scholarly journals Kinematic support modeling in Sage

2020 ◽  
Vol 28 (2) ◽  
pp. 141-153
Author(s):  
Oleg K. Kroytor ◽  
Mikhail D. Malykh ◽  
Sergei P. Karnilovich
Keyword(s):  
Rigid Body ◽  
Computer Algebra ◽  
Computer Algebra System ◽  
Seismic Isolation ◽  
Equations Of Motion ◽  
Classical Mechanics ◽  
Vertical Plane ◽  
Base Plate ◽  
Euler Method ◽  
Point Of View

The article discusses the kinematic support, which allows reducing the horizontal dynamic effects on the building during earthquakes. The model of a seismic isolation support is considered from the point of view of classical mechanics, that is, we assume that the support is absolutely solid, oscillating in a vertical plane above a fixed horizontal solid plate. This approach allows a more adequate description of the interaction of the support with the soil and the base plate of the building. The paper describes the procedure for reducing the complete system of equations of motion of a massive rigid body on a fixed horizontal perfectly smooth plane to a form suitable for applying the finite difference method and its implementation in the Sage computer algebra system. The numerical calculations by the Euler method for grids with different number of elements are carried out and a mathematical model of the support as a perfectly rigid body in the Sage computer algebra system is implemented. The article presents the intermediate results of numerical experiments performed in Sage and gives a brief analysis (description) of the results.

Development of Equations of Motion for a Stewart Platform Under Prescribed Mount Motion

2018 ◽  
Author(s):  
Takeyuki Ono ◽  
Ryosuke Eto ◽  
Junya Yamakawa ◽  
Hidenori Murakami
Keyword(s):  
Rigid Body ◽  
Virtual Work ◽  
Equations Of Motion ◽  
Base Plate ◽  
Stewart Platform ◽  
Euler Method ◽  
Moving Frame ◽  
Real Time Control ◽  
Precise Positioning ◽  
Time Control

Analytical equations of motion are critical for real-time control of translating manipulators, which require precise positioning of various tools for their mission. Specifically, when manipulators mounted on moving robots or vehicles perform precise positioning of their tools, it becomes economical to develop a Stewart platform, whose sole task is stabilizing the orientation and crude position of its top table, onto which various precision tools are attached. In this paper, analytical equations of motion are developed for a Stewart platform whose motion of the base plate is prescribed. To describe the kinematics of the platform, the moving frame method, presented by one of authors [1,2], is employed. In the method the coordinates of the origin of a body attached coordinate system and vector basis are expressed by using 4 × 4 frame connection matrices, which form the special Euclidean group, SE(3). The use of SE(3) allows accurate description of kinematics of each rigid body using (relative) joint coordinates. In kinetics, the principle of virtual work is employed, in which system virtual displacements are expressed through B-matrix by essential virtual displacements, reflecting the connection of the rigid body system [2]. The resulting equations for fixed base plate reduce to those for the top plate, obtained by the Newton-Euler method. A main result of the paper is the analytical equations of motion in matrix form for dynamics analyses of a Stewart platform whose base plate moves. The control applications of those equations will be deferred to subsequent publications.

scholarly journals Building reaction kinetic models for amiloid fibril growth

BIOMATH ◽  
2016 ◽  
Vol 5 (1) ◽  
pp. 1607311 ◽  
Author(s):  
Svetoslav Marinov Markov
Keyword(s):  
Computer Algebra ◽  
Computer Algebra System ◽  
Reaction Kinetic ◽  
Growth Models ◽  
Reaction Network ◽  
Network Models ◽  
Point Of View ◽  
Logistic Growth ◽  
Sigmoidal Functions ◽  
Methodological Aspects

In this work we  discuss some methodological aspects of the creation and formulation of mathematical  models describing the growth of species from the point of view of reaction kinetics. Our discussion is based on familiar examples of growth models such as logistic growth and enzyme kinetics. We   propose several reaction network  models  for  the amiloid fibrillation processes in the citoplasm. The solutions of the models are sigmoidal functions graphically visualized using  the computer algebra system   Mathematica.

scholarly journals The Dynamical Behavior of a Rigid Body Relative Equilibrium Position

2017 ◽  
Vol 2017 ◽  
pp. 1-13 ◽  
Author(s):  
T. S. Amer
Keyword(s):  
Rigid Body ◽  
Degrees Of Freedom ◽  
Numerical Solutions ◽  
Equations Of Motion ◽  
Relative Equilibrium ◽  
Dynamical Behavior ◽  
Vertical Plane ◽  
The Body ◽  
Physical Parameters ◽  
Pendulum Model

In this paper, we will focus on the dynamical behavior of a rigid body suspended on an elastic spring as a pendulum model with three degrees of freedom. It is assumed that the body moves in a rotating vertical plane uniformly with an arbitrary angular velocity. 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 fourth-order Runge-Kutta algorithms through Matlab packages. These solutions are represented graphically in order to describe and discuss the behavior of the body at any instant for different values of the physical parameters of the body. The obtained results have been discussed and compared with some previous published works. Some concluding remarks have been presented at the end of this work. The importance of this work is due to its numerous applications in life such as the vibrations that occur in buildings and structures.

scholarly journals 5. Note on a Singular Problem in Kinetics

1882 ◽  
Vol 11 ◽  
pp. 173-175
Author(s):  
Tait
Keyword(s):  
Kinetic Energy ◽  
Equations Of Motion ◽  
Vertical Plane ◽  
Point Of View ◽  
Singular Problem ◽  
Mixed Potential ◽  
Physical Point ◽  
Peculiar Form ◽  
Subsequent Motion

The following problem presented itself to me nearly thirty years ago. I cannot find any notice of it in books, though it must have occurred to every one who has studied the oscillations of a balance:—Two equal masses are attached to the ends of a cord passing over a smooth pulley (as in Attwood's machine). One of them is slightly disturbed, in a vertical plane, from its position of equilibrium. Find the nature of the subsequent motion of the system.The interest of this case of small motions is twofold. From the peculiar form of the equations of motion, it is of exceptional mathematical difficulty. This is probably the reason for its not having been given as an example in Kinetics. And from the physical point of view it presents a very beautiful example of excessively slow, but continued, transformation of mixed potential and kinetic energy into kinetic energy alone.

Comparison of Selected Methods of Handling Redundant Constraints in Multibody Systems Simulations

2012 ◽  
Vol 8 (2) ◽  
Author(s):  
Marek Wojtyra ◽  
Janusz Frączek
Keyword(s):  
Numerical Methods ◽  
Rigid Body ◽  
Multibody Systems ◽  
Equations Of Motion ◽  
Point Of View ◽  
Redundant Constraints ◽  
Reaction Solution ◽  
Constraints Handling ◽  
Mathematical Operations ◽  
Augmented Lagrangian Formulation

When redundant constraints are present in a rigid body mechanism, only selected (if any at all) joint reactions can be determined uniquely, whereas others cannot. Analytic criteria and numerical methods of finding joints with uniquely solvable reactions are available. In this paper, the problem of joint reactions solvability is examined from the point of view of selected numerical methods frequently used for handling redundant constraints in practical simulations. Three different approaches are investigated in the paper: elimination of redundant constraints; pseudoinverse-based calculations; and the augmented Lagrangian formulation. Each method is briefly summarized; the discussion is focused on techniques of handling redundant constraints and on joint reactions calculation. In the case of multibody systems with redundant constraints, the rigid body equations of motion are insufficient to calculate some or all joint reactions. Thus, purely mathematical operations are performed in order to find the reaction solution. In each investigated method, the redundant constraints are treated differently, which—in the case of joints with nonunique reactions—leads to different reaction solutions. As a consequence, reactions reflecting the redundancy handling method rather than physics of the system are calculated. A simple example of each method usage is presented, and calculated joint reactions are examined. The paper points out the origins of nonuniqueness of constraint reactions in each examined approach. Moreover, it is shown that one and the same method may lead to different reaction solutions, provided that input data are prepared differently. Finally, it is demonstrated that—in case of joints with solvable reactions—the obtained solutions are unique, regardless of the method used for redundant constraints handling.

Control Constraint Realization Applied to Underactuated Aerospace Systems

2011 ◽  
Author(s):  
Pierangelo Masarati ◽  
Marco Morandini ◽  
Alessandro Fumagalli
Keyword(s):  
Position Control ◽  
Feedforward Control ◽  
Equations Of Motion ◽  
Vertical Plane ◽  
Differential Algebraic Equations ◽  
General Purpose ◽  
Point Of View ◽  
Theoretical Point ◽  
Algebraic Equations ◽  
Control Constraint

This paper discusses the problem of control constraint realization applied to the design of maneuvers of complex under-actuated systems modeled as multibody problems. Applications of interest in the area of aerospace engineering are presented and discussed. The tangent realization of the control constraint is discussed from a theoretical point of view and used to determine feedforward control of realistic under-actuated systems. The effectiveness of the computed feedforward input is subsequently verified by applying it to more detailed models of the problems, in presence of disturbances and uncertainties in combination with feedback control. The proposed applications consist in the position control of a complex closed chain mechanism representative of a robotic system, the control of a simplified model of a canard and a conventional air vehicle in the vertical plane, and the angular velocity control of a wind-turbine. In the aeromechanics examples, the tangent realization of the control relies on the availability of the Jacobian matrix of an aeroelastic model. All problems are solved using a free general-purpose multibody software that writes the constrained dynamics of multi-field problems in form of Differential-Algebraic Equations (DAE). The equations are integrated using A/L-stable algorithms. The essential extension to the multibody code consisted in the addition of the capability to write arbitrary constraint equations and apply the corresponding reaction multipliers to arbitrary equations of motion. This allowed to exploit the modeling capabilities of the formulation without any undue restriction on the modeling requirements.

Explaining the importance of Euler’s method in rigid-body dynamics

2020 ◽  
pp. 030641902093412
Author(s):  
Flavius PR Martins ◽  
Agenor T Fleury ◽  
Flavio C Trigo
Keyword(s):  
Rigid Body ◽  
Engineering Students ◽  
Equations Of Motion ◽  
Euler Method ◽  
Rigid Body Dynamics ◽  
Frames Of Reference ◽  
First Year ◽  
Euler’S Method ◽  
Body Dynamics ◽  
Spinning Top

The purpose of this study is essentially pedagogical and aims to provide an additional argument in clarification of a question often raised by first-year undergraduate mechanical engineering students concerning the reason for using two frames of reference—one fixed in space and one fixed in the rigid-body—to describe its motion. The reasoning employed to illustrate the inappropriateness of using a single reference frame entails showing that the equations of motion, thus obtained, are far more complex than the equations resulting from application of the traditional Euler Method. This point is illustrated through the well-known frictionless symmetrical spinning top problem.

Generation of Correlated Logistic-Normal Random Variates for Medical Decision Trees

1998 ◽  
Vol 37 (03) ◽  
pp. 235-238 ◽  
Author(s):  
M. El-Taha ◽  
D. E. Clark
Keyword(s):  
Monte Carlo ◽  
Decision Trees ◽  
Computer Algebra ◽  
Taylor Series ◽  
Computer Algebra System ◽  
Random Variable ◽  
Monte Carlo Analysis ◽  
Normal Random Variable ◽  
Medical Decision ◽  
Theoretical Results

AbstractA Logistic-Normal random variable (Y) is obtained from a Normal random variable (X) by the relation Y = (ex)/(1 + ex). In Monte-Carlo analysis of decision trees, Logistic-Normal random variates may be used to model the branching probabilities. In some cases, the probabilities to be modeled may not be independent, and a method for generating correlated Logistic-Normal random variates would be useful. A technique for generating correlated Normal random variates has been previously described. Using Taylor Series approximations and the algebraic definitions of variance and covariance, we describe methods for estimating the means, variances, and covariances of Normal random variates which, after translation using the above formula, will result in Logistic-Normal random variates having approximately the desired means, variances, and covariances. Multiple simulations of the method using the Mathematica computer algebra system show satisfactory agreement with the theoretical results.

scholarly journals A least action principle for interceptive walking

2021 ◽  
Vol 11 (1) ◽  
Author(s):  
Soon Ho Kim ◽  
Jong Won Kim ◽  
Hyun Chae Chung ◽  
MooYoung Choi
Keyword(s):  
Social Systems ◽  
Equations Of Motion ◽  
Classical Mechanics ◽  
Action Principle ◽  
Lagrange Equations ◽  
Least Action Principle ◽  
Least Action ◽  
The Least Action Principle ◽  
Principle Of Least Action ◽  
Human Participants

AbstractThe principle of least effort has been widely used to explain phenomena related to human behavior ranging from topics in language to those in social systems. It has precedence in the principle of least action from the Lagrangian formulation of classical mechanics. In this study, we present a model for interceptive human walking based on the least action principle. Taking inspiration from Lagrangian mechanics, a Lagrangian is defined as effort minus security, with two different specific mathematical forms. The resulting Euler–Lagrange equations are then solved to obtain the equations of motion. The model is validated using experimental data from a virtual reality crossing simulation with human participants. We thus conclude that the least action principle provides a useful tool in the study of interceptive walking.

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