Linearization of Multibody Dynamic Systems Using Automatic Differentiation (AD) Tools

1993 ◽  
Author(s):  
Tsung-Chieh Lin
Keyword(s):  
Dynamic Systems ◽  
Degrees Of Freedom ◽  
Automatic Differentiation ◽  
Robot Manipulator ◽  
Order Approximation ◽  
General Purpose ◽  
Dynamics Of Multibody Systems ◽  
Closed Chain ◽  
Multibody Dynamic ◽  
First Order Approximation

Abstract This paper presents an automatic method to linearize the dynamics of multibody systems that are modeled through a recursive approach. The first-order approximation of the nonlinear dynamic systems is obtained by the use of an automatic differentiation (AD) tool (GRESS) and a 9,700 lines Fortran model for the dynamics. The efficiency and accuracy of this AD implementation is shown by two examples: a five-bar closed-chain robot manipulator and a 18 degrees of freedom tractor-trailer. This study successfully demonstrates how to create a general-purpose numerical tool that can provide accurate solutions and derivatives for multibody dynamic systems.

Chaotic Amplitude Dynamics for Parametrically Excited Systems With Quadratic Nonlinearities

1995 ◽  
Author(s):  
Bappaditya Banerjee ◽  
Anil K. Bajaj
Keyword(s):  
Harmonic Balance ◽  
Chaotic Dynamics ◽  
Degrees Of Freedom ◽  
Order Approximation ◽  
Amplitude Equations ◽  
Analytical Technique ◽  
First Order ◽  
Numerical Studies ◽  
Quadratic Nonlinearities ◽  
First Order Approximation

Abstract Dynamical systems with two degrees-of-freedom, with quadratic nonlinearities and parametric excitations are studied in this analysis. The 1:2 superharmonic internal resonance case is analyzed. The method of harmonic balance is used to obtain a set of four first-order amplitude equations that govern the dynamics of the first-order approximation of the response. An analytical technique, based on Melnikov’s method is used to predict the parameter range for which chaotic dynamics exist in the undamped averaged system. Numerical studies show that chaotic responses are quite common in these quadratic systems and chaotic responses occur even in presence of damping.

Generalized Modeling of Dynamic Cam-follower Pairs in Mechanisms

1991 ◽  
Vol 113 (2) ◽  
pp. 102-109 ◽  
Author(s):  
B. S. Bagepalli ◽  
T. L. Haskins ◽  
I. Imam
Keyword(s):  
Multibody Systems ◽  
Surface Profile ◽  
General Purpose ◽  
Mechanism Analysis ◽  
Dynamic Effects ◽  
Dynamics Of Multibody Systems ◽  
Multibody Dynamic ◽  
Cam Follower ◽  
2D And 3D ◽  
Multibody Dynamic System

This paper deals with the application of homogeneous (4 × 4) transformations to the generalizing modeling of cam-follower pairs. The procedure adopted is generic, in that it suggests a method of incorporating the modeling of cam-follower pairs in a general purpose program, such as MAP (Mechanism Analysis Program, © General Electric Co.) capable of solving the dynamics of multibody systems, in which any of the bodies could be cams, or followers. This development covers both 2D and 3D cam-follower pairs: XY cams, rotary cams, and drum cams. The cams could be dynamic—with single, or two surfaces (or tracked), with the possibility of impacts between the follower and these surfaces, or, kinematic, with the follower being guided exactly in a slot. This generalized procedure allows one to model several cam-follower pairs in a multibody dynamic system. This is useful in studying, for instance, the dynamic effects that tend to bounce a follower off of a cam surface, the contact force generated, etc. The procedure, also, just as easily, allows for the easy studying of the effects of varying the cam surface profile. Several examples have been tried, and some correlations have been obtained with experimental observations.

An Effective Method for Modeling Stiction in Multibody Dynamic Systems

1996 ◽  
Vol 118 (1) ◽  
pp. 172-176 ◽  
Author(s):  
R. G. Synnestvedt
Keyword(s):  
Complex Systems ◽  
Dynamic Systems ◽  
Degrees Of Freedom ◽  
Dynamic Equations ◽  
Stick Slip ◽  
System A ◽  
Multibody Dynamic ◽  
Model Dynamics ◽  
Spring System ◽  
Mass Spring

This paper presents an effective method for developing dynamic equations which realistically model dynamics of multibody mechanical systems with stiction, or stick-slip friction. This method is used in three examples—a mass-spring system, a top, and a robot linkage—to illustrate the facility with which the method is implemented. The method dynamically partitions sets of dynamic equations to model a system through discontinuities, changes in degrees of freedom and changes in states. Comparisons of this method with others is presented for simple and complex systems.

The method of the averaged Hamiltonian and the two-stream explosive instability

1976 ◽  
Vol 15 (1) ◽  
pp. 1-14 ◽  
Author(s):  
Z. Sedláček
Keyword(s):  
Degrees Of Freedom ◽  
Order Approximation ◽  
Wave Interaction ◽  
Electrostatic Waves ◽  
Explosive Instability ◽  
Complex Wave ◽  
First Order Approximation ◽  
Averaged Hamiltonian ◽  
The Waves ◽  
Action Variables

Second-order perturbation calculation shows that an explosive instability of three resonantly interacting coherent electrostatic waves can be limited, and converted into a multiple-periodic process by nonlinear terms of the same order as those that destabilize the waves in the first-order approximation. No higher- order nonlinearities are necessary. The method used is purely classical, and consists in transforming the Hamiltonian of the waves into angle-action variables, and canonical averaging of the Hamiltonian over the proper angles. The number of degrees of freedom is thus reduced to one, which permits one to analyse the wave interaction in the phase plane without using the usual equations for the complex wave amplitudes.

Nonlinear Free Vibration of a Symmetrically Conservative Two-Mass System With Cubic Nonlinearity

2009 ◽  
Vol 5 (1) ◽  
Author(s):  
T. Pirbodaghi ◽  
S. Hoseini
Keyword(s):  
Free Vibration ◽  
Degrees Of Freedom ◽  
Order Approximation ◽  
Maximum Relative Error ◽  
Nonlinear Free Vibration ◽  
Mass System ◽  
Approximate Analytical Solutions ◽  
Homotopy Analysis ◽  
First Order Approximation ◽  
The Mathematical Model

In this study, the nonlinear free vibration of conservative two degrees of freedom systems is analyzed using the homotopy analysis method (HAM). The mathematical model of such systems is described by two second-order coupled differential equations with cubic nonlinearities. First, novel approximate analytical solutions for displacements and frequencies are established using HAM. Then, the homotopy Padé technique is applied to accelerate the convergence rate of the solutions. Comparison between the obtained results and those available in the literature shows that the first-order approximation of homotopy Padé technique leads to accurate solutions with a maximum relative error less than 0.068 percent for all the considered cases.

Linearization and Jacobian Evaluation of the Dynamics of Closed-Chain Mechanical Systems

1992 ◽  
Author(s):  
Tsung-Chieh Lin ◽  
K. Harold Yae
Keyword(s):  
Dynamic Systems ◽  
Large Scale ◽  
Equations Of Motion ◽  
Mechanical Systems ◽  
Numerical Differentiation ◽  
Large Scale Systems ◽  
Closed Chain ◽  
Multibody Dynamic ◽  
The Finite Difference Method ◽  
Recursive Equations

Abstract This paper presents an analytical/numerical method for linearizing the equations of motion and evaluating the system Jacobian matrices of mechanical systems with closed chains. The linearization algorithm developed here first identifies and linearizes basic recursive kinematic relationships and then applies the chain rule to the derivation of the equations of motion under the framework of recursive formulation. This method can be incorporated into formulating recursive equations of motion for general multibody dynamic systems, to handle large scale systems. Since no numerical differentiation is used in the proposed algorithm, its accuracy is comparable to symbolic, closed-form linearization. Moreover, without the need of repetitious computation to select proper perturbation quantities, this method is computationally more efficient than the finite difference method.

Improved first-order approximation of eigenvalues and eigenvectors

AIAA Journal ◽  
1998 ◽  
Vol 36 ◽  
pp. 1721-1727
Author(s):  
Prasanth B. Nair ◽  
Andrew J. Keane ◽  
Robin S. Langley
Keyword(s):  
Order Approximation ◽  
Eigenvalues And Eigenvectors ◽  
First Order ◽  
First Order Approximation

Analytical solution for unsteady flow behind ionizing shock wave in a rotational axisymmetric non-ideal gas with azimuthal or axial magnetic field

2021 ◽  
Vol 76 (3) ◽  
pp. 265-283
Author(s):  
G. Nath
Keyword(s):  
Magnetic Field ◽  
Shock Wave ◽  
Analytical Solution ◽  
Axial Magnetic Field ◽  
Order Approximation ◽  
Ideal Gas ◽  
Field Region ◽  
First Order ◽  
First Order Approximation ◽  
Flow Variables

Abstract The approximate analytical solution for the propagation of gas ionizing cylindrical blast (shock) wave in a rotational axisymmetric non-ideal gas with azimuthal or axial magnetic field is investigated. The axial and azimuthal components of fluid velocity are taken into consideration and these flow variables, magnetic field in the ambient medium are assumed to be varying according to the power laws with distance from the axis of symmetry. The shock is supposed to be strong one for the ratio C 0 V s 2 ${\left(\frac{{C}_{0}}{{V}_{s}}\right)}^{2}$ to be a negligible small quantity, where C 0 is the sound velocity in undisturbed fluid and V S is the shock velocity. In the undisturbed medium the density is assumed to be constant to obtain the similarity solution. The flow variables in power series of C 0 V s 2 ${\left(\frac{{C}_{0}}{{V}_{s}}\right)}^{2}$ are expanded to obtain the approximate analytical solutions. The first order and second order approximations to the solutions are discussed with the help of power series expansion. For the first order approximation the analytical solutions are derived. In the flow-field region behind the blast wave the distribution of the flow variables in the case of first order approximation is shown in graphs. It is observed that in the flow field region the quantity J 0 increases with an increase in the value of gas non-idealness parameter or Alfven-Mach number or rotational parameter. Hence, the non-idealness of the gas and the presence of rotation or magnetic field have decaying effect on shock wave.

scholarly journals Software Sensor to Enhance Online Parametric Identification for Nonlinear Closed-Loop Systems for Robotic Applications

Sensors ◽  
2021 ◽  
Vol 21 (11) ◽  
pp. 3653
Author(s):  
Lilia Sidhom ◽  
Ines Chihi ◽  
Ernest Nlandu Kamavuako
Keyword(s):  
Degrees Of Freedom ◽  
Closed Loop ◽  
Sliding Mode ◽  
Robot Manipulator ◽  
Parametric Identification ◽  
Recursive Least Squares ◽  
Unknown Parameters ◽  
Closed Loop Identification ◽  
Input Variables ◽  
Robotic Applications

This paper proposes an online direct closed-loop identification method based on a new dynamic sliding mode technique for robotic applications. The estimated parameters are obtained by minimizing the prediction error with respect to the vector of unknown parameters. The estimation step requires knowledge of the actual input and output of the system, as well as the successive estimate of the output derivatives. Therefore, a special robust differentiator based on higher-order sliding modes with a dynamic gain is defined. A proof of convergence is given for the robust differentiator. The dynamic parameters are estimated using the recursive least squares algorithm by the solution of a system model that is obtained from sampled positions along the closed-loop trajectory. An experimental validation is given for a 2 Degrees Of Freedom (2-DOF) robot manipulator, where direct and cross-validations are carried out. A comparative analysis is detailed to evaluate the algorithm’s effectiveness and reliability. Its performance is demonstrated by a better-quality torque prediction compared to other differentiators recently proposed in the literature. The experimental results highlight that the differentiator design strongly influences the online parametric identification and, thus, the prediction of system input variables.

FIRST-ORDER APPROXIMATION OF THE NUMBER-PROJECTED SO(2N) TAMM-DANCOFF EQUATION AND ITS REDUCTION BY THE SCHUR FUNCTION

1999 ◽  
Vol 08 (05) ◽  
pp. 461-483
Author(s):  
SEIYA NISHIYAMA
Keyword(s):  
Wave Function ◽  
Order Approximation ◽  
Approximate Equation ◽  
Variation Principle ◽  
Schur Function ◽  
Soliton Theory ◽  
First Order ◽  
Fermion Systems ◽  
Group Manifold ◽  
First Order Approximation

First-order approximation of the number-projected (NP) SO(2N) Tamm-Dancoff (TD) equation is developed to describe ground and excited states of superconducting fermion systems. We start from an NP Hartree-Bogoliubov (HB) wave function. The NP SO(2N) TD expansion is generated by quasi-particle pair excitations from the degenerate geminals in the number-projected HB wave function. The Schrödinger equation is cast into the NP SO(2N) TD equation by the variation principle. We approximate it up to first order. This approximate equation is reduced to a simpler form by the Schur function of group characters which has a close connection with the soliton theory on the group manifold.

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