Hubert Hahn

Rigid Body Dynamics of Mechanisms 2

Springer-V erlag Berlin Heidelberg GmbH

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Hubert Hahn

Rigid Body Dynamics of Mechanisms 2 Applications

With 228 Figures

'Springer

Professor Dr. Hubert Hahn Universität Gh Kassel Regelungstechnik und Systemdynamik, FB Maschinenbau Mönchebergstraße 7 D-34109 Kassel Germany e-mail: hahn@uni-kassel.de

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To

Elke and Mechthild

Preface

The dynamics of mechanical rigid-body systems is a highly developed disci­pline. The model equations that apply to the tremendous variety of appli­cations of rigid-body systems in industrial practice are based on just a few basic laws of, for example, Newton, Euler, or Lagrange. These basic laws can be written in an extreme compact, symmetrical, and esthetic form, simple enough to be easily learned and kept in mind by students and engineers not only from the area of mechanics, but also from other disciplines like physics, mathematics, or even control, hydraulics, and electronics. This latter aspect is of immense practical importance since mechanisms, machines, robots, and ve­hicles in modern industrial practice (sometimes called mechatronic systems) usually include various subsystems from the areas of hydraulics, electronics, pneumatics, and control and are built by engineers which are trained in quite different disciplines.

Objectives of this monograph

This Volume presents a systematic approach for deriving model equations of many planar and spatial mechanisms:

1. As a first step in DAE form along the systematic approach of Volume I. 2. As a second step in symbolic DE form, as nonlinear and linear state-space

equations, andin transfer-function form.

The objectives of both the theoretical discussions (Volume I) and the practical applications (this volume) are (see Table 1.1 of Chapter 1, Volume I):

1. To prepare the reader for efficiently handling and applications of general­purpose rigid-body programs to complex mechanisms.

2. To set up symbolic mathematical models of mechanisms in DAE form for computer simulations andjor in symbolic DE form as is often required in dynamic analysis and control.

From the point of view of these two objectives this monograph can be consid­ered as an introduction to basic mechanical aspects of mechatronic rigid-body systems.

viii

Organization of the book

In this Volume, the modeling methodology introduced in Volume I, will be applied to various simple and more complex examples of planar and spatial rigid-body mechanisms.

Most of the model equations will be written both in symbolic DAE form and DE form, and sometimes also in nonlinear and linear state-space form, and as transfer-function matrix representation ( compare the examples of the Fig­ures 1.17, 1.18, 1.19 and 1.20 ofVolume I). They include a rich variety of joint models, active constraints, and active and passive force elements. In Chapter 2 of this Volume, the modeling methodology is summarized, and two algo­rithms are briefiy discussed which map the symbolic model equations from the DAE form into the symbolic DE form (in the case that this is feasible). Two applications of planar models of an unconstrained rigid body are dis­cussed in Chapter 3. Several applications of a rigid body under constrained planar motion are presented in Chapter 4. Various applications of planar mechanisms which include two rigid bodies under constraints are discussed in Chapter 5. Three applications of a rigid body under unconstrained spatial motion are collected in Chapter 6, followed by applications of a constrained spatial rigid body in Chapter 7, and by several applications of spatial mech­anisms which include between 2 and 13 constrained rigid bodies in Chapter 8.

U se of the text

The text of this book is intended for use and self-study by practicising in­dustrial engineers (from the areas of mechanics, vibration techniques, vehicle simulation, control, hydraulics, pneumatics, measurement, testing, electro­magnetics, and electronics) which have a bachelor's degree, and by students of undergraduate university courses which are faced with the task to set up theoretical models of rigid-body mechanisms and mechatronic systems which may be used as a basis for computer simulation, analysis, and control.

Spatial mechanics is conceptually more complex and its theoretical modeling provides much lengthier and more unwieldy formal expressions than planar mechanics. To enable the beginner reader to successfully master his or her study of rigid-body dynamics and to keep the amount of notation and for­mal expressions of the applications presented within acceptable limits, only planar mechanisms will be discussed in the Chapters 3, 4, and 5. Teach­ing experience shows that the methodology of modeling rigid-body systems can be basically understood by considering planar systems. Despite the fact that various of the examples presented are simple enough to be directly and briefiy modeled in the traditional DE form, using minimal coordinates and free-body diagrams, all model equations are, as a first step, written in the DAE form using Cartesian coordinates. Having developed confidence and

IX

enough intuition in the basic methods of the theoretical modeling of planar mechanisms, the reader is encouraged to study the applications to spatial rigid-body mechanisms of the Chapters 6, 7, and 8. Basic differences be­tween the model equations of planar and spatial rigid-body mechanisms are summarized in the Appendix A.3 of Volume I.

Acknowledgements

I would like to thank the members of my laboratory and some of my students in Kassel who have helped in many ways in making this book possible.

Especially, I would like to express my sincere gratitude to Dr. Willy Klier für his close cooperation and assistance in deriving various DAE models of complex industrial mechanisms, a.o. the models of the Sections 8.3 and 8.4, and for various helpful comments on this book. I highly appreciate the help of my students Dipl.-Ing. Marco Thiemar and Dipl.-Ing. Reinhard Großheim for their help with proofreading some of the applications.

I am also very grateful to Dr. Willy Klier, Dipl.-Ing Axel Dürrbaum, and Dipl.-Ing. Marco Thiemar for the software packages which they have devel­oped for mapping the symbolic DAEs into symbolic DEs, and especially to Dipl.-Ing Marco Thiemar who has dorre an excellent job in evaluating the symbolic DEs of various applications by means of these programs.

I owe an immense debt of gratitude to Dipl-Ing. Axel Dürrbaum for preparing, writing, handling and respectively correcting the IbTEX document, for typing various of the lengthy mathematical expressions and for drawing some of the figures. This book would have not been possible in this form without his generaus and consistent help.

I would also deeply thank Mr. Ralf Rettberg for his superb drawings of most of the figures and Ms. Michaela Görgl and Ms. Renate Sauerborn for their patience in typing most of the text and some of the mathematical expressions.

I am grateful to the Deutsche Forschungsgemeinschaft (DFG) for supporting my research work in modeling and control of rigid-body mechanisms.

I also thank the unknown copy-editor and the staff of Springer for helpful cooperation and thorough publication of the book.

Finally, on a personal note, I would like to deeply thank my wife, Mechthild, for her generaus understanding and patience during the writing of this book.

Hubert Hahn Sporke /Westfalen Germany J anuary 2003

Contents

Preface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . vii

1. Introduction.............................................. 1 1.1 Purposes of models of rigid-body mechanisms . . . . . . . . . . . . . . 1 1.2 Steps for deriving model equations . . . . . . . . . . . . . . . . . . . . . . . . 2 1.3 Examples and applications presented. . . . . . . . . . . . . . . . . . . . . . 6

2. Model equations in symbolic DAE and DE form . . . . . . . . . . 9 2.1 Model equations in symbolic DAE form . . . . . . . . . . . . . . . . . . . 9 2.2 Model equations in symbolic DE form. . . . . . . . . . . . . . . . . . . . . 11

2.2.1 Global symbolic projection of DAEs into DEs . . . . . . . . 11 2.2.1.1 Global symbolic projection of DAEs with kine­

matic constraint equations into DEs. . . . . . . . . 12 2.2.1.2 Global symbolic projection of DAEs with ac-

tive constraint equations into DEs . . . . . . . . . . 17 2.2.1.3 A global elimination algorithm . . . . . . . . . . . . . 21

2.2.2 Stepwise elimination of the dependent variables . . . . . . 24

3. Planar models of an unconstrained rigid body . . . . . . . . . . . . 31 3.1 Planar airplane model (two tr. DOFs, one rot. DOF)........ 32

3.1.1 Model equations of the airplane . . . . . . . . . . . . . . . . . . . . 32 3.1.2 Nonlinear state-space representation ofthe model equa-

tions............................................ 38 3.1.3 Symbolic Taylor-series linearization of the model equa-

tions............................................ 39 3.2 Planar model of a multi-axis test facility (two tr. DOFs,

one rot. DOF) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46 3.2.1 DE modeling using the Newton-Euler equations . . . . . . 46 3.2.2 Nonlinear state-space representation ofthe model equa-

tions............................................ 61 3.2.3 Linearization of the model equations, linear state-

space equations and transfer-function matrix. . . . . . . . . 63 3.2.4 Different realizations of the mechanism. . . . . . . . . . . . . . 65

xii Contents

4. Planar models of a rigid body under absolute constraints . 67 4.1 Rigid body under pure translational planar motion

(two tr. DOFs)......................................... 67 4.1.1 DAE modeling based on the Newton-Euler equations . 68 4.1.2 DE model obtained by elimination of the dependent

coordinates and Lagrange multipliers . . . . . . . . . . . . . . . 72 4.2 Rack-and-pinion mechanism (one tr.jrot. DOF) . . . . . . . . . . . . 74

4.2.1 DAE-modeling approach based on the Newton-Euler equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 4

4.2.2 Model equations in DE form . . . . . . . . . . . . . . . . . . . . . . . 77 4.2.3 DE modeling approach based on the Lagrange equa-

tions and using a single coordinate . . . . . . . . . . . . . . . . . 80 4.2.4 DAE modeling approach based on the Lagrange equa-

tions in terms of the dependent coordinates. . . . . . . . . . 81 4.3 Mechanical rotor ( one rot. DOF) . . . . . . . . . . . . . . . . . . . . . . . . . 83

4.3.1 DAE approach based on the laws of Newton and Euler 83 4.3.2 Constitutive relations of the external forces: . . . . . . . . . 87 4.3.3 DE modeling by elimination ofthedependent

variables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 96 4.3.3.1 Stepwise elimination of the dependent

variables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 96 4.3.3.2 Global elimination of the dependent

variables ................................. 103

4.3.4 Special cases of the DE model ..................... 106 4.3.5 Technical applications of this example ............... 112 4.3.6 DE modeling of a mechanical rotor based on Euler's

law and using a free-body diagram ................. 112 4.3.6.1 DE modeling approach of a mechanical rotor

based on Euler's law using a single Cartesian coordinate ............................... 113

4.3.6.2 Model equations in linear state-space form and as a transfer function .................. 116

4.3. 7 DE modeling based on the Lagrange equations using a single generalized coordinate . . . . . . . . . . . . . . . . . . . . . 117

4.3.8 DE modeling of a pendulum based on the Lagrange approach ........................................ 118

4.4 Pendulum of a variable length (one tr. DOF, one rot. DOF) . 121 4.4.1 Model equations of the mechanism in DAE form ..... 121 4.4.2 Elimination of the dependent coordinate and Lagrange

multiplier ....................................... 124 4 4 2 1 St . l' . t' f R . R .. R . . . epw1se e 1mma 10n o Xp10 , Xp10 , Xp10 ,

and .-\1 .................................. 124 4.4.2.2 Global elimination of the dependent

variables ................................. 129

Contents xm

4.4.3 Model equations in mixed (relative/absolute) coordi-nates ........................................... 132 4.4.3.1 Forces which act on the body .............. 140 4.4.3.2 Torques which act on the body ............. 144 4.4.3.3 Final model equations in DE form .......... 147

4.4.4 Modeling based on the Lagrange equations .......... 148 4.5 Special wheel suspension (one tr. DOF, one rot. DOF) ...... 155

4.5.1 DAE model of the mechanisms ..................... 156 4.5.2 Model equations in DE form ....................... 164

4.5.2.1 Globalelimination ofthedependent variables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 178

4.5.3 Technical realizations of the mechanism ............. 182 4.6 Vertical vehicle model ( one tr. DOF, one rot. DOF) ......... 182

4.6.1 DAE-modeling approach based on the Newton-Euler equations. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 183

4.6.2 Model equations in DE form ....................... 193 4.6.3 Physical interpretation of some expressions of the model

equations ........................................ 196 4.6.4 Technical realizations of the above mechanism ....... 199 4.6.5 DE model derived from the Newton-Euler approach ... 199

4.7 Airplane under an active constraint (one tr. DOF, one rot. DOF) ................................................. 203 4.7.1 DAE modeling of the airplane ...................... 203 4.7.2 DE model obtained by elimination of the dependent

variables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 205

5. Planar models of two rigid hoclies under constrained motion ................................................... 209 5.1 Cart loaded by a pendulum (one tr. DOF, one rot. DOF) .... 209

5.1.1 DAE modeling approach based on the laws of Newton and Euler ....................................... 210

5.1.2 Model equations in DE form ....................... 218 5.1.2.1 Stepwise Elimination of the dependent coor-

dinates and Lagrange multipliers ............ 218 5.1.2.2 Global elimination of the dependent Coordi-

nates and Lagrange multipliers ............. 225 5.1.3 Technical realizations of the mechanism ............. 227 5.1.4 DE approach based on the Newton-Euler equations ... 228

5.1.4.1 Newton-Euler equations ................... 228 5.1.4.2 Linear DE model ......................... 231 5.1.4.3 Linear state-space equations ................ 232

5.1.5 Model equations based on the Lagrangeapproach .... 233 5.2 Swing with two rigid bodies (one tr. DOF, one rot. DOF) ... 235

5.2.1 Model equations in DAE form ..................... 236

xiv Contents

5.2.2 Model equations in DE form obtained by elimination of the dependent coordinates . . . . . . . . . . . . . . . . . . . . . . . 242

5.2.3 Computation of the applied forces and torques ....... 249 5.2.4 Model equations in DE form written in mixed (relative

and absolute) minimal coordinates .................. 254 5.2.4.1 Global elimination of the dependent Coordi-

nates and Lagrange multiplierB ............. 257 5.3 Pendulum of a variable length (one tr. DOF, one rot. DOF) . 260

5.3.1 DAE modeling approach based on the laws of Newton and Euler . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 260

5.3.2 Global elimination of the dependent coordinates and Lagrange multipliers .............................. 265

5.3.3 Forcesand torques ............................... 267 5.3.4 DE modeling based on the Lagrange formalism ....... 269

5.4 Milling machirre (one tr. DOF, two rot. DOFs) ............. 272 5.4.1 Model equations in DAE form ..................... 274 5.4.2 Stepwise elimination of the dependent coordinates and

Lagrange multiplierB . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 280 5.4.3 Computation of the forces and torques .............. 288 5.4.4 Model equations in DE form represented in hybrid

(relative and absolute) coordinates ................. 292 5.4.4.1 Global elimination of the dependent Coordi-

nates and Lagrange multipliers . . . . . . . . . . . . . 300 5.5 Double pendulum with an "elastic joint" (one tr. DOF, two

rot. DOFs) ............................................ 304 5.5.1 Model equations in DAE-form ..................... 304 5.5.2 Model equations in DE form ....................... 310

5.5.2.1 Global elimination of the dependent coordi-nates and Lagrange multipliers ............. 317

5.5.3 Constraint reaction forces and torques .............. 320 5.5.4 Applied forces and torques ........................ 321

5.6 Excavator (one tr. DOF, two rot. DOFs) .................. 327 5.6.1 DAE modeling of the mechanism ................... 329 5.6.2 Model equations in DE form, written in minimal Co-

ordinates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 335 5.6.3 Model equations in DE form represented in mixed (ab­

solute and relative) minimal coordinates . . . . . . . . . . . . . 343 5.6.4 Forces and torques which act on the hoclies .......... 346 5.6.5 Nonlinear and linear state-space equations ........... 355

5.7 Camera attached to an airplane under active constraints (one rot. DOF) ............................................. 365 5.7.1 DAE modeling of the system using the Newton-Euler

equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 367 5.7.2 DE modeling of the system ........................ 376

Contents xv

5. 7.3 Nonlinear state-space equations .................... 380 5. 7.4 Taylor-series linearization of the nonlinear model equa-

tions ............................................ 382 5. 7.5 Transfer functions and eigenvalues .................. 388

6. Spatial models of an unconstrained rigid body ............ 391 6.1 Rigid body attached to the base by a translational spring-

damper element ........................................ 391 6.1.1 Model equations in DE form ....................... 391 6.1.2 Applied forces and torques ........................ 394 6.1.3 Nonlinear state-space equations .................... 397 6.1.4 Linearmodel equations ........................... 398

6.1.4.1 Linearization of the model equations in an equilibrium point ......................... 399

6.1.4.2 Linear state-space equations ................ 401 6.1.4.3 Matrix of transfer functions . . . . . . . . . . . . . . . . 402

6.2 Spatial servo-pneumatic parallel robot (three tr. DOFs, three rot. DOFs) ............................................ 403 6.2.1 DE modeling based on the Newton-Euler equations ... 405 6.2.2 Inverse kinematic equations associated with the force

elements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 409 6.2.3 Applied forces and torques ........................ 413 6.2.4 Nonlinear state-space equations .................... 415 6.2.5 Linear state-space equations ....................... 416 6.2.6 Different technical realizations of the parallel robot ... 424

6.3 Model equations of a spinning rocket (three tr. DOFs, three rot. DOFs) ............................................ 424 6.3.1 Coordinates and transformation matrices ............ 425 6.3.2 The kinematic DEs ............................... 427 6.3.3 The Newton-Euler equations ....................... 429 6.3.4 Forces and torques which act on the body ........... 432 6.3.5 Linearmodel equations ........................... 435

6.3.5.1 Linear equations of the rocket in the vertical plane .................................... 435

6.3.5.2 Linear coupling of two rotational DOFs ...... 437

7. Spatial models of a rigid body under constrained motion . 439 7.1 Rigid body attached to the base by a spherical joint ........ 439

7.1.1 Model equations in DAE form ..................... 439 7.1.2 Choice ofthedependent Coordinates and computation

of the independent coordinates .................... 442 7.1.3 Computation of the Lagrange multipliers ........... 442 7.1.4 Computation of the forces and torques .............. 444 7.1.5 Stepwise elimination ofthedependent coordinates and

Lagrange multipliers ............................. 448

xvi Contents

7.1.6 Model equations in DE form ....................... 449 7.1.7 Global elimination of the dependent coordinates and

Lagrange multipliers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 452 7.2 Rigid body attached to the base by a revolute joint (one rot.

DOF) ................................................. 454 7.2.1 Model equations in DAE form ..................... 454 7.2.2 Global elimination of the dependent coordinates and

Lagrange multipliers .............................. 458 7.3 Rigid body attached to the base by a universal joint (two rot.

DOFs) ................................................ 462 7.3.1 Model equations in DAE form ..................... 462 7.3.2 Global elimination of the dependent coordinates and

Lagrange multipliers .............................. 465

8. Spatial mechanisms with several rigid bodies ............. 471 8.1 Antenna for flight vehicles (two rot. DOFs) ................ 471

8.1.1 Model equations in DAE form ..................... 471 8.1.2 Computation of the constraint reaction forces and

torques ......................................... 4 78 8.1.3 Stepwise elimination ofthedependent variables ...... 479 8.1.4 Global elimination of the dependent coordinates and

Lagrange multipliers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 492 8.2 Differential gear (two rot. DOFs) ......................... 494

8.2.1 Notations ....................................... 495 8.2.2 Constraint equations .............................. 499 8.2.3 Collection of the kinematic DEs and constraint

equations ........................................ 515 8.2.4 Model equations in DAE form ..................... 522 8.2.5 Simplification of some constraint position equations ... 524 8.2.6 Global elimination of the dependent coordinates and

Lagrange multipliers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 526 8.2. 7 Alternative DAE and DE modeling approach ........ 530 8.2.8 Model equations in DE form ....................... 540

8.3 Platform of an airborne sensor (two rot. DOFs) ............ 545 8.3.1 Mathematical notations ........................... 546 8.3.2 Constraint equations .............................. 550 8.3.3 Model equations in DAE form ..................... 556 8.3.4 Computation of the applied forces and torques on the

platform ........................................ 557 8.3.5 Global elimination of the dependent coordinates and

Lagrange multipliers .............................. 563 8.4 Model of a hexapod with 13 rigid bodies (three trans. DOFs,

three rot. DOFs) ....................................... 572 8.4.1 Notations used in the model equations .............. 572 8.4.2 Model equations in DAE form ..................... 576

Contents xvii

8.4.3 lmplicit form of the constraint equations of the joints . 578 8.4.4 Partitioning of the coordinate vectors and kinematic

DEs ............................................ 581 8.4.5 Constraint position, velocity and acceleration equa­

tions as explicit functions of the independent coordinates ...................................... 581

8.4.6 Computation of the global projector ................ 589 8.4. 7 Construction of the symbolic DE model ............. 592 8.4.8 Elimination of the dependent coordinates of the DEs

by means of the explicit constraint equations. . . . . . . . . 593 8.4.9 Reformulation of the implicit DEs in an explicit form . 595 8.4.10 DEs of the parallel robot for neglected inertia param-

eters of the actuators ............................. 601 8.5 Platform mounted on a test faeility (three trans. DOFs, five

rot. DOFs) ............................................ 601 8.5.1 Mathematieal notations ........................... 602 8.5.2 Constraint equations .............................. 605 8.5.3 Model equations in DAE form ..................... 607 8.5.4 Forees whieh act on the bodies ..................... 609

8.6 Simple vehicle model (seven trans. DOFs, nine rot. DOFs) ... 611 8.6.1 Notations ....................................... 611 8.6.2 Constraint equations .............................. 613 8.6.3 Model equations in DAE form ..................... 620

A. Appendix . ................................................ 623 A.1 Alternative representation of the spring and damper forees of

Beetion 3.2 ............................................ 623 A.2 Auxiliary eomputations and results associated with the meeh-

anism of Section 8.3 .................................... 638 A.2.1 Explieit form of the eonstraint equations of the mass-

less links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 638 A.2.2 Coeffieients of the kinematies of the eleetrieal drives .. 640 A.2.3 Computation of the transformation matrix of the forees

of the eleetrical drives ............................. 642 A.3 Auxiliary eomputations assoeiated with the example of Bee-

tion 8.4 ............................................... 647 A.3.1 Constraint Jaeobian matrix Jt6 used as the transfor­

mation matrix of the aetuator forees . . . . . . . . . . . . . . . . 64 7 A.3.2 Auxiliary eomputations used in Beetion 8.4.9 ......... 651

Referenees . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 655 Index ...................................................... 659 List of Figures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 663