Equations Of Angular Motion Research Articles

Spherical Particle Orbits around a Rotating Black Hole in Massive Gravity

In this paper, we present a rotating de Rham–Gabadadze–Tolley black hole with a positive cosmological constant under massive gravity, achieved by applying a modified Newman–Janis algorithm. The black hole exhibited stable orbits of constant radii, prompting a numerical study of the behavior of the solutions to a nonic equation governing the radii of planar orbits around the black hole. Additionally, we investigated the stability of orbits near the event horizon and provide a comprehensive analytical examination of the solutions to the angular equations of motion. This was followed by a simulation of some spherical particle orbits around the black hole.

Theoretical investigation of braking of tearing mode rotation by resistive walls in ITER

The locking of the 2/1 tearing mode to the resistive wall in the ITER tokamak (15 MA inductive scenario 2) is investigated theoretically using a cylindrical asymptotic matching model. The model takes into account the fact that ITER plasmas will effectively be surrounded by two walls; the inner blanket module layer with a time constant of about 23 ms, and the outer vacuum vessel with a time constant of about 380 ms. The model also takes cognizance of the fact that neither the blanket module layer nor the vacuum vessel can be accurately described as “thin” walls (in the ordinarily accepted sense). The model incorporates changes in both the plasma poloidal and the toroidal angular velocity profiles, in response to the electromagnetic braking torque that develops at the rational surface, because it turns out that neoclassical poloidal flow-damping is not strong enough to completely suppress changes in the poloidal velocity. Finally, the model accurately calculates changes in the poloidal and toroidal plasma angular velocity profiles by evolving the full angular equations of motion, taking the electromagnetic braking torque, plasma inertia, plasma viscosity, and poloidal flow-damping into account. The time required for the 2/1 tearing mode to grow from a small amplitude to a sufficient one to lock to the walls is found to be about 3.5 s. The critical full radial island width at which wall locking is triggered is found to be about 9% of the plasma minor radius.

Modeling and validation of crankshaft speed fluctuations of a single-cylinder four-stroke diesel engine

In this study, a dynamic model of a single-cylinder four-stroke diesel engine has been created, and the crankshaft speed fluctuations have been simulated and validated. The dynamic model of the engine consists of the motion equations of the piston, conrod, and crankshaft. Conrod motion was modeled by two translational and one angular motion equations, by considering the kinetic energy resulted from the mass moment of inertia and conrod mass. Motion equations involve in-cylinder gas pressure forces, hydrodynamic and dry friction, mass inertia moments of moving parts, starter moment, and external load moment. The In-cylinder pressure profile used in the model was obtained experimentally to increase the accuracy of the model. Pressure profiles were expressed mathematically using the Fourier series. The motion equations were solved by using the Taylor series method. The solution of the mathematical model was performed by coding in the MATLAB interface. Cyclic speed fluctuations obtained from the model were compared with experimental results and found compitable. A validated model was used to analyze the effects of in-cylinder pressure, mass moment of inertia of crankshaft and connecting rod, friction, and piston mass. In experiments for 1500, 1800, 2400, and 2700 rpm engine speeds, crankshaft speed fluctuations were observed as 12.84%, 8.04%, 5.02%, and 4.44%, respectively. In simulations performed for the same speeds, crankshaft speed fluctuations were calculated as 10.45%, 7.56%, 4.49%, and 3.65%. Besides, it was observed that the speed fluctuations decreased as the average crankshaft speed value increased. In the simulation for 157.07, 188.49, 219.91, 251.32, and 282.74 rad/s crankshaft speeds, crankshaft speed fluctuations occurred at rates of 10.45%, 7.56%, 5.84%, 4.49%, and 3.65%, respectively. The effective engine power was achieved as 5.25 kW at an average crankshaft angular speed of 219.91 rad/s. The power of friction loss in the engine was determined as 0.68 kW.

An alternative formulation for modeling self-excited oscillations of rotary drilling systems

This paper describes an alternative formulation of the nonlinear equations governing the coupled axial and torsional dynamics of a discrete model of a rotary drilling system equipped with a drag bit. This model assumes a rate-independent bit/rock interface law that accounts for both cutting and frictional contact processes. The regenerative effect associated with the cutting introduces a term with delay, and thus a feedback, in the equations of motion, while the frictional contact results in set-valued contact forces and discontinuities in the boundary conditions, which are responsible for the occurrence of axial and torsional stick-slip oscillations. Inspired by P. Wahi and A. Chatterjee (Self-interrupted regenerative metal cutting in turning, International Journal of Non-Linear Mechanics 43:2, 111–123, 2008), the regenerative effect is captured by a bit trajectory function, whose evolution is governed by a first order partial differential equation. With this approach, the original state-dependent delay differential equations (SDDDEs) governing the dynamics of the discrete model are replaced by the partial differential equation of the bit trajectory function together with the axial and angular equations of motion of the two degrees-of-freedom model. The equation governing the evolution of the bit trajectory function is further approximated by a system of first order ordinary differential equations through application of the Galerkin method. The introduction of the bit trajectory function makes it possible to consider a bit with two state-dependent delays (and in principle multiple delays) without reducing the computational efficiency. The accuracy of this approximation is validated using published results, and the convergence and robustness properties of the proposed method are analyzed using three cases corresponding to different regimes of instability. Finally, we extend the discrete model to a more generic lumped-parameter model with multiple degrees-of-freedom, which captures the propagation of axial and torsional vibrations along the drillstring.

A new collision model for ellipsoidal particles in shear flow

All natural and a growing number of manufactured solid particles are non-spherical. Interesting fluid–particle dynamics applications include the transport of granular material, piling of seeds or grains, inhalation of toxic aerosols, use of nanofluids for enhanced cooling or improved lubrication, and optimal drug-targeting of tumors. A popular approach for computer simulations of such scenarios is the multi-sphere (MS) method, where any non-spherical particle is represented by an assemblage of spheres. However, the MS approach may lead to multiple sphere-to-sphere contact points during collision, and subsequently to erroneous particle transport and deposition. In cases where non-spherical particles can be approximated as ellipsoids with arbitrary aspect ratios, a new theory for particle transport, collision and wall interaction is presented which is more accurate computationally and more efficient than the MS method. In general, with the new ellipsoidal particle interaction (EPI) model, contact points and planes of ellipsoids, rather than spheres, are obtained based on a geometric potential algorithm. Then, interaction forces and torques of the colliding particles are determined via inscribed ‘pseudo-spheres’, employing the soft-particle approach. The off-center forces and moments are then transferred to the mass center of the ellipsoids to solve the appropriate translatory and angular equations of motion. Considering ellipses to illustrate the workings and predictive power of the new collision model, turbulent fluid–particle flow with the EPI model in a 2-D channel is simulated and compared with 3-D numerical benchmark results which relied on the MS method. The 2-D concentrations of micron particles with different aspect ratios matched closely with the 3-D cases. However, interesting differences occurred when comparing the particle-velocity profiles for which the 2-D EPI model generated somewhat larger particle velocities due to out-of-plane collisions, slightly higher particle–wall interactions, and two-way coupling effects.

Some Features of Powered Gyrostabilization of a Large Space Structure Assembled in Orbit

This paper presents the single-axis angular motion equations of a large space structure assembled in orbit from separate elastic construction elements. We introduce a method to calculate the parameters of discretely varying dynamical properties of an assembled structure whose model has variable coefficients and inherent attributes of an elastic multi-frequency vibrating system. In addition, we suggest a parametrically tuneable algorithm for powered gyro control of such objects that guarantees desired dynamics at all steps of robotized assembly. Simulation results demonstrate the efficiency of the suggested algorithm.

Angular motion equations for a satellite with hinged flexible solar panel

Non-linear mathematical model for the satellite with hinged flexible solar panel is presented. Normal modes of flexible elements are used for motion description. Motion equations are derived using virtual work principle. A comparison of normal modes calculation between finite element method and developed model is presented.

An optimal trajectory design for the lunar vertical landing

A research for designing the optimal lunar vertical landing trajectory to reduce the total energy or mass of propellant is addressed in this paper. Most of these problems can be divided into two phases: breaking and approach phase. The optimal landing trajectory in general does not consider the pitch-up motion so that the landing problem has been only solved in the breaking phase. For this reason, there are some attempts to find the optimal trajectory including the final vertical landing phase by including the equations of angular motion of the vehicle. However, the optimal solution using this approach depends on the scale factor of a cost function because the cost function consists of two different mechanical parameters such as the final mass and total control torque. The final control constraints are augmented for vertical lunar landing instead of the equations of angular motion. The obtained optimal trajectory has an additional positive effect of the image acquisition as well as the final vertical landing.

Thermodynamic and dynamic modeling of a single cylinder four stroke diesel engine

In this study the conjugate thermodynamic and dynamic modeling of a single cylinder four-stroke diesel engine was conducted. The gas pressure in cylinder was calculated with the first law of the thermodynamic and the general state equation of the perfect gases. The variation of the heat, given to the working fluid during the heating process of the thermodynamic cycle, was modeled with the Gaussian function. The dynamic model of the engine consists of the motion equations of piston, conrod and crankshaft. Conrod motion was modeled by 2 translational and 1 angular motion equations. In the derivation of the motion equations, the Newton method was used. Motion equations involve hydrodynamic and asperity frictions as well as gas forces. By preparing a heat release rate profile consistent with ones given in the literature, thermal efficiency, knocking, vibration, torque and emission characteristics of the engine were investigated. The counterweight mass and its radial distance were optimized. At full load, if the heat release period is initiated soon after the piston passed the top dead center, the pressure rise rate becomes critical from the knocking point of view however, a couple of degree of retarding of the heat release period avoids the knocking without causing significant loss in thermal efficiency. If the throttling is more than 70%, the temperature of the combustion gas is high enough for NOx formation. At full load the vibrational torque exerting on the crankshaft was determined as about 17 times the engine torque.

Theoretical and Numerical Studies of Dynamic Scaling of a Six-Degree-of-Freedom Laser Propulsion Vehicle

To estimate the flight reactions of a full-scale vehicle from reduced-scale tests, we constructed a scaling theory for the vehicle size, input energy, moment of inertia, and pulse frequency needed to maintain dynamic equivalence between a laboratory-scale and full-scale launch of a laser propulsion vehicle. The dynamic scaling law for a single pulse was constructed using translational and angular equations of motion. The analytical scaling was confirmed for a single-pulse incident using a fluid-orbit coupling simulator for the interaction between the blast wave and the vehicle. Motion equivalence was maintained for multiple pulses by adjusting the repetition frequency of the pulse incident to correct for the effect of aerodynamic drag during the free flight of the pulse-to-pulse interval. The flight of a full-scale vehicle can be estimated for single- and multiple-pulse operations from the flight data for a small-scale vehicle using the proposed scaling theory, which provides correlations between the characteristics of small-scale and large-scale flight systems. Small-scale tests were shown to be useful in estimating the flight of a full-scale vehicle using dynamic scaling theory.

Swerving Orientation of Spin-Stabilized Projectile for Fixed-Cant Canard Control Input

Due to the large launch overload and high spin rate of spin-stabilized projectile, no attitude sensor is adopted in square crossing fixed-cant canard concept, which causes the lack of existing projectile linear theory for the close form solution of swerving motion. This work focuses on swerving orientation prediction with the restricted conditions. By importing the mathematical models of canard force and moment into the projectile angular motion equations, trim angle induced by canard control force is extracted as the analytical solution of angle of attack increment (AOAI). On this basis, analytical orientations of trajectory angular rate increment and swerving increment are obtained via the frozen coefficient method. A series of simulations under different conditions were implemented to validate the expressions in this effort. Results state that increment orientation of swerving motion can be predicted with available trajectory parameters. The analytical orientations indicate trim value of numerical orientations. Deviations between analytical and numerical orientations relate to initial launch angles and control start time, both lower initial launch angle, and the start time which is closer to the end of flight decreases the deviation convergence time.

Methods of the analysis of motion of small space vehicles around the centre of masses at deployment of space tether system

This paper studies analytical and numerical methods of the analysis of spatial motion concerning centre of mass of small space vehicles at deployment of space tether system. Deployment of space tether system is executed from the base space vehicle. The set angular orientation of the base space vehicle concerning a local vertical is provided by means of own system of stabilisation. The small space vehicle (or freight) represents a descent capsule returned to the Earth or the space vehicle transferred into higher orbit. It is supposed that the weight of the base space vehicle is much more than weight of freight. The angular position of freight is described by classical Euler’s angles determined concerning a direction of a tether. Analytical methods of the analysis of angular motion are based on an averaging method (for small angles of nutation) and on a method integral of manifolds which allows to investigate the nonlinear equations of angular motion of freight. The major factors influencing stability of angular motion of freight concerning a direction of a tether are considered. Results of numerical modelling of angular motion of the small space vehicle are presented.

Control of spacecraft turns in scheduled time using rocket engines

We obtained an on-off control law for two spacecraft (SC) rocket engines creating the angular acceleration in the opposite directions considering that SC is controlled by the rocket engines with an exponential increase of thrust at starting and decrease at shutdown. This control law provides the change-over of SC from the initial angular position to the final one in the scheduled time. A special trajectory consisting of eight typical segments is proposed to identify the angular acceleration and delay of its formation in the rocket engines. Both problems are solved by the method of analytical construction of composite trajectories for the angular turn from the typical segments on which the differential equations of angular motion have analytical solutions. The results are applicable for the development of promising space interceptors and design of onboard control systems.

Estimate of Flying Projectile Attitude Based on MEMS Gyroscope Measurements

To estimate the flight attitude angles of Guided Projectile rapidly and accurately to realize the precise control, angular motion equations and MEMS gyroscope measurements were used to build the state equation and the measurement equationcombined withthe ballistic priori knowledge. Based on the unscented kalman filter for nonlinear system, an approach of estimating the attitude angles of Guided Projectile was proposed. Taking a certain type of Guided Projectile as an example, the data of triaxial angular velocities with the measurement noise was considered as the output signal of MEMS gyros, and considered the calculated ballistic data as the ballistic priori knowledge. Through the approach proposed in this paper, the above were used to obtain the estimates for attitude angles. The results indicate that, it’s feasible to obtain the attitude angles by angular motion equations and gyros. On account of the actual random noise signal of gyros, the accuracy of the estimated attitude angles can meet the requirement of trajectory control system for Guided Projectile.

Thrusters Pairing Guidelines for TrajectoryCorrections of Projectiles

Abstract : A novel method for correcting the trajectory of spin- and fin-stabilized projectiles using pairs of impulse thrusters located away from the center of mass is described in this paper. The impulses are generated from the detonation of a very small quantity of explosive. By locating the impulse thrusters away from the center of mass, it is possible to significantly increase the trajectory correction. This is explained by the fact that, in such a case, the deflection results from the aerodynamic jump due to the projectile oscillation in addition to that of the impulses. An optimal combination of thrusters can be achieved by considering pairs of impulses that have the double objective of maximizing the deflection and minimizing the drag due to the oscillations. In this paper, the linear theory of ballistics is used to develop an analytical model representing the motion of a projectile subjected to impulse thrusters. Using the linearized equations of angular motion and considering ideal cases, the optimization scheme for using double impulses is defined by a set of three rules. The latter are then assessed in the case of actual projectiles, using six-degree- of-freedom computations. The course correction process for a 30-mm fin-stabilized air-defense projectile and a standard 105-mm spin-stabilized artillery shell are presented.

Trajectory tracking controller design using neural networks for a tiltrotor unmanned aerial vehicle

This article presents a design of trajectory tracking controller for a tiltrotor unmanned aerial vehicle (UAV). The objective of this design is to control the tiltrotor aircraft with one fixed controller architecture for all flight modes, viz. helicopter mode, conversion mode, and aeroplane mode. The controller is implemented using two time-scale separations, and thus comprises an inner loop for attitude control and an outer loop for trajectory control. The dynamic model inversion (DMI) technique is applied to both the control loops such that the inner-loop and the outer-loop DMIs are driven using the angular equations of motion and the translational equations of motion, respectively. In addition, online adaptive neural networks are employed to compensate for the model inversion errors of the inner loop and the outer loop due to the lack of complete knowledge of tiltrotor aircraft dynamics. The pseudo-control hedging algorithm is adopted to reduce the instability problem caused by the time-delay of control loops, saturation, and rate limit of actuators. The trajectory tracking controller is evaluated using a sophisticated six-degree-of-freedom non-linear simulation programme with an approach and landing scenario including all flight modes of a smart UAV, a tiltrotor UAV.

Dynamically Scaled Generalized Inversion for Asymptotic Stabilization of Underactuated Spacecraft Dynamics

Abstract Novel concept of feedback linearization is introduced for smooth asymptotic stabilization of underactuated spacecraft equipped with one and two degrees of actuation. The concept is based on generalized inversion, and is aimed at asymptotically realizing a perturbation from the unrealizable feedback linearizing transformation. A desired stable second-order linear dynamics in a norm measure of the angular velocity components about the unactuated axes is prescribed. Evaluation of this dynamics along the vector field defined by the underactuated Euler's dynamical equations of angular motion results in a relation that is linear in the control variables. This relation is used to assess realizability of the desired unactuated dynamics, resulting in necessary and sufficient conditions for asymptotic stabilizability of underactuated spacecraft. Generalized inversion of the relation produces a control law that is composed of particular and auxiliary parts. The generalized inverse in the particular part is scaled by a dynamic factor that depends on the spacecraft angular velocity components about the spacecraft actuated axes, such that the generalized inverse converges uniformly to the standard Moore-Penrose generalized inverse as the transient response decays, resulting in asymptotic realization of the desired unactuated stable linear dynamics. The null-control vector in the auxiliary part of the control law is chosen for asymptotic stable perturbed feedback linearization of the actuated subsystem.

Modeling and adaptive attitude control of observation spacecrafts in view of flexible structure

The paper discusses a computerized method for deriving of angular motion equations of observation spacecrafts, whose structural flexibility is determined by large-size solar battery panels. Reorientation dynamics problems that arise because of structural flexibility and inaccurate settings of parameters of the object model are considered. An adaptive system for control of such space objects is suggested. The system provides a robust control and stability with respect to vibrations. Robustness and stability are achieved by using Kalman filtration of flexible modes and by providing optimal phase oscillation conditions for switching of control actions.

Formation of the control strategy for the large space structure assembled in orbit

For the large space structure assembled in orbit of individual building elements, the angular motion equations were derived. Consideration was given to the discrete nature of variations of the mechanical structure and substantial structural elasticity induced by the flexible joint couplings of the framework elements. A procedure was proposed for computeraided analysis of the variations of the dynamic characteristics of the assembled structure as the control plant whose model has variable coefficients and strongly pronounced characteristics of the elastic multifrequency oscillatory system. For such objects, a sequence of variable motion control strategies was formed to provide highly-precise control which is stable to the elastic oscillations at all stages of assembling. For the developing structure, the structural diagrams of the orientation systems realizing the accepted control strategies at the corresponding stages of assembling were presented. The results of modeling corroborating operability of the proposed control algorithms were analyzed.

Calculation and correction of attitude parameters without measuring angular velocity

The conventional methods of estimating the parameters of relative attitude of two coordinate trihedrons, a movable trihedron and an immovable one, using representation of vector quantities in projections on the axes of both trihedrons on a finite time interval provide for application of angular velocity of the movable trihedron. This angular velocity is found either by direct measurement or by integration of dynamic equations of angular motion of the solid body correlated with this trihedron. Both ways of calculation of angular velocity have drawbacks. The approach to processing vector information presented below is intended for estimation of the attitude parameters without using any information about the angular velocity of relative motion of the trihedrons. Two problems of determination of attitude of a low-orbit space vehicle are considered as applications.