Rotational Equations Of Motion Research Articles

Dynamic Stability Analysis of Periodically Time-Varying Rotor System with a Transverse Crack

Fatigue cracks may appear in horizontal rotating machinery due to periodic stresses imposed to its shaft. The investigation of stability behavior of cracked rotors can lead to proper diagnosis of machinery and to prevent possible accidents caused by the rotor failure. In this study, the dynamic stability of a rotor with a transverse crack is investigated. Models of both open and breathing cracks are developed and then used in the model of a cracked Jeffcot (de Laval) rotor. The stability of rotor motion equations represented by differential equations with periodic coefficients is investigated using Floquet theory. While both crack models show instability regions around the first un-damped frequency, sub-harmonic regions are predicted by the breathing crack models. Compared to perturbation methods frequently used to determine the stability regions, the transition matrix approach used in this study can be applied to complex models of rotors and consequently may help in the identification of cracks in rotating machinery.

Stabilizing a Bicycle: A Modeling Project

This article is a project that takes students through the process of forming a mathematical model of bicycle dynamics. Beginning with basic ideas from Newtonian mechanics (forces and torques), students use techniques from calculus and differential equations to develop the equations of rotational motion for a bicycle-rider system as it tips from side to side. Then students are led through a self-discovery process which uses a computer model to test various attempts to “ride” the bicycle so as to keep it upright and traveling along a straight path.

Dynamic simulation of a beta-type Stirling engine with cam-drive mechanism via the combination of the thermodynamic and dynamic models

Dynamic simulation of a beta-type Stirling engine with cam-drive mechanism used in concentrating solar power system has been performed. A dynamic model of the mechanism is developed and then incorporated with the thermodynamic model so as to predict the transient behavior of the engine in the hot-start period. In this study, the engine is started from an initial rotational speed. The torques exerted by the flywheel of the engine at any time instant can be calculated by the dynamic model as long as the gas pressures in the chambers, the mass inertia, the friction force, and the external load have been evaluated. The instantaneous rotation speed of the engine is then determined by integration of the equation of rotational motion with respect to time, which in return affects the instantaneous variations in pressure and other thermodynamic properties of the gas inside the chambers. Therefore, the transient variations in gas properties inside the engine chambers and the dynamic behavior of the engine mechanism should be handled simultaneously via the coupling of the thermodynamic and dynamic models. An extensive parametric study of the effects of different operating and geometrical parameters has been performed, and results regarding the effects of mass moment of inertia of the flywheel, initial rotational speed, initial charged pressure, heat source temperature, phase angle, gap size, displacer length, and piston stroke on the engine transient behavior are investigated.

Runaway transient simulation of a model Kaplan turbine

The runaway transient is a typical transient process of a hydro power unit, where the rotational speed of a turbine runner rapidly increases up to the runaway speed under a working head as the guide vanes cannot be closed due to some reason at the load rejection. In the present paper, the characteristics of the runaway transient of a model Kaplan turbine having ns = 479(m-kW) is simulated by using a time-dependent CFD technique where equation of rotational motion of runner, continuity equation and unsteady RANS equations with RNG k-ϵ turbulence model are solved iteratively. In the calculation, unstructured mesh is used to the whole flow passage, which consists of several sub-domains: entrance, casing, stay vanes + guide vanes, guide section, runner and draft tube. And variable speed sliding mesh technique is used to exchange interface flow information between moving part and stationary part, and three-dimensional unstructured dynamic mesh technique is also adopted to ensure mesh quality. Two cases were treated in the simulation of runaway transient characteristics after load rejection: one is the rated operating condition as the initial condition, and the other is the condition at the maximum head. Regarding the runaway speed, the experimental speed is 1.45 times the initial speed and the calculation is 1.47 times the initial for the former case. In the latter case, the experiment and the calculation are 1.67 times and 1.69 times respectively. From these results, it is recognized that satisfactorily prediction will be possible by using the present numerical method. Further, numerical results show that the swirl in the draft-tube flow becomes stronger in the latter part of the transient process so that a vortex rope will occur in the draft tube and its precession will cause the pressure fluctuations which sometimes affect the stability of hydro power system considerably.

Averaged rotational dynamics of an asteroid in tumbling rotation under the YORP torque

The secular effect of YORP torque on the rotational dynamics of an asteroid in non-principal axis rotation is studied. The general rotational equations of motion are derived and approximated with an illumination function expanded up to second order. The resulting equations of motion can be averaged over the fast rotation angles to yield secular equations for the angular momentum, dynamic inertia and obliquity. We study the properties of these secular equations and compare results to previous research. Finally, an application to several real asteroid shapes is made, in particular we study the predicted rotational dynamics of the asteroid Toutatis, which is known to be in a non-principal axis state.

Stability analysis of rough journal bearings under TEHL with non-Newtonian lubricants

The combined effects of surface roughness and lubricants rheology on stability of a rigid rotor supported on finite journal bearing under thermal elastohydrodynamic lubrication have been investigated using the transient method. The newly derived time dependent modified Reynolds and the adiabatic energy equations were formulated using a non-Newtonian Carreau viscosity model. The simultaneous systems of modified Reynolds equation, elasticity equation, energy equation, and the rotor motion equation with initial conditions were solved numerically using multigrid multi-level method with full approximation technique. From the characteristic equation, the instability threshold is then obtained with various surface roughness parameters and the elastic modulus of the bearing liner materials. The results show that stability of the bearing system deteriorates with decreasing both the power law exponent and the elastic modulus of bearing liner material. The rough surface journal bearing with transverse pattern under TEHL regime exhibits better stability when compared with the rough surface journal bearing with longitudinal pattern.

Numerical Simulation of Autorotation in Forward Flight

Autorotation in forward flight is an important subject for gyroplanes or compound helicopters. By integrating the flapping and rotational equations of motion, we investigate in this work the steady-state autorotation of a three-bladed rotor for a given set of three independent variables: forward velocity, collective pitch angle, and shaft angle. Two-dimensional aerodynamic coefficients as functions of the angle of attack and the Reynolds number from the Navier-Stokes simulation and Pitt/Peters inflow model to determine induced flowfield are adopted to implement the simulation. Transient behavior of the solution from arbitrary initial conditions to a steady-state periodic solution is described. Variations of rotor speed and flapping angle of autorotation with the shaft angle and the collective pitch angle for a given forward velocity are presented and discussed in detail. Drastic changes in the region of autorotation near a specific shaft angle is found to exist, below which the range of collective pitch angle for stable autorotation becomes very limited and the flapping angle changes abruptly with the pitch angle.

Effect of the lubricating layer on rotor stability in sliding bearings

The structure of the forces acting from the lubricating layer on the rotor during nonstationary lubricant flow in sliding bearings is discussed. A closed system of differential nonlinear equations of rotor motion in the sliding bearing is obtained. Based on the closed system of equations, the problem of stabilization of the rotor motion in the bearing is solved.

Rotational Motion Control by Feedback with Minimum L1-Norm

Rotational Motion Control by Feedback with Minimum L1-Norm

Basic Relations for Studying the Influence of Geophysical Processes on the Earth’s Rotation: The Angular Momentum Approach

Relations to study the influence of geophysical processes on the temporally varying rotation of the Earth are considered. Liouville’s equations of rotational motion are derived for a two-component Earth model (consisting of a solid mantle and a fluid core) and suitably simplified for calculations of the influence of mass redistributions on the Earth’s rotational behaviour. Excitation functions, or effective angular momentum functions, describing the influence of mass redistributions on the equations of rotational motion are derived, and their calculation is elucidated by some examples. Relations between temporally varying second degree Stokes coefficients of the gravity field and excitation functions are discussed. Different solutions of the equations of rotational motion are described. The identification of exciting geophysical processes by the kinematics of the inverse calculated excitation function is portrayed.

자동회전의 비정상 수치해법과 2차원 공력계수의 영향

전진 비행중인 자동회전 로터에 대한 비정상 수치해석 기법이 개발되었다. 자동회전 로터의 플래핑과 회전운동 방정식은 주어지는 시간간격에 따라 연속적으로 적분되며 이 때 회전면에서의 유도 속도장은 동적 유도흐름 이론(dynamic inflow theory)에 의해 계속 업데이트 되면서 블레이드의 모든 요소에서 유효 받음각을 결정하여 비정상 상태를 모사한다. 로터의 임의의 초기 회전속도 및 플래핑각에서 정상상태로 천이(transition)되는 과정을 시뮬레이션 하였고 블레이드 에어포일 공력 데이터들을 이용하여 방정식들의 수치해인 자동회전의 정상상태를 예측하였다. 2차원 Navier-Stokes 솔버로 받음각과 레이놀즈수에 따라 해석된 에어포일 데이터를 이용하여 자동회전의 비정상 시뮬레이션을 수행한 해석 결과는 풍동실험 결과와 잘 일치하였다 An unsteady numerical simulation method for an autorotating rotor in forward flight was developed. The flapping and rotational equations of motion of autorotation are continuously integrated for given time steps, meanwhile the induced velocity field at disc plane is obtained by the dynamic inflow theory embodying the unteadiness. The transitions from arbitrary initial states to equilibrium states were simulated. Steady autorotations as numerical solutions of equations were predicted by using two sources of blade airfoil data. The simulations using airfoil data which were obtained by a two dimensional Navier-Stokes solver in terms of angles of attack and Reynolds numbers have shown good agreements with wind tunnel experimental results.

Six-Degree-of-Freedom Trajectory Optimization Using a Two-Timescale Collocation Architecture

Six-degree-of-freedom (6DOF) trajectory optimization of a reentry vehicle is solved using a two-timescale collocation methodology. This class of 6DOF trajectory problems are characterized by two distinct timescales in their governing equations, where a subset of the states have high-frequency dynamics (the rotational equations of motion) while the remaining states (the translational equations of motion) vary comparatively slowly. With conventional collocation methods, the 6DOF problem size becomes extraordinarily large and difficult to solve. Utilizing the two-timescale collocation architecture, the problem size is reduced significantly. The converged solution shows a realistic landing profile and captures the appropriate high-frequency rotational dynamics. A large reduction in the overall problem size (by 55%) is attained with the two-timescale architecture as compared to the conventional single-timescale collocation method. Consequently, optimum 6DOF trajectory problems can now be solved efficiently using collocation, which was not previously possible for a system with two distinct timescales in the governing states.

Rotating magnetic particle microrheometry in biopolymer fluid dynamics: Mucus microrheology

The polymer properties of canine mucus were investigated through the method of rotating magnetic particle microrheometry. Mucus is visualized as a physically entangled biopolymer of low polydispersity in a water-based solution. Mucus was modeled according to the constitutive law of a Doi-Edwards fluid. The magnetic-particle equation of rotational motion is analytically solved in the linear viscoelastic limit rendering theoretical flow profiles which are used to fit the experimental trace signals of the particle remanent-magnetic-field decay. The zero-shear-rate viscosity was found to be 18,000 P and the relaxation time at about 42 s. The molecular weight between entanglements for mucins was estimated at 1.7 MDa rendering an estimation of about seven physical cross-links per molecule. Rheological investigations were extended also to diluted and concentrated rations of the normal mucus simulating the conditions found in more physiological extremes.

Studies on the Internal Fault Simulations of a High-Voltage Cable-Wound Generator

This paper discusses the set up of a mathematical model of the powerformer, a new type of salient-pole synchronous machine, for analyzing internal phase and ground faults in stator windings. The method employs a direct-phase representation considering the cable capacitance. To effectively implement the internal fault simulation, the magnetic axis locations of fault parts are arranged appropriately. Moreover, all machine windings supposed to be sinusoidally distributed in space and the system are magnetically linear. With the above-mentioned assumptions, the current-equivalent equations, voltage-equivalent equations, and the rotor-motion equations are formed and combined to implement the fault simulations. Simulation results showing the fault currents, during a single-phase-to-ground fault, a two-phase-to-ground fault, and a phase-to-phase fault, are presented here. With the data generated by this internal fault simulation model, the protection scheme used for the powerformer can be validated and improved accordingly.

Design of the Starting Device Installed in the Single-Phase Switched Reluctance Motor

The single-phase switched reluctance motor (SRM) generates the positive torque in the restricted section. It cannot start by itself and the torque ripple is heavier than multiphase SRM as the positive torque is not generated in all the sections. For self-starting and fixing rotating direction, the rotor should be placed at the rising inductance slope when stationary. This paper proposes the design process of the starting device which puts the rotor in the starting position before it is started and design the starting device composed of the parking permanent magnet for the prototype single-phase SRM. The equation of the magnetic attractive force required at the acting point is derived from the equation of rotational motion and calculated by the experiment with the prototype. Finally, the proper position and magnetic attractive force of the permanent magnet as well as the starting torque is confirmed through the FEM analysis

A differential-geometric approach for 2D and 3D object grasping and manipulation

This article presents an expository work on a differential-geometric treatment of fundamental problems of 2D and 3D object grasping and manipulation by a pair of robot fingers with multi-joints under holonomic or nonholonomic constraints. First, Lagrange’s equation of motion of a fingers-object system whose motion is confined to a vertical plane is derived under holonomic constraints when rolling contacts between finger-ends and object surfaces are permitted. Then, a class of control signals called “blind grasping” and constructed without knowing the object kinematics or using any external sensing like vision or tactile sensation is shown to realize stable object grasping in a dynamic sense. Stability of motion and its convergence to an equibrium manifold are treated on the basis of differential geometry of solution trajectories of the closed-loop dynamics on the constraint manifolds. Second, a mathematical model of 3D object grasping and manipulation by a pair of multi-joint robot fingers is derived under the assumption that spinning motion of rotation around the opposing axis between contact points does no more arise. It is shown that, differently from the 2D case, the instantaneous axis of rotation of the object is time-varying, which induces a nonholonomic constraint expressed as a linear differential equation of rotational motion of the pinched object. It is shown that there is a class of control signals constructed without knowing the object kinematics or using external sensings that can realize “blind grasping” in a dynamic sense. Finally, it is shown that the proposed differential geometric treatment of stability can naturally cope with redundancy resolution problems of surplus degrees-of-freedom (d.f.) of the overall fingers-object system, which is closely related to Bernstein’s d.f. problem.

Analytical attitude propagation with non-singular variables and gravity gradient torque for spin stabilized satellite

An analytical approach for the spin stabilized satellite attitude propagation is presented using the non-singular canonical variables to describe the rotational motion. Two sets of variables were introduced for Fukushima in 1994 by a canonical transformation and they are useful when the angle between z-satellite axis of a coordinate system fixed in artificial satellite and the rotational angular momentum vector is zero or when the angle between Z-equatorial axis and rotation angular momentum vector is zero. Analytical solutions for rotational motion equations and torque-free motion are discussed in terms of the elliptic functions and by the application of some simplification to get an approximated solution. These solutions are compared with a numerical solution and the results show a good agreement for many rotation periods. When the mean Hamiltonian associated with the gravity gradient torque is included, an analytical solution is obtained by the application of the successive approximations’ method for the satellite in an elliptical orbit. These solutions show that the magnitude of the rotation angular moment is not affected by the gravity gradient torque but this torque causes linear and periodic variations in the angular variables, long and short periodic variations in Z-equatorial component of the rotation angular moment and short periodic variations in x-satellite component of the rotation angular moment. The goal of this analysis is to emphasize the geometrical and physical meaning of the non-singular variables and to validate the approximated analytical solution for the rotational motion without elliptic functions for a non-symmetrical satellite. The analysis can be applied for spin stabilized satellite and in this case the general solution and the approximated solution are coincidence. Then the results can be used in analysis of the space mission of the Brazilian Satellites.

Impulsive control for angular momentum management of tumbling spacecraft

We discuss an angular momentum control of a tumbling spacecraft. The proposed control method is to apply an impulse by a space robot arm, to measure and control the relative position and attitude between the target spacecraft, and then to apply another impulse until the rotational motion of the target spacecraft is well damped. A discrete controller is designed using the simplified equations of rotational motion through appropriate coordinate transformation. The stationary response under contact model uncertainty is investigated and stability condition is analytically derived. Numerical simulations are given to validate the proposed approach.

Flow around spheres by dissipative particle dynamics

The dissipative particle dynamics (DPD) method is used to study the flow behavior past a sphere. The sphere is represented by frozen DPD particles while the surrounding fluids are modeled by simple DPD particles (representing a Newtonian fluid). For the surface of the sphere, the conventional model without special treatment and the model with specular reflection boundary condition proposed by Revenga et al. [Comput. Phys. Commun. 121–122, 309 (1999)] are compared. Various computational domains, in which the sphere is held stationary at the center, are investigated to gage the effects of periodic conditions and walls for Reynolds number (Re)=0.5 and 50. Two types of flow conditions, uniform flow and shear flow are considered, respectively, to study the drag force and torque acting on the stationary sphere. It is found that the calculated drag force imposed on the sphere based on the model with specular reflection is slightly lower than the conventional model without special treatment. With the conventional model the drag force acting on the sphere is in better agreement with experimental correlation obtained by Brown and Lawler [J. Environ. Eng. 129, 222 (2003)] for the case of larger radius up to Re of about 5. The computed torque also approaches the analytical Stokes value when Re<1. For a force-free and torque-free sphere, its motion in the flow is captured by solving the translational and rotational equations of motion. The effects of different DPD parameters (a, γ, and σ) on the drag force and torque are studied. It shows that the dissipative coefficient (γ) mainly affects the drag force and torque, while random and conservative coefficient have little influence on them. Furthermore the settling of a single sphere in square tube is investigated, in which the wall effect is considered. Good agreement is found with the experiments of Miyamura et al. [Int. J. Multiphase Flow 7, 31 (1981)] and lattice-Boltzmann simulation results of Aidun et al. [J. Fluid Mech. 373, 287 (1998)].

Computer simulation of the stochastic dynamics of super-paramagnetic particles in ferrofluids

Several papers have been written on the complex problem of the stochastic dynamics of the magnetic moments of super-paramagnetic particles, simultaneously with the stochastic rotation of these colloidal particles in a ferrofluid [1-3]. None of these works, however, is sufficiently general and conveniently simple and clear to be used in sumulational works to appropriately describe the experimental super-paramagntetic resonance lines. We have a new proposal for the equations of rotational motion, which is appropriate for simulations. Those equations are stochastic differential equations with multiplicative noise. Therefore, they have to be interpreted as Stratonovich-Langevin equations and the roles of stochastic calculus have to be used in the simulations. For this reason we will briefly present the essence of the numerical algorithms used in the solutions of Stratonovich equations. Finally, the simulational results for the magnetic response functions, and the corresponding dynamic susceptibilities, will be shown and their consequences will be analyzed.