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 O 1 , O 2 , and O 3 , respectively. Let the lengths of the bodies be ᐉ 1 , ᐉ 2 , and ᐉ 3 ; let the weights of the bodies be w 1 , w 2 , and w 3 ; 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 β 1 , β 2 , and β 3 . If O 1 is a fixed point in a reference frame R, develop the kinematics of this system using β 1 , β 2 , and β 3 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. FIGURE P20.4.1 A model of the human hand. FIGURE P20.6.1 A model of the arm. n W O 1 2 3 β β β O O n 1 1 2 2 3 3 O1 O2 0593_C20_fm Page 732 Tuesday, May 7, 2002 9:14 AM 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 θ 1 , θ 2 , and θ 3 as in Figure P20.6.4. Repeat Problems P20.6.1, P20.6.2, and P20.6.4 using θ 1 , θ 2 , and θ 3 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 B 1 (the upper arm) on B 2 (the forearm) be represented by a single force F 1/2 passing through O 2 together with a couple with torque M 1/2 . Similarly, let the muscle forces exerted by the forearm on the upper arm be represented by a single force F 2/1 passing through O 2 together with a couple with torque M 2/1 . 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 β 1 , β 2 , and β 3 and the absolute angles θ 1 , θ 2 , and θ 3 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 n ki (k = 1, 2, 3; i = 1, 2, 3) be fixed in the bodies with the n k2 being along the axes of the bodies (and cones) and the n k3 being normal to the plane of motion. Let the n ki 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. FIGURE P20.6.4 A model of the arm with absolute orientation angles. n W O 1 2 3 O O n 1 1 2 2 3 3 O1 O2 θ θ θ FF MM 12 21 12 21 =− =−and IIIII III kk k k k kkk 11 33 3 22 2 12 13 23 0 == = === () () , 0593_C20_fm Page 733 Tuesday, May 7, 2002 9:14 AM 734 Dynamics of Mechanical Systems 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 θ 1 , θ 2 , and θ 3 as shown in Figure P20.6.4 and as shown again in Figure P20.8.3. Find the generalized inertia forces corresponding to θ 1 , θ 2 , and θ 3 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 β 1 , β 2 , and β 3 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 θ 1 , θ 2 , and θ 3 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 (x O , y O) and with radius a). Determine the governing equations of motion for the model with this constraint. Use both relative and absolute orientation angles. FIGURE P20.8.2 A model of the arm. FIGURE P20.8.3 A model of the arm with absolute orientation angles. 0593_C20_fm Page 734 Tuesday, May 7, 2002 9:14 AM 735 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 of Marcel Dekker, New York, 2001. 0593__App I_fm Page 735 Tuesday, May 7, 2002 9:21 AM 736 Dynamics of Mechanical Systems 4. Circle, hoop: II. Surfaces, thin plates, shells 1. Triangle, triangular plate: 2. Rectangle, rectangular plate: x = (a + b)/3 y = c/3 0593__App I_fm Page 736 Tuesday, May 7, 2002 9:21 AM 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 0593__App I_fm Page 737 Tuesday, May 7, 2002 9:21 AM 738 Dynamics of Mechanical Systems 6. Circular segment, circular segment plate: 7. Cylinder, cylindrical shell: 8. Semicylinder, semicylindrical shell: r - = (2r/3) (sin 3 θ )/( θ – sin θ cos θ ) 0593__App I_fm Page 738 Tuesday, May 7, 2002 9:21 AM Appendix I 739 9. Sphere, spherical shell: 10. Hemisphere, hemispherical shell: 11. Cone, conical shell: 0593__App I_fm Page 739 Tuesday, May 7, 2002 9:21 AM 740 Dynamics of Mechanical Systems 12.Half cone, half-conical shell: III Solids, bodies 1. Parallelepiped, block: 2. Cylinder: 3. Half cylinder: 0593__App I_fm Page 740 Tuesday, May 7, 2002 9:21 AM Appendix I 741 4. Cone: 5. Half cone: 6. Sphere: 7. Hemisphere: 0593__App I_fm Page 741 Tuesday, May 7, 2002 9:21 AM [...]... 578, 584 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 Product of inertia, 200, 743 Projectile motion, 251 Projection of a vector, 24, 28, 31 Pure rolling, 107 Pythagorean theorem, 7 R Radius of gyration, 202 Reciprocating machines, balancing of, 520 Reduction of a force system, 171... Ending body, 607-608 Energy, 9, 10 Equality of vectors, 15 Equivalent force systems, 170 Euler angles, 82 Euler parameters, 613, 707-709 Euler torque, 230 Exhaust stroke, 529 Extremity body, 607 Dynamics of Mechanical Systems Free-body diagram, 245-246 Free index, 38 Free vector, 15 Frequency, 440 G Gear drive, 592 Gear glossary, 599-601 Gears, 539, 573 Gear systems, 3 Gear train, 592 Generalized active... orientation angles, 137, 142 Relative velocity, 61, 97 Resultant, 16, 165 Right-hand rule, 29 Rigid body, 3 Ring gear, 594 Rise of cam follower, 543 Robot, 663 Robot arm, 663 Rod pendulum, 255, 380, 396, 418, 425 Rolling, 106 Rolling circular disk, 267 Rolling disk, 107, 357, 385, 399, 421, 488 Root circle, 581 Dynamics of Mechanical Systems Rotating pinned rod, 263 Rotating unit vectors, 63 Rotation,... Minimum moment of inertia, 223 Minimum moments, 175 Minor, 40 Mobile robot, 661 Modes of vibration, 455 Module, 583 Moment, 10, 163 Moment of inertia, 200, 743 Moment of momentum, 282 Momentum, 280 Motion on a circle, 66 Motion on a plane, 68 Multi-arm robot, 662 Multibody system, 258, 605 Muscle forces, 716 N Natural modes of vibration, 456 Newton, I., 241 Newton’s laws, 2, 241, 285, 287 Nonholonomic constraint,... angles, 79 Orientation of a vector, 5, 15 Orientation of bodies, 77, 84 Orthogonal complement arrays, 687, 689 Orthogonal matrix, 40 Orthogonal transformation, 42, 77 Outside unit vector, 80 P M Machine, 3 Magnitude of a vector, 5, 15, 27 Mass, 2, 10, 241 Mass center, 177 Parabolic rise function, 557 Parallel axis theorem, 206, 207 Parallelogram law, 16 Partial angular velocity, 359, 667 Partial angular velocity... 539 Cam profiles, 544 Cams, 5, 15, 539 Cam systems, 3 Cartesian coordinate system, 6, 8 Center of percussion, 298 Centroid, 735 Chord vector, 60 Circular frequency, 440 Circular pitch, 582, 599 Clearance, 583, 599 Closed loops, 606 Coefficient of restitution, 303 Column matrix, 39 Commutative law, 16, 23 Complete elliptic integral, 462 Components of vectors, 16, 19 Compression stroke, 529 Configuration... Kinematics, 1, 2, 57, 241 Kinetic energy, 327 Kinetics, 163 , 241 Kronecker’s delta function, 24, 38, 42 Krylov and Bogoliuboff method, 463 L Lagrange multiplier, 224 Lagrange’s equations, 242, 262, 423, 435 Lagrange’s form of d’Alembert’s principle, 243, 416 Lagrangian, 242, 424 Lanchester balancing mechanism, 525 Law of action and reaction, 717 Law of conjugate action, 576 Linear impulse, 279 Linear momentum,... Direction cosines, 21 Direction of a vector, 5 Distributive law, 17, 25, 32, 40 Dot product, 23 Double-rod pendulum, 258, 381, 396, 418, 426 Driver, 3, 539 Driver gear, 574 Dwell, 542 Dyad, 203 Dyadic, 203 Dynamic balancing, 514 Dynamics, 1 Dynamic unbalance, 516 E Earth rotation effect, 89 Eigen unit vector, 209 Eigenvalue of inertia, 209 Elastic collision, 304 Elements of a matrix, 39 Elliptic integral... shell: I11 = mr2/2 I22 = m(r2/2 + h2/9)/2 12 Half cone, half-conical shell: I11 = mr2(1 – 8/9Π2)/2 I22 = (m/36) (9r2 + 2h2) I33 = mh2/18 + mr2(1 – 16/ 9Π2)/4 I12 = –mrh/9Π 749 0593 AppII_fm Page 750 Tuesday, May 7, 2002 9:21 AM 750 Dynamics of Mechanical Systems III Solids, bodies 1 Parallelepiped, block: I11 = m(a2 + c2)/12 I22 = m(b2 + c2)/12 I33 = m(a2 + b2)/12 2 Cylinder: I11 = mr2/2 I22 = mr2/4... I12 = mc(2a – b)/18 2 Rectangle, rectangular plate: I11 = ma2/12 I22 = mb2/12 745 0593 AppII_fm Page 746 Tuesday, May 7, 2002 9:21 AM 746 Dynamics of Mechanical Systems 3 Circular sector, circular section plate: r = (2r/3) (sinθ)/θ I11 = (mr2/4) (1 + sinθ cosθ/θ – 16sin2θ/9θ2) I22 = (mr2/4) (1 – sinθ cosθ/θ) 4 Semicircle, semicircular plate: r = 4r/3Π I11 = mr2(9Π2 – 64)/36Π2 I22 = mr2/4 0593 AppII_fm . 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. FIGURE P20.4.1 A model of. () , 0593_C20_fm Page 733 Tuesday, May 7, 2002 9:14 AM 734 Dynamics of Mechanical Systems 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. Analysis , pp. 303–310, by courtesy of Marcel Dekker, New York, 2001. 0593__App I_fm Page 735 Tuesday, May 7, 2002 9:21 AM 736 Dynamics of Mechanical Systems 4. Circle, hoop: II. Surfaces,