Principal Axes Of Inertia Research Articles

Inertial and elastic properties of general composite beams

Slender composite structures can be modeled using engineering beam models with properties computed using a cross-sectional analysis program, such as VABS. These properties are given in terms of the mass matrix, stiffness matrix, and compliance matrix in a general coordinate system. The invariance of strain energy and kinetic energy is employed to rigorously transform sectional properties into different coordinate systems with parallel shifts and rotations. Additionally, the computation of commonly used inertial properties (mass center, principal inertial axes, and mass moments of inertia) from the mass matrix, and commonly used elastic properties (extension stiffness, bending stiffness, torsion stiffness, tension center, shear center, principal bending axes, principal shear axes, etc.) from the compliance matrix, is elucidated. The elastic properties are given for both the Timoshenko model and the Euler–Bernoulli model. The definitions for shear center and twist center are clarified and consistently generalized for composite beams. Isotropic homogeneous beams are used to illustrate how to relate commonly used engineering beam properties with the compliance matrix and the stiffness matrix for composite beams. Finally, engineering beam properties necessary for general-purpose aeromechanical analysis programs such as CAMRAD II are derived from the properties computed by VABS.

Dynamics of eugenol included in β-cyclodextrin by nuclear magnetic resonance and molecular simulations.

Eugenol-β-cyclodextrin complex has been widely used because of the enhanced stability and conservation of the properties of eugenol. Applications in food and health sciences have been shown previously, which makes this complex an excellent model to understand and develop methodologies for the analysis and prediction of physical properties. In this work, the dynamics of eugenol incorporated into β-cyclodextrin are presented, using NMR relaxation rates, and the predictive capabilities of molecular dynamics simulations are discussed. Results show a hindered rotation of eugenol around the principal inertial axes when located inside β-cyclodextrin. Moreover, a translational movement of the whole complex is demonstrated.

Analysis of High-Speed Rotor Vibration Failure Due to Sudden Angular Deformation of Bolt Joints

As the efficiency of advanced aero engines improves, the operational speed of their rotors increases. This heightened operational speed makes the rotor dynamics highly sensitive to changes in the rotor’s mass asymmetry state, or unbalance state. During the use of a dual-spool turbofan engine, when its supercritical high-pressure rotor (HPR) exceeds a certain operational speed, the rotor’s vibration spikes and continues to increase with the operational speed until it drops sharply near the maximum operational speed. Analysis of the bolt joints in the faulty rotor reveals various phenomena such as joint interface damage, changes in bolt loosening torque distribution, and alterations in rotor initial unbalance. This paper proposes that at high operational speeds, the bolt joint of the HPR undergoes sudden angular deformation, resulting in the slanting of the principal axis of inertia of the turbine disk. This slant leads to changes in the unbalanced state of the HPR. The additional unbalance causes a sudden rotational inertia load excitation, triggering the rotor vibration failure. Subsequently, a rotor dynamic model that incorporates the angular deformation of the joints is established to simulate how this joint deformation influences the dynamic response of the rotor. The simulation results align well with the observed failure phenomenon and validate the proposed failure mechanism. Finally, troubleshooting measures are proposed and implemented in the faulty engine, effectively mitigating the vibration fault.

INVESTIGATION OF THE INFLUENCE OF THE INFLUENCE OF CONSTANT TORQUE ON EQUILIBRIUM ORIENTATIONS OF A SATELLITE MOVING IN A CIRCULAR ORBIT WITH THE USE OF COMPUTER ALGERRA METHODS

Methods of computer algebra are used to investigate equilibrium orientations of a satellite moving along a circular orbit under the action of gravitational and constant torques. The main focus is placed on the investigation of equilibrium orientations in the cases where the constant torque vector is parallel to the planes formed by the principal central axes of inertia of the satellite. Using methods for Gröbner basis construction, the system of six algebraic equations that determine the equilibrium orientations of the satellite is reduced to one sixth-order algebraic equation in one unknown. Domains with equal numbers of equilibrium solutions are classified using algebraic methods for constructing discriminant hypersurfaces. Bifurcation curves in the space of problem parameters, which define the boundaries of the domains with equal numbers of equilibrium solutions, are constructed. A comparative analysis of the influence of the order of variables in the process of Gröbner basis construction is carried out. Using the proposed approach, it is shown that, under the action of the constant torque, the satellite with unequal principal central moments of inertia has no more than 24 equilibrium orientations in a circular orbit.

Studying the vibrational motion of a rotating symmetrically charged solid body subjected to external forces and moments

This paper studies the rotatory vibrating motion of a dynamically symmetric charged solid body (SB) connected with a spring around a fixed point as a novel model. The body is assumed to be influenced by a Newtonian force field (NFF) besides extrinsic moments like perturbing, restoring, and gyrostatic moment (GM). These moments are guided along the body’s principal axes of inertia. The body is assumed to have a high starting angular velocity in the direction of the dynamic symmetry axis, and the perturbing moments (PM) are thought to be less than the restoring moment. These assumptions provide us with a tiny parameter, to use the averaging technique (AT), and to acquire the averaging system of the regulating one. Several applications are introduced to achieve the solutions of this system separately. These solutions are graphed to reveal the positive influences of the body’s numerical values on the behavior of the body, taking into account the various impacts of the applied moments, the electromagnetic field, and the Newtonian one for all examined applications. The obtained results are considered a generalization of those which were obtained in related previous works. For the projections of the body angular velocity, the proportion of differences as a function of the investigated parameters varies between 17% and 35%, whereas the other parameters are varied between 0.0011% and 0.37%, respectively. Therefore, these outcomes are considered to be significant due to their applications in life specially that based on the theory of vibrating systems in physics and engineering fields.

On the Behavior of Honeycomb, Grid and Triangular PLA Structures under Symmetric and Asymmetric Bending.

Additive manufacturing technologies enable the production of components with lightweight cores, by means of infills with various patterns and densities. Together with reduced mass and material consumption, infill geometries must ensure that strength and stiffness conditions are fulfilled. For the proper correlation of the infill type with the loading case of the part, the mechanical behavior of the infill along all three principal axes of inertia has to be known. In this paper, the behavior in symmetric and asymmetric bending of three infill geometries, commonly used in 3D printing processes (honeycomb, grid and triangles) is analyzed. The variations of deflections as a function of force orientation are presented, showing that honeycomb and triangular structures exhibit similar behaviors along the Y and Z principal axes of inertia. Furthermore, the displacements obtained for the three types of structures are compared, in relation to the consumed volume of material. The larger displacements of the grid structure compared to the honeycomb and triangular structures are highlighted.

ВПЛИВ ДИНАМІЧНОЇ НЕСИМЕТРІЇ НА СТІЙКІСТЬ ОБЕРТАННЯ У СЕРЕДОВИЩІ З ОПОРОМ ТВЕРДОГО ТІЛА ПІД ДІЄЮ ПОСТІЙНОГО МОМЕНТУ У ІНЕРЦІАЛЬНІЙ СИСТЕМІ ВІДЛІКУ

Under the assumption that the center of mass of an asymmetric rigid body is located on the third principal axis of inertia of a rigid body, the previously obtained conditions for the asymptotic stability of uniform rotation in a medium with resistance of a dynamically asymmetric rigid body are investigated. A rigid body rotates around a fixed point, is under the action of gravity, dissipative moment and constant moment in an inertial frame of reference. The stability conditions are presented as a system of three inequalities. The first and second inequalities have the first degree relative to the dynamic unbalance, and the third inequality has the third degree. The first and third inequalities are of the second degree with respect to the overturning or restoring moment, and the second inequality is of the first degree. The first and third inequalities are of the fourth degree with respect to the constant moment, and the second inequality is of the second degree. The third inequality is the most difficult to study. Analytical studies of the influence of dynamic unbalance, restoring and overturning moments on the conditions of asymptotic stability are carried out. Conditions for the asymptotic stability of uniform rotation in a medium with resistance to an asymmetric rigid body are obtained for sufficiently small values of dynamic unbalance. Sufficient stability conditions are written out up to the second order of smallness with respect to the constant moment and the first order of smallness with respect to the restoring and overturning moments. Instability conditions are obtained for sufficiently large dynamic unbalance. The effect of dynamic unbalance on the stability conditions for the rotation of a rigid body around the center of mass is studied. It is shown that in the absence of dissipative asymmetry, it is sufficient for asymptotic stability that the axial moment of inertia of a rigid body be greater than the double equatorial moment and that the well-known necessary stability condition for a symmetric rigid body be satisfied.

Modeling a semi-optimal deceleration of a rigid body rotational motion in a resisting medium

This paper studies the shortest time of slowing rotation of a free dynamically asymmetric rigid body (RB), analogous to Euler’s case. This body is influenced by a rotatory moment of a tiny control torque with closer coefficients but not equal, a gyrostatic moment (GM) due to the presence of three rotors, and in the presence of a modest slowing viscous friction torque. Therefore, this problem can be regarded as a semi-optimal one. The controlling optimal decelerating law for the rotation of the body is constructed. The trajectories that are quasi-stationary are examined. The obtained new results are displayed to identify the positive impact of the GM. The dimensionless form of the regulating system of motion is obtained. The functions of kinetic energy and angular momentum besides the square module are drawn for various values of the GM’s projections on the body’s principal axes of inertia. The effect of control torques on the body's motion is investigated in a case of small perturbation, and the achieved results are compared with the unperturbed one. For the case of a lack of GM, the comparison between our results and those of the prior ones reveals a high degree of consistency, in which the deviations between them are examined. As a result, these outcomes generalized those that were acquired in previous studies. The significance of this research stems from its practical applications, particularly in the applications of gyroscopic theory to maintain the stability and determine the orientation of aircraft and undersea vehicles.

Stability Conditions for Permanent Rotations of a Heavy Gyrostat with Two Constant Rotors

In this paper, we consider the motion of an asymmetric heavy gyrostat, when its center of mass lies along one of the principal axes of inertia. We determine the possible permanent rotations and, by means of the Energy-Casimir method, we give sufficient stability conditions. We prove that there exist permanent stable rotations when the gyrostat is oriented in any direction of the space, by the action of two spinning rotors, one of them aligned along the principal axis, where the center of mass lies. We also derive necessary stability conditions that, in some cases, are the same as the sufficient ones.

Existential Properties of Algebraic Integrals of a Rigid Body

In this article, we consider kinematical considerations of a rigid body rotating around a given fixed point in a Newtonian force field exerted by an attractive center with a rotating couple about their principal axes of inertia. The kinematic equations and their well-known three elementary integrals of the problem are introduced. The existence properties of the algebraic integrals are considered. Besides, we search as a special case of the fourth algebraic integral for the problem of the rigid body’s motion around a fixed point under the action of a Newtonian force field with an orbiting couple. Lagrange’s case and Kovalevskaya’s one are obtained. The large parameter is used for satisfying the existing conditions of the algebraic integrals. The comparison between the obtained results and the previous ones is arising. The numerical solutions of the regulating system of motion are obtained utilizing the fourth-order Runge-Kutta method and are plotted in some figures to illustrate the positive impact of the imposed forces and torques on the behavior of the body at any time.

Secular perturbations of the translational-rotational motion in a nonstationary three-body problem

A non-stationary three-body problem with axisymmetric dynamic structure, shape, and variable compression is considered. The Newtonian interaction force is characterized by an approximate expression of the force function accurate to the second harmonic. The masses of bodies change isotropically at different rates. The axes of inertia of the proper coordinate system of nonstationary axisymmetric three bodies coincide with the principal axes of inertia of the bodies, and it is assumed that their relative orientations remain unchanged during the evolution. Differential equations of translational and rotational motion of three non-stationary axisymmetric bodies with variable masses and dimensions in the relative coordinate system with the origin in the center of the more massive body are presented. The analytical expression for the Newtonian force function of the interaction of three bodies with variable masses and dimensions is given. The canonical equations of translational-rotational motion of three bodies in the Delaunaye-Andouye analogues have been obtained. The equations of secular perturbations of translational-rotational motion of unsteady axisymmetric three-bodies in the analogue of the Delaunaye-Andouye osculating elements have been obtained. The problem is investigated by methods of perturbation theory.

Conformer Selection by Electrostatic Hexapoles: A Theoretical Study on 1-Chloroethanol and 2-Chloroethanol

The electrostatic hexapole is a versatile device that has been used for many years in gas-phase experiments. Its inhomogeneous electric field has been employed for many purposes such as the selection of rotational states, the selection of clusters, the focusing of molecular beams, and molecular alignment as a precursor for molecular orientation. In the last few years, the hexapolar electric field has been demonstrated to be able to control the conformer composition of molecular beams. The key point is that conformers, where the component of the permanent electric dipole moment with respect to the largest of the principal axes of inertia is close to zero, require more intense hexapolar electric fields to be focused with respect to the other conformers. Here, we simulated the focusing curves of the conformers of 1-chloroethanol and 2-chloroethanol under hypothetical beam conditions, identical for all conformers, in a hypothetical and realistic experimental setup with three different hexapole lengths: 0.5, 1, and 2 m. The objective was to characterize this selection process to set up collision experiments on conformer-selected beams that provide information on the van der Waals clusters formed in collision processes.

Classical and Quantum Controllability of a Rotating Asymmetric Molecule

We study both the classical and quantum rotational dynamics of an asymmetric top molecule, controlled through three orthogonal electric fields that interact with its dipole moment. The main difficulties in studying the controllability of these infinite-dimensional quantum systems are the presence of severe spectral degeneracies in the drift Hamiltonian and the nonsolvability of the stationary free Schrödinger equation, which lead us to apply a perturbative Lie algebraic approach. In this paper we show that, while the classical equations given by the Hamiltonian system on $$\mathrm{SO}(3)\times {\mathbb {R}}^3$$ are controllable for all values of the rotational constants and all dipole configurations, the Schödinger equation for the quantum evolution on $$L^2(\mathrm{SO}(3){,{\mathbb {C}}})$$ is approximately controllable for almost all values of the rotational constants if and only if the dipole is not parallel to any of the principal axes of inertia of the asymmetric rigid body.

Ellipsoidal equilibrium figure and Cassini states of rotating planets and satellites deformed by a tidal potential in the spatial case

The equilibrium figure of an inviscid tidally deformed body is the starting point for the construction of many tidal theories such as Darwinian tidal theories or the hydrodynamical Creep tide theory. This paper presents the ellipsoidal equilibrium figure when the spin rate vector of the deformed body is not perpendicular to the plane of motion of the companion. We obtain the equatorial and the polar flattenings as functions of the Jeans and the Maclaurin flattenings, and of the angle $\theta$ between the spin rate vector and the radius vector. The equatorial vertex of the equilibrium ellipsoid does not point toward the companion, which produces a torque perpendicular to the rotation vector, which introduces terms of precession and nutation. We find that the direction of spin may differ significantly from the direction of the principal axis of inertia $C$, so the classical approximation $\mathsf{I}\vec{\omega}\approx C\vec{\omega}$ only makes sense in the neighborhood of the planar problem. We also study the so-called Cassini states. Neglecting the short-period terms in the differential equation for the spin direction and assuming a uniform precession of the line of the orbital ascending node, we obtain the same differential equation as that found by Colombo (1966). That is, a tidally deformed inviscid body has exactly the same Cassini states as a rotating axisymmetric rigid body, the tidal bulge having no secular effect at first order.

Automated Solving Algorithm and Application for Oriented Bounding Box Based on Principal Axis of Inertia

In mechanical design and manufacture, the minimum size of raw material of each part should be measured before processing. Generally, this size could be equal to the size of minimum bounding box. When the assembled parts of a product reach an enormous quantity, the measurement would be a rough work. This paper introduced an automated solving algorithm for the oriented bounding box based on principal axis of inertia, combined with CATIA structural tree recursive traversal algorithm according to CAA V5 Automation API, to implement an automated solution of the engineering acceptable minimum oriented bounding box of CATIA model.

On the equilibriums stability in an approximate problem of the dynamics of a rigid body with a suspension point vibrating along an inclined straight line

We consider the heavy rigid body dynamics under the assumption that one of the body points (the suspension point) performs the specified high-frequency vibrations of small amplitude along an inclined straight line, and the body mass centre lies on the principal axis of inertia for the suspension point. In the framework of an approximate autonomous system of canonical equations of motion, the question of existence, number, and stability of the body relative equilibrium positions is solved. It is shown that for all such equilibria, the mass centre radius-vector lies in the vertical plane containing the vibration axis. The number of equilibria is four, six or eight depending on the vibration intensity. Sufficient and necessary conditions for their stability are found. The existence of high-frequency periodic motions of the initial non-autonomous system, which are generated by the investigated equilibria, is justified using Poincare’s method. Conclusions about stability (in linear approximation) of these periodic motions are drawn.

Controlling rotation in the molecular frame with an optical centrifuge

We computationally demonstrate a new method for coherently controlling the rotation-axis direction in asymmetric top molecules with an optical centrifuge. Appropriately chosen electric-field strengths and the centrifuge's acceleration rate allow to generate a nearly arbitrary rotational wavepacket. For D$_2$S and 2H-imidazole (C$_3$H$_4$N$_2$) we created wavepackets at large values of the rotational quantum number $J$ with the desired projections of the total angular momentum onto two of the molecules' principal axes of inertia. One application of the new method is three-dimensional alignment with a molecular axis aligned along the laser's wave vector, which is important for the three-dimensional imaging of molecules yet not accessible in standard approaches. The simultaneous orientation of the angular momentum in the laboratory frame and in the molecular frame could also be used in robust control of scattering experiments.

Effect of some Disturbance Factors on Falling Point Distribution of Unguided Rocket

The paper investigates the individual and combined effects of disturbance factors including thrust misalignment, the offset of the centre of gravity and misalignment of the principal axes of inertia on falling point distribution of a type of unguided rocket (fin-stabilized rockets with single-stage solid-propellant rocket engines). The mathematical model used in the paper is developed from an available rocket motion model and solved for a representative rocket which is BM-21 rocket. The obtained results show dependencies of falling point deviation on disturbance parameters. These dependencies agree with the standard data given in the firing table; in addition, they are fuller and more insight ful than the previous research.

Effect of some Disturbance Factors on Falling Point Distribution of Unguided Rocket

The paper investigates the individual and combined effects of disturbance factors including thrust misalignment, the offset of the centre of gravity and misalignment of the principal axes of inertia on falling point distribution of a type of unguided rocket (fin-stabilized rockets with single-stage solid-propellant rocket engines). The mathematical model used in the paper is developed from an available rocket motion model and solved for a representative rocket which is BM-21 rocket. The obtained results show dependencies of falling point deviation on disturbance parameters. These dependencies agree with the standard data given in the firing table; in addition, they are fuller and more insight ful than the previous research.

Model based estimation of inertial parameters of a rigid rotor having dynamic unbalance on Active Magnetic Bearings in presence of noise

Unbalance in rotors is known as the primary source of excitation, the state of unbalance changes over time and, therefore, industrial rotors require online or smart identification and attenuation of unbalance for smooth operation. To begin with, this paper proposes the use of Active Magnetic Bearings (AMBs) to identify the unbalance. Unbalance is defined as the location of centre of mass of the rotor and orientation of the axes of principal mass moments of inertia, both with respect to the spin-axis, assumed as the geometric axis of symmetry of rotor. Non-uniform distribution of mass makes a rotor dynamically unbalanced. The centre of mass shifts from the spin-axis and none of the principal axes of inertia is aligned with it. Dynamic behaviour of such a rotor is modelled by a set of differential equations with time varying coefficients. A rigid rotor is assumed to be supported on two AMBs with all known characteristics. A methodology based on modelling and simulation is proposed to determine the eccentricity and the orientation of the principal axes of inertia (angles of its skewness) by utilizing the force and response of the rotor recorded by each AMB. The median of simulated inertial properties are shown to be in close agreement with their actual values. The proposed methodology is also found robust enough in the presence of different types of simulated noise used to pollute signals from the bearings. Therefore, the methodology is suitable for online identification of inertial properties of a rotor.