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Journal articles on the topic 'Equations of motion'

Author: Grafiati

Published: 4 June 2021

Last updated: 19 October 2024

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1

Loide, R.-K., and P. Suurvarik. "SUPERFIELD EQUATIONS OF MOTION." Proceedings of the Academy of Sciences of the Estonian SSR. Physics. Mathematics 34, no. 3 (1985): 248. http://dx.doi.org/10.3176/phys.math.1985.3.02.

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2

Wilson, C. R. "Discrete polar motion equations." Geophysical Journal International 80, no. 2 (February 1, 1985): 551–54. http://dx.doi.org/10.1111/j.1365-246x.1985.tb05109.x.

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3

Robinson, Enders, and Dean Clark. "Elasticity: Equations of motion." Leading Edge 9, no. 7 (July 1990): 24–27. http://dx.doi.org/10.1190/1.1439758.

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4

Bao, Dehai, and Z. Y. Zhu. "Quantization from motion equations." International Journal of Theoretical Physics 32, no. 8 (August 1993): 1409–22. http://dx.doi.org/10.1007/bf00675202.

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5

Chang, Liang-Wey, and James F. Hamilton. "A Sequential Integration Method." Journal of Dynamic Systems, Measurement, and Control 110, no. 4 (December 1, 1988): 382–88. http://dx.doi.org/10.1115/1.3152700.

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This paper presents a method for simulating systems with two inertially coupled motions, i.e., a slow motion and a fast motion. The equations of motion are separated into two sets of coupled nonlinear ordinary differential equations. For each time step, the two sets of equations are integrated sequentially rather than simultaneously. Explicit integration methods are used for integrating the slow motion since the stability of the integration is not a problem and the explicit methods are very convenient for nonlinear equations. For the fast motion, the equations are linear and the implicit integrations can be used with guaranteed stability. The size of time step only needs to be chosen to provide accuracy of the solution for the modes that are excited. The interaction between the two types of motion must be treated such that secular terms do not appear due to the sequential integration method. A lumped model of a flexible pendulum will be presented in this paper to illustrate the application of the method. Numerical results for both simultaneous and sequential integration are presented for comparison.

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6

Burov, A. A. "Motion of a Variable Body with a Fixed Point in a Time-dependent Force Field." Прикладная математика и механика 87, no. 6 (November 1, 2023): 984–94. http://dx.doi.org/10.31857/s0032823523060024.

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The problem of motion around a fixed point of a variable body in a time-dependent force field is considered. The conditions under which the equations of motion are reduced to the classical Euler–Poisson equations describing the motions of a rigid body in the field of attraction are indicated. The problems of the existence of the first integrals and the stability of steady motions are discussed.

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7

Jung, Soo-jin, and Eric Yee. "Compatible Ground Motion Models for South Korea Using Moderate Earthquakes." Applied Sciences 14, no. 3 (January 31, 2024): 1182. http://dx.doi.org/10.3390/app14031182.

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Due to a heightened interest in the field of earthquakes after two moderately sized earthquakes occurred in Gyeongju and Pohang, this study explores which ground motion prediction equations are compatible for the South Korea region. Due to data availability, ground motions from five earthquakes of moderate magnitude were used for comparing against selected ground motion models. Median rotated response spectral ordinates at a period of 0.2 s were extracted from these ground motions, which served as a basis for comparison. Twelve ground motion models were considered from the Next Generation Attenuation West, West2, and East programs due to their extensive databases and robust analytical techniques. A comparison of relative residuals, z-score, and each event found that the subset of Next Generation Attenuation—East ground motion prediction equations did not perform as well as the suite of Next Generation Attenuation—West2 ground motion prediction equations, most likely due to the regional simulations involved in developing the database. Interestingly, the ground motion models that performed relatively well were from the set designed for rock conditions.

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8

Chang, Liang-Wey, and J. F. Hamilton. "Dynamics of Robotic Manipulators With Flexible Links." Journal of Dynamic Systems, Measurement, and Control 113, no. 1 (March 1, 1991): 54–59. http://dx.doi.org/10.1115/1.2896359.

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This paper presents a dynamic model for the robotic manipulators with flexible links by means of the Finite Element Method and Lagrange’s formulation. By the concept of the Equivalent Rigid Link System (ERLS), the generalized coordinates are selected to represent the total motion as a large motion and a small motion. Two sets of coupled nonlinear equations are obtained where the equations representing small motions are linear with respect to the small motion variables. An example is presented to illustrate the importance of the flexibility effects.

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9

Tleubergenov, Marat, and Gulmira Ibraeva. "ON THE CLOSURE OF STOCHASTIC DIFFERENTIAL EQUATIONS OF MOTION." Eurasian Mathematical Journal 12, no. 2 (2021): 82–89. http://dx.doi.org/10.32523/2077-9879-2021-12-2-82-89.

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10

NAMSRAI, KH, YA HULREE, and N. NJAMTSEREN. "AN OVERVIEW OF THE APPLICATION OF THE LANGEVIN EQUATION TO THE DESCRIPTION OF BROWNIAN AND QUANTUM MOTIONS OF A PARTICLE." International Journal of Modern Physics A 07, no. 12 (May 10, 1992): 2661–77. http://dx.doi.org/10.1142/s0217751x92001198.

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A simple scheme of unified description of different physical phenomena by using the Langevin type equations is reviewed. Within this approach much attention is being paid to the study of Brownian and quantum motions. Stochastic equations with a white noise term give all characteristics of the Brownian motion. Some generalization of the Langevin type equations allows us to obtain nonlinear equations of particles' motion, which are formally equivalent to the Schrödinger equation. Thus, we establish Nelson's stochastic mechanics on the basis of the Langevin equation.

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11

Chelnokov, Yu N. "Quaternion Regularization of Singularities of Astrodynamic Models Generated by Gravitational Forces (Review)." Прикладная математика и механика 87, no. 6 (November 1, 2023): 915–53. http://dx.doi.org/10.31857/s0032823523060036.

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The article presents an analytical review of works devoted to the quaternion regularization of the singularities of differential equations of the perturbed three-body problem generated by gravitational forces, using the four-dimensional Kustaanheimo–Stiefel variables. Most of these works have been published in leading foreign publications. We consider a new method of regularization of these equations proposed by us, based on the use of two-dimensional ideal rectangular Hansen coordinates, two-dimensional Levi-Civita variables, and four-dimensional Euler (Rodrigues–Hamilton) parameters. Previously, it was believed that it was impossible to generalize the famous Levi-Civita regularization of the equations of plane motion to the equations of spatial motion. The regularization proposed by us refutes this point of view and is based on writing the differential equations of the perturbed spatial problem of two bodies in an ideal coordinate system using two-dimensional Levi-Civita variables to describe the motion in this coordinate system (in this coordinate system, the equations of spatial motion take the form of equations of plane motion) and based on the use of the quaternion differential equation of the inertial orientation of the ideal coordinate system in the Euler parameters, which are the osculating elements of the orbit, as well as on the use of Keplerian energy and real time as additional variables, and on the use of the new independent Sundmann variable. Reduced regular equations, in which Levi-Civita variables and Euler parameters are used together, have not only the well-known advantages of equations in Kustaanheimo–Stiefel variables (regularity, linearity in new time for Keplerian motions, proximity to linear equations for perturbed motions), but also have their own additional advantages: 1) two-dimensionality, and not four-dimensionality, as in the case of Kustaanheimo-Stiefel, a single-frequency harmonic oscillator describing in new time in Levi-Civita variables the unperturbed elliptic Keplerian motion of the studied (second) body, 2) slow change in the new time of the Euler parameters, which describe the change in the inertial orientation of the ideal coordinate system, for perturbed motion, which is convenient when using the methods of nonlinear mechanics. This work complements our review paper [1].

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12

Mitiguy, Paul C., and Thomas R. Kane. "Motion Variables Leading to Efficient Equations of Motion." International Journal of Robotics Research 15, no. 5 (October 1996): 522–32. http://dx.doi.org/10.1177/027836499601500507.

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13

NAM, SOON-KWON, and KI-BUM KIM. "Stability of an electron beam in a two-frequency wiggler with a self-generated field." Journal of Plasma Physics 77, no. 2 (April 27, 2010): 257–63. http://dx.doi.org/10.1017/s0022377810000206.

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AbstractWe investigate the relativistic electron motions in a two-frequency wiggler magnetic field with self-generated fields. The equations of motion are derived from the Hamiltonian which include the self-generated field, and we find the steady-state orbit from the equations of motion. The stability of electron motion in a two-frequency wiggler is examined by the numerical simulation. We analyze the a dynamical systems using the fast Fourier transformation and the Poincarè surface of section to find the critical value which have the periodical electron motion and to optimize the two-frequency wiggler.

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14

Constantin, Fetecau. "On the Governing Equations for Velocity and Shear Stress of some Magnetohydrodynamic Motions of Rate-type Fluids and their Applications." IgMin Research 2, no. 1 (January 31, 2024): 045–47. http://dx.doi.org/10.61927/igmin144.

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The governing equations for the shear stress corresponding to some magnetohydrodynamic (MHD) motions of a large class of rate-type fluids are brought to light. In rectangular domains, the governing equations of velocity and shear stress are identical as form. The provided governing equations can be used to solve motion problems of such fluids when shear stress is prescribed on the boundary. For illustration, the motion in an infinite circular cylinder with shear stress on the boundary is discussed.

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15

Atanackovic, T. M., and L. J. Cveticanin. "Dynamics of Plane Motion of an Elastic Rod." Journal of Applied Mechanics 63, no. 2 (June 1, 1996): 392–98. http://dx.doi.org/10.1115/1.2788877.

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Plane motion of an elastic rod, subjected to a compressive force is analyzed. Equations of motions are derived for the case when deformations are not small. It is assumed that the compressive force has a periodic component, so that parametric instability is possible. Stability boundary is estimated analytically and determined numerically. In the special case of static loading a new formula for the slenderness ratio, below which there is no buckling, is obtained. In deriving the differential equations of motion a generalized constitutive equations taking in to account compressibility and shear stresses are used.

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16

Thresher, R. W., A. D. Wright, and E. L. Hershberg. "A Computer Analysis of Wind Turbine Blade Dynamic Loads." Journal of Solar Energy Engineering 108, no. 1 (February 1, 1986): 17–25. http://dx.doi.org/10.1115/1.3268046.

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The flapping motion of a single wind turbine rotor blade has been analyzed and equations describing the flapping motion have been developed. The analysis was constrained to allow only flapping motions for a cantilevered blade, and the equations of motion are linearized. A computer code, called FLAP (Force and Loads Analysis Program), to solve the equations of motion and compute the blade loads, has been completed and compared to measured loads for a 3-bladed downwind turbine with stiff blades. The results of the program are presented in tabulated form for equidistant points along the blade and equal azimuth angles around the rotor disk. The blade deflection, slope and velocity, flapwise shear and moment, edgewise shear and moment, blade tension, and blade torsion are given. The deterministic excitations considered in the analysis include wind shear, tower shadow, gravity, and a prescribed yaw motion.

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17

Nguyen, Tung Lam, Trong Hieu Do, and Hong Quang Nguyen. "Vibration Suppression Control of a Flexible Gantry Crane System with Varying Rope Length." Journal of Control Science and Engineering 2019 (February 11, 2019): 1–8. http://dx.doi.org/10.1155/2019/9640814.

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The paper presents a control approach to a flexible gantry crane system. From Hamilton’s extended principle the equations of motion that characterized coupled transverse-transverse motions with varying rope length of the gantry is obtained. The equations of motion consist of a system of ordinary and partial differential equations. Lyapunov’s direct method is used to derive the control located at the trolley end that can precisely position the gantry payload and minimize vibrations. The designed control is verified through extensive numerical simulations.

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18

Allan, A. Peter, and Miles A. Townsend. "Motions of a Constrained Spherical Pendulum in an Arbitrarily Moving Reference Frame: The Automobile Seatbelt Inertial Sensor." Shock and Vibration 2, no. 3 (1995): 227–36. http://dx.doi.org/10.1155/1995/932938.

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A common automatic seatbelt inertial sensor design, comprised of a constrained spherical pendulum, is modeled to study its motions and possible unintentional release during vehicle emergency maneuvers. The kinematics are derived for the system with the most general inputs: arbitrary pivot motions. The influence of forces due to gravity and constraint torque functions is developed. The equations of motion are then derived using Kane's method. The equations of motion are used in a numerical simulation with both actual and hypothetical automobile crash data.

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19

WAGNER, DAVID H. "SYMMETRIC-HYPERBOLIC EQUATIONS OF MOTION FOR A HYPERELASTIC MATERIAL." Journal of Hyperbolic Differential Equations 06, no. 03 (September 2009): 615–30. http://dx.doi.org/10.1142/s0219891609001940.

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We offer an alternate derivation for the symmetric-hyperbolic formulation of the equations of motion for a hyperelastic material with polyconvex stored energy. The derivation makes it clear that the expanded system is equivalent, for weak solutions, to the original system. We consider motions with variable as well as constant temperature. In addition, we present equivalent Eulerian equations of motion, which are also symmetric-hyperbolic.

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20

BEJANCU, AUREL. "EQUATIONS OF MOTION FOR SPACE-TIME-MATTER THEORY." International Journal of Geometric Methods in Modern Physics 10, no. 06 (April 30, 2013): 1350018. http://dx.doi.org/10.1142/s0219887813500187.

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The purpose of this paper is to present, in a covariant form and in their full generality, the equations of motion for space-time-matter (STM) theory. The whole study is based on the new approach of STM theory developed in our first paper [1] of this series. We show that the theory of geodesics in a general Kaluza–Klein space is best presented and explained by splitting the set of all geodesics into horizontal and non-horizontal geodesics. It is noteworthy that the horizontal geodesics (respectively, non-horizontal geodesics) project on the base manifold on motions which generalize the motions from general relativity theory (respectively, motions from Lorentz force equations).

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21

Ershkov, Sergey V. "Stability of the Moons Orbits in Solar System in the Restricted Three-Body Problem." Advances in Astronomy 2015 (2015): 1–7. http://dx.doi.org/10.1155/2015/615029.

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We consider the equations of motion of three-body problem in aLagrange form(which means a consideration of relative motions of 3 bodies in regard to each other). Analyzing such a system of equations, we consider in detail the case of moon’s motion of negligible massm3around the 2nd of two giant-bodiesm1,m2(which are rotating around their common centre of masses on Kepler’s trajectories), the mass of which is assumed to be less than the mass of central body. Under assumptions of R3BP, we obtain the equations of motion which describe the relative mutual motion of the centre of mass of 2nd giant-bodym2(planet) and the centre of mass of 3rd body (moon) with additional effective massξ·m2placed in that centre of massξ·m2+m3, whereξis the dimensionless dynamical parameter. They should be rotating around their common centre of masses on Kepler’s elliptic orbits. For negligible effective massξ·m2+m3it gives the equations of motion which should describe aquasi-ellipticorbit of 3rd body (moon) around the 2nd bodym2(planet) for most of the moons of the planets in Solar System.

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22

Sankar, N., V. Kumar, and Xiaoping Yun. "Velocity and Acceleration Analysis of Contact Between Three-Dimensional Rigid Bodies." Journal of Applied Mechanics 63, no. 4 (December 1, 1996): 974–84. http://dx.doi.org/10.1115/1.2787255.

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During manipulation and locomotion tasks encountered in robotics, it is often necessary to control the relative motion between two contacting rigid bodies. In this paper we obtain the equations relating the motion of the contact points on the pair of contacting bodies to the rigid-body motions of the two bodies. The equations are developed up to the second order. The velocity and acceleration constraints for contact, for rolling, and for pure rolling are derived. These equations depend on the local surface properties of each contacting body. Several examples are presented to illustrate the nature of the equations.

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23

Ider, S. K., and F. M. L. Amirouche. "Nonlinear Modeling of Flexible Multibody Systems Dynamics Subjected to Variable Constraints." Journal of Applied Mechanics 56, no. 2 (June 1, 1989): 444–50. http://dx.doi.org/10.1115/1.3176103.

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This paper presents the geometric stiffening effects and the complete nonlinear interaction between elastic and rigid body motion in the study of constrained multibody dynamics. A recursive formulation (or direct path approach) of the equations of motion based on Kane’s equations, finite element method and modal analysis techniques is presented. An extended matrix formulation of the partial angular velocities and partial velocities for flexible (elastic) bodies is also developed and forms the basis for our analysis. Closed loops and kinematical constraints (specified motions) are allowed and their corresponding Jacobian matrices are fully developed. The constraint equations are appended onto the governing equations of motion by representing them in a minimum dimension form using an innovative method called the Pseudo-Uptriangular Decomposition method. Examples are presented to illustrate the method and procedures proposed.

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24

Karlgaard, Christopher D., and Frederick H. Lutze. "Second-Order Relative Motion Equations." Journal of Guidance, Control, and Dynamics 26, no. 1 (January 2003): 41–49. http://dx.doi.org/10.2514/2.5013.

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25

Hoover, William G. "Constant-pressure equations of motion." Physical Review A 34, no. 3 (September 1, 1986): 2499–500. http://dx.doi.org/10.1103/physreva.34.2499.

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26

Rajput, B. S. "Quantum equations from Brownian motion." Canadian Journal of Physics 89, no. 2 (February 2011): 185–91. http://dx.doi.org/10.1139/p10-111.

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The Schrödinger free particle equation in 1+1 dimension describes second-order effects in ensembles of lattice random walks, in addition to its role in quantum mechanics, and its solutions represent the continuous limit of a property of ensembles of Brownian particles. In the present paper, the classical Schrödinger and Dirac equations have been derived from the Brownian motions of a particle, and it has been shown that the classical Schrödinger equation can be transformed into the usual Schrödinger quantum equation on applying the Heisenberg uncertainty principle between position and momentum, while the Dirac quantum equation follows from its classical counterpart on applying the Heisenberg uncertainty principle between energy and time, without applying any analytical continuation.

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27

Loide, R. K., and P. Suurvarik. "Supersymmetry: superfield equations of motion." Journal of Physics: Conference Series 532 (September 10, 2014): 012016. http://dx.doi.org/10.1088/1742-6596/532/1/012016.

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28

Botham, C. P., S. A. Blundell, A.-M. Mårtensson-Pendrill, and P. G. H. Sandars. "PNC Equations of Motion method." Physica Scripta 36, no. 3 (September 1, 1987): 481–84. http://dx.doi.org/10.1088/0031-8949/36/3/017.

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29

Minglibayev, M. Zh, and A. B. Kosherbayeva. "EQUATIONS OF PLANETARY SYSTEMS MOTION." SERIES PHYSICO-MATHEMATICAL 6, no. 334 (December 15, 2020): 53–60. http://dx.doi.org/10.32014/2020.2518-1726.97.

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The study of the dynamically evolution of planetary systems is very actually in relation with findings of exoplanet systems. free spherical bodies problem is considered in this paper, mutually gravitating according to Newton's law, with isotropically variable masses as a celestial-mechanical model of non-stationary exoplanetary systems. The dynamic evolution of planetary systems is learned, when evolution's leading factor is the masses' variability of gravitating bodies themselves. The laws of the bodies' masses varying are assumed to be known arbitrary functions of time. When doing so the rate of varying of bodies' masses is different. The planets' location is such that the orbits of planets don't intersect. Let us treat this position of planets is preserve in the evolution course. The motions are researched in the relative coordinates system with beginning in the center of the parent star, axes that are parallel to corresponding axes of the absolute coordinates system. The canonical perturbation theory is used on the base aperiodic motion over the quasi-canonical cross-section. The bodies evolution is studied in the osculating analogues of the second system of canonical Poincare elements. The canonical equations of perturbed motion in analogues of the second system of canonical Poincare elements are convenient for describing the planetary systems dynamic evolution, when analogues of eccentricities and analogues of inclinations of orbital plane are sufficiently small. It is noted that to obtain an analytical expression of the perturbing function main part through canonical osculating Poincare elements using computer algebra is preferably. If in expansions of the main part of perturbing function is constrained with precision to second orders including relatively small quantities, then the equations of secular perturbations will obtained as linear non-autonomous system. This circumstance meaningful makes much easier to study the non-autonomous canonical system of differential equations of secular perturbations of considering problem.

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30

Basile, Anthony G., and Veit Elser. "Equations of motion for superfluids." Physical Review E 51, no. 6 (June 1, 1995): 5688–94. http://dx.doi.org/10.1103/physreve.51.5688.

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31

Reid, Thomas F., and Stephen C. King. "Pendulum Motion and Differential Equations." PRIMUS 19, no. 2 (March 6, 2009): 205–17. http://dx.doi.org/10.1080/10511970701693942.

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32

Yamaleev, R. M. "Generalized Newtonian Equations of Motion." Annals of Physics 277, no. 1 (October 1999): 1–18. http://dx.doi.org/10.1006/aphy.1999.5929.

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33

Hu, Ping, Li Xin Wang, and Liang Min Guo. "Analysis of an Airdrop Motion Coupling N-S Equations with 6 DOF Motion Equations." Advanced Materials Research 1016 (August 2014): 506–10. http://dx.doi.org/10.4028/www.scientific.net/amr.1016.506.

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An airdrop model with aerial carrier is carried out to simulate an airdrop test. A coupled CFD/6 DOF is designed to compute the orientation and trajectory of the airdrop model. The results show that the CFD/6 DOF coupled method is appropriate to simulate this problem with a sufficient precision. The airdrop’s motion is strongly coupled with the aerodynamic force after leaving the sliding rail and the trajectory of its CG is mainly in the symmetry of the aerial carrier. The variation of its orientation mainly rolls in X axis direction and it swings as an approximate periodic oscillation which amplitude decreases over time. The maximum angle of orientation in X axis direction is-116°in 5 seconds. The similar variation of the airdrop motion is present in different angle of attack and sideslip of the aerial carrier.

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34

Desloge, Edward A. "A comparison of Kane’s equations of motion and the Gibbs–Appell equations of motion." American Journal of Physics 54, no. 5 (May 1986): 470–72. http://dx.doi.org/10.1119/1.14566.

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35

ZHANG, Yi, and Junling XIA. "Gauss Principle of Least Compulsion for Relative Motion Dynamics and Differential Equations of Motion." Wuhan University Journal of Natural Sciences 29, no. 3 (June 2024): 273–83. http://dx.doi.org/10.1051/wujns/2024293273.

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This paper focuses on Gauss principle of least compulsion for relative motion dynamics and derives differential equations of motion from it. Firstly, starting from the dynamic equation of the relative motion of particles, we give the Gauss principle of relative motion dynamics. By constructing a compulsion function of relative motion, we prove that at any instant, its real motion minimizes the compulsion function under Gaussian variation, compared with the possible motions with the same configuration and velocity but different accelerations. Secondly, the formula of acceleration energy and the formula of compulsion function for relative motion are derived because the carried body is rigid and moving in a plane. Thirdly, the Gauss principle we obtained is expressed as Appell, Lagrange, and Nielsen forms in generalized coordinates. Utilizing Gauss principle, the dynamical equations of relative motion are established. Finally, two relative motion examples also verify the results' correctness.

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36

Blanchet, Luc. "Relativistic equations of motion of massive bodies." Proceedings of the International Astronomical Union 5, S261 (April 2009): 102. http://dx.doi.org/10.1017/s1743921309990226.

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AbstractHighly relativistic equations of motions will play a crucial role for the detection and analysis of gravitational waves emitted by inspiralling compact binaries in detectors LIGO/VIRGO on ground and LISA in space. Indeed these very relativistic systems (with orbital velocities of the order of half the speed of light in the last orbital rotations) require the application of a high-order post-Newtonian formalism in general relativity for accurate description of their motion and gravitational radiation [1]. In this contribution the current state of the art which has reached the third post-Newtonian approximation for the equations of motion [2–6] and gravitational waveform [7–9] has been described (see [10] for an exhaustive review). We have also emphasized the successful matching of the post-Newtonian templates to numerically generated predictions for the merger and ring-down in the case of black-hole binaries [11].

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37

Ismail, A. I. "New Vertically Planed Pendulum Motion." Mathematical Problems in Engineering 2020 (December 28, 2020): 1–6. http://dx.doi.org/10.1155/2020/8861738.

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This article is concerned about the planed rigid body pendulum motion suspended with a spring which is suspended to move on a vertical plane moving uniformly about a horizontal X-axis. This model depends on a system containing three generalized coordinates. The three nonlinear differential equations of motion of the second order are obtained to the elastic string length and the oscillation angles φ 1 and φ 2 which represent the freedom degrees for the pendulum motions. It is assumed that the body moves in a rotating vertical plane uniformly with an arbitrary angular velocity ω . The relative periodic motions of this model are considered. The governing equations of motion are obtained using Lagrange’s equations and represent a nonlinear system of second-order differential equations that can be solved in terms of generalized coordinates. The numerical solutions are investigated using the approximated fourth-order Runge–Kutta method through programming packages. These solutions are represented graphically to describe and discuss the behavior of the body at any instant for different values of the different physical parameters of the body. The obtained results have been discussed and compared with some previously published works. Some concluding remarks have been presented at the end of this work. The value of this study comes from its wide applications in both civil and military life. The main findings and objectives of the current study are obtaining periodic solutions for the problem and satisfying their accuracy and stabilities through the numerical procedure.

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38

Liang, Bin, and Mao Sun. "Nonlinear flight dynamics and stability of hovering model insects." Journal of The Royal Society Interface 10, no. 85 (August 6, 2013): 20130269. http://dx.doi.org/10.1098/rsif.2013.0269.

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Current analyses on insect dynamic flight stability are based on linear theory and limited to small disturbance motions. However, insects' aerial environment is filled with swirling eddies and wind gusts, and large disturbances are common. Here, we numerically solve the equations of motion coupled with the Navier–Stokes equations to simulate the large disturbance motions and analyse the nonlinear flight dynamics of hovering model insects. We consider two representative model insects, a model hawkmoth (large size, low wingbeat frequency) and a model dronefly (small size, high wingbeat frequency). For small and large initial disturbances, the disturbance motion grows with time, and the insects tumble and never return to the equilibrium state; the hovering flight is inherently (passively) unstable. The instability is caused by a pitch moment produced by forward/backward motion and/or a roll moment produced by side motion of the insect.

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39

Liu, Ping, Senyue Lou, and Lei Peng. "Second-Order Approximate Equations of the Large-Scale Atmospheric Motion Equations and Symmetry Analysis for the Basic Equations of Atmospheric Motion." Symmetry 14, no. 8 (July 27, 2022): 1540. http://dx.doi.org/10.3390/sym14081540.

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In this paper, symmetry properties of the basic equations of atmospheric motion are proposed. The results on symmetries show that the basic equations of atmospheric motion are invariant under space-time translation transformation, Galilean translation transformations and scaling transformations. Eight one-parameter invariant subgroups and eight one-parameter group invariant solutions are demonstrated. Three types of nontrivial similarity solutions and group invariants are proposed. With the help of perturbation method, we derive the second-order approximate equations for the large-scale atmospheric motion equations, including the non-dimensional equations and the dimensional equations. The second-order approximate equations of the large-scale atmospheric motion equations not only show the characteristics of physical quantities changing with time, but also describe the characteristics of large-scale atmospheric vertical motion.

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40

Bae, Junhyeok, Juhwan Cha, Min-Guk Seo, Kangsu Lee, Jaeyong Lee, and Namkug Ku. "Experimental Study on Development of Mooring Simulator for Multi Floating Cranes." Journal of Marine Science and Engineering 9, no. 3 (March 20, 2021): 344. http://dx.doi.org/10.3390/jmse9030344.

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In this study, the coupled motion of a mooring system and multifloating cranes were analyzed. For the motion analysis, the combined equations of motions of the mooring line and multifloating cranes were introduced. The multibody equations for floating cranes were derived from the equations of motion. The finite element method (FEM) was used to derive equations to solve the stretchable catenary problem of the mooring line. To verify the function of mooring simulator, calculation results were compared with commercial mooring software. To validate the analysis results, we conducted an experimental test for offshore operation using two floating crane models scaled to 1:40. Two floating crane models and a pile model were established for operation of uprighting flare towers. During the model test, the motion of the floating cranes and tensions of the mooring lines were measured. Through the model test, the accuracy of the mooring analysis program developed in this study was verified. Therefore, if this mooring analysis program is used, it will be possible to perform a mooring analysis simulation at the same time as a maritime work simulation.

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Meeker, David C., and Miles A. Townsend. "Motions of a Powered Top with a Spherical Tip on a Curved Surface." Shock and Vibration 2, no. 1 (1995): 23–32. http://dx.doi.org/10.1155/1995/697496.

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The purpose of this article is to model the dynamics of a top with a finite radius tip on a curved basin in a gravitational field without (and with) energy addition and dissipation. This is an extension of a very general and classical problem and requires development of a method for treating the dynamical interactions between the two curved surfaces. The full nonlinear equations of motion are indicated; however, these equations are complex and do not show the dominant mechanisms that define the system motions. A novel method of “partial linearization” is employed that reduces the equations of motion to a relevant and tractable form in which these mechanisms are clearly exposed. The model and related results are compared with relevant examples from the literature. The movement of the top is simulated by an integration of the fully nonlinear equations of motion and compared with the partially linearized results.

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Kleiza, Vytautas, and Rima Šatinskaitė. "Modeling of constrained motion." Lietuvos matematikos rinkinys 62 (December 20, 2021): 43–49. http://dx.doi.org/10.15388/lmr.2021.25225.

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This paper presents an investigation of modeling and solving of differential equations in the study of mechanical systems with holonomic constraints. The 2D and 3D mathematical models of constrained motion are made. The structure of the models consists of nonlinear first or second order differential equations. Cases of free movement and movement with resistance are investigated. Solutions of the Cauchy problem of obtained differential equations were obtained by Runge–Kutta method.

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Stryczniewicz, Kamila, and Przemysław Drężek. "CFD Approach to Modelling Hydrodynamic Characteristics of Underwater Glider." Transactions on Aerospace Research 2019, no. 4 (December 1, 2019): 32–45. http://dx.doi.org/10.2478/tar-2019-0021.

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Abstract Autonomous underwater gliders are buoyancy propelled vehicles. Their way of propulsion relies upon changing their buoyancy with internal pumping systems enabling them up and down motions, and their forward gliding motions are generated by hydrodynamic lift forces exerted on a pair of wings attached to a glider hull. In this study lift and drag characteristics of a glider were performed using Computational Fluid Dynamics (CFD) approach and results were compared with the literature. Flow behavior, lift and drag forces distribution at different angles of attack were studied for Reynolds numbers varying around 105 for NACA0012 wing configurations. The variable of the glider was the angle of attack, the velocity was constant. Flow velocity was 0.5 m/s and angle of the body varying from −8° to 8° in steps of 2°. Results from the CFD constituted the basis for the calculation the equations of motions of glider in the vertical plane. Therefore, vehicle motion simulation was achieved through numeric integration of the equations of motion. The equations of motions will be solved in the MatLab software. This work will contribute to dynamic modelling and three-dimensional motion simulation of a torpedo shaped underwater glider.

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Zak, Michail. "Post-Instability Behavior of Solids." Transactions of the Canadian Society for Mechanical Engineering 9, no. 4 (December 1985): 200–209. http://dx.doi.org/10.1139/tcsme-1985-0027.

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The necessity of model reformulation in elasticity results from the failure of hyperbolicity of the governing equations of motion for classical models. The reformulation is based upon the introduction of additional kinematical microstructures in the form of multivalued displacement and velocity field (or fractal functions) which arc generated by the mechanism of the instability. The small scale motions describing this microstructure interact with the original large scale motion and restore the hyperbolicity of new governing equations of motion. The applications of the reformulated models to the problem of vibrational control and impact energy absorption are discussed.

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Farhang, K., A. Midha, and A. Bajaj. "A Higher-Order Analysis of Basic Linkages for Harmonic Motion Generation." Journal of Mechanisms, Transmissions, and Automation in Design 109, no. 3 (September 1, 1987): 301–7. http://dx.doi.org/10.1115/1.3258794.

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In an earlier work, a perturbation technique was first presented to obtain approximate simple harmonic equations for describing the output motions of rudimentary linkages, i.e., a crank-rocker and a slider-crank, with relativley small input cranks. The technique involved consideration of a small motion excursion about a so-called “mean linkage configuration.” These equations were facilitated through truncation of the binomial series expansion of the output motions, expressed in terms of the input crank angle. Assuming a small crank to ground link length ratio, terms containing second or higher powers of this ratio were neglected. This paper retains terms containing higher powers in an effort to improve upon (i) the definition of the mean linkage configuration, and (ii) the harmonic motion content representation of the output motions. The improvements made due to the “modified” equations, relative to the “original” ones, are pictorially presented as being significant.

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Bradley, Brendon A. "Empirical Correlations between Peak Ground Velocity and Spectrum-Based Intensity Measures." Earthquake Spectra 28, no. 1 (February 2012): 17–35. http://dx.doi.org/10.1193/1.3675582.

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Empirical correlation equations between peak ground velocity ( PGV) and several spectrum-based ground motion intensity measures are developed. The intensity measures examined in particular were: peak ground acceleration ( PGA), 5% damped pseudo-spectral acceleration ( SA), acceleration spectrum intensity ( ASI), and spectrum intensity ( SI). The computed correlations were obtained using ground motions from active shallow crustal earthquakes and four ground motion prediction equations. Results indicate that PGV is strongly correlated (i.e., a correlation coefficient of [Formula: see text]) with SI, moderately correlated with medium to long-period SA (i.e., [Formula: see text] for vibration periods 0.5-3.0 seconds), and also moderately correlated with short period SA, PGA and ASI ([Formula: see text]). A simple example is used to illustrate one possible application of the developed correlation equations for ground motion selection.

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LOZINSKI, ALEXEI, and MICHEL V. ROMERIO. "MOTION OF GAS BUBBLES, CONSIDERED AS MASSLESS BODIES, AFFORDING DEFORMATIONS WITHIN A PRESCRIBED FAMILY OF SHAPES, IN AN INCOMPRESSIBLE FLUID UNDER THE ACTION OF GRAVITATION AND SURFACE TENSION." Mathematical Models and Methods in Applied Sciences 17, no. 09 (September 2007): 1445–78. http://dx.doi.org/10.1142/s0218202507002340.

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A model allowing to describe motion and coalescence of gas bubbles in a liquid under the action of gravitation and surface tension is proposed. The shape of the bubbles is described by a pre-defined family of mappings, for example ellipsoids with a fixed volume and the effects of the gas motions inside the bubbles are neglected. The motion of a bubble is obtained in a Lagrangian form using the D'Alembert principle of virtual works. The set of equations is numerically solved with the help of the fictitious domain technique in which the Navier–Stokes equations in the domain formed by both fluid and gas are considered. The equations governing the bubbles motion are imposed by introducing Lagrange multipliers on the bubbles boundaries. Numerical results in 2D and 3D are presented.

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Ibrahim, R. A., and I. M. Grace. "Modeling of Ship Roll Dynamics and Its Coupling with Heave and Pitch." Mathematical Problems in Engineering 2010 (2010): 1–32. http://dx.doi.org/10.1155/2010/934714.

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In order to study the dynamic behavior of ships navigating in severe environmental conditions it is imperative to develop their governing equations of motion taking into account the inherent nonlinearity of large-amplitude ship motion. The purpose of this paper is to present the coupled nonlinear equations of motion in heave, roll, and pitch based on physical grounds. The ingredients of the formulation are comprised of three main components. These are the inertia forces and moments, restoring forces and moments, and damping forces and moments with an emphasis to the roll damping moment. In the formulation of the restoring forces and moments, the influence of large-amplitude ship motions will be considered together with ocean wave loads. The special cases of coupled roll-pitch and purely roll equations of motion are obtained from the general formulation. The paper includes an assessment of roll stochastic stability and probabilistic approaches used to estimate the probability of capsizing and parameter identification.

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Banerjee, A. K., and M. E. Lemak. "Multi-Flexible Body Dynamics Capturing Motion-Induced Stiffness." Journal of Applied Mechanics 58, no. 3 (September 1, 1991): 766–75. http://dx.doi.org/10.1115/1.2897262.

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This paper presents a multi-flexible-body dynamics formulation incorporating a recently developed theory for capturing motion-induced stiffness for an arbitrary structure undergoing large rotation and translation accompanied by small vibrations. In essence, the method consists of correcting dynamical equations for an arbitrary flexible body, unavoidably linearized prematurely in modal coordinates, with generalized active forces due to geometric stiffness corresponding to a system of 12 inertia forces and 9 inertia couples distributed over the body. Computation of geometric stiffness in this way does not require any iterative update. Equations of motion are derived by means of Kane’s method. A treatment is given for handling prescribed motions and calculating interaction forces. Results of simulations of motions of three flexible spacecraft, involving stiffening during spinup motion, dynamic buckling, and a slewing maneuver, demonstrate the validity and generality of the theory.

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50

Sanh, Do. "A method of constructing programmed motions of a mechanical system." Vietnam Journal of Mechanics 20, no. 3 (September 30, 1998): 46–57. http://dx.doi.org/10.15625/0866-7136/10027.

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In the paper the method for constructing programmed motion is represented. The requirements for the programmed motion are treated as ideal constraints in analytical mechanics. The programmed motions expressed in Lagrange coordinates and in quasi coordinates are investigated. By applying the form of equations of motion of a constrained mechanical system [7], a schema for calculating programmed motions has been established. By this schema the errors of realizing programmed motions have been reduced and controlled. For illustration of the method some examples have been investigated.

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