Dynamics and Control of an Elastic Dumbbell Spacecraft in a Central Gravitational Field


 Veronica Welch
 2 years ago
 Views:
Transcription
1 Dynamics Control of an Elastic Dumbbell Spacecraft in a Central Gravitational Field Amit K. Sanyal, Jinglai Shen, N. Harris McClamroch 1 Department of Aerospace Engineering University of Michigan Ann Arbor, MI {asanyal, jinglais, Abstract The dynamics of a dumbbell shaped spacecraft a modeled as two identical mass particles connected by a linear elastic spring. The equations of motion of the spacecraft in a planar orbit in a central gravitational field a psented. The equations of motion characterize orbit, attitude, shape or elastic deformation deges of fedom the coupling between them. Relative equilibria, corsponding to circular planar orbits, a obtained from these equations of motion. Linear equations of motion that describe perturbations from these lative equilibria a psented. New dynamics control problems a introduced for these linear equations. Controllability sults a psented for various actuation assumptions, based on the linear equations. individual mass particle of the dumbbell spacecraft. The diffential gravity effects about the center of mass play a crucial role in the overall spacecraft dynamics. Although the dumbbell spacecraft model is exceedingly simple, it is effective in demonstrating complex dynamics control phenomena that can arise when a multibody spacecraft is in orbit about a massive central spherical body. It provides a framework for studying the orbital, attitude, shape deges of fedom, the coupling between them. The dynamics of such multibody systems psent significant analytical challenges. In this paper, we introduce new orbital, attitude, shape control problems that have not been pviously studied in the published literatu. We demonstrate that the spacecraft is completely controllable, based on the linear equations of motion, for several diffent actuation assumptions. 1 Introduction This paper models a dumbbellshaped spacecraft moving in a fixed orbital plane in a central gravitational field. The equations of motion describe the translational or orbit dynamics, the rotational dynamics, the shape or elastic deformation dynamics. The spacecraft model consists of two ideal mass particles of identical mass connected by a linear elastic spring. The spring is assumed to be massless, rigid in the transverse diction, flexible along its longitudinal axis. The dumbbell spacecraft can rotate translate in a plane. The shape of the spacecraft is given by the distance of each mass particle from its center of mass. We make the following assumptions throughout the paper: an inertial frame is chosen such that its origin is at the center of a large idealized spherical central body, e.g. the Earth. A gravitational force acts on each 1 This search has been supported in part by NSF under grants ECS ECS The dumbbell spacecraft can also be viewed as a model of a tethed spacecraft. The levant published literatu tats the dynamics control of tethed spacecraft. Deployment, station keeping, trieval of tethers have been studied in [1]. Attitude control issues for tethed spacecraft have been tated in []. Orbital dynamics control issues for tethed spacecraft have been tated in [3, 4]. The use of attitudeorbit coupling to alter spacecraft orbital motions by controlling the attitude was dealt with in [5, 6, 7]. The authors a not awa of any published papers that tat orbital, attitude shape dynamics issues in a unified way. This paper makes a contribution to this problem for the simplified dumbbell spacecraft. The psent paper can also be viewed as an extension of [8]. In that prior paper the coupling between translation, rotation, shape deges of fedom was studied, new control problems that exploit the coupling between them we suggested. However, [8] did not include a central body gravity field, so the sults in that paper a not applicable to the problems con
2 sided he. Finally, we contrast the sults in this paper with sults in our earlier papers that tat control problems involving translation, rotation, shape deges of fedom. Examples of our prior work in this diction include [9, 10, 11]. In those papers, gravitational effects we completely ignod, the coupling between translation, attitude shape was inhently nonlinear. In this paper, the coupling between translation or orbit, attitude shape deges of fedom is effectively linear, since it arises from the central body gravity assumption. The diffence between linear nonlinear coupling effects is very important in actually exploiting these mechanisms for control purposes. Equations of Motion Let r be the radial distance of the spacecraft center of mass from the origin of the inertial frame, φ be the angle between this radial vector the principal inertial axis, q be the distance of each mass particle from the spacecraft center of mass, θ be the attitude angle of the dumbbell spacecraft with spect to the radial diction. Let m be the mass of each mass particle in the dumbbell spacecraft. We approximate the gravitational potential energy up to second order in q r 1 as V g = m r q r 1 3cos θ. 1 The elastic potential energy is given by V s = kq l, whe l is half the distance between the two mass particles when the spring force is zero. Thus, the Lagrangian is given by L = m q +ṙ + q θ +q θ φ + q φ + r φ + q r r 1 3cos θ kq l. 3 Note that the Lagrangian is independent of φ, this corsponds to conservation of the total angular momentum L φ =mr + q φ + q θ, in the absence of external forces along the local horizontal. from the above Lagrangian, a: m r m r φ r + 3q r 1 4 3cos θ = f r, 4 mr φ + q φ + q θ +q q θ +q q φ +rṙ φ =fh, 5 mq θ + q 6m φ +q θ q +q q φ+ r 3 q cos θ sin θ = τ, 6 m q m q θ +4q θ φ +q φ q r 1 3 3cos θ +kq l =f s, 7 for the local radial, local horizontal, attitude, shape deges of fedom, spectively. He f r, f h, τ, f s a the generalized forces corsponding to the generalized coordinates r, φ, θ, q, spectively. 3 Dynamics Control of Orbit, Attitude Shape In this section, we study the dynamics control of orbit, attitude, shape deges of fedom of the dumbbell spacecraft. Relative equilibria a identified. Linear equations of motion for the orbit, attitude shape dynamics a obtained that describe the perturbations from a lative equilibria. Several controllability sults a psented based on the linear perturbation equations. 3.1 Relative Equilibria for the Orbit, Attitude Shape Dynamics We first identify natural lative equilibria that corspond to circular orbits for this dumbbell spacecraft. At a lative equilibrium: r =ṙ =0, φ =0, θ =0, q =0, fr = f h = τ = f s =0. We use the subscript e to denote quantities evaluated at a lative equilibrium. The local horizontal orbit equation of motion is trivially satisfied at a lative equilibrium. Equations 4, 6 7 give the following algebraic equations for lative equilibria: r e φ e = 6qe 3 3q e 4 1 3cos θ e, 8 cos θ e sin θ e =0, 9 q e φ e q e 3 1 3cos θ e = k m q e l. 10 We assume that q e 0. The EulerLagrange equations of motion, obtained From equations 8, 9, 10 we obtain the follow
3 ing two categories of lative equilibria: θ e = nπ, n Z, φ e = 3 + 3q e 5, 3q e 3 = k m q e l, 11 θ e =n + 1 π, n Z, φ e = 3 3q e 5, q e = l. 1 The class of lative equilibria given by equation 11 corsponds to the dumbbell spacecraft aligned with the local radial. The class of lative equilibria given by equation 1 corsponds to the dumbbell spacecraft aligned with the local horizontal. 3. Linearized Orbit, Attitude Shape Dynamics We now derive linear equations of motion for perturbations from a lative equilibrium. Let x =[δr δφ δθ δq] T denote the vector of configuration perturbations, let u = [f r f h τ f s ] denote the vector of inputs. We can expss the linear equations of motion about a lative equilibrium given by 11 with θ e =0as whe C 1 = M 1 ẍ + C 1 ẋ + K 1 x = Bu, 13 M 1 = m 0 0 m + qe mqe 0 0 mqe mqe 0 0 m 0 4mr eω 1 4mr eω 1 4mq eω 1 0 4mq eω 1 0 4mq eω 1 4mq eω 1 0 { m K 1 = ω 1 + } + 1q e mq e r 4 e 6mq e, 14, mq... e { }, k m ω1 + 3 He M 1 is symmetric positive definite, C 1 is skewsymmetric, K 1 is symmetric. The matrix B Now we provide the linear equations of motion for perturbations from a lative equilibrium given by 1 with θ e = π. We can expss the linearized equations of motion in the form whe M ẍ + C ẋ + K x = Bu, 17 C = K = M = m 0 0 m + qe mqe 0 0 mqe mqe 0 0 m 0 4mr eω 4mr eω 4mq eω 0 4mq eω 0 4mq eω 4mq eω 0 { m ω1 + 3 ω = 6q e r 5 e } 6mq e r 4 e mq e r 4 e 3q e 6mq e k He M is symmetric positive definite, C is skewsymmetric, K is symmetric. The matrix B Note that the linearized equations about both classes of lative equilibria form linear gyroscopic systems. The study of system theotic properties of such systems has been dealt with in prior literatu, a sample of which can be found in [1, 13, 14] the fences thein. We now study controllability under various actuation assumptions for these equations. 3.3 Controllability of the Linear Orbit, Attitude Shape Equations of Motion The linear equations of motion a expssed in a stard linear second order control vector form [13] Mẍ + Cẋ + Kx = Bu 1, 18, 19, 0 ω 1 = + 3q e whe x R n, M is a symmetric positive definite matrix, C is a skewsymmetric matrix flecting gyroscopic effects, K is a symmetric matrix. Necessary sufficient conditions for complete controllability for
4 such systems a developed in [14]. Equation 1 is completely controllable if only if the controllability rank condition rank[λ M + λc + K, B] =n holds for all λ that satisfies det[λ M + λc + K] =0. We fer to the matrix [λ M + λc + K, B] asthe controllability matrix. This controllability sult is now used to give the following sults. Proposition 1. If f h = 0, then the linear orbit, attitude shape equations of motion a not completely controllable. Proof. Suppose the is no orbital actuation along the local horizontal, but the other the deges of fedom a actuated. Since the second row of both K 1 K a zero, it follows that the controllability matrix has a row of zeros for the second row when λ = 0. The sult follows. The above sult is consistent with the fact that the angular momentum of the spacecraft in orbit about the central body is conserved if the is no orbital actuation in the horizontal diction. Proposition. The linear orbit, attitude shape equations of motion a completely controllable if the orbit deges of fedom a actuated. Proof. In this case the controllability matrix has rank 4 for all values of λ. Proposition 3. The linear orbit, attitude shape equations of motion a completely controllable if the horizontal orbit dege of fedom the attitude dege of fedom a actuated. Proof. In this case, the controllability matrix has rank 4 for all values of λ. Proposition 4. The linear orbit, attitude shape equations of motion a completely controllable if the horizontal orbit dege of fedom the shape dege of fedom a actuated. fedom is actuated. This is possible due to the linear coupling between the orbit, attitude shape deges of fedom. Although these coupling effects may be weak, they may still be used to control the spacecraft using established linear control methods. 4 Dynamics Control of Orbit Attitude In this section, we study the dynamics control of the orbit attitude deges of fedom of the dumbbell spacecraft. The shape of the dumbbell spacecraft is assumed fixed. Relative equilibria for the orbit attitude dynamics a identified. Linear equations of motion a obtained that describe the perturbations from a lative equilibria. Several controllability sults a psented based on the linear perturbation equations. 4.1 Relative Equilibria for the Orbit Attitude Dynamics When the spacecraft has a fixed shape q e 0, the nonlinear equations of motion 47 become: m r m r φ r + 3q e r 4 1 3cos θ = f r, mr φ + q e φ + q e θ +rṙ φ =fh, 3 mqe θ 6m + φ+ r 3 q e cos θ sin θ = τ, 4 mq e θ + φ q e r 3 1 3cos θ +kq e l =f s. 5 Equation 5 defines the generalized shape force f s quid to maintain a constant shape. In the mainder of this section, equation 5 is assumed to be satisfied, it is not consided further. In this case, the lative equilibria satisfy r =ṙ =0, φ =0, θ =0, fr = f h = τ =0. The lative equilibria a thus characterised by equations 8 9. As in the general case consided in section 3, the a two categories of lative equilibria: Proof. In this case, the controllability matrix has rank 4 for all values of λ. θ e = nπ, n Z, φ e = + 3q e 5, 6 Propositions, 3 4 show that the dumbbell spacecraft is completely controllable for perturbations about its natural lative equilibria under various cases of underactuation, provided the horizontal orbit dege of θ e =n + 1 π, n Z, φ e = 3q e 5, 7 They corspond to the dumbbell spacecraft aligned with the local radial the local horizontal, spec
5 tively. 4. Linearized Orbit Attitude Dynamics We first consider a lative equilibrium with θ e =0, which belongs to the first category given by equation 6. Let x =[δr δφ δθ] T denote the vector of configuration perturbations, let u =[f r f h τ] denote the vector of control inputs. We can expss the linearized equations of motion as M 1 ẍ + C 1 ẋ + K 1 x = Bu, 8 whe M 1 = C 1 = K 1 = ω 1 = m 0 m + qe mqe 0 mqe mqe 0 4mr eω 1 0 4mr eω 1 0 m ω q e, 9, q e 5 0 6mq e,31 He M 1 is symmetric positive definite, C 1 is skewsymmetric, K 1 is symmetric. The matrix B Now we consider a lative equilibrium given by 7 with θ e = π, which belongs to the second category of lative equilibria. We can expss the linearized equations of motion in the form M ẍ + C ẋ + K x = Bu, 3 whe M = C = K = m 0 m + qe mqe 0 mqe mqe 0 4mr eω 0 4mr eω 0 { m ω 1 +, 33, 34 } 6q e 5 0 6mq e,35 He M is symmetric positive definite, C is skewsymmetric, K is symmetric. The matrix B The linear perturbation equations about both classes of lative equilibria have the form of linear gyroscopic systems, as in the case when the shape is not fixed. We study controllability under various actuation assumptions for these equations. 4.3 Controllability of the Linear Orbit Attitude Equations of Motion We now study controllability of the linear orbit attitude equations of motion given by Corollary 1. If f h = 0, then the linear orbit attitude equations of motion a not completely controllable. This is a corollary of Proposition 1. As in the case when the shape is not fixed, it is necessary to actuate the horizontal orbital dege of fedom. In the following we look at two cases with such actuation. Corollary. The linear orbit attitude equations of motion a completely controllable if the orbit deges of fedom a actuated. This is a corollary of Proposition. Corollary 3. The linear orbit attitude equations of motion a completely controllable if the horizontal orbit dege of fedom the attitude dege of fedom a actuated. This is a corollary of Proposition 3. Corollaries 3 show that the dumbbell spacecraft with fixed shape is completely controllable for perturbations about its lative equilibria under some cases of underactuation, provided the horizontal orbit dege of fedom is actuated. This is possible due to the linear coupling between the orbital attitude deges of fedom, which may be used to control the spacecraft, using established linear control methods. ω = 3q e 5 Conclusion Equations of motion of an elastic dumbbell spacecraft in a planar orbit about a massive spherical central body
6 have been developed. Although the dumbbell spacecraft model, consisting of an elastic interconnection of two mass particles, is extmely simple, the model does illustrate the coupling between orbit deges of fedom, attitude deges of fedom, shape deges of fedom. Relative equilibria, corsponding to circular orbits, a determined, linear equations of motion that describe the perturbations from the lative equilibria a obtained. Our development tats the general case of orbit, attitude shape coupling. For completeness, we also tat the special case that the shape is constrained to be fixed. This leads to equations of motion that describe the orbit attitude dynamics of a dumbbell spacecraft of fixed shape. The control inputs consist, in the most general case, of horizontal vertical forces on the dumbbell spacecraft, a moment about an axis normal to the orbital plane, a force between the two mass particles of the dumbbell spacecraft. The full actutation assumption is that horizontal vertical control forces a used to control the orbit, a control moment is used to control the attitude, a shape control force is used to control the shape. However, we identify several meaningful control problems corsponding to underactuation assumptions. Our controllability analysis suggests that in many cases effective control can be achieved using coupling between the orbit, attitude shape deges of fedom. Many of the key coupling terms, such as the gravity gradient moment, a small their control influence only acts over long time periods. Thus, the practical utility of exploiting these coupling mechanisms, at least in some cases, mains to be demonstrated. Dynamics control problems have been suggested based on the models for a dumbbell spacecraft that have been developed. The linear controllability sults imply that orbital actuation is necessary to achieve complete controllability. If the is no orbital actuation, the models studied in this paper cannot be completely controllable since angular momentum is necessarily conserved. However, it is important to mention that, if the is no orbital actuation, a simplified model, ferd to as the Routh duced model, can be developed that incorporates the conservation law. This duced model also leads to new dynamics control problems that differ from the problems consided in this paper. Control problems for Routh duced models a of significant practical importance, they a tated in our forthcoming publications. Refences [1] A. K. Misra, V. J. Modi, A Survey on the Dynamics Control of Tethed Satellite Systems, Advances in Astronautical Science, vol. 6, pp , [] S. B. Cho, N. H. McClamroch, Attitude Control of a Tethed Spacecraft, Proceedings of the American Control Confence, pp , Denver, CO, 003. [3] V. J. Modi, A. K. Misra, Orbital Perturbations of Tethed Satellite Systems, Journal of Astronautical Sciences, pp , [4] S. B. Cho, N. H. McClamroch, Optimal Orbit Transfer of a Spacecraft with Fixed Tether Length, submitted for publication. [5] J.P.Moran, EffectsofPlaneLibrationsontheOrbital Motion of a Dumbbell Satellite, ARS Journal, vol. 31, no. 8, pp , [6] K.Y. Lian, L.S. Wang, L.C. Fu, Controllability of Spacecraft Systems in a Central Gravitational Field, IEEE Transactions on Automatic Control, vol. 39, no. 1, pp , [7] D. J. Schees, Stability in the Full TwoBody Problem, Celestial Mechanics Dynamical Astronomy, vol. 83, pp , 00. [8] S. Cho, N. H. McClamroch M. Reyhanoglu, Dynamics of multibody vehicle their formulation as nonlinear control systems, Proceedings of the American Control Confence, pp , Chicago, IL, 000. [9] M. Reyhanoglu N. H. McClamroch, Reorientation Maneuvers of Planar Multibody Systems in Space using Internal Controls, AIAA Journal of Guidance, Control Dynamics, vol. 15, no. 6, pp , 199. [10] C. Rui, I. Kolmanovsky, N. H. McClamroch, Nonlinear Attitude Shape Control of Spacecraft with Articulated Appendages Reaction Wheels, IEEE Transactions on Automatic Control, vol. 45, no. 8, pp , 000. [11] J. Shen N. H. McClamroch, Translational Rotational Maneuvers of an Underactuated Space Robot using Prismatic Actuators, International Journal of Robotics Research, vol. 1, no. 56, pp , 00. [1] J. E. Marsden, T. S. Ratiu, Introduction to Mechanics Symmetry, nd. ed., SpringerVerlag Inc., New York, [13] R. E. Skelton, Dynamic Systems Control: Linear Systems Analysis Synthesis, John Wiley & Sons Inc., New York, [14] A.J. Laub W.F. Arnold, Controllability Observability Criteria for Multivariable Linear Second Order Models, IEEE Transactions on Automatic Control, Vol. 9, No., pp , Acknowledgement The authors thank Dr. Sangbum Cho for help in formulating the concepts tated in this paper. Our discussions with him we instrumental in the development of the psent paper.
Dynamics and Control of an Elastic Dumbbell Spacecraft in a Central Gravitational Field
Dynamics and Control of an Elastic Dumbbell Spacecraft in a Central Gravitational Field Amit K. Sanyal Jinglai Shen N. Harris McClamroch Department of Aerospace Engineering University of Michigan Conference
More informationAMIT K. SANYAL. 20012004 Ph.D. in Aerospace Engineering, University of Michigan, Ann Arbor, MI. Date of completion:
AMIT K. SANYAL Office Home 305 Holmes Hall 3029 Lowrey Avenue Mechanical Engineering Apartment # N2211 University of Hawaii at Manoa Honolulu, HI 96822 Honolulu, HI 96822 4806038938 8089562142 aksanyal@hawaii.edu
More informationPhysics 9e/Cutnell. correlated to the. College Board AP Physics 1 Course Objectives
Physics 9e/Cutnell correlated to the College Board AP Physics 1 Course Objectives Big Idea 1: Objects and systems have properties such as mass and charge. Systems may have internal structure. Enduring
More informationHomework 4. problems: 5.61, 5.67, 6.63, 13.21
Homework 4 problems: 5.6, 5.67, 6.6,. Problem 5.6 An object of mass M is held in place by an applied force F. and a pulley system as shown in the figure. he pulleys are massless and frictionless. Find
More informationLecture L18  Exploring the Neighborhood: the Restricted ThreeBody Problem
S. Widnall 16.07 Dynamics Fall 008 Version 1.0 Lecture L18  Exploring the Neighborhood: the Restricted ThreeBody Problem The ThreeBody Problem In Lecture 1517, we presented the solution to the twobody
More informationLecture L222D Rigid Body Dynamics: Work and Energy
J. Peraire, S. Widnall 6.07 Dynamics Fall 008 Version.0 Lecture L  D Rigid Body Dynamics: Work and Energy In this lecture, we will revisit the principle of work and energy introduced in lecture L3 for
More informationOrbital Mechanics. Angular Momentum
Orbital Mechanics The objects that orbit earth have only a few forces acting on them, the largest being the gravitational pull from the earth. The trajectories that satellites or rockets follow are largely
More informationSymmetric planar non collinear relative equilibria for the Lennard Jones potential 3 body problem with two equal masses
Monografías de la Real Academia de Ciencias de Zaragoza. 25: 93 114, (2004). Symmetric planar non collinear relative equilibria for the Lennard Jones potential 3 body problem with two equal masses M. Corbera,
More informationDYNAMICS OF A TETRAHEDRAL CONSTELLATION OF SATELLITESGYROSTATS
7 th EUROMECH Solid Mechanics Conference J. Ambrosio et.al. (eds.) Lisbon, Portugal, 7 11 September 2009 DYNAMICS OF A TETRAHEDRAL CONSTELLATION OF SATELLITESGYROSTATS Alexander A. Burov 1, Anna D. Guerman
More informationDynamics. Basilio Bona. DAUINPolitecnico di Torino. Basilio Bona (DAUINPolitecnico di Torino) Dynamics 2009 1 / 30
Dynamics Basilio Bona DAUINPolitecnico di Torino 2009 Basilio Bona (DAUINPolitecnico di Torino) Dynamics 2009 1 / 30 Dynamics  Introduction In order to determine the dynamics of a manipulator, it is
More informationAttitude Control and Dynamics of Solar Sails
Attitude Control and Dynamics of Solar Sails Benjamin L. Diedrich A thesis submitted in partial fulfillment of the requirements for the degree of Master of Science in Aeronautics & Astronautics University
More informationStability Of Structures: Basic Concepts
23 Stability Of Structures: Basic Concepts ASEN 3112 Lecture 23 Slide 1 Objective This Lecture (1) presents basic concepts & terminology on structural stability (2) describes conceptual procedures for
More informationDiscrete mechanics, optimal control and formation flying spacecraft
Discrete mechanics, optimal control and formation flying spacecraft Oliver Junge Center for Mathematics Munich University of Technology joint work with Jerrold E. Marsden and Sina OberBlöbaum partially
More informationMechanics 1: Conservation of Energy and Momentum
Mechanics : Conservation of Energy and Momentum If a certain quantity associated with a system does not change in time. We say that it is conserved, and the system possesses a conservation law. Conservation
More informationCentral configuration in the planar n + 1 body problem with generalized forces.
Monografías de la Real Academia de Ciencias de Zaragoza. 28: 1 8, (2006). Central configuration in the planar n + 1 body problem with generalized forces. M. Arribas, A. Elipe Grupo de Mecánica Espacial.
More informationOrbits of the LennardJones Potential
Orbits of the LennardJones Potential Prashanth S. Venkataram July 28, 2012 1 Introduction The LennardJones potential describes weak interactions between neutral atoms and molecules. Unlike the potentials
More informationSpacecraft Dynamics and Control. An Introduction
Brochure More information from http://www.researchandmarkets.com/reports/2328050/ Spacecraft Dynamics and Control. An Introduction Description: Provides the basics of spacecraft orbital dynamics plus attitude
More informationGravity Field and Dynamics of the Earth
Milan Bursa Karel Pec Gravity Field and Dynamics of the Earth With 89 Figures SpringerVerlag Berlin Heidelberg New York London Paris Tokyo HongKong Barcelona Budapest Preface v Introduction 1 1 Fundamentals
More informationLecture 17. Last time we saw that the rotational analog of Newton s 2nd Law is
Lecture 17 Rotational Dynamics Rotational Kinetic Energy Stress and Strain and Springs Cutnell+Johnson: 9.49.6, 10.110.2 Rotational Dynamics (some more) Last time we saw that the rotational analog of
More informationProblem 6.40 and 6.41 Kleppner and Kolenkow Notes by: Rishikesh Vaidya, Physics Group, BITSPilani
Problem 6.40 and 6.4 Kleppner and Kolenkow Notes by: Rishikesh Vaidya, Physics Group, BITSPilani 6.40 A wheel with fine teeth is attached to the end of a spring with constant k and unstretched length
More information(Most of the material presented in this chapter is taken from Thornton and Marion, Chap. 7)
Chapter 4. Lagrangian Dynamics (Most of the material presented in this chapter is taken from Thornton and Marion, Chap. 7 4.1 Important Notes on Notation In this chapter, unless otherwise stated, the following
More informationState Newton's second law of motion for a particle, defining carefully each term used.
5 Question 1. [Marks 28] An unmarked police car P is, travelling at the legal speed limit, v P, on a straight section of highway. At time t = 0, the police car is overtaken by a car C, which is speeding
More informationE X P E R I M E N T 8
E X P E R I M E N T 8 Torque, Equilibrium & Center of Gravity Produced by the Physics Staff at Collin College Copyright Collin College Physics Department. All Rights Reserved. University Physics, Exp 8:
More informationG U I D E T O A P P L I E D O R B I T A L M E C H A N I C S F O R K E R B A L S P A C E P R O G R A M
G U I D E T O A P P L I E D O R B I T A L M E C H A N I C S F O R K E R B A L S P A C E P R O G R A M CONTENTS Foreword... 2 Forces... 3 Circular Orbits... 8 Energy... 10 Angular Momentum... 13 FOREWORD
More informationRotation: Moment of Inertia and Torque
Rotation: Moment of Inertia and Torque Every time we push a door open or tighten a bolt using a wrench, we apply a force that results in a rotational motion about a fixed axis. Through experience we learn
More informationPhysics in the Laundromat
Physics in the Laundromat Kirk T. McDonald Joseph Henry Laboratories, Princeton University, Princeton, NJ 08544 (Aug. 5, 1997) Abstract The spin cycle of a washing machine involves motion that is stabilized
More informationIMPORTANT NOTE ABOUT WEBASSIGN:
Week 8 homework IMPORTANT NOTE ABOUT WEBASSIGN: In the WebAssign versions of these problems, various details have been changed, so that the answers will come out differently. The method to find the solution
More informationCONTRIBUTIONS TO THE AUTOMATIC CONTROL OF AERIAL VEHICLES
1 / 23 CONTRIBUTIONS TO THE AUTOMATIC CONTROL OF AERIAL VEHICLES MINH DUC HUA 1 1 INRIA Sophia Antipolis, AROBAS team I3SCNRS Sophia Antipolis, CONDOR team Project ANR SCUAV Supervisors: Pascal MORIN,
More informationOptimal Reconfiguration of Formation Flying Satellites
Proceedings of the th IEEE Conference on Decision and Control, and the European Control Conference 5 Seville, Spain, December 5, 5 MoA.6 Optimal Reconfiguration of Formation Flying Satellites Oliver Junge
More informationLecture L303D Rigid Body Dynamics: Tops and Gyroscopes
J. Peraire, S. Widnall 16.07 Dynamics Fall 2008 Version 2.0 Lecture L303D Rigid Body Dynamics: Tops and Gyroscopes 3D Rigid Body Dynamics: Euler Equations in Euler Angles In lecture 29, we introduced
More informationCHAPTER 2 ORBITAL DYNAMICS
14 CHAPTER 2 ORBITAL DYNAMICS 2.1 INTRODUCTION This chapter presents definitions of coordinate systems that are used in the satellite, brief description about satellite equations of motion and relative
More informationTHEORETICAL MECHANICS
PROF. DR. ING. VASILE SZOLGA THEORETICAL MECHANICS LECTURE NOTES AND SAMPLE PROBLEMS PART ONE STATICS OF THE PARTICLE, OF THE RIGID BODY AND OF THE SYSTEMS OF BODIES KINEMATICS OF THE PARTICLE 2010 0 Contents
More informationA Python Project for Lagrangian Mechanics
The 3rd International Symposium on Engineering, Energy and Environments 1720 November 2013, Pullman King Power Hotel, Bangkok A Python Project for Lagrangian Mechanics Peter Slaets 1, Pauwel Goethals
More informationGeometric Camera Parameters
Geometric Camera Parameters What assumptions have we made so far? All equations we have derived for far are written in the camera reference frames. These equations are valid only when: () all distances
More informationIsaac Newton s (16421727) Laws of Motion
Big Picture 1 2.003J/1.053J Dynamics and Control I, Spring 2007 Professor Thomas Peacock 2/7/2007 Lecture 1 Newton s Laws, Cartesian and Polar Coordinates, Dynamics of a Single Particle Big Picture First
More informationThe TwoBody Problem
The TwoBody Problem Abstract In my short essay on Kepler s laws of planetary motion and Newton s law of universal gravitation, the trajectory of one massive object near another was shown to be a conic
More informationAPPLIED MATHEMATICS ADVANCED LEVEL
APPLIED MATHEMATICS ADVANCED LEVEL INTRODUCTION This syllabus serves to examine candidates knowledge and skills in introductory mathematical and statistical methods, and their applications. For applications
More informationKyuJung Kim Mechanical Engineering Department, California State Polytechnic University, Pomona, U.S.A.
MECHANICS: STATICS AND DYNAMICS KyuJung Kim Mechanical Engineering Department, California State Polytechnic University, Pomona, U.S.A. Keywords: mechanics, statics, dynamics, equilibrium, kinematics,
More informationPhysics 9 Fall 2009 Homework 2  Solutions
Physics 9 Fall 009 Homework  s 1. Chapter 7  Exercise 5. An electric dipole is formed from ±1.0 nc charges spread.0 mm apart. The dipole is at the origin, oriented along the y axis. What is the electric
More informationMODELLING A SATELLITE CONTROL SYSTEM SIMULATOR
National nstitute for Space Research NPE Space Mechanics and Control Division DMC São José dos Campos, SP, Brasil MODELLNG A SATELLTE CONTROL SYSTEM SMULATOR Luiz C Gadelha Souza gadelha@dem.inpe.br rd
More informationLecture L17  Orbit Transfers and Interplanetary Trajectories
S. Widnall, J. Peraire 16.07 Dynamics Fall 008 Version.0 Lecture L17  Orbit Transfers and Interplanetary Trajectories In this lecture, we will consider how to transfer from one orbit, to another or to
More informationHalliday, Resnick & Walker Chapter 13. Gravitation. Physics 1A PHYS1121 Professor Michael Burton
Halliday, Resnick & Walker Chapter 13 Gravitation Physics 1A PHYS1121 Professor Michael Burton II_A2: Planetary Orbits in the Solar System + Galaxy Interactions (You Tube) 21 seconds 131 Newton's Law
More informationA. 81 2 = 6561 times greater. B. 81 times greater. C. equally strong. D. 1/81 as great. E. (1/81) 2 = 1/6561 as great.
Q12.1 The mass of the Moon is 1/81 of the mass of the Earth. Compared to the gravitational force that the Earth exerts on the Moon, the gravitational force that the Moon exerts on the Earth is A. 81 2
More informationState Newton's second law of motion for a particle, defining carefully each term used.
5 Question 1. [Marks 20] An unmarked police car P is, travelling at the legal speed limit, v P, on a straight section of highway. At time t = 0, the police car is overtaken by a car C, which is speeding
More informationDynamic Analysis. Mass Matrices and External Forces
4 Dynamic Analysis. Mass Matrices and External Forces The formulation of the inertia and external forces appearing at any of the elements of a multibody system, in terms of the dependent coordinates that
More informationCentripetal Force. This result is independent of the size of r. A full circle has 2π rad, and 360 deg = 2π rad.
Centripetal Force 1 Introduction In classical mechanics, the dynamics of a point particle are described by Newton s 2nd law, F = m a, where F is the net force, m is the mass, and a is the acceleration.
More informationLecture 16. Newton s Second Law for Rotation. Moment of Inertia. Angular momentum. Cutnell+Johnson: 9.4, 9.6
Lecture 16 Newton s Second Law for Rotation Moment of Inertia Angular momentum Cutnell+Johnson: 9.4, 9.6 Newton s Second Law for Rotation Newton s second law says how a net force causes an acceleration.
More informationWorldwide, space agencies are increasingly exploiting multibody dynamical structures for their most
Coupled OrbitAttitude Dynamics in the ThreeBody Problem: a Family of OrbitAttitude Periodic Solutions Davide Guzzetti and Kathleen C. Howell Purdue University, Armstrong Hall of Engineering, 71 W. Stadium
More informationPhysics Notes Class 11 CHAPTER 5 LAWS OF MOTION
1 P a g e Inertia Physics Notes Class 11 CHAPTER 5 LAWS OF MOTION The property of an object by virtue of which it cannot change its state of rest or of uniform motion along a straight line its own, is
More informationLet s first see how precession works in quantitative detail. The system is illustrated below: ...
lecture 20 Topics: Precession of tops Nutation Vectors in the body frame The free symmetric top in the body frame Euler s equations The free symmetric top ala Euler s The tennis racket theorem As you know,
More informationPHY411. PROBLEM SET 3
PHY411. PROBLEM SET 3 1. Conserved Quantities; the RungeLenz Vector The Hamiltonian for the Kepler system is H(r, p) = p2 2 GM r where p is momentum, L is angular momentum per unit mass, and r is the
More information11. Rotation Translational Motion: Rotational Motion:
11. Rotation Translational Motion: Motion of the center of mass of an object from one position to another. All the motion discussed so far belongs to this category, except uniform circular motion. Rotational
More informationSimulation of Trajectories and Comparison of Joint Variables for Robotic Manipulator Using Multibody Dynamics (MBD)
Simulation of Trajectories and Comparison of Joint Variables for Robotic Manipulator Using Multibody Dynamics (MBD) Jatin Dave Assistant Professor Nirma University Mechanical Engineering Department, Institute
More informationKINEMATICS OF PARTICLES RELATIVE MOTION WITH RESPECT TO TRANSLATING AXES
KINEMTICS OF PRTICLES RELTIVE MOTION WITH RESPECT TO TRNSLTING XES In the previous articles, we have described particle motion using coordinates with respect to fixed reference axes. The displacements,
More informationChapter 2. Derivation of the Equations of Open Channel Flow. 2.1 General Considerations
Chapter 2. Derivation of the Equations of Open Channel Flow 2.1 General Considerations Of interest is water flowing in a channel with a free surface, which is usually referred to as open channel flow.
More informationPS 320 Classical Mechanics EmbryRiddle University Spring 2010
PS 320 Classical Mechanics EmbryRiddle University Spring 2010 Instructor: M. Anthony Reynolds email: reynodb2@erau.edu web: http://faculty.erau.edu/reynolds/ps320 (or Blackboard) phone: (386) 2267752
More informationExemplar Problems Physics
Chapter Eight GRAVITATION MCQ I 8.1 The earth is an approximate sphere. If the interior contained matter which is not of the same density everywhere, then on the surface of the earth, the acceleration
More information= = GM. v 1 = Ωa 1 sin i.
1 Binary Stars Consider a binary composed of two stars of masses M 1 and We define M = M 1 + and µ = M 1 /M If a 1 and a 2 are the mean distances of the stars from the center of mass, then M 1 a 1 = a
More informationTorque Analyses of a Sliding Ladder
Torque Analyses of a Sliding Ladder 1 Problem Kirk T. McDonald Joseph Henry Laboratories, Princeton University, Princeton, NJ 08544 (May 6, 2007) The problem of a ladder that slides without friction while
More informationGeometric Adaptive Control of Quadrotor UAVs Transporting a CableSuspended Rigid Body
Geometric Adaptive Control of Quadrotor UAVs Transporting a CableSuspended Rigid Body Taeyoung Lee Abstract This paper is focused on tracking control for a rigid body payload that is connected to an arbitrary
More informationLecture L293D Rigid Body Dynamics
J. Peraire, S. Widnall 16.07 Dynamics Fall 2009 Version 2.0 Lecture L293D Rigid Body Dynamics 3D Rigid Body Dynamics: Euler Angles The difficulty of describing the positions of the bodyfixed axis of
More informationUnit 21 Influence Coefficients
Unit 21 Influence Coefficients Readings: Rivello 6.6, 6.13 (again), 10.5 Paul A. Lagace, Ph.D. Professor of Aeronautics & Astronautics and Engineering Systems Have considered the vibrational behavior of
More informationStructural Axial, Shear and Bending Moments
Structural Axial, Shear and Bending Moments Positive Internal Forces Acting Recall from mechanics of materials that the internal forces P (generic axial), V (shear) and M (moment) represent resultants
More informationLab 7: Rotational Motion
Lab 7: Rotational Motion Equipment: DataStudio, rotary motion sensor mounted on 80 cm rod and heavy duty bench clamp (PASCO ME9472), string with loop at one end and small white bead at the other end (125
More informationHalliday, Resnick & Walker Chapter 13. Gravitation. Physics 1A PHYS1121 Professor Michael Burton
Halliday, Resnick & Walker Chapter 13 Gravitation Physics 1A PHYS1121 Professor Michael Burton II_A2: Planetary Orbits in the Solar System + Galaxy Interactions (You Tube) 21 seconds 131 Newton's Law
More informationA Simulation Study on Joint Velocities and End Effector Deflection of a Flexible Two Degree Freedom Composite Robotic Arm
International Journal of Advanced Mechatronics and Robotics (IJAMR) Vol. 3, No. 1, JanuaryJune 011; pp. 90; International Science Press, ISSN: 09756108 A Simulation Study on Joint Velocities and End
More informationAP1 Gravity. at an altitude equal to twice the radius (R) of the planet. What is the satellite s speed assuming a perfectly circular orbit?
1. A satellite of mass m S orbits a planet of mass m P at an altitude equal to twice the radius (R) of the planet. What is the satellite s speed assuming a perfectly circular orbit? (A) v = Gm P R (C)
More informationRotational inertia (moment of inertia)
Rotational inertia (moment of inertia) Define rotational inertia (moment of inertia) to be I = Σ m i r i 2 or r i : the perpendicular distance between m i and the given rotation axis m 1 m 2 x 1 x 2 Moment
More informationIMU Components An IMU is typically composed of the following components:
APN064 IMU Errors and Their Effects Rev A Introduction An Inertial Navigation System (INS) uses the output from an Inertial Measurement Unit (IMU), and combines the information on acceleration and rotation
More informationChapter 11 Equilibrium
11.1 The First Condition of Equilibrium The first condition of equilibrium deals with the forces that cause possible translations of a body. The simplest way to define the translational equilibrium of
More informationOrigins of the Unusual Space Shuttle Quaternion Definition
47th AIAA Aerospace Sciences Meeting Including The New Horizons Forum and Aerospace Exposition 58 January 2009, Orlando, Florida AIAA 200943 Origins of the Unusual Space Shuttle Quaternion Definition
More informationEngineering Mechanics I. Phongsaen PITAKWATCHARA
2103213 Engineering Mechanics I Phongsaen.P@chula.ac.th May 13, 2011 Contents Preface xiv 1 Introduction to Statics 1 1.1 Basic Concepts............................ 2 1.2 Scalars and Vectors..........................
More informationMidterm Solutions. mvr = ω f (I wheel + I bullet ) = ω f 2 MR2 + mr 2 ) ω f = v R. 1 + M 2m
Midterm Solutions I) A bullet of mass m moving at horizontal velocity v strikes and sticks to the rim of a wheel a solid disc) of mass M, radius R, anchored at its center but free to rotate i) Which of
More informationPHYSICS 111 HOMEWORK SOLUTION #10. April 8, 2013
PHYSICS HOMEWORK SOLUTION #0 April 8, 203 0. Find the net torque on the wheel in the figure below about the axle through O, taking a = 6.0 cm and b = 30.0 cm. A torque that s produced by a force can be
More informationUnit 4 Practice Test: Rotational Motion
Unit 4 Practice Test: Rotational Motion Multiple Guess Identify the letter of the choice that best completes the statement or answers the question. 1. How would an angle in radians be converted to an angle
More informationVELOCITY, ACCELERATION, FORCE
VELOCITY, ACCELERATION, FORCE velocity Velocity v is a vector, with units of meters per second ( m s ). Velocity indicates the rate of change of the object s position ( r ); i.e., velocity tells you how
More informationModeling Mechanical Systems
chp3 1 Modeling Mechanical Systems Dr. Nhut Ho ME584 chp3 2 Agenda Idealized Modeling Elements Modeling Method and Examples Lagrange s Equation Case study: Feasibility Study of a Mobile Robot Design Matlab
More informationPHYS 211 FINAL FALL 2004 Form A
1. Two boys with masses of 40 kg and 60 kg are holding onto either end of a 10 m long massless pole which is initially at rest and floating in still water. They pull themselves along the pole toward each
More informationNEWTON S LAWS OF MOTION
NEWTON S LAWS OF MOTION Background: Aristotle believed that the natural state of motion for objects on the earth was one of rest. In other words, objects needed a force to be kept in motion. Galileo studied
More informationState of Stress at Point
State of Stress at Point Einstein Notation The basic idea of Einstein notation is that a covector and a vector can form a scalar: This is typically written as an explicit sum: According to this convention,
More informationNotes on Elastic and Inelastic Collisions
Notes on Elastic and Inelastic Collisions In any collision of 2 bodies, their net momentus conserved. That is, the net momentum vector of the bodies just after the collision is the same as it was just
More informationThe elements used in commercial codes can be classified in two basic categories:
CHAPTER 3 Truss Element 3.1 Introduction The single most important concept in understanding FEA, is the basic understanding of various finite elements that we employ in an analysis. Elements are used for
More informationCHAPTER 11. The total energy of the body in its orbit is a constant and is given by the sum of the kinetic and potential energies
CHAPTER 11 SATELLITE ORBITS 11.1 Orbital Mechanics Newton's laws of motion provide the basis for the orbital mechanics. Newton's three laws are briefly (a) the law of inertia which states that a body at
More informationLectures notes on orthogonal matrices (with exercises) 92.222  Linear Algebra II  Spring 2004 by D. Klain
Lectures notes on orthogonal matrices (with exercises) 92.222  Linear Algebra II  Spring 2004 by D. Klain 1. Orthogonal matrices and orthonormal sets An n n realvalued matrix A is said to be an orthogonal
More informationDynamics of Iain M. Banks Orbitals. Richard Kennaway. 12 October 2005
Dynamics of Iain M. Banks Orbitals Richard Kennaway 12 October 2005 Note This is a draft in progress, and as such may contain errors. Please do not cite this without permission. 1 The problem An Orbital
More informationSample Questions for the AP Physics 1 Exam
Sample Questions for the AP Physics 1 Exam Sample Questions for the AP Physics 1 Exam Multiplechoice Questions Note: To simplify calculations, you may use g 5 10 m/s 2 in all problems. Directions: Each
More informationFigure 3.1.2 Cartesian coordinate robot
Introduction to Robotics, H. Harry Asada Chapter Robot Mechanisms A robot is a machine capable of physical motion for interacting with the environment. Physical interactions include manipulation, locomotion,
More informationMathematics Course 111: Algebra I Part IV: Vector Spaces
Mathematics Course 111: Algebra I Part IV: Vector Spaces D. R. Wilkins Academic Year 19967 9 Vector Spaces A vector space over some field K is an algebraic structure consisting of a set V on which are
More informationUNIVERSITETET I OSLO
UNIVERSITETET I OSLO Det matematisknaturvitenskapelige fakultet Exam in: FYS 310 Classical Mechanics and Electrodynamics Day of exam: Tuesday June 4, 013 Exam hours: 4 hours, beginning at 14:30 This examination
More informationDIRECT ORBITAL DYNAMICS: USING INDEPENDENT ORBITAL TERMS TO TREAT BODIES AS ORBITING EACH OTHER DIRECTLY WHILE IN MOTION
1 DIRECT ORBITAL DYNAMICS: USING INDEPENDENT ORBITAL TERMS TO TREAT BODIES AS ORBITING EACH OTHER DIRECTLY WHILE IN MOTION Daniel S. Orton email: dsorton1@gmail.com Abstract: There are many longstanding
More informationC B A T 3 T 2 T 1. 1. What is the magnitude of the force T 1? A) 37.5 N B) 75.0 N C) 113 N D) 157 N E) 192 N
Three boxes are connected by massless strings and are resting on a frictionless table. Each box has a mass of 15 kg, and the tension T 1 in the right string is accelerating the boxes to the right at a
More informationPrerequisites 20122013
Prerequisites 20122013 Engineering Computation The student should be familiar with basic tools in Mathematics and Physics as learned at the High School level and in the first year of Engineering Schools.
More informationPES 1110 Fall 2013, Spendier Lecture 33/Page 1. [kg m 2 /s] (dropped radians)
PES 1110 Fall 2013, Spendier Lecture 33/Page 1 Today:  Conservation o Angular Momentum (11.11)  Quiz 5, next Friday Nov 22nd (covers lectures 2933,HW 8) angular moment or point particles: L r p mr v
More informationTesting dark matter halos using rotation curves and lensing
Testing dark matter halos using rotation curves and lensing Darío Núñez Instituto de Ciencias Nucleares, UNAM Instituto Avanzado de Cosmología A. González, J. Cervantes, T. Matos Observational evidences
More informationCenter of Gravity. We touched on this briefly in chapter 7! x 2
Center of Gravity We touched on this briefly in chapter 7! x 1 x 2 cm m 1 m 2 This was for what is known as discrete objects. Discrete refers to the fact that the two objects separated and individual.
More informationReview D: Potential Energy and the Conservation of Mechanical Energy
MSSCHUSETTS INSTITUTE OF TECHNOLOGY Department of Physics 8.01 Fall 2005 Review D: Potential Energy and the Conservation of Mechanical Energy D.1 Conservative and Nonconservative Force... 2 D.1.1 Introduction...
More informationUniversal Law of Gravitation
Universal Law of Gravitation Law: Every body exerts a force of attraction on every other body. This force called, gravity, is relatively weak and decreases rapidly with the distance separating the bodies
More information2. Dynamics, Control and Trajectory Following
2. Dynamics, Control and Trajectory Following This module Flying vehicles: how do they work? Quick refresher on aircraft dynamics with reference to the magical flying space potato How I learned to stop
More informationProblem Set V Solutions
Problem Set V Solutions. Consider masses m, m 2, m 3 at x, x 2, x 3. Find X, the C coordinate by finding X 2, the C of mass of and 2, and combining it with m 3. Show this is gives the same result as 3
More informationChapter 2. Mission Analysis. 2.1 Mission Geometry
Chapter 2 Mission Analysis As noted in Chapter 1, orbital and attitude dynamics must be considered as coupled. That is to say, the orbital motion of a spacecraft affects the attitude motion, and the attitude
More information