General Equations Of Motion Research Articles

Automated Translational and Rotational Control of Biological Cells With a Robot-Aided Optical Tweezers Manipulation System

Research and biomedical applications in cell surgery require transportation and rotation of biological cells. In these cell manipulation tasks, the cell of interest must be translated and oriented properly such that the desired component, such as the polar body or other organelles, can be imaged with optical microscopy. This paper presents a holographic optical tweezers (HOT) based system to carry out automated translational control in the plane, and rotational control about one rotational axes of a suspended cell. Based on the proposed general equations of motion of the cell, held in an optical trap, two controllers, one for cell translational and one for rotational control, are developed to translate and orient the cells to the desired position and orientation in a sequential manner. Experiments are performed to demonstrate the effectiveness of the proposed approach.

A more general model equation of nonlinear Rayleigh waves and their quasilinear solutions

A more general two-dimensional wave motion equation with consideration of attenuation and nonlinearity is proposed to describe propagating nonlinear Rayleigh waves of finite amplitude. Based on the quasilinear theory, the numerical solutions for the sound beams of fundamental and second harmonic waves are constructed with Green’s function method. Compared with solutions from the parabolic approximate equation, results from the general equation have more accuracy in both the near distance of the propagation direction and the far distance of the transverse direction, as quasiplane waves are used and non-paraxial Green’s functions are obtained. It is more effective to obtain the nonlinear Rayleigh sound beam distributions accurately with the proposed general equation and solutions. Brief consideration is given to the measurement of nonlinear parameter using nonlinear Rayleigh waves.

Experimental Study of Active Vibration Control of Planar 3-RRR Flexible Parallel Robots Mechanism

An active vibration control experiment of planar 3-RRR flexible parallel robots is implemented in this paper. Considering the direct and inverse piezoelectric effect of PZT material, a general motion equation is established. A strain rate feedback controller is designed based on the established general motion equation. Four control schemes are designed in this experiment: three passive flexible links are controlled at the same time, only passive flexible link 1 is controlled, only passive flexible link 2 is controlled, and only passive flexible link 3 is controlled. The experimental results show that only one flexible link controlled scheme suppresses elastic vibration and cannot suppress the elastic vibration of the other flexible links, whereas when three passive flexible links are controlled at the same time, they are able to effectively suppress the elastic vibration of all of the flexible links. In general, the experiment verifies that a strain rate feedback controller is able to effectively suppress the elastic vibration of the flexible links of plane 3-RRR flexible parallel robots.

Physical variational principle in dissipative media

The physical variational principle (PVP) is a physical principle which is implied in the thermodynamic laws. For a conservative system, the PVP is implied in the first thermodynamic law and gives the motion equation. But for a dissipative system, PVP is implied in the extended Gibbs equation, which is the result of the first and second thermodynamic laws. The precision of the PVP in a dissipative system is in the same order of the Gibbs equation. The dissipative work and its converted internal irreversible heat are simultaneously included in the PVP to get the governing equation and the boundary condition of the dissipated variables. The “generalized motion equations” including governing equations of the mechanical momentum and thermoelastic, thermal viscoelastic, thermal elastoplastic, linear thermoelastic diffusive and linear electromagnetic thermoelastic materials etc. can be derived by the PVP of dissipative media in this paper. The conservative system is the special case of the dissipative system. Other than the mathematical variational principle, which is obtained by a known governing equation, the PVP is used to deduce the governing equation. The PVPs including the hyperbolic temperature wave equation with a finite phase speed are also discussed shortly.

Vibration Analysis of Flexible Structural Components

This article deals with the vibration analysis of flexible structural or mechanical components. The purpose of this article is to derive the general nonlinear differential equations of motion regarding the free vibration of flexible members of uniform and/or variable cross section along their length. Unique solution methodologies are developed that simplify the solution of very complex problems.

Gauge Physics of Spin Hall Effect.

Spin Hall effect (SHE) has been discussed in the context of Kubo formulation, geometric physics, spin orbit force, and numerous semi-classical treatments. It can be confusing if the different pictures have partial or overlapping claims of contribution to the SHE. In this article, we present a gauge-theoretic, time-momentum elucidation, which provides a general SHE equation of motion, that unifies under one theoretical framework, all contributions of SHE conductivity due to the kinetic, the spin orbit force (Yang-Mills), and the geometric (Murakami-Fujita) effects. Our work puts right an ambiguity surrounding previously partial treatments involving the Kubo, semiclassical, Berry curvatures, or the spin orbit force. Our full treatment shows the Rashba 2DEG SHE conductivity to be instead of −, and Rashba heavy hole instead of −. This renewed treatment suggests a need to re-derive and re-calculate previously studied SHE conductivity.

Spin-axion coupling

We establish a new covariant phenomenological model, which describes an influence of pseudoscalar (axion) field on spins of test massive particles. The model includes general relativistic equations of particle motion and spin evolution in background pseudoscalar (axion), electromagnetic and gravitational fields. It describes both the direct spin-axion coupling of the gradient type and indirect spin-axion interaction mediated by electromagnetic fields. Special attention is paid to the direct spin-axion coupling caused by the gradient of the pseudoscalar (axion) field. We show that it describes a spin precession, when the pseudoscalar (axion) field is inhomogeneous and/or non-stationary. Applications of the model, which correspond to the three types of four-vectors attributed to the gradient of the pseudoscalar (axion) field (time-like, space-like, and null), are considered in detail. These are the spin precessions induced by relic cosmological axions, axions distributed around spherically symmetric static objects, and axions in a gravitational wave field, respectively. We discuss features of the obtained exact solutions and some general properties of the axionically induced spin rotation.

Modal Analyses of a Thin Shell with Constrained Layer Damping (CLD) Based on Rayleigh-Ritz Method

Foreign Object Debris (FOD) is debris or article alien, which may This paper presents analytical results of natural frequencies and loss factors of node-diameter and node-circumferential modes of a thin shell treated with constrained layer damping (CLD) based on Rayleigh-Ritz method. General differential equations of motion of the thin CLD shell are derived firstly by following the Donnell-Mushtari shell theory. By taking the beam characteristic functions as the admissible functions, the Rayleigh-Ritz method is employed to deduce the higher degree equations of natural frequencies of the thin CLD shell under different boundary conditions. It is confirmed that the present method is accurate and convenient so that it is applicable to the thin CLD shell compared with classic analytic method or transfer matrix method. Several examples are achieved and compared to illustrate the effects of the viscoelastic material (VEM) and constrain layer’s thickness ratios on the natural frequencies and modal loss factors, besides the effect of boundary conditions. The results show that the thickness ratio of VEM affects sensitively the modal frequencies and the total damping capacity of the thin CLD shell.

Nonlinear dynamics of nano-resonators: an analytical approach

Prior to the design and fabrication of MEMS/NEMS devices, analysis of static and dynamic behaviors of these systems is necessary. In the present study, the nonlinear dynamic behavior of micro- and nano-mechanical resonators is investigated and classified based on the resonator's physical parameters for first time. The Galerkin method is used to convert the distributed-parameter model to a nonlinear ordinary differential equation where mid-plane stretching, axial stress, DC electrostatic and AC harmonic voltages are taken into account. To obtain the analytical frequency response of the micro resonator near its primary resonance, the second order multiple scales method is applied to the general equation of motion with cubic, quadratic and parametric nonlinearities. It is demonstrated that variation of the micro resonator's physical parameters strongly affects its dynamic behavior by changing equilibrium points and their stability properties and complex behaviors appear in its frequency and phase responses. The global dynamics of the resonator is classified into four different categories in terms of the system parameters in this paper. The dynamic characteristics and frequency response of each class are analyzed numerically as well as analytically. Comparison of the obtained closed-form solution with the numerical simulation results confirms its validity. A striking point of the obtained closed-form solution is that it predicts some complex nonlinear behaviors of the resonator. This paper presents a quick and efficient method for determining the global dynamics of the micro-resonators and can be useful in design and analyses of these devices.

Application of the Udwadia–Kalaba approach to tracking control of mobile robots

Udwadia–Kalaba approach which presents a new, general and explicit equation of motion for constrained mechanical systems with holonomic or nonholonomic constraints is applied to the trajectory tracking control of the mobile robot in this paper. Unlike any other nonlinear control methods, the inspiration for this methodology which does not make any linearization or approximations comes from a different, though closely allied, field, namely analytical dynamics. The control torques required to control the mobile robot so that it precisely satisfies the trajectory requirements which are represented by an arbitrary (sufficiently smooth) function of time are obtained explicitly and in closed form by solving Udwadia–Kalaba equation. Numerical simulations are performed to show the simplicity, efficacy and accuracy of this closed-form method.

Relative attitude and position estimation for a tumbling spacecraft

This paper presents a novel relative attitude and position estimation approach for a tumbling spacecraft. It is assumed that the tumbling chief spacecraft is in failure or out of control and there is no a priori rotation rate information. The Euler's rotational dynamics is used to propagate the chief angular velocity, and the unknown inertia parameter circumstance is also considered. The integrated sensor suit comprises a rate-integrating gyro and a vision-based navigation system on the deputy spacecraft. Two relative quaternions that map the chief Local Vertical Local Horizontal (LVLH) frame to the deputy and chief body frames are involved to construct the line-of-sight observations. Therefore, the assumption that the chief body frame coincides with its LVLH frame in the traditional algorithm can be released. The general relative equations of motion for eccentric orbits are used to describe the positional dynamics. An extended Kalman filter is derived to estimate the relative quaternions, relative position and velocity, deputy gyro bias, as well as chief angular velocity and inertia ratios. Simulation results verify the validity and feasibility of the proposed algorithm.

Unscented Kalman Filtering for Relative Spacecraft Attitude and Position Estimation

A novel relative spacecraft attitude and position estimation approach based on an Unscented Kalman Filter (UKF) is derived. The integrated sensor suite comprises the gyro sensors on each spacecraft and a vision-based navigation system on the slave spacecraft. In the traditional algorithm, an assumption that the master's body frame coincides with its Local Vertical Local Horizontal (LVLH) frame is made to construct the line-of-sight observations for convenience. To solve this problem, two relative quaternions that map the master's LVLH frame to the slave and master body frames are involved. The general relative equations of motion for eccentric orbits are used to describe the positional dynamics. The implementation equations for the UKF are derived. A modified version of the UKF is presented based on the averaging-quaternion algorithm. Simulation results indicate that the proposed filters provide more accurate estimates of relative attitude and position than the Extended Kalman Filter (EKF).

Why can a free-falling cat always manage to land safely on its feet?

Udwadia–Kalaba equation is a simple, aesthetic and thought-provoking description of the world at a very fundamental level. It is about the way systems of bodies move. We creatively apply the Udwadia–Kalaba approach to study falling cat’s movements. The cat is modeled as a constrained discrete dynamical system. In an alternative way, Udwadia–Kalaba formulation is used for analysis of the falling cat’s dynamics. With this novel approach, we can easily obtain the dynamical model and get the explicit analytic form of the general equations of motion of the falling cat. The surprise phenomenon (that a cat when dropped at rest with its feet pointing up can always manage to right itself and land safely on its feet) is observed through numerical simulation based on the constructed dynamical model. Unmatched ease, clarity and elegance of the Udwadia–Kalaba formulation for solving the falling cat problem (constrained discrete dynamical system or multibody system) are presented.

Free vibration analysis of size-dependent shear deformable functionally graded cylindrical shell on the basis of modified couple stress theory

In this paper, size-dependent equations of motion for functionally graded cylindrical shell were developed using shear deformation model and rotation inertia. Material properties of the shell were assumed as continuously variable along thickness, and consistent with the variation in the component’s volume fraction based on power law distribution. To consider the size effect, modified couple stress theory in conjunction with first order shear deformation shell model were used, and general equations of motion and classical and non-classical boundary conditions were derived based on Hamilton’s principle. Finally, in the special case, using the Navier procedure, the free vibrations of simply supported functionally graded cylindrical nanoshell were obtained, and the effects of parameters such as dimensionless length scale parameter, distribution of FG properties, thickness, and length on the natural frequency were identified and was compared with the classical theory. Results obtained through the modified couple stress theory are indicative of the considerable effect of the size parameter, particularly in bigger thicknesses and shorter lengths of nanotubes, on the natural frequency.

Experimental and Numerical of Sloshing Effect on Heave and Pitch Motions of FLNG Vessel

Recently the demand of sloshing analysis is rising for building FLNG (Floating Liquefied Natural Gas) vessel. This study considers the experimental and numerical observations on strongly nonlinear sloshing flows in ship motion. The FLNG (Floating Liquefied Natural Gas) vessel was considered to be rigid body supported by non-permanent pole with distributed spring damper. Based on the general equation of the ship motion in waves, and various wave amplitude, various heading, wave period and critical fluid level on the cargo tank period governing equation induced by sloshing were derived. Several physical issues are introduced in the analysis of sloshing flows, and the corresponding numerical models are described. To study the sloshing effects on ship motion, a ship motion program based on impulsive response function (IRF) is coupled with the developed numerical models for sloshing analysis. The results show that the nonlinearity of sloshing-induced forces and moments plays a critical role in the coupling effects.

Orbital dynamics of the gravitational field of stringy black holes

In this paper, we study the orbital dynamics of the gravitational field of stringy black holes by analyzing the effective potential and the phase plane diagram. By solving the equation of Lagrangian, the general relativistic equations of motion in the gravitational field of stringy black holes are given. It is easy to find that the motion of test particles depends on the energy and angular momentum of the test particles. Using the phase plane analysis method and combining the conditions of the stability, we discuss different types of the test particles' orbits in the gravitational field of stringy black holes. We get the innermost stable circular orbit which occurs at rmin= 5.47422 and when the angular momentum b ≤ 4.3887 the test particles will fall into the black hole.

Cubature Kalman filtering for relative spacecraft attitude and position estimation

A novel relative spacecraft attitude and position estimation approach based on cubature Kalman filter is derived. The integrated sensor suit comprises the gyro sensors on each spacecraft and a vision-based navigation system on the deputy spacecraft. In the traditional algorithm, an assumption that the chief׳s body frame coincides with its Local Vertical Local Horizontal (LVLH) frame is made to construct the line-of-sight observations for convenience. To solve this problem, two relative quaternions that map the chief׳s LVLH frame to the deputy and chief body frames are involved. The general relative equations of motion for eccentric orbits are used to describe the positional dynamics. The implementation equations for the cubature Kalman filter are derived. Simulation results indicate that the proposed filter provides more accurate estimates of relative attitude and position over than the extended Kalman filter.

Dissipation in Josephson tunneling junctions at low temperatures

It is important to know the decoherence mechanism of a qubit based on Josephson junctions. At low temperatures, as quasiparticle concentration becomes exponentially small, one needs to consider energy transfer from tunneling electrons to other degrees of freedom to find dissipation in Josephson junctions and decoherence in qubits. Here we discuss the energy transfer to two-level systems, i.e. the transitions between two different configurations of ions inside insulating layer separated by a potential barrier. We derive a general equation of motion for the phase difference between two superconducting electrodes and we find a retarded dissipation term due to electromagnetic mechanism and also contribution due to electron tunneling mechanism. Using the equation of motion we calculate the decay of Rabi oscillations and frequency shift in qubits due to the presence of the two-level systems. In the long time limit our results coincide with those obtained by Martinis et al. [J. M. Martinis et al.. Phys. Rev. Lett. 95, 210503 (2005)] within the Fermi's Golden rule approach up to a numerical factor.

Analytical method of determination of the general equation of absolute rotational speed of planetary gear trains

This is a simple analytical method for determining the general equation of absolute rotational speed of planetary gear trains. This method is based upon the general equation of motion of planetary gear trains (Krstich, 1985a, 1987b). The basic absolute rotational speeds for simple and compound planetary differential conical (bevel) and cylindrical (spur and helical) gear trains have been derived, and then the general equation of absolute rotational speed of planetary gear trains. This equation can be applied to any types of simple and complex arrangement of planetary gear trains by using only train values (gear ratios) of the planetary differential two–gear train or the planetary differential pair of gears (Krstich, 1985b).

Free vibration of FGM Lévy conical panels

Free vibration analysis of open conical panels made of through-the-thickness functionally graded materials (FGMs) is analyzed in this research. Mechanical properties of the shell are distributed across the thickness based on the power law function. First order shear deformation theory of shells accompanied with the Donnell type of kinematic assumptions are used to establish the general motion equations and the associated boundary conditions. Considering the Lévy type of conical shells, which are simply supported on straight edges, a semi-analytical solution based on the trigonometric expansion through the circumferential direction combined with generalized differential quadrature (GDQ) discretization in meridional direction is developed. Due to the special configuration of the conical panel, free vibration of flat rectangular plates, flat annular and circular sectoral plates, and singly curved cylindrical panels may also be studied. Solution method is useful for any arbitrary type of free, clamped, and simply supported boundary conditions along the two other curved edges of the shell. After establishment of eigenvalue problem, a series of comparison studies are conducted to assure the validity and accuracy of the present solution. Afterwards, parametric studies are developed to examine the influences of boundary conditions, semi-vertex angle of the cone, subtended angle of the panel, power law index, and thickness to radius ratio. Influences of the aforementioned parameters on natural frequencies and the associated mode shapes are discussed.