Analysis of equations of motion for second-order systems using generalized velocity components

2007 ◽  
Vol 221 (2) ◽  
pp. 205-212 ◽  
Author(s):  
P Herman
Keyword(s):  
Differential Equations ◽  
System Dynamics ◽  
Equations Of Motion ◽  
Second Order ◽  
Model Based ◽  
Interesting Insight ◽  
Disturbance Model ◽  
Second Order Systems ◽  
Multi Body ◽  
Insight Into

An analysis of equations of motion expressed in terms of generalized velocity components (GVCs) for two second-order systems is presented in this paper. The two-link planar manipulator and the cart-pendulum are considered. It was shown that using GVC, two firstorder differential equations are obtained, which give some useful information about the system. The analysis allows one to detect several interesting properties concerning the system. In addition, A friction or disturbance model based on GVC can be considered, which gives an interesting insight into the system dynamics. The results of the analysis can be extended to multi-body systems.

Equations of motion with the identity mass matrix for holonomic systems

2004 ◽  
Vol 218 (7) ◽  
pp. 731-736
Author(s):  
P Herman
Keyword(s):  
Differential Equations ◽  
Mass Matrix ◽  
Equations Of Motion ◽  
Dynamical Equations ◽  
First Order ◽  
Interesting Insight ◽  
Holonomic Systems ◽  
Insight Into ◽  
First Order Differential Equations

Some consequences concerning holonomic systems described in terms of the inertial quasi-velocities (IQV) are discussed in this note. Introducing the IQV vector into Lagrange's formulation leads to first-order equations with the identity mass matrix of the system. The first-order differential equations give an interesting insight into dynamics and some important properties. The two examples that are provided use the dynamical equations in terms of IQV.

Some Further Insight Into Self-Adjoint Second-Order Systems

1999 ◽  
Vol 5 (2) ◽  
pp. 237-252 ◽  
Author(s):  
S.D. Garvey ◽  
M.I. Friswell ◽  
J.E.T. Penny
Keyword(s):  
Second Order ◽  
Second Order Systems ◽  
Insight Into

On the instability in second-order systems for acoustic VTI and TTI media

Geophysics ◽  
2012 ◽  
Vol 77 (5) ◽  
pp. T171-T186 ◽  
Author(s):  
Kenneth P. Bube ◽  
Tamas Nemeth ◽  
Joseph P. Stefani ◽  
Ray Ergas ◽  
Wei Liu ◽  
...  
Keyword(s):  
Wave Equation ◽  
Dispersion Relation ◽  
Differential Equations ◽  
Shear Wave ◽  
Transversely Isotropic ◽  
Second Order ◽  
Wave Speeds ◽  
Tti Media ◽  
Second Order Systems ◽  
Vti Media

We studied second-order wave propagation systems for vertical transversely isotropic (VTI) and tilted transversely isotropic (TTI) acoustic media with variable axes of symmetry that have their shear-wave speeds set to zero. Acoustic TTI systems are commonly used in reverse-time migration, but these second-order systems are susceptible to instablities appearing as nonphysical stationary noise growing linearly in time, particularly in variable-tilt TTI media. We found an explanation of the cause of this phenomenon. The instabilities are not caused only by the numerical schemes; they are inherent to the differential equations. These instabilities are present even in homogeneous VTI media. These instabilities are caused by zero wave speeds at a wide variety of wavenumbers — a direct consequence of setting the shear-wave speeds to zero — coupled with the second time derivative in these systems. Although the second-order isotropic wave equation allows smooth time-growing solutions, a larger class of time-growing solutions exists for the second-order acoustic TI systems, including nonsmooth solutions. Boundary conditions appear to be less effective in controlling these time-growing solutions than they are for the isotropic wave equation. These systems conserve an incomplete energy that does not prevent the instabilities. The corresponding steady-state systems are no longer elliptic differential equations and can have nonsmooth solutions that are related to the instabilities. We started initially with homogeneous VTI media, and then extended these results to heterogeneous variable-tilt TTI media. We also developed a second-order acoustic system for heterogeneous variable-tilt TTI media derived directly from the full-elastic system for heterogeneous variable-tilt TTI media. All second-order systems with a dispersion relation obtained by setting the shear-wave speeds to zero in the elastic dispersion relation allowed these nonphysical time-growing solutions; however, knowing the cause of these instabilities, it may be possible to prevent or control the activation of these solutions.

Performance Study of a Bat Searching Algorithm From System Dynamics Perspective

2019 ◽  
Author(s):  
Haopeng Zhang ◽  
Nathan Schutte
Keyword(s):  
System Dynamics ◽  
Optimization Problems ◽  
Second Order ◽  
Point Of View ◽  
Great Success ◽  
Performance Study ◽  
Step Size ◽  
Discrete Time Systems ◽  
Second Order Systems ◽  
Time Systems

Abstract In this paper, the performance of a bat searching algorithm is studied from system dynamics point of view. Bat searching algorithm (BA) is a recently developed swarm intelligence based optimization algorithm which has shown great success when solving complicated optimization problems. Each bat in the BA has two main states: velocity and position. The position represents the solution of the optimization problems while the velocity represents the searching direction and step size during each iteration. Due to the nature of the update equations, the dynamics of the bats are formulated as a group of second-order discrete-time systems. In this paper, the performance of the algorithm is analyzed based on the nature of the responses in the second-order systems. The over-damped response, under-damped responses are studied and the parameters requirements are derived. Moreover, unstable scenarios of the bats are also considered when examining the performance of the algorithm. Numerical evaluations are conducted to test different choices of the parameters in the BA.

scholarly journals New Vertically Planed Pendulum Motion

2020 ◽  
Vol 2020 ◽  
pp. 1-6
Author(s):  
A. I. Ismail
Keyword(s):  
Differential Equations ◽  
Numerical Solutions ◽  
Equations Of Motion ◽  
Vertical Plane ◽  
Numerical Procedure ◽  
Second Order ◽  
The Body ◽  
Physical Parameters ◽  
Pendulum Motion ◽  
Generalized Coordinates

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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