scholarly journals Solving gyrokinetic systems with higher-order time dependence

2020 ◽  
Vol 86 (4) ◽  
Author(s):  
A. Y. Sharma ◽  
B. F. McMillan
Keyword(s):  
Singular Perturbation ◽  
Degrees Of Freedom ◽  
Equations Of Motion ◽  
Time Derivative ◽  
Additional Term ◽  
Plasma Phys ◽  
Singular Perturbation Theory ◽  
Field Equations ◽  
Lagrange Equations ◽  
Explicit Equation

We discuss theoretical and numerical aspects of gyrokinetics as a Lagrangian field theory when the field perturbation is introduced into the symplectic part. A consequence is that the field equations and particle equations of motion in general depend on the time derivatives of the field. The most well-known example is when the parallel vector potential is introduced as a perturbation, where a time derivative of the field arises only in the equations of motion, so an explicit equation for the fields may still be written. We will consider the conceptually more problematic case where the time-dependent fields appear in both the field equations and equations of motion, but where the additional term in the field equations is formally small. The conceptual issues were described by Burby (J. Plasma Phys., vol. 82 (3), 2016, 905820304): these terms lead to apparent additional degrees of freedom to the problem, so that the electric field now requires an initial condition, which is not required in low-frequency (Darwin) Vlasov–Maxwell equations. Also, the small terms in the Euler–Lagrange equations are a singular perturbation, and these two issues are interlinked. For well-behaved problems the apparent additional degrees of freedom are spurious, and the physically relevant solution may be directly identified. Because we needed to assume that the system is well behaved for small perturbations when deriving gyrokinetic theory, we must continue to assume that when solving it, and the physical solutions are thus the regular ones. The spurious nature of the singular degrees of freedom may also be seen by changing coordinate systems so the varying field appears only in the Hamiltonian. We then describe how methods appropriate for singular perturbation theory may be used to solve these asymptotic equations numerically. We then describe a proof-of-principle implementation of these methods for an electrostatic strong-flow gyrokinetic system; two basic test cases are presented to illustrate code functionality.

Elements of Classical Field Theory

2021 ◽  
pp. 24-34
Author(s):  
J. Iliopoulos ◽  
T.N. Tomaras
Keyword(s):  
Field Theory ◽  
Degrees Of Freedom ◽  
Equations Of Motion ◽  
Classical Mechanics ◽  
Classical Field Theory ◽  
Classical Field ◽  
Lagrange Equations ◽  
Lie Group Of Transformations ◽  
Infinitesimal Transformations ◽  
Conserved Currents

The purpose of this chapter is to recall the principles of Lagrangian and Hamiltonian classical mechanics. Many results are presented without detailed proofs. We obtain the Euler–Lagrange equations of motion, and show the equivalence with Hamilton’s equations. We derive Noether’s theorem and show the connection between symmetries and conservation laws. These principles are extended to a system with an infinite number of degrees of freedom, i.e. a classical field theory. The invariance under a Lie group of transformations implies the existence of conserved currents. The corresponding charges generate, through the Poisson brackets, the infinitesimal transformations of the fields as well as the Lie algebra of the group.

Comparison between Analytical and Vectorial Methods of the Synthesis of Equations of Motion

1983 ◽  
Vol 197 (4) ◽  
pp. 265-274
Author(s):  
Z J Goraj
Keyword(s):  
Angular Momentum ◽  
Analytical Method ◽  
Degrees Of Freedom ◽  
Equations Of Motion ◽  
Discrete Systems ◽  
Momentum Equation ◽  
Lagrange Equations ◽  
Weak Points ◽  
Complicated Geometry ◽  
Boltzmann Hamel Equations

In this paper the advantages and weak points of the analytical and vectorial methods of the derivation of equations of motion for discrete systems are considered. The analytical method is discussed especially with respect to Boltzmann-Hamel equations, as generalized Lagrange equations. The vectorial method is analysed with respect to the momentum equation and to the generalized angular momentum equation about an arbitrary reference point, moving in an arbitrary manner. It is concluded that, for the systems with complicated geometry of motion and a large number of degrees of freedom, the vectorial method can be more effective than the analytical method. The combination of the analytical and vectorial methods helps to verify the equations of motion and to avoid errors, especially in the case of systems with rather complicated geometry.

Automatic Generation of Equations of Motion of Mechanical Systems

2014 ◽  
Vol 611 ◽  
pp. 40-45
Author(s):  
Darina Hroncová ◽  
Jozef Filas
Keyword(s):  
Computer Simulation ◽  
Degrees Of Freedom ◽  
Equations Of Motion ◽  
Mechanical Systems ◽  
Automatic Generation ◽  
Lagrange Equations ◽  
Kinematic Chains ◽  
Transformation Matrices

The paper describes an algorithm for automatic compilation of equations of motion. Lagrange equations of the second kind and the transformation matrices of basic movements are used by this algorithm. This approach is useful for computer simulation of open kinematic chains with any number of degrees of freedom as well as any combination of bonds.

scholarly journals Dynamical analysis of a 2-degrees of freedom spatial pendulum

2018 ◽  
Vol 184 ◽  
pp. 01003 ◽  
Author(s):  
Stelian Alaci ◽  
Florina-Carmen Ciornei ◽  
Sorinel-Toderas Siretean ◽  
Mariana-Catalina Ciornei ◽  
Gabriel Andrei Ţibu
Keyword(s):  
Differential Equations ◽  
Degrees Of Freedom ◽  
Nonlinear Differential Equations ◽  
Equations Of Motion ◽  
Dynamical Analysis ◽  
Analysis Software ◽  
Lagrange Equations ◽  
Moments Of Inertia ◽  
Dynamical Simulation ◽  
Principal Moments Of Inertia

A spatial pendulum with the vertical immobile axis and horizontal mobile axis is studied and the differential equations of motion are obtained applying the method of Lagrange equations. The equations of motion were obtained for the general case; the only simplifying hypothesis consists in neglecting the principal moments of inertia about the axes normal to the oscillation axes. The system of nonlinear differential equations was numerically integrated. The correctness of the obtained solutions was corroborated to the dynamical simulation of the motion via dynamical analysis software. The perfect concordance between the two solutions proves the rightness of the equations obtained.

scholarly journals Minimization of dynamic joint reaction forces of the 2-DOF serial manipulators based on interpolating polynomials and counterweights

2015 ◽  
Vol 42 (4) ◽  
pp. 249-260 ◽  
Author(s):  
Slavisa Salinic ◽  
Marina Boskovic ◽  
Radovan Bulatovic
Keyword(s):  
Degrees Of Freedom ◽  
Equations Of Motion ◽  
Lagrange Equations ◽  
Symbolic Form ◽  
Serial Manipulators ◽  
Joint Reaction Forces ◽  
Reaction Forces ◽  
Inertia Forces ◽  
Joint Reaction ◽  
Selection Of

This paper presents two ways for the minimization of joint reaction forces due to inertia forces (dynamic joint reaction forces) in a two degrees of freedom (2-DOF) planar serial manipulator. The first way is based on the optimal selection of the angular rotations laws of the manipulator links and the second one is by attaching counterweights to the manipulator links. The influence of the payload carrying by the manipulator on the dynamic joint reaction forces is also considered. The expressions for the joint reaction forces are obtained in a symbolic form by means of the Lagrange equations of motion. The inertial properties of the manipulator links are represented by dynamical equivalent systems of two point masses. The weighted sum of the root mean squares of the magnitudes of the dynamic joint reactions is used as an objective function. The effectiveness of the two ways mentioned is discussed.

Limit Cycle Stability Reversal via Singular Perturbation and Wing-Flap Flutter

2002 ◽  
Author(s):  
Daniele Dessi ◽  
Franco Mastroddi
Keyword(s):  
Singular Perturbation ◽  
Limit Cycles ◽  
Degrees Of Freedom ◽  
Perturbation Technique ◽  
Equations Of Motion ◽  
Control Surface ◽  
System Response ◽  
Singular Perturbation Technique ◽  
Nonlinear Springs ◽  
Time Marching

A three degrees of freedom aeroelastic typical section with control surface is theoretically modeled including nonlinear springs and augmented states for linear unsteady aerodynamic description. The system response is determined by time marching of the governing equations by using a standard Runge-Kutta algorithm in conjunction with a ‘shooting method’ to find out stable and unstable limit cycles along with stability reversal in the neighborhood of the Hopf bifurcation. Furthermore, the equations of motion are analyzed by a singular perturbation technique, specifically, by using a normal form method. Approximate analytical expressions for amplitudes and frequencies of limit cycles are obtained and the terms which are responsible of the nonlinear system behavior are identified.

Bending-Bending Mode of a Rotating Tapered-Twisted Turbomachine Blade Including Rotatory Inertia and Shear Deformation

1969 ◽  
Vol 91 (4) ◽  
pp. 1017-1024 ◽  
Author(s):  
R. M. Krupka ◽  
A. M. Baumanis
Keyword(s):  
Shear Deformation ◽  
Equations Of Motion ◽  
Torsional Vibrations ◽  
Parameter System ◽  
Field Equations ◽  
Lagrange Equations ◽  
Rotatory Inertia ◽  
Lumped Parameter ◽  
Bending Mode ◽  
Definition Of

This paper presents the effect on natural frequency and mode shape of the inclusion of terms that are present in the general equations of motion to describe phenomena associated with Rotatory Inertia and Shear Deformation. The coupling that exists between the flexural and torsional vibrations is not considered. Carnegie’s formulation of the Lagrange Equations of motion is used and the set of field equations solved using Myklestad’s adaptation of the Holzer method. The definition of the lumped parameter system used and the derivation of the associated discrete “difference equations,” which are utilized in the computer approach to the boundary value problem considered, constitute an extension of the Carnegie work.

Effects of Temperature and Cross Section Variations on the Vibrational Behavior of Finished Wire Blocks

2014 ◽  
Vol 622-623 ◽  
pp. 95-102
Author(s):  
Mohammadzaman Savari ◽  
Paul Josef Mauk ◽  
Oberrat Bernhardt Weyh
Keyword(s):  
Product Quality ◽  
Torsional Vibration ◽  
Degrees Of Freedom ◽  
Equations Of Motion ◽  
Equation System ◽  
Rolling Process ◽  
Essential Elements ◽  
Lagrange Equations ◽  
Cross Sectional ◽  
Coupled Equations

Because of the great speed range in a finishing block of a wire rod mill, the reduction of torsional vibrations makes it possible to achieve closer rolling tolerances. The components transmitting the torque like gear box, coupling etc. can generate non-smooth or alternating torques which affect the product quality. To study the influences of torsion vibrations on the product quality, the dynamic interactions of the block and rolling process are simultaneously analyzed by a simulation model. As an example, a three stand arrangement is considered. The real transmission system is idealized as a structurally discrete torsional vibration model. The generalized rotational coordinates with a large number of rotational degrees of freedom can be reduced among others constants by means of gear ratios. Euler-Lagrange equations are applied to create the coupled equations of motion, which together constitute an ordinary differential equation of order 28. The rolling and main drive torques are defined as excitation for vibration on the right side of the equation system. A DC motor is selected as main drive and the voltage circuit equation of motor is integrated into the system of differential equations. The armature current and its interaction are consequently simulated. The rotational speed of rolls and motor, as well as roll torques, longitudinal stresses in rod and section widths are shown in diagrams as the result and thereby the torsional vibration of the essential elements of the system are studied for different temperatures and cross-sectional variations.

scholarly journals A Semihydrostatic Theory of Gravity-Dominated Compressible Flow

2014 ◽  
Vol 71 (12) ◽  
pp. 4621-4638 ◽  
Author(s):  
Thomas Dubos ◽  
Fabrice Voitus
Keyword(s):  
Acoustic Waves ◽  
Degrees Of Freedom ◽  
Equations Of Motion ◽  
Potential Temperature ◽  
Normal Modes ◽  
Additional Term ◽  
Internal Gravity Waves ◽  
Least Action Principle ◽  
Vertical Coordinate ◽  
Hydrostatic Balance

Abstract From Hamilton’s least-action principle, compressible equations of motion with density diagnosed from potential temperature through hydrostatic balance are derived. Slaving density to potential temperature suppresses the degrees of freedom supporting the propagation of acoustic waves and results in a soundproof system. The linear normal modes and dispersion relationship for an isothermal state of rest on f and β planes are accurate from hydrostatic to nonhydrostatic scales, except for deep internal gravity waves. Specifically, the Lamb wave and long Rossby waves are not distorted, unlike with anelastic or pseudoincompressible systems. Compared to similar equations derived by A. Arakawa and C. S. Konor, the semihydrostatic system derived here possesses an additional term in the horizontal momentum budget. This term is an apparent force resulting from the vertical coordinate not being the actual height of an air parcel but its hydrostatic height (the hypothetical height it would have after the atmospheric column it belongs to has reached hydrostatic balance through adiabatic vertical displacements of air parcels). The Lagrange multiplier λ introduced in Hamilton’s principle to slave density to potential temperature is identified as the nonhydrostatic vertical displacement (i.e., the difference between the actual and hydrostatic heights of an air parcel). The expression of nonhydrostatic pressure and apparent force from λ allow the derivation of a well-defined linear symmetric positive definite problem for λ. As with hydrostatic equations, vertical velocity is diagnosed through Richardson’s equation. The semihydrostatic system has therefore precisely the same degrees of freedom as the hydrostatic primitive equations, while retaining much of the accuracy of the fully compressible Euler equations.

scholarly journals Quadrotor UAV Control for Transportation of Cable Suspended Payload

2019 ◽  
Vol 26 (2) ◽  
pp. 77-84 ◽  
Author(s):  
Tom Kusznir ◽  
Jarosław Smoczek
Keyword(s):  
Degrees Of Freedom ◽  
Sliding Mode ◽  
Equations Of Motion ◽  
Lagrange Equations ◽  
Wide Range ◽  
Time Required ◽  
Interest In Research ◽  
Emergency Equipment ◽  
Pendulum Dynamics ◽  
4 Degrees Of Freedom

Abstract Payload transportation with UAV’s (Unmanned Aerial Vehicles) has become a topic of interest in research with possibilities for a wide range of applications such as transporting emergency equipment to otherwise inaccessible areas. In general, the problem of transporting cable suspended loads lies in the under actuation, which causes oscillations during horizontal transport of the payload. Excessive oscillations increase both the time required to accurately position the payload and may be detrimental to the objects in the workspace or the payload itself. In this article, we present a method to control a quadrotor with a cable suspended payload. While the quadrotor itself is a nonlinear system, the problem of payload transportation with a quadrotor adds additional complexities due to both input coupling and additional under actuation of the system. For simplicity, we fix the quadrotor to a planar motion, giving it a total of 4 degrees of freedom. The quadrotor with the cable suspended payload is modelled using the Euler-Lagrange equations of motion and then partitioned into translation and attitude dynamics. The design methodology is based on simplifying the system by using a variable transformation to decouple the inputs, after which sliding mode control is used for the translational and pendulum dynamics while a feedback linearizing controller is used for the rotational dynamics of the quadrotor. The sliding mode parameters are chosen so stability is guaranteed within a certain region of attraction. Lastly, the results of the numerical simulations created in MATLAB/Simulink are presented to verify the effectiveness of the proposed control strategy.

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