《ADVANCED DYNAMICS VOLⅡ》

CHAPTER Ⅰ:MOMENTS AND PRODUCTS OF INERTIA1

1．Definitions1

EXERCISES Ⅰ12

2．Given the moments and product of inertia of a lamina about rectangular axes Ox,Oy in its plane,to find the moment of inertia about any axis through O in its plane14

3．Equimomental particles for a lamina19

EXERCISES Ⅱ21

4．Moment of inertia about any axes in space22

EXERCISES Ⅲ32

CHAPTER Ⅱ:PLANE KINEMATICS33

1．Two-dimensional or plane motion33

2．Translation and rotation33

3．Composition of translations33

4．Composition of finite rotations about parallel axes34

5．Composition of angular velocities about parallel axes35

6．General displacement of a lamina in its own plane37

EXERCISES Ⅳ41

7．The instantaneous centre and the centrodes42

EXERCISES Ⅴ45

8．Acceleration of points of a rigid body having plane motion50

9．The circle of inflexions56

EXERCISES Ⅵ59

CHAPTER Ⅲ:D’ALEMBERT’S PRINCIPLE AND THE EQUATIONS OF MOTION62

1．Application of Newton’s Second Law of Motion to a rigid body62

2．D’Alembert’s Principle63

3．The equations of motion of a rigid body acted on by finite forces63

4．Motion of the centre of mass64

5．The independence of translation and rotation64

6．Final form of the two-dimensional equations of rotation for finite forces65

7．Summary of results66

8．Motion about a fixed axis under finite forces67

EXERCISES Ⅶ．(Motion about a fixed axis．)70

9．Equations of motion of a body acted on by impulsive forces80

10．Motion of the centre of mass81

11．The independence of trans-lation and rotation81

12．Final form of the rotational equation in two-dimensional impulsive motion81

13．The fundamental equa-tions of impulsive motion in two dimensions82

14．Motion about a fixed axis．Impulsive forces83

EXERCISES Ⅷ85

15．Historical Notes:(1)Huygens,(2)D’Alembert87

CHAPTER Ⅳ:TWO-DIMENSIONAL MOTION OF A RIGID BODY(FINITE FORCES)90

1．Methods of setting out equations of motion90

2．Typical examples90

3．Pure rolling94

4．The equation of energy for a rigid body in two dimensions97

5．Moments of momentum must be about a fixed axis100

EXERCISES Ⅸ101

EXERCISES Ⅹ106

7．Discontinuous friction109

EXERCISES Ⅺ110

8．Principles of conservation of linear and angular momentum118

EXERCISES Ⅻ124

CHAPTER Ⅴ:TWO-DIMENSIONAL MOTION OF A RIGID BODY(IMPULSIVE FORCES)129

1．Motion under impulsive forces129

EXERCISES ⅩⅢ135

EXERCISES ⅩⅣ145

EXERCISES ⅩⅤ150

2．Moments about the instantaneous centre151

3．Initial motions154

EXERCISES ⅩⅥ155

EXERCISES ⅩⅦ168

4．Small oscillations170

EXERCISES ⅩⅧ175

5．Tendency to break177

EXERCISES ⅩⅨ183

CHAPTER Ⅵ:DIMENSIONS AND DYNAMICAL SIMILARITY184

1．Theory of Dimensions184

2．Theory of mechanical similitude185

3．Illustrations of the foregoing principles186

4．The effect of friction187

CHAPTER Ⅶ:MOTION IN THREE DIMENSIONS190

1．Degrees of freedom190

2．Translation and rotation190

3．Com-position of small rotations about intersecting axes196

4．Composi-tion of angular velocities197

5．General motion of a rigid body198

6．The motion of the body being given as in § 5,to find the equivalent screw199

7．The instantaneous axis of rotation when one point of the body is fixed200

EXERCISES ⅩⅩ201

8．Moving axes in three dimensions202

9．Use of moving axis in solid geometry206

10．Euler’s geometrical equations209

CHAPTER Ⅷ:MOTION IN THREE DIMEN-SIONS(continued)212

1．Equations of motion of a rigid body acted upon by finite forces212

2．Euler’s equations for the motion of a rigid body about a fixed point213

3．Moments about the instantaneous axis214

4．Kinetic energy of a rigid body215

5．The momentum of a rigid body216

6．Impulsive motion218

7．Illustrative exercises for finite forces222

EXERCISES ⅩⅪ235

EXERCISES ⅩⅫ239

CHAPTER Ⅸ:LAGRANGE’S EQUATIONS242

1．Generalised Coordinates242

2．Lagrange’s equations242

EXERCISES ⅩⅩⅢ244

3．Deduction of the equation of energy from Lagrange’s equations249

4．Use of Lagrange’s equations in finding the small oscillations of a conservative system about a position of equilibrium251

EXERCISES ⅩⅩⅣ254

5．Lagrange’s equations for impulsive forces260

EXERCISES ⅩⅩⅤ262

6．Use of Lagrange’s equations in determining initial motions267

EXERCISES ⅩⅩⅥ267

7．Ignoration of coordinates．The modified Lagrangian function269

8．Case in which the number of coordinates exceeds the number of degrees of freedom．Method of undetermined multipliers271

9．Reduction to a system with fewer degrees of freedom by use of the equation of energy273

10．Some cases of integration of Lagrange’s equations276

11．Holonomous and non-holonomous systems278

EXERCISES ⅩⅩⅦ280

12．Historical Note:J．L．Lagrange281

CHAPTER Ⅹ:MOTION UNDER NO FORCES282

1．A heavy body supported at its centre of gravity and left to itself282

2．Euler’s equations282

3．Poinsot’s geometrical representa-tion of the motion283

4．Body with two principal moments at the centre of inertia equal284

EXERCISES ⅩⅩⅧ288

5．The Polhode and the Herpolhode in the general case292

6．The Herpolhode294

7．Axes of permanent rotation294

8．Integration of Euler’s equations in the general case when A,B,C are unequal294

9．The Poinsot geometrical interpretation for this case300

10．Sylvester’s theorem on the measure of the time302

EXERCISES ⅩⅩⅨ304

11．Historical note:Euler304

CHAPTER Ⅺ:MOTION OF A TOP OR GYROSTAT305

1．Definitions305

2．The phenomenon of precession and a simple dynamical explanation305

3．Gyrostatic resistance307

4．The steady motion of a top308

EXERCISES ⅩⅩⅩ309

5．The general motion of a top,whose peg O is a fixed point,deduced from the Principles of Energy and Momentum310

6．Rise and fall of the top312

EXERCISES ⅩⅩⅪ315

7．Top spinning on a smooth horizontal plane316

8．The small oscillations of a top318

EXERCISES ⅩⅩⅫ320

9．The gyroscopic compass322

EXERCISES ⅩⅩⅩⅢ328

10．Motion of a solid of revolution on a rough horizontal plane330

EXERCISES ⅩⅩⅩⅣ336

11．Motion of a cone rolling on a rough plane or on another fixed cone338

EXERCISES ⅩⅩⅩⅤ343

12．Ellipsoid spinning on a smooth horizontal plane345

EXERCISES ⅩⅩⅩⅥ350

CHAPTER Ⅻ:HAMILTON’S EQUATIONS;GEN-ERAL THEOREMS ON IMPULSES,ETC355

1．Notation for generalised coordinates355

2．Hamilton’s equations357

EXERCISES ⅩⅩⅩⅦ360

3．Application of the calculus of variations363

4．The solution of the general equations of motion368

EXERCISES ⅩⅩⅩⅧ369

5．The differential equations for S and A371

EXERCISES ⅩⅩⅩⅨ372

6．Jacobi’s Complete Integral372

EXERCISES ⅩL375

7．General theorems on impulses376

EXERCISES ⅩLⅠ379

8．Proofs of the theorems of Bertrand and Kelvin in generalised coordinates383

9．Gauss’Principle of Least Constraint385

EXERCISES ⅩLⅡ388

10．Historical Notes:(1)Jacobi,(2)Hamilton(Sir William Rowan)390

CHAPTER ⅩⅢ:THEORY OF SMALL OSCILLA-TIONS(continued from Chapter Ⅷ)392

1．Introductory392

EXERCISES ⅩLⅢ399

2．Cases of the use of principal coordinates399

EXERCISES ⅩLⅣ401

3．Stability of steady motion402

EXERCISES ⅩLⅤ407

INDEX417