Elliptic Path Research Articles

Study of different theories of thermoelasticity under the propagation of Rayleigh waves in thermoelastic medium

The present research is on the propagation of Rayleigh waves in a homogenous thermoelastic solid half-space by considering the compact form of six different theories of thermoelasticity. The medium is subjected to an insulated boundary surface that is free from normal stress, tangential stress, and a temperature gradient normal to the surface. After developing a mathematical model, a dispersion equation is obtained with irrational terms. To apply the algebraic method, this equation must be converted into a rational polynomial equation. From this, only those roots are filtered out, which has satisfied both of the above equations for the propagation of waves decaying with depth. With the help of these roots, different characteristics are computed numerically, like phase velocity, attenuation coefficient, and path of particles. Various particular cases are compared graphically by using phase velocity and attenuation coefficient. The elliptic path of surface particles in Rayleigh wave propagation is also presented for the different theories using physical constants of copper material for different depths and thermal conductivity.

The Ultimate Speed of an accelerated Electron without Infinite Mass

Failure to accelerate electrons beyond the speed of light, led to special relativity, where mass increases with speed, becoming infinitely large at the speed of light. Invoking aberration of electric field, this paper introduces radiative electrodynamics where an electron is accelerated to the speed of light at constant mass and with emission of radiation. The accelerating force on an electron moving in an electric field, is less than the force on a stationary one, relative to an observer. This difference is radiation reaction force, something akin to frictional force, against which work done appears as radiation. An accelerated electron moves in an electric field, with constant mass as the rest mass, emitting radiation equal to the difference between change in potential energy and change in kinetic energy. Circular revolution of an electron, in an atom, round the central force of attraction of a nucleus, as in the Rutherford’s nuclear model of the hydrogen atom, is shown to be without radiation and inherently stable. Emission of radiation takes place if the electron in an atom is dislodged from a circular orbit. It then moves in an elliptic path, emitting radiation, at the frequency of revolution, with fine structure, before reverting to a circular orbit. Describing aberration of electric field, introducing radiative electrodynamics, obtaining the ultimate of speed of light outside special relativity, and getting radiation from accelerated electrons outside quantum mechanics, are notable results of this paper.

Motion and Radiation for Charged Particles Revolving Round Centre of Force of Attraction

Nuclear and non-nuclear models of hydrogen atom are proposed, each with N equal to 42 or 43 orbits of revolution. Aberration of electric field is invoked to describe radiation from nuclear model with particle of electronic charge -e and mass nm of electronic mass m, revolving in nth orbit, with angular momentum nL, at radius nr1 from a nucleus of charge +Ne and mass (Nm/2)(1 + N), where n = 1, 2, 3...N, and r1 is radius of first orbit. The revolving particles have total charge -Ne and same total mass as the nucleus. The non-nuclear model has two particles of same mass nm and charges +e and -e, revolving diametrically in nth orbit, at radius nr1 from a centre, with particles of total mass Nm(1 + N). Distinct masses nm gives discrete coplanar circular orbits. A particle dislodged from a circular orbit, revolves in an elliptic path with emission of radiation, at the frequency of revolution, in fine structure, before reverting to a stable circular orbit. The nuclear atom is identified with solid or liquid state of hydrogen and non-nuclear atom with gaseous state

Contouring Control of an Innovative Manufacturing System Based on Dual-Arm Robot

In this article, an innovative manufacturing system based on a dual-arm robot, each arm with five degree-of-freedom (DOF), is proposed. Although the proposed system can work in the automation mode or machining mode, only the function of machining is studied here. In this case, the system is equivalent to a five-axis machine tool. The method of equivalent errors is employed to design a robust contouring controller. It is shown that reducing the equivalent contour error in the joint space is equivalent to reducing both position contour error and tool orientation error in the task space. As a comparison, a conventional distributed controller is also designed. Several cases of experiments have been conducted, including a circular path with four different feedrates and an elliptic path. The experimental results show that the proposed method can achieve much better results in all cases, and demonstrate the feasibility of adopting the dual-arm robotic system for machining task. <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">Note to Practitioners</i> —This article was motivated by the traditional manufacturing system that is a cooperation between computer numerical controlled (CNC) machine tool(s) and robot arm(s). This article proposes a novel usage of a dual-arm robot with a fixed spindle and cutting tool in order to integrate the main functions of the CNC machine and robot arm together. This system alone can perform machining and automation tasks, which can significantly reduce the overall cost of a manufacturing system. The existing dual-arm robots have seldom been designed to perform machining tasks. For machining, a contouring controller is necessary and is proposed in this article. This model-based contouring controller relies on the accurate system parameters to achieve high precision machining. For example, the mass and inertia changes of a workpiece during machining should be considered. This issue will be addressed in our future research.

Thermoelastic interactions on the propagation of surface waves in a piezoelectric half-space: A comparative analysis of GN-III type and three-phase-lag model

This study elucidates the propagation of surface (Rayleigh type) waves in a homogeneous, transversely isotropic, piezothermoelastic half-space subjected to stress free, electrically open or shorted, and thermally insulated or isothermal boundary conditions, based on GN-III type and three-phase-lag thermoelastic models. Plane harmonic wave solutions are employed to find the mechanical displacements, electrical potential, and temperature change. With the aid of these expressions, stresses, electrical displacement, and temperature gradient are derived. Based on different boundary conditions, four secular equations are derived in the considered half-space. Path of the surface particles traces an elliptic path in vertical plane parallel to the direction of wave propagation and the eccentricity of the ellipse is calculated. Particle path degenerates a straight line when there is no phase difference between vertical and horizontal components of displacements. A pre-established analysis is discussed as a particular case of this study. Effect of various characteristics of waves like phase velocity, attenuation coefficient, and specific loss is demonstrated graphically for the GN-III type and three-phase-lag thermoelastic models engaging cadmium selenide (6 mm class) material of hexagonal symmetry. This mathematical framework may be utilized to design and develop temperature sensors, and other piezoelectric surface acoustic wave (SAW) devices.

Modeling and Stability Analysis for the Vibrating Motion of Three Degrees-of-Freedom Dynamical System Near Resonance

The focus of this article is on the investigation of a dynamical system consisting of a linear damped transverse tuned-absorber connected with a non-linear damped-spring-pendulum, in which its hanged point moves in an elliptic path. The regulating system of motion is derived using Lagrange’s equations, which is then solved analytically up to the third approximation employing the approach of multiple scales (AMS). The emerging cases of resonance are categorized according to the solvability requirements wherein the modulation equations (ME) have been found. The stability areas and the instability ones are examined utilizing the Routh–Hurwitz criteria (RHC) and analyzed in line with the solutions at the steady state. The obtained results, resonance responses, and stability regions are addressed and graphically depicted to explore the positive influence of the various inputs of the physical parameters on the rheological behavior of the inspected system. The significance of the present work stems from its numerous applications in theoretical physics and engineering.

Stellar Aberration from Earth and from a Satellite

In his endeavor to find a concrete evidence in favor of the Copernican picture of the solar system, the English astronomer James Bradley made a series of astronomical observations during the period (1725-1728) aiming to detect a stellar parallax. His findings, which manifested indeed an annual apparent cyclic motion of a star, were however at conflict with what is expected in a parallax. To his surprise, the result of every measurement obtained corresponded to what he expected to get in a measurement done three months earlier. Bradley realized that he was witnessing a new physical effect, and he presented an explanation that conceived light as a corpuscular stream travelling at finite velocity. Despite that Bradley’s explanation of the stellar aberration effect was inadequate, the equation which he derived to quantify the aberration angle, predicted a better estimation of light velocity, and the aberration phenomenon itself was a concrete support of heliocentrism. Stellar aberration as well as some other optical experiments, whose explanations posed challenges to the existing physical theories in the late nineteenth century paved the way for the emergence of the special theory of relativity. In the current work we employ the theory of universal space and time to show that a given direction in a frame of reference is tilted when observed in a moving frame by an angle that depends on the direction itself and the velocity of the moving frame. The latter fact is utilized to explain stellar aberration, determine the deviation of a star’s vision direction from its true one, and deduce its apparent position at any instant as a function of its latitude and time. The novel concept of aberration correction vector is employed to derive the apparent elliptic path of an observed celestial object at any time. The concept of graded inertial frames is introduced and utilized to deal with aberration when observed from a satellite in a similar way to its treatment when observed from Earth. The transformation matrix between a geocentric frame and a satellite’s non-rotating frame is derived and used to transform temporary Earthly vision directions to the satellite’s frame. Furthermore, the transformed vectors are adopted as transient fixed directions relative to which the vision directions of a star from the satellite are specified throughout one revolution. Satellites connective matrices are constructed to make geometric information regarding the celestial sphere in one frame immediately usable by observers on Earth and in all other satellites.

Treating the Solid Pendulum Motion by the Large Parameter Procedure

In this paper, we consider the dynamical description of a pendulum model consists of a heavy solid connection to a nonelastic string which suspended on an elliptic path in a vertical plane. We suppose that the dimensions of the solid are large enough to the length of the suspended string, in contrast to previous works which considered that the dimensions of the body are sufficiently small to the length of the string. According to this new assumption, we define a large parameter ε and apply Lagrange’s equation to construct the equations of motion for this case in terms of this large parameter. These equations give a quasi-linear system of second order with two degrees of freedom. The obtained system will be solved in terms of the generalized coordinates θ and φ using the large parameter procedure. This procedure has an advantage over the other methods because it solves the problem in a new domain when fails all other methods for solving the problem in such a domain under these conditions. It is one of the most important applications, when we study the slow spin motion of a rigid body in a Newtonian field of force under an external moment or the rotational motion of a heavy solid in a uniform gravity field or the gyroscopic motions with a sufficiently small angular velocity component about the major or the minor axis of the ellipsoid of inertia. There are many applications of this technique in aerospace science, satellites, navigations, antennas, and solar collectors. This technique is also useful in all perturbed problems in physics and mechanics, for example, the perturbed pendulum motions and the perturbed mechanical systems. The results of this paper also are useful in moving bridges and the swings. For satisfying the validation of the obtained solutions, we consider numerical considerations by one of the numerical methods and compare the obtained analytical and numerical solutions.

Analysis of dissipation energy and ratchetting behaviors on multiaxial fatigue of adhesively bonded joints

The multiaxial fatigue experiments were conducted on the adhesive bonded hollow cylinder butt-joints under the stress-controlled mode. The effect of loading path, equivalent stress amplitude, equivalent mean stress and period on the multiaxial fatigue behaviors were discussed. The multiaxial fatigue damages of adhesively bonded joints were also analyzed from the evolution of ratchetting and dissipative energy behavior. The results showed that the loading paths had a great influence on dissipated energy and ratchetting strain, and the effects of non-proportional loading path on the ratchetting strain and dissipation energy were greater than that of the proportional loading path. Moreover, the equivalent stress amplitude, equivalent mean stress and period all had some influence on the dissipation energy and equivalent ratchetting strain. In addition, the dissipative energy and the ratchetting strain increased gradually with the number of cycle increase, which resulted in the fatigue life decreases with the increase of equivalent stress amplitude and mean stress, and the decreasing period. The effects of period on the multiaxial fatigue life under the elliptic loading path were insignificant for the adhesive bonding butt-joints.

Experimental investigation on multiaxial ratchetting behaviour and fatigue life of silicone seal adhesive bonding butt-joints

The multiaxial fatigue experiments were conducted on the silicone seal adhesive bonding hollow cylinder butt-joints under the stress-controlled mode at laboratory ambient temperature. The effect of loading path, equivalent stress amplitude, equivalent mean stress and cycle time on the multiaxial ratchetting behaviors and fatigue life were discussed. The experimental results showed that the effects of non-proportional loading path on the ratchetting strain were greater than that of the proportional loading path. Moreover, the effects of the normal stress on the ratchetting strain were relatively larger than that of the equivalent shear stress under the multiaxial loading paths. In addition, the ratchetting strain increases and fatigue life decreases with the increase of equivalent stress amplitude and mean stress, and the decreasing cycle time. The effects of cycle time on the multiaxial ratchetting strain and fatigue life under the elliptic loading path were insignificant for the silicone seal adhesive bonding butt-joints.

Entropy Analysis of an MHD Synthetic Cilia Assisted Transport in a Microchannel Enclosure with Velocity and Thermal Slippage Effects

The magnitude of shear stress at the ciliated wall is considered as the measure of efficiency of cilia beatings as it describes the momentum transfer between the medium and the cilia. Under high shear rate, some non-Newtonian fluids behave as visco-inelastic fluids. We consider here a ciliated channel coated with Prandtl fluid, a visco-inelastic fluid, with Hartmann layer under momentum and thermal slip effects. The flow in the channel is produced due to beatings of cilia that obey an elliptic path of motion in the flow direction. An entropy analysis of the flow is also conducted in wave frame. After introducing lubrication approximations in the governing equation, the perturbation solutions are calculated. The data for pressure rise per metachronal wavelength and frictional force at the ciliated wall are obtained by numerical integration. The analysis reveals that the higher values of cilia length and velocity slip parameters support fluid flow near the channel wall surface. Fluid temperature is an increasing function of thermal slip but a decreasing function of cilia length and slip parameters. Entropy in the channel can be minimized with an increase in cilia length and slip effect at the boundary. The magnitude of the heat transfer coefficient decreases by taking the substantial slippage and tiny cilia in length at the microchannel wall.

Second Law Analysis of Ciliary Pumping Transport in an Inclined Channel Coated with Carreau Fluid under a Magnetic Field

A complete thermal analysis is performed for the propulsion of cilia in an inclined channel. Coating around the channel walls is provided by a Carreau fluid under a uniform magnetic field. Uniformly grown cilia produce propulsive metachronal waves by moving in a coordinated rhythm along the channel surface and adapt an elliptic path along the direction of flow. Using lubrication approximations, the governing equations, formulated in the wave frame of reference, are solved by the perturbation method. Validation of the analytic solution is provided by computing the solution numerically with the shooting method. This study is concerned with the parametric consequences on pertinent flow and heat transfer quantities, such as streamlines, velocity profile, temperature profile, entropy lines and the Bejan number. The results reveal that large cilia propel the axial velocity near the channel wall but put hindrance to the axial velocity and the temperature profile in the central part of the channel. The entropy production in the channel reduces for large cilia and a high Hartmann number.

Peristaltic Flow with Heat Transfer on Sisko Fluid in a Ciliated Arteries

The flow of blood through arteries is an important physiological problem. In the present investigation, we carried out to study the peristaltic of non-Newtonian incompressible blood flow with heat transfer through ciliated arteries. The blood flow is characterized by the generalized Sisko model. The nonlinear partial differential equations of the problem are simplified by using an approximation of long wavelength and low Reynolds number. The differential equations are solved analytically by using the perturbation method. We find that Sisko fluid parameter and the power index effects the behavior of the velocity where the velocity increase in the arteries then decreases near the wall, but the Sisko parameter give opposite behavior where the velocity decrease then increases near the wall of arteries. The velocity increase in arteries with the increase of cilia length and elliptic path. The temperature profile increases then decreases near the wall of arteries with the increase of power index, Sisko fluid parameter and Grashof number, while the temperature decrease then increase near the wall with increase of Sisko parameter. The effect of increase in the cilia length give an increase of the temperature. The pressure gradient increases with the increase of power index and elliptic path, while the pressure gradient decrease with an increase of elliptic path, Sisko parameter. The pressure gradient increases and decreases in a different interval with the increase the cilia length. Our results are illustrated through a set of Figures.

Metachronal propulsion of a magnetised particle-fluid suspension in a ciliated channel with heat and mass transfer

Biologically-inspired pumping systems are of great interest in modern engineering since they achieve enhanced efficiency and circumvent the need for moving parts and maintenance. Industrial applications also often feature two-phase flows. In this article, motivated by these applications, the pumping of an electrically-conducting particle-fluid suspension due to metachronal wave propulsion of beating cilia in a two-dimensional channel with heat and mass transfer under a transverse magnetic field is investigated theoretically. The governing equations for mass and momentum conservation for fluid- and particle-phases are formulated by ignoring the inertial forces and invoking the long wavelength approximation. The Jeffrey viscoelastic model is employed to simulate non-Newtonian characteristics. The normalised resulting differential equations are solved analytically. Symbolic software is employed to evaluate the results and simulate the influence of different parameters on flow characteristics. Results are visualised graphically with carefully selected and viable data. With increasing wave number (β) fluid velocity is accelerated in the core region whereas it is decelerated near the channel wall, for the Newtonian case. With increasing eccentricity of cilia elliptic path (α), a similar response is computed as for the wave number. The size of the bolus is enhanced (and quantity of boluses is reduced) with increasing eccentricity of the cilia elliptic path (α) and Hartmann (magnetic) number (M) whereas bolus size is decreased (and quantity of boluses is increased) with increasing wave number (β) and particle volume fraction (C). It is also noted that increasing Schmidt number (Sc) and Soret number (Sr) diminish the concentration magnitudes. Furthermore, Brinkman number (which represents viscous heating effects) significantly boosts the temperature magnitudes. The current analysis provides a useful benchmark for more general computational simulations.

On the motion of a harmonically excited damped spring pendulum in an elliptic path

In this work the problem of a nonlinear damped spring pendulum in which the motion of its pivot point in an elliptical path is investigated. The second end of the spring is connected with the body. A linear force acting along the pendulum arm besides two anticlockwise moments; one at the suspension point of the body with the damped spring and the other at the pivot point. One of the important perturbation techniques called the multiple scales (MS) technique is utilized to obtain the approximate solutions of the governing equations of motion till the third approximation. The modulation equations and the solvability conditions are obtained in view of the emerging resonance cases. The time history and the resonances curves are performed in some plots to show the good effect of the physical parameters on the behavior of the considered dynamical model.

Kepler’s Ellipse Generated by the Trigonometrically Organized Gravitons

Johannes Kepler made his great breakthrough when he discovered the elliptical path of the planet Mars around the Sun located in one focus of that ellipse (on the 11th October in 1605 in a letter to Fabricius). The first generation of researchers in the 17th century intensively discussed about the possible mechanism needed for the generation of that elliptical orbit and about the function of the empty focus of that ellipse. First generations of researchers proposed an interplay between attractive and repulsive forces that might guide the planet on its elliptical orbit. Isaac Newton made a giant mathematical progress in his Principia and introduced the concept of the attractive gravitational force between the Sun and planets. However, Newton did not propose a possible mechanism behind this attractive force. Albert Einstein in 1915 left the concept of attractive and repulsive forces and introduced his Theory based on the elastic spacetime. In his concept gravity itself became fictitious force and the attraction is explained via the elastic spacetime. In our proposed model we try to re-open the discussion of Old Masters on the existence of attractive and repulsive forces. The guiding principle for our trigonometrical model is the generation of the ellipse discovered by one of the last ancient Greek mathematicians – Anthemius of Tralles – who generated the ellipse by the so-called gardener’s method (one string and two pins fixed to the foci of that ellipse). Frans van Schooten in 1657 invented a series of original simple mechanisms for generating ellipses, hyperbolas, and parabolas. Schooten’s antiparallelogram might simulate the interplay of attractive and repulsive forces creating the elliptical path. We propose a model with trigonometrically organized Solar and planet gravitons. In this model the Solar and planet gravitons are reflected and refracted in predetermined directions so that their joint momentum transferred on the planet atoms guides the planet on an elliptical path around the Sun. At this stage we cannot directly measure the gravitons but we can use the analogy with behavior of photons. We propose to observe paths of photons emitted from one focus of the ellipse towards the QUARTER-silvered elliptical mirror. 1/4 of photons will be reflected towards to the second empty focus and the ¾ of photons might be reflected and refracted into the trigonometrically expected directions. (Until now we have experimental data only for the FULLY–silvered elliptical mirrors). The observed behavior of photons with the quarter-silvered elliptic mirror might support this concept or to exclude this model as a wrong model. The quantitative values of attractive and repulsive forces could be found from the well-known geometrical properties of the ellipse. The characteristic lengths of distances will be inserted into the great formula of Isaac Newton - the inverse square law. (In order to explain some orbit anomalies, we can use Paul Gerber’s formula derived for the Pierre Fermat principle). We have found that the Kant’s ellipse rotating on the Keppler’s ellipse might express the co-operation of attractive and repulsive forces to guide the planet on its elliptic path. Finally, we have derived a new formula inspired by Bradwardine - Newton - Tan - Milgrom that might contribute to the MOND gravitational model. We have found that the Kepler ellipse is the very elegant curve that might still keep some hidden secrets waiting for our future research.

On the vibrational analysis for the motion of a harmonically damped rigid body pendulum

The present work outlines the investigation of a damped harmonically excited spring pendulum which moves in an elliptic path with constant angular velocity. A rigid body is attached with a damped spring and has a linear force acted along the pendulum arm. The multiple scales technique was utilized to obtain the asymptotic solutions for the governing equations of motion up to the third approximation. Some resonance cases have been classified in view of the attained modulation equations. The solvability conditions for the steady-state solutions are obtained. The time history of the attained solutions is represented graphically and compared with the numerical solutions of the governing equations of motion for suitable physical parameters of the considered dynamical model. Moreover, the resonance curves for these solutions are plotted for the same parameters.

Analysis of the dynamic stress path under obliquely incident P-waves and its influencing factors

The dynamic stress path of a rock and soil mass under seismic action has a crucial influence on its catastrophic behavior. In soil dynamics, earthquakes are commonly simplified as vertically incident shear waves and the seismic stresses in soil are estimated based on rigid foundation models. However, the great effect of P-waves should not be overlooked in strong earthquakes, which have happened frequently in recent years. The characteristics of the dynamic stress path under longitudinal waves with significant oblique incidence are still unclear. Analytical formulas for the seismic stresses at any depth of a semi-infinite elastic space under obliquely incident P-waves are derived, which degenerate into the traditional rigid foundation method in soil dynamics when both the incident angle and Poisson’s ratio are taken as zero. Here, we reveal the fundamental characteristics of a dynamic stress path under obliquely incident P-waves. The stress path is proved mathematically to be an oblique ellipse in the plane of normal stress difference and horizontal shear stress. We identify factors affecting the stress path, including the incident angle, Poisson’s ratio, and depth corresponding to unit wavelength. The possible variation in the range of an oblique elliptic stress path is systematically analyzed, which lays a theoretical foundation for further study of the dynamic response of sites under obliquely incident seismic waves.

Velocity-curvature patterns limit human-robot physical interaction.

Physical human-robot collaboration is becoming more common, both in industrial and service robotics. Cooperative execution of a task requires intuitive and efficient interaction between both actors. For humans, this means being able to predict and adapt to robot movements. Given that natural human movement exhibits several robust features, we examined whether human-robot physical interaction is facilitated when these features are considered in robot control. The present study investigated how humans adapt to biological and non-biological velocity patterns in robot movements. Participants held the end-effector of a robot that traced an elliptic path with either biological (two-thirds power law) or non-biological velocity profiles. Participants were instructed to minimize the force applied on the robot end-effector. Results showed that the applied force was significantly lower when the robot moved with a biological velocity pattern. With extensive practice and enhanced feedback, participants were able to decrease their force when following a non-biological velocity pattern, but never reached forces below those obtained with the 2/3 power law profile. These results suggest that some robust features observed in natural human movements are also a strong preference in guided movements. Therefore, such features should be considered in human-robot physical collaboration.

On the Harmonic Oscillations for the Motion of a Dynamical System

This work touches two important cases for the motion of a pendulum called Sub and Ultra-harmonic cases. The small parameter method is used to obtain the approximate analytic periodic solutions of the equation of motion when the pivot point of the pendulum moves in an elliptic path. Moreover, the fourth order Runge-Kutta method is used to investigate the numerical solutions of the considered model. The comparison between both the analytical solution and the numerical ones shows high consistency between them.