Advanced Engineering Dynamics 2010 Part 16 pot

294 Appendix 3 which are the scale factors which relate to length, area and volume respectively. In these terms the general expressions are div(F) = V-'(VT{ [x](F)} curl(F) = [AII-'[VI~{[[ZI(F)J grad(f) = [~I-'(v)(F) where For spherical co-ordinates corresponding to Fig. A3.5(a) 1 a2 a rcosa dr ae div (F) = (-(r cos0l;;) + -(r 4) + 1 af 1 af -q + - -e0 af gradcf) = -er + - ar rcoso ae r a0 For spherical co-ordinates corresponding to Fig. A3.5(b) 1 a4 + 1 a(:Q 1 a(sinO&) 7 ar rsine ae r sine 80 div(F) = - + - a r sine (a ae a0 curl (F) = - -(sine F,) - -(&))er gradcf) = -e, af + - 1 e, af + - 1 af ar r ae r sine In cylindrical co-ordinates Appendix3 295 la la a r ar r% az div (F) = (rE) + (8) + -(Q grad0 = -e, a! + - 1 q a! + -q a! ar r a0 az Strain In cylindrical co-ordinates For spherical co-ordinates corresponding to Fig. A3.5(b) Stress For any orthogonal co-ordinate system the stresses in an isotropic linear solid are related to the strains by = + 9J + ekk) + 2pe44 + 5J + 'kk) + 2pq, O), = 296 Appendix 3 and og = Peg (T. Jk = @ 0, = peki where h and p are the Lame constants. k = z. In spherical co-ordinates i = r, j = 8 and k = 0. In Cartesian co-ordinates i = x,j = y and k = z. For cylindrical co-ordinates i = r,j = 8 and Bibliography Arnold and Maunder, L., 1961: Gyrodynamics. Academic Press. Clough, R.W. and Penzien, J., 1975: Dynamics of structures. McGraw-Hill. Dugas, R., 1986: A history of mechanics. Dover. Einstein, A., 1967: The meaning of relativity, 6th edn. Chapman and Hall (or Princeton Uni- Ellis, J.R., 1969: Vehicle dynamics. London Business Books. Fu, K.S., Gonzalez, R.C. and Lee, C.S.G., 1987: Robotics. McGraw-Hill. Ginsberg, J.H. 1995: Advanced engineering dynamics, 2nd edn. Cambridge University Goldsmith, W. 1960: Impact. Edward Arnold. Goldstein, H. 1980: Classical mechanics, 2nd edn. Addison-Wesley. Hamson, H.R. and Nettleton, T., 1994: Principles of engineering mechanics, 2nd edn. Houghton, E.L. and Carruthers, N.B. 1982: Aerodynamics, 3rd edn. Edward Arnold. Hunter, S.C. 1983: Mechanics ofcontinuous media. Simon and Schuster. Johnson, W. 1972: Impact strength of materials. Edward Arnold. Kolsky, H. 1953: Stress waves in solids. Oxford University Press. Longair, M.S. 1984: Theoretical concepts in physics. Cambridge University Press. Mot -oy, D. and Harris, M. 1986: Robotics: An introduction. Open University Press. Meii-ovitch, L. 1967: Analytical methods in vibrations. McGraw-Hill. Meirovitch, L. 1970: Methods of analytical dynamics. McGraw-Hill. Rao, S.11 . 1986: Mechanical vibrations. Addison-Wesley. Redwood, M., 1960: Mechanical waveguides. Pergamon. Routh, E.J., 1891/1892: Dynamics of a system of rigid bodies, Part lrPart 2. Macmillan. Snowdon, 1964: Shock and vibration in damped mechanical systems. Wiley. Symon, K.R., 197 1 : Mechanics. 3rd edn. Addison-Wesley. Thomson, W.T. 1986: Introduction to space dynamics. Dover. Udwadia, F.E., and Kalaba, R.E. 1996: Analytical dynamics, a new approach. Cambridge versity Press, 5th edn, 1955). Press. Edward Arnold. University Press. Index Acceleration 7, 21 8 Aerodynamic mean chord 113 Aircraft Stability 271 Angular velocity 58 Apsides 93 Archimedes 1 Aristotle 1 Aspect Ratio I12 Bandpass Filter 163 Bending Waves 155 Bertrand 93 Binet Diagram 73 Car, Stability 270 Centre of Mass 15 Characteristic Impedance 130 Chasle’s theorem 56 Co-ordinates 7 Cartesian 7 Curvilinear 289 Cylindrical 8 Path 9 Spherical 8 Coefficient of Restitution 136 Collision 4 Conic sections 94 Conservation Laws 3 1 of Momentum 250, 17 Constraints 28 I Coriolis’s Theorem 14 Cross Mamx 274 Curl 290 Cyclic Co-ordinates 3 1 D’ Aiembert’s Principle 19 Degeneracy 279 Denavit Hartenberg Representation 208 Dispersion 125, 149 Diagram 151 Dissipation function 27 Divergence 290 Doppler Effect Light 244 Sound 242 Drag I 12 Dyad 212 Dyadic 273 Eigenvalue 279,62 Eigenvector 279.62 Einstein A 235 Einstein’s Summation Convention 272 Elastic Modulii 177 End Effector 185 Roll Pitch and Yaw 196 in Wave 132 Equivalence - Principle of 5 Euler’s Angles 75 Transverse Waves in Beams I57 for Rigid Body Rotation 64 theorem of Rotation 56 Eulerian Co-ordinates 129 Evanescent Waves 162 Event 236 Force I, 2, 5 Central 90 Relativistic 252 Four Velocity 248 Frame of Reference, Rotating 35 Galilean Transformation 238 Galilea 2 Gauss’s Principle 283 300 Index Generalized Coordinates 23 Momentum 32 Gibbs Appell Equations 284 Gradient 292 Gravitational Potential 85 Gravity 1 Universal Constant 3 Group Velocity 15 I Gyroscope 26 1 Gyroscopic behaviour 83 Hamilton’s Equation 33 Principle 47 Hamiltonian 33 Helical Spring 168 Herpolhode 71 Hertz Theory of Contact 139 Holonomic constraints 24 Homogeneous Co-ordinates 205 Hopkinson Bar 144 Relativistic 258 Ignorable Co-ordinates 3 1 Impact 133 Impulse 12 Inverse Kinematic Problem 2 14 Relativistic 254 Jet Damping 105 Kepler 1 Kepler’s 1 st and 3rd Laws 99 2nd Law 97 Kinetic Energy 13 Compementary 48 of a Rigid Body 65 Kronecker Delta 277 Lagrange’s Equations 2 1 from Hamilton’s Principle 51 Impulsive forces 43 Moving Co-ordinates 39 Relativistic 258 Rotating Frame 35 in Robotics 223 Lagrange’s Undetermined Multipliers 41 Lagrangian 21 Lagrangian Electro magnetic 37 Coordinates 129 Lame Constants I77 Lateral Force Coeff., Tyres 1 I8 Lift Coefficient 1 I 1 Lorenz Factor 239 Transformation 239 Manipulator 2 10 Mass 3 active and passive 3 Invarient 25 1 Reduced 89 Maxwell 235 Moment of Force 12 Inertia 61 Ellipsoid 67 Principal 63 Momentum 12 Momentum 1 I Moore Penrose Inverse 286 Neutral Steer Point 122 Newton’s Laws 2 Third Law, weak 18 Non-Holonomic constraints 24 systems 4 1 Oblateness of Earth 100 Orbits, stability of 91 Orthogonality 279 Oversteer 124 Periodic Structures 161 Phase Velocity I5 1 Phugoid Oscillation I 17 Pitch 110 Pitching Moment Coefficient 112 Poinsot Ellipsoid 69 Polhode 71 Potential Pseudo 91 Energy 13 Power 6 Precession 71 Forced 80 Principle of Equivalence 259, 5 Pulse Peak Velocity 153 Rayleigh Waves 189 Rear wheel steer 120 Reflection, At Plane Surface 186 One dimensional Wave 130 Revolute Robot Arm 185 Rheonomous 24 Rigid Body, Torque free motion 67 Rise time 141 Robot Co-ordinates 194 Minimover 267 Index 301 Puma 269 Stanford Type 266,268 Discussion Example 227 Data Sheet 233 Robotics Rocket 103,271 Roll 110 Rotation 55 about Arbitrary Axis 202 about Body Axes 204 Finite 200 Matrix 201 Routh-Hurwitz 116 Satellite 93 Scleronomous 24,30 Serret-Frenet Formulae 11 Simultanaeity 241 Snell’s Law 187 Space 2 Spacecraft, Lunar Mission 261 Stability De Spinning 107 Aircraft 109 of rotating flexible body 73 Static Margin Aircraft 115 Cars 121 Strain 172,295 Plane 184 Shear 175 Tensile 175 Waves 125 Stress 176, 295 Symmetrical Body - Top Gyroscope 76 Tensor Alternating 274 Rank 272 Diagonalization 278 Tides, effect of 74 Time 2 Proper 240 Dilation 240 Timoshenko Equation - Transverse Waves in Beams 159 Top, Sleeping 79 Transformation Matrix (4x4) 206 Twin Paradox 249 Tycho Brahe 1 Understeer 124 Vectors, Axial and Polar 277 Velocity 6 angular 58 Relativistic 246 Virtual Work 18,28 1 Wave Equation 128 in String 52 Torsional 263 Speed 128 Wavenumber 149 Waves Dilatatational 183 Equivoluminal 183 Irrotational 183 One dimensional 125 Potential functions 18 I Reflection One Dimensional 265 Seismic 183 Shear 183 Three Dimensional 179 Transverse in Beam with Tension 265 Work 5 Yaw 110 . J.R., 1969: Vehicle dynamics. London Business Books. Fu, K.S., Gonzalez, R.C. and Lee, C.S.G., 1987: Robotics. McGraw-Hill. Ginsberg, J.H. 1995: Advanced engineering dynamics, 2nd edn r,j = 8 and Bibliography Arnold and Maunder, L., 1961: Gyrodynamics. Academic Press. Clough, R.W. and Penzien, J., 1975: Dynamics of structures. McGraw-Hill. Dugas, R., 1986: A history. Hamson, H.R. and Nettleton, T., 1994: Principles of engineering mechanics, 2nd edn. Houghton, E.L. and Carruthers, N.B. 1982: Aerodynamics, 3rd edn. Edward Arnold. Hunter, S.C. 1983: