Dynamic Behavior of Spatial Linkages: Part 1—Exact Equations of Motion, Part 2—Small Oscillations About Equilibrium

1969 ◽  
Vol 91 (1) ◽  
pp. 251-265 ◽  
Author(s):  
J. J. Uicker
Keyword(s):  
Differential Equations ◽  
Equations Of Motion ◽  
Three Dimensional ◽  
Dynamic Force ◽  
Lagrange Equations ◽  
Oscillation Analysis ◽  
Restricted Problems ◽  
The Matrix ◽  
Laplace Transformations ◽  
External Forces

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.

scholarly journals Re-Evaluating the Classical Falling Body Problem

Mathematics ◽  
2020 ◽  
Vol 8 (4) ◽  
pp. 553 ◽  
Author(s):  
Essam R. El-Zahar ◽  
Abdelhalim Ebaid ◽  
Abdulrahman F. Aljohani ◽  
José Tenreiro Machado ◽  
Dumitru Baleanu
Keyword(s):  
Differential Equations ◽  
Body Problem ◽  
Inertial Frame ◽  
Analytic Solution ◽  
Equations Of Motion ◽  
Three Dimensional ◽  
Three Dimensions ◽  
Rotating Frame ◽  
Dimensional Model ◽  
Three Dimensional Model

This paper re-analyzes the falling body problem in three dimensions, taking into account the effect of the Earth’s rotation (ER). Accordingly, the analytic solution of the three-dimensional model is obtained. Since the ER is quite slow, the three coupled differential equations of motion are usually approximated by neglecting all high order terms. Furthermore, the theoretical aspects describing the nature of the falling point in the rotating frame and the original inertial frame are proved. The theoretical and numerical results are illustrated and discussed.

scholarly journals Nonlinear interactions in deformable container cranes

2015 ◽  
Vol 230 (1) ◽  
pp. 5-20 ◽  
Author(s):  
Andrea Arena ◽  
Walter Lacarbonara ◽  
Matthew P Cartmell
Keyword(s):  
Rigid Body ◽  
Internal Resonance ◽  
Equations Of Motion ◽  
Three Dimensional ◽  
Higher Order ◽  
Mode Shapes ◽  
Lagrange Equations ◽  
Internal Resonances ◽  
Container Cranes ◽  
Bending Mode

Nonlinear dynamic interactions in harbour quayside cranes due to a two-to-one internal resonance between the lowest bending mode of the deformable boom and the in-plane pendular mode of the container are investigated. To this end, a three-dimensional model of container cranes accounting for the elastic interaction between the crane boom and the container dynamics is proposed. The container is modelled as a three-dimensional rigid body elastically suspended through hoisting cables from the trolley moving along the crane boom modelled as an Euler-Bernoulli beam. The reduced governing equations of motion are obtained through the Euler-Lagrange equations employing the boom kinetic and stored energies, derived via a Galerkin discretisation based on the mode shapes of the two-span crane boom used as trial functions, and the kinetic and stored energies of the rigid body container and the elastic hoisting cables. First, conditions for the onset of internal resonances between the boom and the container are found. A higher order perturbation treatment of the Taylor expanded equations of motion in the neighbourhood of a two-to-one internal resonance between the lowest boom bending mode and the lowest pendular mode of the container is carried out. Continuation of the fixed points of the modulation equations together with stability analysis yields a rich bifurcation behaviour, which features Hopf bifurcations. It is shown that consideration of higher order terms (cubic nonlinearities) beyond the quadratic geometric and inertia nonlinearities breaks the symmetry of the bifurcation equations, shifts the bifurcation points and the stability ranges, and leads to bifurcations not predicted by the low order analysis.

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.

Dynamic Force Analysis of Spatial Linkages

1967 ◽  
Vol 34 (2) ◽  
pp. 418-424 ◽  
Author(s):  
J. J. Uicker
Keyword(s):  
Dynamic Force ◽  
Force Analysis ◽  
Lagrange Equations ◽  
Single Degree Of Freedom ◽  
Digital Computation ◽  
Mass Distributions ◽  
Single Degree ◽  
The Matrix ◽  
Spatial Linkages ◽  
Matrix Expression

The matrix method of linkage analysis is extended to the analysis of the bearing forces and torques which result from the inertia of the moving links when a single-loop, single-degree-of-freedom linkage is driven with known input velocity and acceleration. The method is well suited to digital computation and has been tested on several examples of spatial linkages, one of which is presented. A 4 × 4 “inertia matrix” is defined to describe the mass distributions of the links, and a matrix expression is derived for their kinetic energy. The dynamic bearing reactions are found by using the Lagrange equations with a varying constraint.

scholarly journals Nonlinear Dynamical Behavior of Axially Accelerating Beams: Three-Dimensional Analysis

2016 ◽  
Vol 11 (1) ◽  
Author(s):  
Mergen H. Ghayesh ◽  
Hamed Farokhi
Keyword(s):  
Differential Equations ◽  
Fourier Transforms ◽  
Equations Of Motion ◽  
Nonlinear Partial Differential Equations ◽  
Dynamical Behavior ◽  
Three Dimensional ◽  
Global Dynamics ◽  
Nonlinear Dynamical ◽  
Time Histories ◽  
Lateral Displacements

The three-dimensional (3D) nonlinear dynamics of an axially accelerating beam is examined numerically taking into account all of the longitudinal, transverse, and lateral displacements and inertia. Hamilton’s principle is employed in order to derive the nonlinear partial differential equations governing the longitudinal, transverse, and lateral motions. These equations are transformed into a set of nonlinear ordinary differential equations by means of the Galerkin discretization technique. The nonlinear global dynamics of the system is then examined by time-integrating the discretized equations of motion. The results are presented in the form of bifurcation diagrams of Poincaré maps, time histories, phase-plane portraits, Poincaré sections, and fast Fourier transforms (FFTs).

Characteristic Forms of Differential Equations for Wave Propagation in Nonlinear Media

1981 ◽  
Vol 48 (4) ◽  
pp. 743-748 ◽  
Author(s):  
T. C. T. Ting
Keyword(s):  
Wave Propagation ◽  
Differential Equations ◽  
Constitutive Equations ◽  
Equations Of Motion ◽  
Three Dimensional ◽  
Compressible Fluids ◽  
Elastic Solids ◽  
Nonlinear Media ◽  
Nonlinear Materials ◽  
One Dimensional

Characteristic forms of differential equations for wave propagation in nonlinear media are derived directly from equations of motion and equations which combine the constitutive equations and the equations of continuity. Both Lagrangian coordinates and Eulerian coordinates are considered. The constitutive equations considered here apply to a large class of nonlinear materials. The characteristic forms derived here clearly show which components of the stress and velocity are involved in the differentiation along the bicharacteristics. Moreover, the reduction to one-dimensional cases from three-dimensional problems is obvious for the characteristic forms obtained here. Examples are given and compared with the known solution in the literature for wave propagation in linear isotropic elastic solids and isentropic compressible fluids.

Methods for Automated Dynamic Analysis of Discrete Mechanical Systems

1973 ◽  
Vol 40 (3) ◽  
pp. 809-811 ◽  
Author(s):  
Y. O. Bayazitoglu ◽  
M. A. Chace
Keyword(s):  
Differential Equations ◽  
Dynamic Analysis ◽  
Ordinary Differential Equations ◽  
Mechanical System ◽  
Dynamic Systems ◽  
Equations Of Motion ◽  
Three Dimensional ◽  
Constrained Systems ◽  
Lower Pair ◽  
Linear And Nonlinear

The equations of motion for any discrete, lower pair mechanical system can be obtained by analyzing a branched, three-dimensional compound pendulum of indefinite length. In this paper, a set of expressions which provides the equations of motion of arbitrary mechanical dynamic systems directly as ordinary differential equations are presented. These expressions and the associated technique is applicable to linear and nonlinear unconstrained dynamic systems, kinematic systems and multidegree-of-freedom constrained systems.

Development and Application of a Generalized d’Alembert Force for Multifreedom Mechanical Systems

1971 ◽  
Vol 93 (1) ◽  
pp. 317-326 ◽  
Author(s):  
M. A. Chace ◽  
Y. O. Bayazitoglu
Keyword(s):  
Dynamic Analysis ◽  
Dynamic Systems ◽  
Equations Of Motion ◽  
Three Dimensional ◽  
Mechanical Systems ◽  
Lagrange Equations ◽  
Second Order Differential Equations ◽  
Dimensional Version ◽  
Mechanical Networks ◽  
Geometric Effects

A set of expressions termed the generalized d’Alembert force is determined for application to two and three-dimensional dynamic analysis of discrete, nonlinear, multifreedom, constrained, mechanical dynamic systems. These expressions greatly simplify the task of developing a correct set of second order differential equations of motion for mechanical systems which are nonlinear because of large deflections or other geometric effects. They apply to both constrained and unconstrained mechanical systems via the method of Lagrange equations with constraint. The two-dimensional version of the expressions has been successfully applied in a type-varient computer program for the dynamic analysis of mechanical networks, and example problems simulated with this program are discussed.

Quantum Mechanics and a Completely Integrable Dynamical System

1987 ◽  
Vol 42 (8) ◽  
pp. 819-824 ◽  
Author(s):  
W.-H. Steeb ◽  
A. J. van Tonder
Keyword(s):  
Differential Equations ◽  
Symmetric Matrix ◽  
Energy Levels ◽  
Equations Of Motion ◽  
Integrable Dynamical System ◽  
Finite Dimensional ◽  
Open Questions ◽  
The Matrix ◽  
Constants Of Motion ◽  
Energy Dependent

From the eigenvalue equation (H0 + λV) \ ψn(λ)) = En (λ) | ψn( λ ) ) one can derive an autonomous system of first order differential equations for the eigenvalues En (λ) and the matrix elements Vmn(X) = ( ψm(λ) \ V \ ψn(λ), where λ is the independent variable. If the initial values En (λ = 0) and ψn (λ = 0) are known the differential equations can be solved. Thus one finds the “motion” of the energy levels En(λ). Here we give two applications of this technique. Furthermore we describe the connection with the stationary state perturbation theory. We also derive the equations of motion for the extended case H = H0 + λ\ Vt + λ2 V2. Finally we investigate the case where the Hamiltonian is given by a finite dimensional symmetric matrix and derive the energy dependent constants of motion. Several open questions are also discussed.

scholarly journals 3D Device for Forces in Swimming Starts and Turns

2019 ◽  
Vol 9 (17) ◽  
pp. 3559 ◽  
Author(s):  
Karla de Jesus ◽  
Luis Mourão ◽  
Hélio Roesler ◽  
Nuno Viriato ◽  
Kelly de Jesus ◽  
...  
Keyword(s):  
Degrees Of Freedom ◽  
Force Plate ◽  
Three Dimensional ◽  
Wheatstone Bridge ◽  
Vertical Axis ◽  
Dynamic Force ◽  
Resonance Frequencies ◽  
Bridge Circuits ◽  
External Forces ◽  
Development And Validation

Biomechanical tools capable of detecting external forces in swimming starts and turns have been developed since 1970. This study described the development and validation of a three-dimensional (six-degrees of freedom) instrumented block for swimming starts and turns. Seven force plates, a starting block, an underwater structure, one pair of handgrips and feet supports for starts were firstly designed, numerically simulated, manufactured and validated according to the Fédération Internationale de Natation rules. Static and dynamic force plate simulations revealed deformations between 290 to 376 µε and 279 to 545 µε in the anterior-posterior and vertical axis and 182 to 328.6 Hz resonance frequencies. Force plates were instrumented with 24 strain gauges each connected to full Wheatstone bridge circuits. Static and dynamic calibration revealed linearity ( R 2 between 0.97 and 0.99) and non-meaningful cross-talk between orthogonal (1%) axes. Laboratory and ecological validation revealed the similarity between force curve profiles. The need for discriminating each upper and lower limb force responses has implied a final nine-force plates solution with seven above and two underwater platforms. The instrumented block has given an unprecedented contribution to accurate external force measurements in swimming starts and turns.

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