Dynamics of Mechanical Systems 2009 Part 16 ppt

Finally, let the orientation of the bodies be defined by the relative angles β1, β2, and β3.. If O1 is a fixed point in a reference frame R, develop the kinematics of this system using β

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732 Dynamics of Mechanical Systems

P20.4.2: See Problem P20.4.1 Construct a table analogous to Table 20.4.1 for the hand model, showing the body numbers, the body names, the generalized speeds, and the variable names.

Section 20.5 Kinematics: Velocities and Accelerations

P20.5.1: Review the analysis of Section 20.5 relating to the example for the kinematics of the right hand Develop an analogous analysis for the kinematics of the right foot.

P20.5.2: See Problems P20.3.1, P20.3.2, P20.4.1, and P20.4.2 Consider again the model of the hand shown in Figure 20.2.3 and again in Figure P20.4.1 Develop the kinematics of the hand model analogous to the gross-motion development in Section 20.5.

Section 20.6 Kinetics: Active Forces

P20.6.1: Consider a simple planar model of the arm as shown in Figure P20.6.1 It consists

of three bodies representing the upper arm, the forearm, and the hand, labeled and numbered as 1, 2, and 3, respectively, as shown Let the shoulder, elbow, and wrist joints

be O1, O2, and O3, respectively Let the lengths of the bodies be 1 , 2 , and 3 ; let the

weights of the bodies be w1, w2, and w3; and let the mass centers each be one third of the

body length distal from the upper joint Let the hand support a mass with weight W at

the finger tips as shown Finally, let the orientation of the bodies be defined by the relative angles β1, β2, and β3 If O1 is a fixed point in a reference frame R, develop the kinematics

of this system using β1, β2, and β3 as generalized coordinates.

P20.6.2: See Problem P20.6.1 Determine the contribution to the generalized active forces

of the weights of the bodies.

P20.6.3: See Problem P20.6.1 Determine the contribution to the generalized active forces

of the weight of the mass at the finger tips.

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Application with Biosystems, Human Body Dynamics 733

P20.6.4: See Problem P20.6.1 and Figure P20.6.1 Let the movement of the system be defined

by the absolute angles θ1, θ2, and θ3 as in Figure P20.6.4 Repeat Problems P20.6.1, P20.6.2, and P20.6.4 using θ1, θ2, and θ3 as generalized coordinates.

Section 20.7 Kinetics: Muscle and Joint Forces

P20.7.1: See Problems P20.6.1 to P20.6.4 Let the forces exerted between the bodies due to the muscles be equivalent to and represented by single forces passing through the joint centers together with couples For example, at a typical joint — say, the elbow (joint 2) —

let the muscle forces exerted by B1 (the upper arm) on B2 (the forearm) be represented by

a single force F1/2 passing through O2 together with a couple with torque M1/2 Similarly, let the muscle forces exerted by the forearm on the upper arm be represented by a single

force F2/1 passing through O2 together with a couple with torque M2/1 Let these forces and moments be negative to one another; that is,

Determine the contribution of the muscle forces to the generalized active forces using both the relative angles β1, β2, and β3 and the absolute angles θ1, θ2, and θ3 as generalized coordinates.

Section 20.8 Kinetics: Inertia Forces

P20.8.1: Verify again the validity of Eq (20.8.11).

P20.8.2: Consider again the model of the arm of Problem P20.6.1 shown in Figure P20.6.1 and as shown again in Figure P20.8.2 Let the bodies be modeled by frustrums of cones

and let mutually perpendicular unit vectors nki (k = 1, 2, 3; i = 1, 2, 3) be fixed in the bodies

with the nk2 being along the axes of the bodies (and cones) and the nk3 being normal to

the plane of motion Let the nki be parallel to principal inertia axes of the bodies, and let the corresponding moments and products of inertia then be:

Determine the generalized inertia forces on the system.

θ θ

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P20.8.3: See Problems P20.6.4 and P20.8.2 Let the movement of the arm model be defined

by the absolute angles θ1, θ2, and θ3 as shown in Figure P20.6.4 and as shown again in Figure P20.8.3 Find the generalized inertia forces corresponding to θ1, θ2, and θ3 using the inertia data of Problem P20.8.2.

Section 20.9 Dynamics, Equations of Motion

P20.9.1: See Problems P20.6.1, P20.6.2, P20.6.3, P20.7.1, and P20.8.2 Using the results obtained in these problems, determine the governing equations of motion for the arm model using the relative orientation angles β1, β2, and β3 as generalized coordinates.

P20.9.2: See Problems P20.6.4, P20.7.1, and P20.8.3 Using the results obtained in these problems, determine the governing equations of motion for the arm model using the absolute orientation angles θ1, θ2, and θ3 as generalized coordinates.

Section 20.10 Constrained Motion

P20.10.1: See Problems P20.9.1 and P20.9.2 Let the finger tips (extremity of body 3) have

a desired motion — say, movement on a circle at constant speed v (let the circle be in the vertical plane with center (xO, yO) and with radius a) Determine the governing

equations of motion for the model with this constraint Use both relative and absolute orientation angles.

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Appendix I*

Centroid and Mass Center Location for Commonly Shaped Bodies with Uniform Mass Distribution

I Curves, wires, thin rods

1 Straight line, rod:

2 Circular arc, circular rod:

3 Semicircular arc, semicircular rod:

* Reprinted from Huston, R L., and Liu, C Q., Formulas for Dynamic Analysis, pp 303–310, by courtesy ofMarcel Dekker, New York, 2001

0593 App I_fm Page 735 Tuesday, May 7, 2002 9:21 AM

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4 Circle, hoop:

II Surfaces, thin plates, shells

1 Triangle, triangular plate:

2 Rectangle, rectangular plate:

x = (a + b)/3

y = c/3

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Appendix I 737

3 Circular sector, circular section plate:

4 Semicircle, semicircular plate:

5 Circle, circular plate:

r- = (2r/3) (sin θ )/ θ

r- = 4r/ Π

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6 Circular segment, circular segment plate:

7 Cylinder, cylindrical shell:

8 Semicylinder, semicylindrical shell:

r- = (2r/3) (sin3θ )/( θ – sin θ cos θ )

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Appendix I 739

9 Sphere, spherical shell:

10 Hemisphere, hemispherical shell:

11 Cone, conical shell:

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12 Half cone, half-conical shell:

III Solids, bodies

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Appendix II*

Inertia Properties (Moments and Products of Inertia) for Commonly Shaped Bodies with Uniform Mass Distribution

I Curves, wires, thin rods

1 Straight line, rod:

2 Circular arc, circular rod:

* Reprinted from Huston, R L., and Liu, C Q., Formulas for Dynamic Analysis, pp 303–310, by courtesy ofMarcel Dekker, New York, 2001

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3 Semicircular arc, semicircular rod:

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Appendix II 745

II Surfaces, thin plates, shells

1 Triangle, triangular plate:

2 Rectangle, rectangular plate:

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3 Circular sector, circular section plate:

4 Semicircle, semicircular plate:

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Appendix II 747

5 Circle, circular plate:

6 Circular segment, circular segment plate:

I11 = mr2/4

r- = (2r/3) (sin3θ )/( θ – sin θ cos θ )

I11 = mr2(9 θ2 + 9sin2θ cos2θ – 36sin4θ cos4θ – 18 θ sin θ cos θ + 36sin3θ cos θ – 8sin6θ )/18( θ – sin θ cos θ )2

I = mr2[1 – 2sin3θ cos θ /3( θ – sin θ cos θ )]/4

0593 AppII_fm Page 747 Tuesday, May 7, 2002 9:21 AM

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7 Cylinder, cylindrical shell:

8 Semicylinder, semicylindrical shell:

9 Sphere, spherical shell:

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Appendix II 749

10 Hemisphere, hemispherical shell:

11 Cone, conical shell:

12 Half cone, half-conical shell:

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III Solids, bodies

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6 Sphere:

7 Hemisphere:

I11 = 83mr2/320

I22 = 2mr2/5

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Addition theorem for angular velocity, 90Adjoining bodies, 609

Adjoint, 40Amplitude, 440Angle, 9Angle between two vectors, 23, 28Angle of action, 584, 599

Angle of approach, 584, 599Angle of contact, 584, 599Angular acceleration, 83, 93, 640Angular impulse, 280

Angular momentum, 282Angular speed, 83Angular velocity, 9, 83, 85, 635, 667Anticyclic indices, 30

Antisymmetric matrix, 39Applied forces, 177, 677Articulation angles, 662, 688Associative law, 18, 40Axial pitch, 599

B

Backlash, 582, 599Balancing, 513Ball-and-socket joint, 4Base circle, 578, 599Basic rack, 591Bevel gears, 596, 599Biceps, 716, 717Biosystem, 701Boltzmann-Hamel equations, 242, 243Branching body, 608

Buridan, John, 241

C

Cam-pair, 539Cam profiles, 544Cams, 5, 15, 539Cam systems, 3Cartesian coordinate system, 6, 8Center of percussion, 298Centroid, 735

Chord vector, 60Circular frequency, 440Circular pitch, 582, 599Clearance, 583, 599Closed loops, 606Coefficient of restitution, 303Column matrix, 39

Commutative law, 16, 23Complete elliptic integral, 462Components of vectors, 16, 19Compression stroke, 529Configuration graphs, 79, 614, 664Conformable matrix, 39

Conjugate action, 575, 600Connection configuration, 605Conservation of momentum, 294, 301Constrained motion, 722

Constraint equations, 125, 126, 684Constraint forces, 684

Constraint matrix, 683Constraints, 3, 125, 353Contact forces, 180Contact ratio, 585, 600Coordinates, 6, 125, 353, 628Couples, 170

Cross product, 29Cyclic indices, 30Cycloidal rise function, 563Cylindrical coordinate system, 7

D

d’Alembert’s principle, 185, 242, 243, 262, 279, 290Damped linear oscillator equation, 442

Dedendum, 581, 600Degrees of freedom, 3, 125, 353, 628, 661Derivative of transformation matrices, 612, 668

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Determinant, 40Dextral rotation, 43Dextral vectors, 30Diagonal matrix, 39Diametral pitch, 583Dirac’s delta function, 554Directed line segment, 5Direct impact, 306Direction cosines, 21Direction of a vector, 5Distributive law, 17, 25, 32, 40Dot product, 23

Double-rod pendulum, 258, 381, 396, 418, 426Driver, 3, 539

Driver gear, 574Dwell, 542Dyad, 203Dyadic, 203Dynamic balancing, 514Dynamics, 1

Dynamic unbalance, 516

E

Earth rotation effect, 89Eigen unit vector, 209Eigenvalue of inertia, 209Elastic collision, 304Elements of a matrix, 39Elliptic integral 460End effector, 661, 674, 692, 698Ending body, 607-608Energy, 9, 10Equality of vectors, 15Equivalent force systems, 170Euler angles, 82

Euler parameters, 613, 707-709Euler torque, 230

Exhaust stroke, 529Extremity body, 607

F

Fillet radius, 583Finite segment model, 142Firing order of internal combustion engines, 530First integral, 459

First moments, 182Fixed stars, 244Fixed vector, 15Follower, 3, 539Follower gear, 574Force, 2, 5, 9, 163Forced vibration, 446, 449Forcing function, 442Four-bar linkage, 136Four-stroke engines, balancing of, 528

Free-body diagram, 245-246Free index, 38

Free vector, 15Frequency, 440

G

Gear drive, 592Gear glossary, 599-601Gears, 539, 573Gear systems, 3Gear train, 592Generalized active force, 363Generalized applied force, 363Generalized coordinates, 242, 353Generalized forces, 360

Generalized inertia forces, 360, 377Generalized passive force, 377Generalized speeds, 628General plane motion, 129, 130Gibbs equations, 243

Gibbs function, 243Gravity forces, 177Gripper, 661Gross-motion model, 702

H

Hamilton’s principle, 242Helical gears, 595Helix joint, 4Holonomic constraint, 357Human body dynamics, 702Human body model, 704Hypoid gears, 597

I

Identity matrix, 39Imbalance, 513Impact, 303Impulse, 279Impulse-momentum, 242Incomplete elliptic integral, 462Inertia, 1, 2, 199, 241

Inertia coefficients, 651Inertia ellipsoid, 228Inertia forces, 177, 184, 243, 244, 248, 648, 680, 719Inertial reference frame, 2, 185, 244

Inertia properties (common shapes), 743Inertia torque, 228, 249-250

Infinitesimal stability, 479Inside unit vector, 80Instant center of zero acceleration, 150Instant center of zero velocity, 133, 147

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Kronecker’s delta function, 24, 38, 42

Krylov and Bogoliuboff method, 463

Lanchester balancing mechanism, 525

Law of action and reaction, 717

Law of conjugate action, 576

Linear impulse, 279

Linear momentum, 280

Linear oscillator equation, 246, 439

Linear rise function, 551

Loop closure equation, 137

Lower body arrays, 605

Matrix inverse, 40Maximum moment of inertia, 223Mechanism, 3

Mesh, 575Minimum moment of inertia, 223Minimum moments, 175Minor, 40

Mobile robot, 661Modes of vibration, 455Module, 583

Moment, 10, 163Moment of inertia, 200, 743Moment of momentum, 282Momentum, 280

Motion on a circle, 66Motion on a plane, 68Multi-arm robot, 662Multibody system, 258, 605Muscle forces, 716

N

Natural modes of vibration, 456Newton, I., 241

Newton’s laws, 2, 241, 285, 287Nonholonomic constraint, 357Nonlinear vibrations, 458Normal pitch, 585N-rod pendulum, 260, 433

O

Oblique impact, 306Open-chain system, 606Open-tree system, 606Orientation angles, 79Orientation of a vector, 5, 15Orientation of bodies, 77, 84Orthogonal complement arrays, 687, 689Orthogonal matrix, 40

Orthogonal transformation, 42, 77Outside unit vector, 80

P

Parabolic rise function, 557Parallel axis theorem, 206, 207Parallelogram law, 16Partial angular velocity, 359, 667Partial angular velocity array, 639Partial velocity, 359

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Partial velocity arrays, 651

Principal axis of inertia, 208, 215

Principal moment of inertia, 208, 209

Principal unit vector, 209

Principle of angular momentum, 289

Principle of linear momentum, 285

Reciprocating machines, balancing of, 520

Reduction of a force system, 171

S

Scalar, 5, 9, 15Scalar product, 23, 27Scalar triple product, 33Screw joint, 4

Second-moment vectors, 199Sense of a vector, 5, 15Simple angular velocity, 83, 87, 665Simple chains, 4

Simple pendulum, 245, 324, 333, 365, 379, 395, 417,

424, 445, 459, 479Singularity functions, 553Singular matrix, 39Sinistral vectors, 30Sinusoidal rise function, 560Sliding joint, 4

Sliding vector, 15Solver, 688Space, 2, 3Speed, 63, 64Spherical coordinate system, 7Spherical joint, 4

Spiral angle, 597Spiral bevel gears, 597Spring forces, 178Spur gear, 4, 581Stability, 479Static balancing, 513Statics, 1

Stress, 10Stroke, 528Substitution symbol, 38Summation convention, 37Sun gear, 593

Symmetric matrix, 39System of forces, 165

T

Tensor, 204Thrust bearing, 110Time, 2

Torque, 10, 170Transformation matrices, 78, 609, 663, 704Transformation matrix derivatives, 612, 668Translation, 129

Transmission, 540, 573Transpose of a matrix, 39

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Weight, 177Wheel rolling over a step, 302, 341Whole-body model, 701, 703Work, 321

Work-energy, 242Work-energy principles, 329Worm gears, 597

Worm wheel, 598Wrench, 173

Z

Zero force systems, 170Zero vector, 6, 15

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