Lagrangian Equation of Coupled Spring-Pendulum System

Analytical dynamics of mechanisms—a computer oriented overview

In this paper, a model for transverse web vibration in a roll-to-roll system is presented. Web axial tension and web axial speed, decisive parameters in the equation of motion that describes web vibration, are rigorously obtained by considering the two rolls-web coupled system’s dynamics, coupled with the equation of motion. According to the present analysis, the idealized simply-supported boundary conditions, commonly used in studies on vibrations of axially moving structures are not needed. Instead, a mathematical model comprised of the governing equation of web transverse vibration and the roll angular velocity – web axial tension relationship is solved as a coupled system. A finite-difference based algorithm is used for solving the coupled system of differential equations. It is worth noting that the web axial speed and web-transmitted tension are not constants when a certain amount of the web material is transferred from the unwinding roll to the winding roll; they vary nonlinearly after a short transient period. The transverse vibration response at selected points on the web span shows higher (lower) frequency fluctuations corresponding to lower (higher) transport axial speed. This behavior is significantly different from that of a vibrating web under constant axial speed and tension.

The quaternion equations of disturbed motion of an artificial Earth satellite (AES) in Earth's gravitational field (with regard to its zonal, tesseral, and sectorial harmonics) are obtained in four-dimensional Kustaanheimo–Stiefel variables and in modified four-dimensional variables. In the latter variables the equations of satellite motion have a simpler and more symmetric structure, as compared to the equations of motion in the Kustaanheimo–Stiefel variables. The obtained equations are linear for undisturbed Keplerian motion and, in general, have the form of equations of motion of a disturbed four-dimensional oscillator. These equations, in contrast to classical equations, are regular (they do not contain specific points of singularity) for satellite motion in the central gravitational field of Earth under an effect of disturbing forces, the description of which does not contain negative, higher-than-first powers of the satellite distance to Earth's center. In these equations, the basic variables are Kustaanheimo–Stiefel variables, or modified four-dimensional variables proposed by the author, as well as satellite motion energy and time. The new independent variable is associated with time by the differential relation containing the satellite distance to Earth's center of mass. The equations are convenient for applying nonlinear mechanics methods and for high-precision numerical calculations. In the case of artificial satellite motion in Earth's gravitational field, the description of which does not take into account the tesseral and sectorial harmonics, but takes into account its zonal harmonics, the first integrals of the equations of satellite motion are found, and the replacements of variables and transformations of these equations are proposed, which allowed one to get, for studying satellite motion, closed systems of differential equations of lower dimension, in particular, the system of equations of fourth and third orders.

An alternative approach for the analysis of nonlinear vibrations of pipes conveying fluid

Dynamics and Control of Machines

<div class="htmlview paragraph">This paper is to propose the equations of motion in time of steady-state cornering motion, which are derived from the Lagranges's equations. The proposed equation is the nonlinear algebraic equation analyzable the handling torque characteristic, and applicable also to a large bank angle conditions.</div> <div class="htmlview paragraph">The amount of this paper is explained under the following four points. 1. The equations of motion in time of steady-state motorcycle cornering motion are derived from applying the Lagranges's equations of motorcycles R.S. Sharp proposed. The experiment value was used in the process to derive from. This new proposed equations include the additional new items which are not used in existing equations of motion in time of steady-state cornering motion, based on the equations of the straight-running stability. 2. The proposed new equations are considered the influence of tire section radius. 3. The equations of motion in time of steady-state cornering motion are derived as the nonlinear algebraic equation. 4. The handling torque characteristic over the large bank angle was analyzable by deriving the nonlinear algebraic equations to the linear equations for arbitrary bank angle around there.</div>

Part 1: Over the past several years, the matrix method of linkage analysis has been developed to give the kinematic, static and dynamic force, error, and equilibrium analyses of three-dimensional mechanical linkages. This two-part paper is an extension of these methods to include some aspects of dynamic analysis. In Part 1, expressions are developed for the kinetic and potential energies of a system consisting of a multiloop, multi-degree-of-freedom spatial linkage having springs and damping devices in any or all of its joints, and under the influence of gravity as well as time varying external forces. Using the Lagrange equations, the exact differential equations governing the motion of such a system are derived. Although these equations cannot be solved directly, they form the basis for the solution of more restricted problems, such as a linearized small oscillation analysis which forms Part 2 of the paper. Part 2: This paper is a direct extension of Part 1 and it is assumed that the reader has a thorough knowledge of the previous material. Assuming that the spatial linkage has a stable position of static equilibrium and oscillates with small displacements and small velocities about this position, the general differential equations of motion are linearized to describe these oscillations. The equations lead to an eigenvalue problem which yields the resonant frequencies and associated damping constants of the system for the equilibrium position. Laplace transformations are then used to solve the linearized equations. Digital computer programs have been written to lest these methods and an example solution dealing with a vehicle suspension is presented.

Hamiltonian systems, information transmittal and special relativity

Analytical Mechanics. Foundations of Engineering Mechanics Series

Saint-Venant end effects for plane deformations of linear piezoelectric solids

With the recent flurry of research on satellite formation flying, a need has become apparent for a set of linearized equations of relative motion that capture the effect of the J2 geopotential disturbance force. Typically, Hill's linearized equations of relative motion have been used for this analysis, but they fail to capture the effect of the J2 disturbance force on a satellite cluster. In this paper, a new set of constant coefficient, linearized differential equations of motion is derived. These equations are similar in form to Hill's equations, but they capture the effects of the J2 disturbance force. A numerical simulator is employed to check the fidelity of the equations. It is shown that with the appropriate initial conditions, the new lin earized equations of motion have periodic errors (on the order of centimeters) that do not grow in time. The new linearized equations of motion also allow for insight into the effects of the J2 disturbance on a satellite cluster. This includes 'tumbling', the period of the relative orbit, and satellite separation due to differential J2 effects. Overall, a new high fidelity set of linearized equations are produced that are well suited to model satellite relative motion in the presence of the J2 disturbance force.

Unified Hamiltonian theory of relativistic particle equations

Modeling and Control of a Flexible Orbiting Spacecraft

In the current paper an asymptotic method to the problem of establishing the optimal program trajectory and optimal control in the movement of a sucker rod pumping unit of oscillatory systems with liquid damper is considered, where the plunger is inside the Newtonian fluid. In this case the mass of the head is large enough. From the properties of the plunger motion, the boundary conditions are taken to be periodic and the transition of motion from one mode to the second is described by impulse systems. By means of expedient transformations, the given equation of motion with fractional derivatives is reduced to the equation of fractional order containing a small parameter (the inverse of the mass of the head). Using the method of discretization of oscillatory systems with liquid dampers, a system of the first-order difference equations is reduced to the two-dimensional system. Using the given static data, the definition of fractional derivatives in subordinate terms is considered, a quadratic functional is constructed and this problem is investigated by the least squares method. Constructing the extended functional the discrete Euler-Lagrange equations are obtained. The control actions and the corresponding optimal program trajectory is found from the obtained system of discrete Euler-Lagrange equations using the Matlab software package and the algorithm of the calculation process is proposed. The results are illustrated with a specific, simple numerical example from practice and the graphs of optimal control and optimal program trajectory are given. Keywords: asymptotic method; Newtonian fluid; Euler-Lagrange equations; algorithm of the calculation.

This paper is concerned with the application of perturbation method to the dynamic analysis of floating flexible body. In dealing with the dynamics of free-floating body, the rigid-body motions and elastic vibrations are analyzed separately. However, the rigid-body motions cause vibrations and elastic vibrations also affect rigid-body motions in turn, which indicates that the rigid-body motions and elastic vibrations are coupled in nature. The resulting equations of motion are hybrid and nonlinear. We can discretize the equations of motion by means of admissible functions but still we have to cope with nonlinear equations. In the previous paper, we proposed the use of perturbation method to the coupled equations of motion and derived zero-order and first-order equations of motion. The derivation process was lengthy and tedious. Hence, in this paper, we propose a new approach to the same problem by applying the perturbation method to the Lagrange's equations, thus providing a systematic approach to the addressed problem. Theoretical derivations show the efficacy of the proposed method.