meriam0i - Absolute measurements 2 119 Absolute motion 18...

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Unformatted text preview: Absolute measurements, 2, 119 Absolute motion, 18, 90, 334, 348, 370 Absolute system of units, 5, 120 Acceleration: absolute, 91, 119, 386 angular, 324, 326, 521 average, 19, 40 constant, 21, 324 Coriolis, 119, 387 cylindrical components of, 81 function of displacement, 22 function of time, 21 function of velocity, 22 graphical determination of, 19 due to gravity, 7, 8, 120 instantaneous, 19, 40 normal components of, 55 polar components of, 68 rectangular components of, 41, 81 relative to rotating axes, 386, 534 relative to translating axes, 91, 244, 370, 533 spherical components of, 82 tangential component of, 55 vector representation of, 40 from work-energy principle, 479 Acceleration-displacement diagram, 20 Acceleration-time diagram, 20 Accelerometer, 619 Action and reaction, principle of, 4 Active-force diagram, 161, 408, 463 Addition of vectors, 3, 690 Amplitude of vibration, 596 Angular acceleration, 324, 326, 521 Angular displacement, 324 Angular impulse, 209, 491 Angular momentum: applied to fluid streams, 289 conservation of, 211, 230, 276, 492 of a particle, 208, 247 relative, 247, 274 of a rigid body, 409, 490, 546 of a system, 272 units of, 208 vector representation of, 208 Angular motion: of a line, 323 vector representation of, 326, 518 Angular velocity, 324, 326, 518, 522 absolute, 348, 370 of the earth, 119, 704 vector representation of, 326, 520, 522 Apogee velocity, 234 Area moments of inertia, 655 Associative law, 690 Astronomical frame of reference, 2, 118 Axes: rotating, 383, 534 translating, 90, 244, 346, 533 Balancing in rotation, 561 Base units, 5, 120 Bodies, interconnected, 270, 413, 479, 492 Body, rigid, 3, 267, 321, 407, 517 Body centrode, 363 Body cone, 521, 577 Cajori, F., 1 Center: of curvature, 55 of mass, motion of, 269 of percussion, 428 Central-force motion, 230 Centrifugal force, 246 Centrode, body and space, 363 Centroids, table of, 705 Circular frequency, natural, 596 Circular motion, 56, 69 713 714 INDEX Index m Coefficient: ratio, 597 of friction, 703 viscous or fluid, 597 of restitution, 219, 493 Dashpot, 597 of viscous damping, 597 Degrees of freedom, 100, 123, 593 Commutative law, 690, 691 Densities, table of, 703 Complementary solution, 615 Derivative: Computer-oriented problems, 12, 114, 262, 403, 514, table of, 693 653, 684 transformation of, 385, 535 Cone, body and space, 521, 57 7 of a vector, 39, 692 Conservation: Descartes, R., 16 of energy, 179, 275, 638 Diagram: of momentum, 195, 211, 219, 230, 276, 492, 577 acceleration-displacement, 20 Conservative force, 179 acceleration-time, 20 Conservative system, 275, 462, 638 active-force, 161, 408, 463 Constant of gravitation, 6, 704 displacement-time, 20 Constrained motion, 17, 100, 123, 413, 442 force-displacement, 158 Constraint, equations of, 480 force-time, 195 Coordinates: free—body, 11, 124, 408, 463, 491, 594 cartesian, 18 kinetic, 409, 411, 416, 428, 442 choice of, 41, 90, 109, 122, 123, 658 velocity-displacement, 20 cylindrical, 81 velocity-time, 20 normal and tangential, 54 Dimensions, homogeneity of, 9 polar, 67 Direction cosines, 690 rectangular, 41, 81 Discrete or lumped-parameter model, 593 rotating, 383, 534 Displacement: spherical, 82 angular, 324 transformation of, 83, 109 in curvilinear motion, 38 translating, 90, 244, 346, 533 graphical determination of, 20 Coriolis, G., 2, 387 linear, 18 Coriolis acceleration, 119, 387 virtual, 480 Couple: Displacement meter, 618 gyroscopic, 572 Displacement-time diagram, 20 resultant, 479, 560 Distance, 38 work of, 460 Distributed-parameter system, 593 Critical frequency, 615 Distributive law, 691 Cross or vector product, 208, 326, 691 Dot or scalar product, 157, 160, 691 Curvature: Dynamical energy, conservation of, 179, 275 center of, 55 Dynamic balance in rotation, 561 radius of, 54, 696 Dynamic equilibrium, 245 Curvilinear motion: Dynamics, 1 in cylindrical coordinates, 81 in normal and tangential coordinates, 54, 140 Earth, angular velocity of, 119, 7 04 of a particle, 38, 81, 140 Earth satellites, equations of motion for, 230 in polar coordinates, 67, 140 Efficiency, 162 in rectangular coordinates, 41, 81, 140 Einstein, A., 2, 120 in spherical coordinates, 82 Elastic impact, 220 Curvilinear translation, 323, 415, 518 Elastic potential energy, 177, 462 Electric circuit analogy, 619 D’Alembert, J., 2, 245 Energy: D’Alembert’s principle, 245 conservation of, 179, 275, 638 Damped forced vibration, 616 kinetic, 160, 271, 462, 550 Damped free vibration, 597 potential, 176, 462 Damping: in satellite motion, 233 coefficient, 597 in vibration, 638 critical, 598 Equations of constraint, 480 Index Equations of motion: for fixed-axis rotation, 428 for particles, 125, 140 for plane motion, 409, 442 in polar coordinates, 140 for rectilinear and curvilinear translation, 415 for a rigid body, 408, 559 for rotation about point, 412 for a system of particles, 269 Equilibrium, dynamic, 245 Euler, L., 2 Euler’s equations, 560 Fluid damping, 597 Fluid streams, momentum equations for, 288, 289 Foot, 5 Force: centrifugal, 246 concept of, 3 conservative, 179 external, 268 inertia, 245 internal, 268 gravitational, 6, 121 resultant, 4, 125, 160, 193, 271, 409, 479, 560 units of, 5 work of, 157, 460 Force-displacement diagram, 158 Forced vibration, 613, 627 damped, 616 equation for, 614 frequency ratio for, 615 magnification factor for, 615, 616 resonant frequency of, 615 steady-state, 615, 616 undamped, 614 Force field, conservative, 179 Force-time diagram, 195 Forcing functions, 613 Formulation of problems, 10 Frame of reference, 4, 90, 244, 246, 383, 533, 534 Free-body diagram, 11, 124, 408, 463, 491, 594 Freedom, degrees of, 100, 123, 593 Free vibration: damped, 597 energy solution for, 638 equations for, 595, 597 undamped, 594 vector representation of, 596 Frequency: critical, 615 damped, 599 natural and circular, 596 Frequency ratio, 615 INDEX Friction: coefficients of, 703 work of, 463 Galileo, 1 Gradient, 180 Graphical representation, 12, 20, 158, 325, 348, 371, 596 Gravitation: constant of, 6, 704 law of, 6 Gravitational force, 6, 9, 121 Gravitational potential energy, 176, 462 Gravitational system of units, 5, 120 Gravity: acceleration due to, 7, 8, 120 International Formula for, 7 Gyration, radius of, 659 Gyroscope, 570 Gyroscopic couple, 572 Gyroscopic motion, equation of, 571 Harmonic motion, simple, 25, 595 Hertz (unit), 596 Hodograph, 41 Horsepower, 162 Huygens, C., 1 Impact, 218, 493 classical theory of, 220 direct central, 218 elastic, 220 energy loss in, 220 inelastic or plastic, 220 oblique, 220 Impulse: angular, 209, 491 linear, 194, 490 Impulse-momentum equation, 194, 209, 247 Inertia, 3, 118 area moments of, see Moments of inertia of area mass moments of, see Moments of inertia of mass principal axes of, 548, 676 products of, 547, 675 Inertia force, 245 Inertial system, 2, 90, 91, 119 Inertia tensor or matrix, 548, 676 Instantaneous axis of rotation, 362, 520 Instantaneous center, of zero velocity, 362 Integrals, table of selected, 694 Integration, numerical techniques for, 699, 701 of vectors, 693 Interconnected bodies, 270, 413, 479, 492 International Gravity Formula, 7 International System of units, 4 715 716 INDEX Index Joule (unit), 158 Kepler, J., 230 Kepler’s laws of motion, 230, 232 Kilogram, 5, 120 Kinematics, 1, 15, 321, 518 of angular motion, 324, 518 of curvilinear motion, 38 of rectilinear motion, 18 of relative motion, 18, 90, 244, 346, 370, 383, 533 of rigid bodies, 321, 517 Kinetic diagram, 409, 411, 416, 428, 442 Kinetic energl: ofa particle, 160 of plane motion, 462 of rotation, 461, 550 of space motion, 550 of a system of particles, 271, 549 of translation, 461 units of, 160 Kinetic friction, coefficient of, 703 Kinetics, 1, 17, 117, 267,407, 546 of particles, 117 of rigid bodies, in plane motion, 407, 561 in rotation, 428, 546 in space motion, 546 Kinetic system of units, 120 Lagrange, J. L., 2 Lagrange’s equations, 414 Laplace, P., 2 Law: associative, 690 commutative, 690, 691 of conservation of dynamical energy, 179, 276, 638 distributive, 691 of gravitation, 6 Laws of motion: Kepler’s, 230, 232 Newton’s, 4, 117, 244, 269 Light, speed of, 2, 120 Line, angular motion of, 323 Linear displacement, 18 Linear impulse, 194, 490 Linear momentum: applied to fluid streams, 288 conservation of, 195, 219, 276, 492 moment of, 208 of a particle, 193 relative, 247 of a rigid body, 489 of a system, 271 Logarithmic decrement, 600 Lumped-parameter or discrete model, 593 Magnification factor, 615, 616 Mass, 3, 118 steady flow of, 287 unit of, 5, 121 variable, 303 Mass center, motion of, 269 Mass flow, equations of motion for, 288, 289 Mass moments of inertia, see Moments of inertia of mass Mathematical model, 10 Mathematics, selected topics in, 687 Matrix, inertia, 548, 676 Measurements: absolute, 2, 119 relative, 90, 244, 246, 346, 383, 533 Mechanics, 1 Meter, 5 Metric units, 4, 120 Moment center, choice of, 412, 416, 442 Moment equation of motion, 209, 272, 273, 274, 275, 412, 559 Moment of linear momentum, 208 Moments of inertia of area, 655, 661, 705 Moments of inertia of mass, 408, 547, 657 choice of element of integration for, 658 for composite bodies, 661 about any prescribed axis, 676 principal axes for, 548, 676 radius of gration for, 659 table of, 707 transfer of axes for, 659 Momentum: angular, 208, 247, 272, 289, 490, 546 conservation of, 195, 211, 219, 230, 276, 492, 577 equations for mass flow, 288, 289 linear, 193, 271, 489 moment of, 208 rate of change of, 4, 193, 209, 271, 272, 490, 491, 559 vector representation of, 193, 208, 489, 490, 549 Motion: absolute, 18, 90, 334, 347, 370 angular, 323, 326, 518, 521, 522 central-force, 230 circular, 56, 69 constrained, 17, 100, 123, 413, 443 curvilinear, 38, 81, 140 in cylindrical coordinates, 81 general space, 533 graphical representation of, 12, 20, 158, 325, 348, 371, 596 gyroscopic, 570 of mass center, 269 Newton’s laws of, 4, 117, 244, 269, 489 in normal and tangential coordinates, 54 parallel-plane, 519, 561 Index INDEX 71 7 plane, 18, 38, 323, 407, 442 planetary and satellite, 230 in polar coordinates, 67 in rectangular coordinates, 41, 81 rectilinear, 18, 125 relative, 18, 90, 244, 347, 370, 383, 533 rotational, 323, 428, 518, 519 simple harmonic, 25, 595 in spherical coordinates, 82 of a system of particles, 267 Natural frequency, 596 Newton, Isaac, 1 Newton (unit), 5 Newtonian frame of reference, 246, 268, 272 Newtonian mechanics, 119, 268 Newton’s laws, 4, 117, 244, 269, 489 Newton’s method, 697 Notation for vectors, 3, 39, 690 Numerical integration, 699, 701 Nutation, 573 Oblique central impact, 229 Orbit, elliptical, 232 Osculating plane, 18 Parallel-axis theorems, for mass moments of inertia, 659 Parallel-plane motion, 519, 561 Particles, 3, 15 curvilinear motion of, 38, 81, 140 equations of motion of, 125, 140 kinematics of, 17 kinetics of, 117 motion of system of, 269 Particle vibration, 594 Particular solution, 615, 616 Path variables, 18, 54 Percussion, center of, 428 Perigee velocity, 234 Period: of orbital motion, 232, 235 of vibration, 596, 599 Phase angle, 617 Plane motion, 18, 38, 323, 407 curvilinear, 38 equations of motion for, 409, 442 general, 322, 442 kinematics of, 38, 90, 323 kinetic energy of, 462 kinetics of, 407, 561 Planetary motion: Kepler’s laws of, 230, 232 period of, 232 Poinsot, L., 2 Polar moment of inertia, 661 Position vector, 38 Potential energy, 176, 462 Potential function, 180 Pound force, 6, 120, 122 Pound mass, 6, 121, 122 Power, 161, 463 Precession: defined, 522, 570 direct and retrograde, 577 steady, 570, 572, 577 velocity of, 570 with zero moment, 577 Primary inertial system, 2, 90, 118 Principal axes of inertia, 548, 676 Principia, 1 Principle: of action and reaction, 4 of conservation of momentum, 195, 211, 219, 230, 276, 492, 577 D’Alembert’s, 245 of motion of mass center, 269 Products of inertia, 547, 675 Products of vectors, 157, 208, 691 Projectile motion, 42 Propulsion, rocket, 305 Radius: of curvature, 54, 696 of gyration, 659 Rectilinear motion of a particle, 18, 125 Rectilinear translation, 323, 415, 518 Reference frame, 4, 90, 244, 246, 383, 533, 534 Relative acceleration, rotating axes, 386, 534 translating axes, 91, 244, 370, 533 Relative angular momentum, 247, 274 Relative linear momentum, 247 Relative motion, 18, 90, 244, 347, 370, 383, 533 Relative velocity: rotating axes, 384, 534 translating axes, 91, 346 Relativity, theory of, 119, 120 Resonance, 615 Restitution, coefficient of, 219, 493 Resultant: couple, 479, 560 force, 4, 125, 160, 193, 271, 409, 479, 560 Right-hand rule, 208, 691 Rigid bodies: kinematics of, 321, 517 kinetics of, 407, 546 Rigid body, 3, 267, 321, 407, 517 Rigid-body motion, general moment equations for, 409, 412 Rigid-body Vibration, 627 718 INDEX Index W Rocket propulsion, 305 of densities, 703 Rotating axes, 383, 534 of derivatives, 693 Rotation: of mass moments of inertia, 707 equations of motion for, 412, 428 of mathematical relations, 687 finite, 520 of solar-system constants, 704 fixed-axis, 323, 325, 428, 518 of units, 5 fixed-point, 519 Tensor, inertia, 548, 676 infinitesimal, 520 Theory of relativity, 119, 120 instantaneous axis of, 362, 520 Thrust, rocket, 305 kinematics of, 323, 325, 522 Time, 3, 5, 120 kinetic energy of, 461, 550 Time derivative, transformation of, 385, 535 of a line, 323 Transfer of axes: ofa rigid body, 323, 428, 518, 519 for moments of inertia, 659 Rotational unbalance, 561 for products of inertia, 675 Transformation of derivative, 385, 535 Satellite, motion of, 230 Transient solution, 615, 616 Scalar, 3 Translating axes, 90, 244, 346, 533 Scalar or dot product, 157, 160, 691 Translation, rectilinear and curvilinear, 323, 415, 518 Second, 5 Triple scalar product, 550, 692 Series, selected expansions, 693 Triple vector product, 692 Simple harmonic motion, 25, 595 Two-body problem, 234 SI units, 4, 6 Slug, 5, 120, 122 Unconstrained motion, 17, 123, 442 Solar system constants, 704 Units, 4, 6, 120 Solution, method of, 11 kinetic system of, 120 Space, 2 Unit vectors, 41, 54, 67, 82, 690 Space centrode, 363 derivative of, 54, 67, 383, 534 Space cone, 521, 577 US customary units, 4, 6, 120 Space motion, general, 533 Speed, 39 Variable mass, force equation of, 304 Spin axis, 570 Vectors, 3, 690 Spin velocity, 570 addition of, 3, 690 Spring: cross or vector product of, 208, 326, 691 constant or stiffness of, 158, 594 derivative of, 39, 692 potential energy of, 177 dot or scalar product of, 157, 160, 691 work done by, 158 integration of, 693 Standard conditions, 8, 121 notation for, 3, 39, 690 Static friction, coefficient of, 703 subtraction of, 690 Steady mass flow, force and moment equations for, 288, triple scalar product of, 692 289 triple vector product of, 692 Steady-state vibration, 615, 616 unit, 41, 54, 67, 82, 690 Subtraction of vectors, 690 Velocity: System: absolute, 91 conservative, 275, 462, 638 angular, 324, 326, 518, 522 of interconnected bodies, 270, 413, 479, 492 average, 18, 39 of particles: cylindrical components of, 82 angular momentum of, 272 defined, 18, 39 equation of motion for, 269 graphical determination of, 20 kinetic energy of, 271, 549 instantaneous, 18, 39 linear momentum of, 271 instantaneous axis or center of, 362, 519 of units, 4, 6, 120 in planetary motion, 234 polar components of, 68 Table: rectangular components of, 41, 81 of area moments of inertia, 705 relative to rotating axes, 384, 534 of centroids, 705 relative to translating axes, 91, 346 of coefficients of friction, 7 03 spherical components of, 83 m Index . .. INDEX. 719 tangential component of, 54 vector representation of, 39 Velocity-displacement diagram, 20 Velocity—time diagram, 20 Vibration: amplitude of, 596 damped, 597 energy in, 638 forced, 613, 627 free, 594 frequency of, 596, 599 over- and underdamped, 598, 599 period of, 596, 599 reduction of, 617 simple harmonic, 595 steady-state, 615, 616 transient, 615 work-energy solution for, 638 Virtual displacement, 480 Virtual work, 157, 480 Viscous damping coefficient, 597 Watt, 162 Weight, 6, 9, 121 Work, 157, 460 of a couple, 460 an exact differential, 180 of a force, 157, 460 of friction, 463 graphical representation of, 158 on a spring, 158 units of, 158 virtual, 157 , 480 Work-energy equation, 160, 161, 178, 246, 270, 462, 561 ...
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