be applied to other under-actuated mechanisms.

Nonlinear double-mass pendulum for vibration-based energy harvesting

To enhance the performance of a vibration-based energy harvester, typical approaches employ frequency-matching strategies by either using nonlinear broadband or frequency-tunable harvesters. This study systematically analyzes the nonlinear dynamics and energy harvesting performance of a recently emerging tunable low-frequency vibration-based energy harvester, namely, a double-mass pendulum (DMP) energy harvester. This energy harvester can, to some extent, eliminate frequency dependence on pendulum length but exhibit vibration-amplitude-dependent softening nonlinearity. The natural frequency of the DMP structure is theoretically derived, showing several unique characteristics compared with the typical simple pendulum. The DMP energy harvester exhibits alternate single-period, multiple-period, and chaotic vibration behaviors with increase in excitation amplitudes. The analysis of gross output power indicates that the rotating motion, regardless of chaotic or periodic rolling motions, improves the energy harvesting performance in terms of power leap and broader bandwidth. Based on the parameter space analysis, the rotating motions usually occur at the shift-left locations of frequency ratios 1 and 2; a smaller damping ratio corresponds to a lower on-demand excitation amplitude for the rotating-motion occurrence. Numerical results confirm that the DMP is suitable for low-frequency energy harvesting scenarios, suggesting the realization of rotating motion for improving energy harvesting performance. Moreover, a shake table test was performed, and the experimental results validated the accuracy and effectiveness of the DMP modeling analysis. Practical issues related to DMP energy harvesters under different types of excitations are finally discussed. Although the analysis is for the DMP, the corresponding conclusions may shed light on other pendulum-type energy harvesters.

Sedimentation of two circular particles with different sizes in a vertical channel at low Reynolds numbers

In this study, the generalized finite-difference with singular value decomposition method for fluid–structure interaction problems is used to simulate the sedimentation of the two circular particles with different sizes in a vertical channel. The effects of the Reynolds number (8 ≤ Re ≤ 70) and the size difference (0 ≤ β ≤ 0.1) on the final motions of the two particles are analyzed. Over the ranges of the parameters investigated, three modes in the final state of the two-particle system are identified, i.e., the steady state, the periodic oscillation state, and the period-doubling bifurcation (PDB) state. Depending on the importance of the inertial force, the steady state can be classified as the steady state I and the steady state II. Similarly, the periodic oscillation state can be categorized into the periodic motion I (PMI) and the periodic motion II (PMII) based on the influence of the wake between the two particles. The directions of the limit cycles corresponding to PMI and PMII are counterclockwise and clockwise, respectively. In PMI, the limit cycle at 8 ≤ Re ≤ 9 decreases in size with increasing β, while the limit cycle at 12 ≤ Re < 70 behaves oppositely. The limit cycle in PMII always increases in size with β. PDB, characterized by the limit cycle with two branches, mainly appears at 14 ≤ Re ≤ 30.

Stabilization of Laminars in Chaos Intermittency

Chaos intermittency is composed of a laminar regime, which exhibits almost periodic motion, and a burst regime, which exhibits chaotic motion; it is known that in chaos intermittency, switching between these regimes occurs irregularly. In the laminar regime of chaos intermittency, the periodic solution before the saddle node bifurcation is closely related to its generation, and its behavior becomes periodic in a short time; the laminar is not, however, a periodic solution, and there are no unstable periodic solutions nearby. Most chaos control methods cannot be applied to the problem of stabilizing a laminar response to a periodic solution since they refer to information about unstable periodic orbits. In this paper, we demonstrate a control method that can be applied to the control target with laminar phase of a dynamical system exhibiting chaos intermittency. This method records the time series of a periodic solution prior to the saddle node bifurcation as a pseudo-periodic orbit and feeds it back to the control target. We report that when this control method is applied to a circuit model, laminar motion can be stabilized to a periodic solution via control inputs of very small magnitude, for which robust control can be obtained.

The motion of a circular foil in the field of a fixed point singularity: Integrability and asymptotic behavior

A finite-dimensional model is developed, which describes the motion of a balanced circular foil with proper circulation in the field of a fixed vortex source. The motion of the foil has been studied in two special cases: that of a fixed vortex and that of a fixed source. It is shown that in the absence of proper circulation, the fixed vortex and the fixed source have the same impact on the motion of the foil. However, adding nonzero proper circulation leads to qualitative differences in the foil's dynamics. For a fixed vortex, there exist three types of motions: the fall on a vortex in finite time, periodic and quasiperiodic motion around the vortex. The investigation of this case reduces to analysis of a Hamiltonian system with one degree of freedom. Typical phase portraits and graphs of the effective potential of the system are plotted vs the distance between the geometric center of the foil and the vortex. For a fixed source, two types of motions are possible: the fall on the source in finite time and unbounded escape from the source. For small intensities of the source, the asymptotics of escape to infinity is constructed.

Powering an autonomous clock with quantum electromechanics

We theoretically analyse an autonomous clock comprising a nanoelectromechanical system, which undergoes self-oscillations driven by electron tunnelling. The periodic mechanical motion behaves as the clockwork, similar to the swinging of a pendulum, while induced oscillations in the electrical current can be used to read out the ticks. We simulate the dynamics of the system in the quasi-adiabatic limit of slow mechanical motion, allowing us to infer statistical properties of the clock’s ticks from the current auto-correlation function. The distribution of individual ticks exhibits a tradeoff between accuracy, resolution, and dissipation, as expected from previous literature. Going beyond the distribution of individual ticks, we investigate how clock accuracy varies over different integration times by computing the Allan variance. We observe non-monotonic features in the Allan variance as a function of time and applied voltage, which can be explained by the presence of temporal correlations between ticks. These correlations are shown to yield a precision advantage for timekeeping over the timescales that the correlations persist. Our results illustrate the non-trivial features of the tick series produced by nanoscale clocks, and pave the way for experimental investigation of clock thermodynamics using nanoelectromechanical systems.

Planar Walking of a Five-Link Biped Robot over a Stepped Surface with Obstacles of Different Heights and Lengths

A problem of modeling biped locomotion is an actual and complex problem of modern control theory. In the present paper, we deal with a special case when a five-link biped robot with knees moves along a given stepped surface. We describe the robot motion by a hybrid system which consists of a system of ordinary differential equations at the single support phase and an algebraic relation at the double support phase. The main contribution of this paper is the construction of a periodic motion that satisfies a set of conditions inherent in human walking. The results of the paper will be useful to those who are interested in control theory applications.

Nonlinear dynamics of a circular curved cantilevered pipe conveying pulsating fluid based on the geometrically exact model

Due to the novel applications of flexible pipes conveying fluid in the field of soft robotics and biomedicine, the investigations on the mechanical responses of the pipes have attracted considerable attention. The fluid-structure interaction (FSI) between the pipe with a curved shape and the time-varying internal fluid flow brings a great challenge to the revelation of the dynamical behaviors of flexible pipes, especially when the pipe is highly flexible and usually undergoes large deformations. In this work, the geometrically exact model (GEM) for a curved cantilevered pipe conveying pulsating fluid is developed based on the extended Hamilton’s principle. The stability of the curved pipe with three different subtended angles is examined with the consideration of steady fluid flow. Specific attention is concentrated on the large-deformation resonance of circular pipes conveying pulsating fluid, which is often encountered in practical engineering. By constructing bifurcation diagrams, oscillating shapes, phase portraits, time traces, and Poincare maps, the dynamic responses of the curved pipe under various system parameters are revealed. The mean flow velocity of the pulsating fluid is chosen to be either subcritical or supercritical. The numerical results show that the curved pipe conveying pulsating fluid can exhibit rich dynamical behaviors, including periodic and quasi-periodic motions. It is also found that the preferred instability type of a cantilevered curved pipe conveying steady fluid is mainly in the flutter of the second mode. For a moderate value of the mass ratio, however, a third-mode flutter may occur, which is quite different from that of a straight pipe system.

Dynamic analysis of slant cracked rotor system considering nonlinear oil film force.

In this paper, considering the combined effects of nonlinear oil film forces and cracks on the rotor-bearing system, the differential equations of motion with 4 degrees of freedom are established by Lagrangian method. Then, the Lundgren-Kutta method is used to solve them and the results of the model are compared with the experimental data. The study demonstrate that the cracked rotor-bearing system is relatively stable at subcritical speeds, mostly in the period-1 motion. But near 1/3 of the critical speed, there is an inner loop in its whirl orbit and a significant increase in the 2x frequency component. When the system speed rises to the region near 1/2 of the critical speed, though the bifurcation motion and a relatively high 2x frequency can be observed, there are no other reliable fault characteristics. The study proves that the rotor crack fault diagnosis method based on the whirl orbits is convincing for slant cracked rotors.

On the collision dynamics in a molecular model

We study the problem of the hydrogen atom submitted to a circularly polarized microwave field. This problem, analyzed from a classical mechanics approach, can be modeled by an autonomous Hamiltonian depending on one parameter K>0. The paper is focused on the so called n-ejection–collision orbits (n-EC orbits), that is orbits that the electron describes when it ejects from the nucleus and collides with it at the n relative minimum in the distance with respect to the nucleus. In this work, we analyze the evolution of the families of n-EC orbits. We conduct a comprehensive numerical analysis of the bifurcations, which involves multiple precision computations, to characterize the successive bifurcation families that emerge. Additionally, we examine the periodic and quasi-periodic motion of the n-EC orbits belonging to these bifurcation families.

Vibration stability and bifurcation analysis of spur gear systems based on time-varying backlash and random load

Gear backlash and random load affect the stability of gear transmission, which are the main factors causing meshing impact. This study presents the establishment of a six-degree-of-freedom model for the spur gear system, employing the lumped parameter theory and D'Alembert's principle. The model comprehensively incorporates the effects of time-varying meshing stiffness (TVMS), static transmission error (STE), time-varying gear backlash, and random load. The system's equations of motion are solved using the fourth-order Runge-Kutta method. Subsequently, the dynamic response of the constant backlash constant load model is compared with that of the time-varying backlash constant load model and the constant backlash random load model. This research quantitatively dissects the impact of the nonlinear characteristics of backlash and load on the dynamic response of the gear system. This analysis is conducted utilizing a variety of methods, including bifurcation diagrams, Floquet characteristic multipliers, Lyapunov exponents, phase diagrams, and Poincaré sections. The study demonstrates that fluctuations in backlash and load during meshing compromise the stability of the periodic solution, thereby escalating the complexity of predicting system motion. Moreover, observations include the jump phenomena, quasi-periodic and periodic motions, and the trajectory of the system transitioning into a chaotic state via bifurcation. This investigation serves as a valuable reference for future explorations into spur gear systems.

Human aging and perception of time in view of the Earth’s motion events

Here we consider human aging and time perception by operating with time, as a physical quantity, linked to periodic motions of the Earth. Taking into account the periodicity of Earth’s revolutions, the model of aging in a form like Fermat’s spiral has been proposed. This model demonstrates the progressive extension of the paces of aging implying accelerated accumulation of adverse changes over life, which is, as a vector, directed to the center of the spiral. In this regard, an internal clock underpinning time perception that was evolved with life under the permanent influence of the Earth’s motion events can accelerate with aging presumably due to progressive extension of the paces of aging.

Rayleigh–Bénard convection with an immersed floating body

The paper presents the results of an experimental and numerical study of turbulent thermal convection in a rectangular box containing an extended immersed free-floating plate. Varying the values of control parameters, such as Rayleigh number, aspect ratio and vertical position of the plate, provides a wide range of possible modes, from immobile and purely periodic to stochastic. We have shown that stable periodic motions occur when the plate floats close to one of the heat exchangers. An increase in the distance between the plate and the heat exchanger breaks the periodic motion and (at moderate Rayleigh numbers) leads to a pronounced asymmetry, when the plate stays close to one of the walls most of the time, makes rare excursions to the opposite wall and immediately returns. As the Rayleigh number increases, the plate motions from one edge of the box to the other reappear, but always have an irregular character. Regarding the dependence of the system behaviour on the geometry of the box, both lower and upper limits of periodic plate motions were found in the experiments. In the numerical simulations, the upper limit was not achieved – the plate moves quasi-periodically through the chain of vortices of different signs even at the largest aspect ratio being considered. The heat-insulating floating plate provides the spatial and temporal variation of the heat flux and reduces the integral heat flux, but the reduction in heat flux depends significantly on the vertical position of the plate.

A new capsule-intestine model for the capsule robot self-propelling in the lower gastrointestinal tract

Circular and haustral folds in the lower gastrointestinal tract (small and large intestines) are the major obstructions impeding the locomotion of the capsule robots for endoscopic diagnosis. Understanding the interactions between the capsule and these folds is critical for design and control of these robots to reach the areas of clinical interest. This paper proposes a new mathematical model of capsule-intestine interaction based on our previous work (Yan et al. 2022) by introducing capsule’s rotation during fold crossing. The resisting force of the fold predicted by the new model is more consistent with our finite element and experimental results compared to our previous model. It is found that the obstructive effect of the fold is stronger for a higher and thinner fold with a thinner and stiffer intestine. For the capsule robot, which is actuated by a periodically driven inner mass, a stronger excitation force is required to overcome the fold with a larger resisting force. Moreover, our bifurcation analysis reveals that a small excitation force always incurs a simple period-1 motion for the robot, while a large excitation force may result in various complex dynamics before the fold crossing. The findings of this work may help capsule robotics engineers to evaluate their designs in terms of propulsion and understand the locomotion of their robots in the lower gastrointestinal tract.

Defected twisted ring topology for autonomous periodic flip-spin-orbit soft robot.

Periodic spin-orbit motion is ubiquitous in nature, observed from electrons orbiting nuclei to spinning planets orbiting the Sun. Achieving autonomous periodic orbiting motions, along circular and noncircular paths, in soft mobile robotics is crucial for adaptive and intelligent exploration of unknown environments-a grand challenge yet to be accomplished. Here, we report leveraging a closed-loop twisted ring topology with a defect for an autonomous soft robot capable of achieving periodic spin-orbiting motions with programmed circular and re-programmed irregular-shaped trajectories. Constructed by bonding a twisted liquid crystal elastomer ribbon into a closed-loop ring topology, the robot exhibits three coupled periodic self-motions in response to constant temperature or constant light sources: inside-out flipping, self-spinning around the ring center, and self-orbiting around a point outside the ring. The coupled spinning and orbiting motions share the same direction and period. The spinning or orbiting direction depends on the twisting chirality, while the orbital radius and period are determined by the twisted ring geometry and thermal actuation. The flip-spin and orbiting motions arise from the twisted ring topology and a bonding site defect that breaks the force symmetry, respectively. By utilizing the twisting-encoded autonomous flip-spin-orbit motions, we showcase the robot's potential for intelligently mapping the geometric boundaries of unknown confined spaces, including convex shapes like circles, squares, triangles, and pentagons and concaves shapes with multi-robots, as well as health monitoring of unknown confined spaces with boundary damages.

Experimental investigation of instantaneous lift on NACA 0018 airfoil section due to a non-harmonic pitching motion

In this study, we explore the problem of an airfoil in a harmonic and arbitrary periodic pitching motion and seek to understand how theoretical estimation with first-order potential theory match experiments. We adopt Jones’ approximation for the Wagner function for lift computations in the time domain and Theodorsen’s model in the frequency domain. We experimentally investigate a two-dimensional symmetric NACA 0018 airfoil in pitching motion under attached flow with reduced frequencies up to 0.25 and Reynolds number varying from 1.5×105 to 3×105. To eliminate the influence of the laminar separation bubble, free-stream turbulence intensity was elevated above the baseline level by placing turbulence generating grid, thus, obtaining better compatibility between the flow conditions in the experiments and the theoretical assumptions. Time-resolved sectional lift is determined by simultaneous static pressure measurement with miniature pressure transducers. The transient response is captured with accuracy in the time domain by quantifying the non-circulatory contribution. Excellent agreement is achieved in the time and frequency domains due to the highly accurate measurement of the instantaneous magnitude and phase of the time-resolved surface pressures. We further explore the applicability of the theory for high-pitching amplitudes, which results in a temporary deviation of the measured lift from the theoretical predictions. As the frequency domain unsteady theory is more widely studied than its time domain counterpart, this study provides a unique opportunity to highlight the significance of time-domain analysis in estimating instantaneous lift in non-harmonic motion.

An augmented hybrid multibaseline and referenceless MR thermometry motion compensation algorithm for MRgHIFU hyperthermia.

A hybrid principal component analysis and projection onto dipole fields (PCA-PDF) MR thermometry motion compensation algorithm was optimized with atlas image augmentation and validated. Experiments were conducted on a 3T Philips MRI and Profound V1 Sonalleve high intensity focused ultrasound (high intensity focused ultrasound system. An MR-compatible robot was configured to induce motion on custom gelatin phantoms. Trials with periodic and sporadic motion were introduced on phantoms while hyperthermia was administered. The PCA-PDF algorithm was augmented with a predictive atlas to better compensate for larger sporadic motion. During periodic motion, the temperature SD in the thermometry was improved from to C with both the original and augmented PCA-PDF application. For large sporadic motion, the augmented atlas improved the motion compensation from the original PCA-PDF correction from to C. The PCA-PDF algorithm improved temperature accuracy to <1°C during periodic motion, but was not able to adequately address sporadic motion. By augmenting the PCA-PDF algorithm, temperature SD during large sporadic motion was also reduced to <1°C, greatly improving the original PCA-PDF algorithm.

Mechanical switches integrated sandwich triboelectric nanogenerator with high instantaneous output power

Triboelectric nanogenerator (TENG) has emerged as a promising energy harvester that converts random mechanical energy in the environment into electricity to enable self-powered system construction. However, the conventional TENG exhibits a large internal resistance resulting in a low output power across external loads for practical application. In this work, the travel mechanical switches were integrated into the TENG to boost output power, where the switches were turned on and off by the periodic motion between the triboelectric layers. Moreover, the triboelectric layers of TENG were developed as contact separation sandwich structures to work with mechanical switches, allowing the device to release the co-accumulated charges from the dual triboelectric layer to yield giant instantaneous discharge. Compared to a conventional TENG, the device offers ~2.8 times more transfer charge in one cycle and ~106 times more instantaneous power at an external load of 500 Ω, which renders the internal impedance match almost negligible. As a result, the high-power source demonstrates the ability to illuminate the commercial lamps in parallel (total power: 50 W). Such a unique TENG with high output power for low resistance load could greatly enrich the practical application fields of self-powered systems.

Solutions of the KdV-MKdV equations arising in non-linear elastic rods under fractal dimension

A prediction of rod wave type with great precision is extremely important in theoretical analysis and practical applications. Besides the well-known periodic motion and resonance, this paper studies the rod wave in a fractal space, and a fractal solitary wave is unlocked by the variational approach, the results reveal that the rod strain non-linearity and fractal dimensions affect greatly the wave travelling properties. This paper offers a new window for identifying a solitary wave from periodic motion easily and accurately.