DOCSLIB.ORG

Dynamics of a 3D Elastic String Pendulum

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
Dynamics of a 3D Elastic String Pendulum Joint 48th IEEE Conference on Decision and Control and ThA06.5 28th Chinese Control Conference Shanghai, P.R. China, December 16-18, 2009 Dynamics of a 3D Elastic String Pendulum Taeyoung Lee, Melvin Leok∗, and N. Harris McClamrochy Abstract— This paper presents an analytical model and a We show that the governing equations of motion can be geometric numerical integrator for a rigid body connected to an developed according to Hamilton’s variational principle. elastic string, acting under a gravitational potential. Since the The second part of this paper deals with a geometric point where the string is attached to the rigid body is displaced from the center of mass of the rigid body, there exist nonlinear numerical integrator for the 3D elastic string pendulum. Ge- coupling effects between the string deformation and the rigid ometric numerical integration is concerned with developing body dynamics. A geometric numerical integrator, referred to as numerical integrators that preserve geometric features of a a Lie group variational integrator, is developed to numerically system, such as invariants, symmetry, and reversibility [9]. preserve the Hamiltonian structure of the presented model A geometric numerical integrator, referred to as a Lie group and its Lie group configuration manifold. These properties are illustrated by a numerical simulation. variational integrator, has been developed for a Hamiltonian system on an arbitrary Lie group in [10]. I. INTRODUCTION A 3D elastic string pendulum is a Hamiltonian system, and its configuration manifold is expressed as the product of the The dynamics of a body connected to a string are of special orthogonal group SO(3) and the space of connected relevance in several engineering problems such as cable curve segments on R3. This paper develops a Lie group cranes, towed underwater vehicles, and tethered spacecraft. variational integrator for a 3D elastic string pendulum based It has been shown that gravitational forces acting along a on the results presented in [10]. The proposed geometric string can alter the tension of the string and significantly numerical integrator preserves symplecticity and momentum influence the connected rigid body dynamics [1]. Therefore, maps, and exhibits desirable energy conservation properties. it is important to accurately model the string dynamics, the It also respects the Lie group structure of the configuration dynamics of the body, and their interaction. manifold, and avoids the singularities and computational Several dynamic and numerical models have been devel- complexities associated with the use of local coordinates. oped. Lumped mass models, where the string is spatially In summary, this paper develops an analytical model and discretized into connected point masses, were developed a geometric numerical integrator for a 3D elastic string pen- in [2], [3], [4]. Finite difference methods in both the spatial dulum. These provide a mathematical model and a reliable domain and the time domain were applied in [5], [6]. Finite numerical simulation tool that characterizes the nonlinear element discretizations of the weak form of the equations of coupling between the string dynamics and the rigid body motion were applied in [6], [7]. dynamics accurately. It can be used to study non-local, large The goal of this paper is to develop an analytical model maneuvers of the 3D elastic string pendulum accurately over and a numerical simulation tool for a rigid body connected to a long time period. a string acting under a gravitational potential. This dynamic model is referred to as a 3D elastic string pendulum. It II. 3D ELASTIC STRING PENDULUM generalizes the 3D pendulum model introduced in [8] to include the effects of string deformations, and extends the Consider a rigid body that is attached to an elastic string. model of a string pendulum with a point mass bob [6]. The other end of the string is fixed to a pivot point. We This paper provides a realistic and accurate analytical assume that the rigid body can freely translate and rotate in model for the 3D elastic string pendulum. We assume that a three dimensional space, and the string is extensible and the point where the string is attached to the rigid body flexible. The bending stiffness of the string is not considered is displaced from the center of mass of the rigid body as the diameter of the string is assumed to be negligible so that there exist nonlinear coupling effects between the compared to its length. The point where the string is attached string deformation dynamics and the rigid body dynamics. to the rigid body is displaced from the center of mass of the rigid body so that the dynamics of the rigid body is coupled Taeyoung Lee, Mechanical and Aerospace Engineering, Florida Institute to the string deformations and displacements. This is referred of Technology, Melbourne, FL 39201 [email protected] to as a 3D elastic string pendulum. Melvin Leok, Mathematics, Purdue University, West Lafayette, IN 47907 [email protected] We choose a global reference frame and a body-fixed N. Harris McClamroch, Aerospace Engineering, University of Michigan, frame. The origin of the body-fixed frame is located at the Ann Arbor, MI 48109 [email protected] end of the string where the string is attached to the rigid body. ∗This research has been supported in part by NSF under grants DMS- 0714223, DMS-0726263, DMS-0747659. Since the string is extensible, we need to distinguish between yThis research has been supported in part by NSF under grant CMS- the arc length for the stretched deformed configuration and 0555797. the arc length for the unstretched reference configuration. 978-1-4244-3872-3/09/$25.00 ©2009 IEEE 3347 ThA06.5 A. Lagrangian Kinetic energy: The total kinetic energy is composed of the kinetic energy of the string Tstr and the kinetic energy s(s; t) of the rigid body Trb. Let r_(s; t) be the partial derivative of s r(s; t) with respect to t. The kinetic energy of the string is r(s; t) Z l 1 2 P Tstr = µ kr_(s)k ds: (2) 0 2 P Let ρ 2 R3 be the vector from the center of mass of the rigid R(t) body to a mass element of the rigid body represented in the body fixed frame. The location of the mass element is given ρc ρc by r(l)+R(ρc +ρ) in the global reference frame. Therefore, the kinetic energy of the rigid body can be written as Reference configuration Deformed configuration Z 1 2 Trb = kr_(l) + RΩ(^ ρc + ρ)k dM(ρ) Fig. 1. 3D Elastic String Pendulum B 2 1 1 = Mr_(l) · r_(l) + Ω · JΩ + Mr_(l) · RΩ^ρ ; (3) 2 2 c Define where B denotes the region enclosed by the rigid body R l 2 R Total length of the unstretched string surface, and we used the following properties: B ρ dM = 0; R ^ 2 s 2 [0; l] Length of the string from the pivot to a ma- xy^ = −yx^ ; J = − B((ρ + ρc) ) dM. terial point P for the unstretched reference Potential Energy: The strain of the string at a material configuration point located at r(s) is given by + s(s; t) 2 R Length of the string from the pivot to a ∆s(s) − ∆s material point P for the stretched deformed = lim = s0(s) − 1; ∆s!0 ∆s configuration 3 where ()0 denote the partial derivative with respect to . The r(s; t) 2 R Vector from the pivot to a material point P s in the global reference frame tangent vector at the material point is given by R 2 SO(3) Rotation matrix from the body-fixed frame @r(s) @r(s) @s r0(s) et = = = : to the reference frame @s @s @s(s) s0(s) Ω 2 R3 Angular velocity of the rigid body repre- Since this tangent vector has the unit length, we have s0(s) = sented in the body-fixed frame 0 0 3 kr (s)k. Therefore, the strain is given by = kr (s)k − 1. ρc 2 R Vector from the origin of the body fixed frame to the center of mass of the rigid body The potential energy of the string is composed of the elastic represented in the body fixed frame potential and the gravitational potential: + Z l µ 2 R Mass density of the string per unit un- 1 0 2 stretched length Vstr = EA(kr (s)k − 1) − µgr(s) · e3 ds; (4) 0 2 M 2 + Mass of the rigid body R where E and A denote the Young’s modulus and the sectional J 2 3×3 Inertia matrix of the rigid body represented R area of the string, respectively, and the unit vector e in the body fixed frame 3 represents the gravity direction. A configuration of this system can be described by the Since the location of the center of mass of the rigid body locations of all the material points of the string, r(s; t) for is r(l) + Rρc in the global reference frame, the gravitational s 2 [0; l], and the attitude of the rigid body R(t) with respect potential energy of the rigid body is to the reference frame. So, the configuration manifold is 1 3 1 3 V = −Mg(r(l) + Rρ ) · e : (5) G = C ([0; l]; R ) × SO(3), where C ([0; l]; R ) denotes rb c 3 3 the space of smooth connected curve segments on R and From (2)-(5), the Lagrangian of the 3D elastic string 3×3 T SO(3) = fR 2 R j R R = I; det[R] = 1g. pendulum is given by III. CONTINUOUS-TIME ANALYTICAL MODEL L = Tstr − Vstr + Trb − Vrb: (6) In this section, we develop continuous-time equations B.
Load more

Recommended publications
  • The Swinging Spring: Regular and Chaotic Motion
    References The Swinging Spring: Regular and Chaotic Motion Leah Ganis May 30th, 2013 Leah Ganis The Swinging Spring: Regular and Chaotic Motion References Outline of Talk I Introduction to Problem I The Basics: Hamiltonian, Equations of Motion, Fixed Points, Stability I Linear Modes I The Progressing Ellipse and Other Regular Motions I Chaotic Motion I References Leah Ganis The Swinging Spring: Regular and Chaotic Motion References Introduction The swinging spring, or elastic pendulum, is a simple mechanical system in which many different types of motion can occur. The system is comprised of a heavy mass, attached to an essentially massless spring which does not deform. The system moves under the force of gravity and in accordance with Hooke's Law. z y r φ x k m Leah Ganis The Swinging Spring: Regular and Chaotic Motion References The Basics We can write down the equations of motion by finding the Lagrangian of the system and using the Euler-Lagrange equations. The Lagrangian, L is given by L = T − V where T is the kinetic energy of the system and V is the potential energy. Leah Ganis The Swinging Spring: Regular and Chaotic Motion References The Basics In Cartesian coordinates, the kinetic energy is given by the following: 1 T = m(_x2 +y _ 2 +z _2) 2 and the potential is given by the sum of gravitational potential and the spring potential: 1 V = mgz + k(r − l )2 2 0 where m is the mass, g is the gravitational constant, k the spring constant, r the stretched length of the spring (px2 + y 2 + z2), and l0 the unstretched length of the spring.
    [Show full text]
  • Dynamics of the Elastic Pendulum Qisong Xiao; Shenghao Xia ; Corey Zammit; Nirantha Balagopal; Zijun Li Agenda
    Dynamics of the Elastic Pendulum Qisong Xiao; Shenghao Xia ; Corey Zammit; Nirantha Balagopal; Zijun Li Agenda • Introduction to the elastic pendulum problem • Derivations of the equations of motion • Real-life examples of an elastic pendulum • Trivial cases & equilibrium states • MATLAB models The Elastic Problem (Simple Harmonic Motion) 푑2푥 푑2푥 푘 • 퐹 = 푚 = −푘푥 = − 푥 푛푒푡 푑푡2 푑푡2 푚 • Solve this differential equation to find 푥 푡 = 푐1 cos 휔푡 + 푐2 sin 휔푡 = 퐴푐표푠(휔푡 − 휑) • With velocity and acceleration 푣 푡 = −퐴휔 sin 휔푡 + 휑 푎 푡 = −퐴휔2cos(휔푡 + 휑) • Total energy of the system 퐸 = 퐾 푡 + 푈 푡 1 1 1 = 푚푣푡2 + 푘푥2 = 푘퐴2 2 2 2 The Pendulum Problem (with some assumptions) • With position vector of point mass 푥 = 푙 푠푖푛휃푖 − 푐표푠휃푗 , define 푟 such that 푥 = 푙푟 and 휃 = 푐표푠휃푖 + 푠푖푛휃푗 • Find the first and second derivatives of the position vector: 푑푥 푑휃 = 푙 휃 푑푡 푑푡 2 푑2푥 푑2휃 푑휃 = 푙 휃 − 푙 푟 푑푡2 푑푡2 푑푡 • From Newton’s Law, (neglecting frictional force) 푑2푥 푚 = 퐹 + 퐹 푑푡2 푔 푡 The Pendulum Problem (with some assumptions) Defining force of gravity as 퐹푔 = −푚푔푗 = 푚푔푐표푠휃푟 − 푚푔푠푖푛휃휃 and tension of the string as 퐹푡 = −푇푟 : 2 푑휃 −푚푙 = 푚푔푐표푠휃 − 푇 푑푡 푑2휃 푚푙 = −푚푔푠푖푛휃 푑푡2 Define 휔0 = 푔/푙 to find the solution: 푑2휃 푔 = − 푠푖푛휃 = −휔2푠푖푛휃 푑푡2 푙 0 Derivation of Equations of Motion • m = pendulum mass • mspring = spring mass • l = unstreatched spring length • k = spring constant • g = acceleration due to gravity • Ft = pre-tension of spring 푚푔−퐹 • r = static spring stretch, 푟 = 푡 s 푠 푘 • rd = dynamic spring stretch • r = total spring stretch 푟푠 + 푟푑 Derivation of Equations of Motion
    [Show full text]
  • A New Pendulum Motion with a Suspended Point Near Infinity
    www.nature.com/scientificreports OPEN A new pendulum motion with a suspended point near infnity A. I. Ismail In this paper, a pendulum model is represented by a mechanical system that consists of a simple pendulum suspended on a spring, which is permitted oscillations in a plane. The point of suspension moves in a circular path of the radius (a) which is sufciently large. There are two degrees of freedom for describing the motion named; the angular displacement of the pendulum and the extension of the spring. The equations of motion in terms of the generalized coordinates ϕ and ξ are obtained using Lagrange’s equation. The approximated solutions of these equations are achieved up to the third order of approximation in terms of a large parameter ε will be defned instead of a small one in previous studies. The infuences of parameters of the system on the motion are obtained using a computerized program. The computerized studies obtained show the accuracy of the used methods through graphical representations. Te pendulum motions are studied in many works 1–7. Te motion of the pendulum on an ellipse is studied in 8. Te supported point of this pendulum moves on an ellipse path while the end moves with arbitrary angular dis- placements. Te equation of motion is obtained and solved for one degree of freedom ϕ . In9, the relative periodic solutions of a rigid body suspended on an elastic string in a vertical plane are considered. Te equations of motion are obtained and solved in terms of the small parameter ε .
    [Show full text]
  • Chaos Physics in Secondary School a Material Applicable in Online Teaching Ildikó Bajkó1 1Szent István High School Budapest, 1146 Budapest, Ajtósi Dürer 15
    Chaos Physics in Secondary School A material applicable in online teaching Ildikó Bajkó1 1Szent István High School Budapest, 1146 Budapest, Ajtósi Dürer 15. Motto”But if you stir backward, the jam will not come together again.” Tom Stoppard: Arcadia Abstract Chaotic systems are not only subject for researchers, but we all encounter them in our everyday life, when, e.g., mixing cream in coffee. This paper addresses the issues of teaching chaos physics in high school within both extracurricular framework, and by incorporating it into high school physics curriculum. The teaching material is based on mechanical processes and has been tested in different student groups. The module was specially elaborated to conceptualize some basic aspects of chaos theory, like predictability, chaos, complex and chaotic motions, irreversibility, determinism and mixing. The post-tests, administered to students after completing the module, have shown significantly higher scores on conceptual questions, as compared to the results of the pre-tests, indicating a deeper understanding of the enumerated concepts. The curriculum is suitable for online teaching, too. Keywords: chaos physics, high school Introduction Chaotic processes can be experienced in almost every branch of science. They are present not only in physical sciences, but also in many other disciplines, ranging from populations dynamics, via chemical reactions, to cardiac fluctuations. Many books available in the chaos literature give an overview of several chaotic systems (Gleick, 1987, Lorenz, 1993, Tél & Gruiz, 2006, Argyris et al., 2015), some others concentrate only on specific fields like, e.g. astronomy (Diacu & Holmes, 1996) or oceanic plankton patterns (Neufeld et al., 2003).
    [Show full text]
  • 3 the Van Der Pol Oscillator 19 3.1Themethodofaveraging
    Lecture Notes on Nonlinear Vibrations Richard H. Rand Dept. Theoretical & Applied Mechanics Cornell University Ithaca NY 14853 [email protected] http://www.tam.cornell.edu/randdocs/ version 45 Copyright 2003 by Richard H. Rand 1 R.Rand Nonlinear Vibrations 2 Contents 1PhasePlane 4 1.1ClassificationofLinearSystems............................ 4 1.2 Lyapunov Stability ................................... 5 1.3 Structural Stability ................................... 7 1.4Examples........................................ 8 1.5Problems......................................... 8 1.6 Appendix: Lyapunov’s Direct Method ........................ 10 2 The Duffing Oscillator 13 2.1Lindstedt’sMethod................................... 14 2.2 Elliptic Functions . ................................... 15 2.3Problems......................................... 17 3 The van der Pol Oscillator 19 3.1TheMethodofAveraging............................... 19 3.2HopfBifurcations.................................... 23 3.3HomoclinicBifurcations................................ 24 3.4 Relaxation Oscillations ................................. 27 3.5 The van der Pol oscillator at Infinity . ........................ 29 3.6Example......................................... 32 3.7Problems......................................... 32 4 The Forced Duffing Oscillator 34 4.1TwoVariableExpansionMethod........................... 35 4.2CuspCatastrophe.................................... 38 4.3Problems......................................... 39 5 The Forced van der Pol Oscillator 40 5.1Entrainment......................................
    [Show full text]
  • Asymptotic Smooth Stabilization of the Inverted 3D Pendulum Nalin A
    IEEE TRANSACTIONS ON AUTOMATIC CONTROL 1 Asymptotic Smooth Stabilization of the Inverted 3D Pendulum Nalin A. Chaturvedi, N. Harris McClamroch, Fellow, IEEE, and Dennis S. Bernstein, Fellow, IEEE Abstract—The 3D pendulum consists of a rigid body, supported Pendulum models are useful for both pedagogical and at a fixed pivot, with three rotational degrees of freedom; it is research reasons. They represent simplified versions of me- acted on by gravity and it is fully actuated by control forces. The chanical systems arising in robotics and spacecraft. In addition 3D pendulum has two disjoint equilibrium manifolds, namely a hanging equilibrium manifold and an inverted equilibrium to their role in demonstrating the foundations of nonlinear manifold. The contribution of this paper is that two fundamental dynamics and control, pendulum models have motivated re- stabilization problems for the inverted 3D pendulum are posed search in nonlinear dynamics and nonlinear control. In [19], and solved. The first problem, asymptotic stabilization of a controllers for pendulum problems with applications to control specified equilibrium in the inverted equilibrium manifold, is of oscillations have been presented using the speed-gradient solved using smooth and globally defined feedback of angular velocity and attitude of the 3D pendulum. The second problem, method. asymptotic stabilization of the inverted equilibrium manifold, is The 3D pendulum is a rigid body supported at a fixed solved using smooth and globally defined feedback of angular pivot point with three rotational degrees of freedom. It is velocity and a reduced attitude vector of the 3D pendulum. acted on by a uniform gravity force and, perhaps, by control These control problems for the 3D pendulum exemplify attitude and disturbance forces.
    [Show full text]
  • The Dynamics of the Elastic Pendulum
    MATH MODELING MIDTERM REPORT MATH 485, Instructor: Dr. Ildar Gabitov, Mentor: Joseph Gibney THE DYNAMICS OF THE ELASTIC PENDULUM A group project proposal with preliminary research by Corey Zammit, Nirantha Balagopal, Zijun Li, Shenghao Xia, and Qisong Xiao 1. An Introduction to the System Considered The system of the elastic pendulum consists of a spring, connected to a pivot, suspending a mass. This spring can have many different properties among these are stiffness which can be considered a constant in some practical cases, so the spring has a linear reaction force when extended and compressed. Typically a spring that one would find in real-life applications is either an extension spring or a compression spring (and this will be discussed in greater detail in section 5) however for simplified models and preliminary research we find it practical to consider the case where this spring behaves in a Hookian manor in both extension and compression. This assumption, however, does beg this question among others: Does the spring bend as shown in Figure 1 when compressed ever, and how would this effect the behavior of the system? One way to answer this question is to acknowledge that assumptions such as the ones that follow need to be made to analyze this system, and in any case, behaviors such as these are not easy to analyze and would involve making many more assumptions that could be equally unsatisfying. We will be considering two regimes of this system in our preliminary research with the same assumptions and will pose questions for further research with suggestions for different assumptions.
    [Show full text]
  • Control of a Flexible Inverse Pendulum Based on the Singular Perturbation Method
    Journal of Physics: Conference Series PAPER • OPEN ACCESS Control of a flexible inverse pendulum based on the singular perturbation method To cite this article: Zainib Hatif Abbas 2020 J. Phys.: Conf. Ser. 1679 022009 View the article online for updates and enhancements. This content was downloaded from IP address 170.106.34.90 on 24/09/2021 at 14:33 APITECH II IOP Publishing Journal of Physics: Conference Series 1679 (2020) 022009 doi:10.1088/1742-6596/1679/2/022009 Control of a flexible inverse pendulum based on the singular perturbation method Zainib Hatif Abbas1,2 1 Mathematics Department, Voronezh State Technical University, XX-letiya Oktyabrya st. 84, 394006 Voronezh, Russia 2 Institute of Genetic Engineering, University of Baghdad, Al-Jadriya, Baghdad, Iraq E-mail: [email protected] Abstract. In this article, a partial differential equation model for a flexible inverted pendulum system is derived by using the Hamilton principle. Specifically, problems of stabilization and the optimization of such a system are considered. In addition, the singular perturbation method has been used to divide the partial differential equation model for a fast and a slow subsystem. For a fast subsystem stabilization, the control algorithm proposed a boundary force applied at the free end of the beam which proved that the closed-loop subsystem is appropriate and exponentially stable. To stabilize the slow subsystem, a sliding mode control method was used to design the controller, while the method of linear matrix inequality was used in designing the sliding surface. In conclusion, it was shown that the slow subsystem is exponentially stable.
    [Show full text]
  • Chaotic Modelling and Simulation: Analysis of Chaotic Models, Attractors and Forms / Christos H
    Chaotic Modelling and Simulation Analysis of Chaotic Models, Attractors and Forms Christos H. Skiadas Charilaos Skiadas Chapman & Hall/CRC Taylor & Francis Group 6000 Broken Sound Parkway NW, Suite 300 Boca Raton, FL 33487-2742 © 2009 by Taylor & Francis Group, LLC Chapman & Hall/CRC is an imprint of Taylor & Francis Group, an Informa business No claim to original U.S. Government works Printed in the United States of America on acid-free paper 10 9 8 7 6 5 4 3 2 1 International Standard Book Number-13: 978-1-4200-7900-5 (Hardcover) This book contains information obtained from authentic and highly regarded sources. Reasonable efforts have been made to publish reliable data and information, but the author and publisher can- not assume responsibility for the validity of all materials or the consequences of their use. The authors and publishers have attempted to trace the copyright holders of all material reproduced in this publication and apologize to copyright holders if permission to publish in this form has not been obtained. If any copyright material has not been acknowledged please write and let us know so we may rectify in any future reprint. Except as permitted under U.S. Copyright Law, no part of this book may be reprinted, reproduced, transmitted, or utilized in any form by any electronic, mechanical, or other means, now known or hereafter invented, including photocopying, microfilming, and recording, or in any information storage or retrieval system, without written permission from the publishers. For permission to photocopy or use material electronically from this work, please access www.copy- right.com (http://www.copyright.com/) or contact the Copyright Clearance Center, Inc.
    [Show full text]
  • The Pendulum Plain and Puzzling
    The Pendulum Plain and Puzzling Chris Sangwin School of Mathematics University of Edinburgh April 2017 Chris Sangwin (University of Edinburgh) Pendulum April 2017 1 / 38 Outline 1 Introduction and motivation 2 Geometrical physics 3 Classical mechanics 4 Driven pendulum 5 The elastic pendulum 6 Chaos 7 Conclusion Chris Sangwin (University of Edinburgh) Pendulum April 2017 2 / 38 2 The pendulum illustrates the concerns of each age. 1 Divinely ordered world of classical mechanics 2 Catastrophe theory 3 Chaos 3 Contemporary examples relevant to A-level mathematics/FM. 4 Physics apparatus Illustrated by examples. (Equations on request!) Connections within and beyond mathematics The pendulum is a paradigm. 1 Almost every interesting dynamic phenomena can be illustrated by a pendulum. Chris Sangwin (University of Edinburgh) Pendulum April 2017 3 / 38 3 Contemporary examples relevant to A-level mathematics/FM. 4 Physics apparatus Illustrated by examples. (Equations on request!) Connections within and beyond mathematics The pendulum is a paradigm. 1 Almost every interesting dynamic phenomena can be illustrated by a pendulum. 2 The pendulum illustrates the concerns of each age. 1 Divinely ordered world of classical mechanics 2 Catastrophe theory 3 Chaos Chris Sangwin (University of Edinburgh) Pendulum April 2017 3 / 38 4 Physics apparatus Illustrated by examples. (Equations on request!) Connections within and beyond mathematics The pendulum is a paradigm. 1 Almost every interesting dynamic phenomena can be illustrated by a pendulum. 2 The pendulum illustrates the concerns of each age. 1 Divinely ordered world of classical mechanics 2 Catastrophe theory 3 Chaos 3 Contemporary examples relevant to A-level mathematics/FM.
    [Show full text]
  • Three Approaches to the Dynamic Problem of a Pendulum and a Rotating Flexible Beam
    Procedings of COBEM 2007 19th International Congress of Mechanical Engineering Copyright c 2007 by ABCM November 5-9, 2007, Brasília, DF THREE APPROACHES TO THE DYNAMIC PROBLEM OF A PENDULUM AND A ROTATING FLEXIBLE BEAM Fernando Buezas, [email protected] Department of Physics, Universidad Nacional del Sur, Alem 1253, 8000 Bahía Blanca, Argentina CONICET, Argentina Rubens Sampaio, [email protected] Department of Mechanical Engineering, Pontificia Universidade Catolica do Rio de Janeiro, Rua Marques de Sao Vicente, 225, 22453- 900, Rio de Janeiro, Brasil Carlos P. Filipich, cfi[email protected] Mechanical Systems Analysis Group, FRBB, Universidad Tecnológica Nacional, 11 de Abril 451, 8000, Bahía Blanca, Argentina Department of Engineering, Universidad Nacional del Sur, Alem 1253, 8000 Bahía Blanca, Argentina Marta B. Rosales, [email protected] Department of Engineering, Universidad Nacional del Sur, Alem 1253, 8000 Bahía Blanca, Argentina CONICET, Argentina Abstract. The dynamic characteristics of a rotating beam play a significant role in the overall performance and design of various engineering systems, such as turbo-machinery, wind turbines, etc. In this work the dynamic problem of a flexible pendulum and a beam rotating around an axis perpendicular to its plane is addressed taking into account the gravitational effect. The dynamics of pendulums and rotating beams is studied according to three approaches, two of them related with the Strength of Materials theory (SM) and the other one with nonlinear Theory of Elasticity (TE). The stiffening effect due to the centrifugal forces is taken into account only in the case of the rotating beam subjected to high rotating speeds and neglected in the case of a pendulum under self-weight due to the low values of the speed, when addressed with SM theory.
    [Show full text]
  • Near the Resonance Behavior of a Periodicaly Forced Partially Dissipative Three‐Degrees‐Of‐Freedom Mechanical System
    Original article Near the resonance behavior of a periodicaly forced partially dissipative three‐degrees‐of‐freedom mechanical system Abstract a In this paper, a nonlinear three‐degrees‐of‐freedom dynamical system con‐ Paweł Pietrzak a sisting of a variable‐length pendulum mass attached by a massless spring to Marta Ogińska the forced slider is investigated. Numerical solution is preceded by applica‐ Tadeusz Krasuskia tion of Euler‐Lagrange equation. Various techniques like time histories, Kévin Figueiredoa phase planes, Poincaré maps and resonance plots are used to observe and Paweł Olejnikb* identify the system responses. The results show that the variable‐length spring pendulum suspended from the periodically forced slider can exhibit a International Faculty of Engineering, Lodz Uni‐ quasi‐periodic, and in a resonance state, even chaotic motions. It was con‐ versity of Technology, 36 Żwirki Str., 90‐001 cluded that near the resonance the influence of coupling of bodies on the Lodz, Poland. E‐mail: pawel‐pietrzak@out‐ system dynamics can lead to unpredictable dynamical behavior. look.com, [email protected], kra‐ [email protected], figueiredo.kevin@out‐ Keywords look.fr. Euler‐Lagrange equation, time history, phase plane, Poincaré map, reso‐ b Department of Automation, Biomechanics and nance plot, dynamical analysis, quasi‐periodic motion, chaos. Mechatronics, Faculty of Mechanical Enginee‐ ring, Lodz University of Technology, 1/15 Stefa‐ nowski Str., 90‐924 Lodz, Poland. E‐mail: pa‐ [email protected]. *Corresponding author http://dx.doi.org/10.1590/1679-78254423 Received August 23, 2017 In revised form December 19, 2017 Accepted December 19, 2017 Available online February 02, 2018 1 INTRODUCTION The existence of the resonance phenomena both external and internal occurs in vibrating structures as an increased amplitude of vibrations.
    [Show full text]