MACHINE MOTION 433 U.S. Customary SI/Metric Step 2. Substitute the velocity (v A) found in step 1 and the given radius (r) of the rolling wheel in Eq. (10.51) to determine the angular velocity (ω) as Step 2. Substitute the velocity (v A) found in step 1 and the given radius (r) of the rolling wheel in Eq. (10.51) to determine the angular velocity (ω) as ω = vA rev = 88 ft/s 0.67 ft = 132 rad s × 1 rev 2π rad × 60 s 1 min = 1,260 rpm ω = vA rev = 26.8 m/s 0.2 m = 134 rad s × 1 rev 2πrad × 60 s 1 min = 1,280 rpm From the principles of relative motion, the velocity of any other point on the wheel will be the velocity (v A), which has a magnitude of (rω), plus an additional velocity equal to (rω) except directed perpendicular to the line connecting the point with the center of the wheel and is in the direction of the angular velocity (ω). Fig. 10.22 shows the velocities of three special points B, C, and D, and why the velocity of point P is in fact zero. B vA = rw r w vB A vA vA vA vA rw C vP = 0 D rw rw rw vC vD FIGURE 10.22 Velocity of special points on a rolling wheel. Therefore, the velocity at the top of the wheel, point B, has a magnitude vB = vA + rω = vA + vA = 2 vA (10.52) which is twice the velocity of the center of the wheel(v A) and directed to the right as shown. Also, the velocity of the instantaneous contact point P is zero as the velocity (v A) to the right is canceled by the velocity (rω) to the left. The velocity (vC) at the left side of the wheel, point C, has a magnitude given by the pythagorean theorem as vC = (vA)2 + (rω)2 = (vA)2 + (vA)2 = √ 2 vA (10.53) and directed upward at 45◦ relative to the horizontal as shown. Similarly, the velocity (vD) at the right side of the wheel, point D, has a magnitude given by the pythagorean theorem as vD = (vA)2 + (rω)2 = (vA)2 + (vA)2 = √ 2 vA (10.54) and directed downward at 45◦ relative to the horizontal as shown.
434 APPLICATION TO MACHINES Fig. 10.23 shows the velocities of four additional points E, F, G, and H. F vA = rw r w vF A vA vA vE vA vA rw E G rw rw rw H vG vH 45∞ FIGURE 10.23 Velocity of four additional points on a rolling wheel. Choosing point F in Fig. 10.23, its velocity has a magnitude given by the Pythagorean theorem as vF = (vA + rω cos 45◦)2 + (−rω sin 45◦)2 = (vA + vA cos 45◦)2 + (−vA sin 45◦)2 (10.55) = v2 A[(1 + cos 45◦)2 + (− sin 45◦)2] = v2 A[3.414] = (1.85) vA and its direction is downward from the horizontal, a negative angle (θ) given by the expression tan θ = −rω sin 45◦ vA + rω cos 45◦ = −vA sin 45◦ vA + vA cos 45◦ = − sin 45◦ 1 + cos 45◦ = −0.414 (10.56) θ = −22.5◦ The magnitude of the other three velocities is the same as that given by Eq. (10.55); however, each velocity is at a different angle relative to the horizontal. U.S. Customary SI/Metric Example 2. Determine the velocity, both its magnitude and direction, of point H on the rolling wheel shown in Fig. 10.23, where v A = 60 mph = 88 ft/s Example 2. Determine the velocity, both its magnitude and direction, of point H on the rolling wheel shown in Fig. 10.23, where v A = 96.5 kph = 26.8 m/s solution solution Step 1. Substitute the given velocity (vA) of the center of the wheel in Eq. (10.55) to determine the magnitude of the velocity (vH ) as Step 1. Substitute the given velocity (vA) of the center of the wheel in Eq. (10.55) to determine the magnitude of the velocity (vH ) as vH = (1.85)vA = (1.85)(88 ft/s) = 162.8 ft/s vH = (1.85)vA = (1.85)(26.8 m/s) = 49.6 m/s
MACHINE MOTION 435 U.S. Customary SI/Metric Step 2. Substitute the given velocity (vA) of the center of the wheel in Eq. (10.56) to determine the angle (θ) as Step 2. Substitute the given velocity (vA) of the center of the wheel in Eq. (10.56) to determine the angle (θ) as tan θ = rω sin 45◦ vA − rω cos 45◦ = vA sin 45◦ vA − vA cos 45◦ = sin 45◦ 1 − cos 45◦ = 2.414 θ = 67.5◦ tan θ = rω sin 45◦ vA − rω cos 45◦ = vA sin 45◦ vA − vA cos 45◦ = sin 45◦ 1 − cos 45◦ = 2.414 θ = 67.5◦ The velocity (vH ) is to the right at the magnitude calculated in step 1 at the angle (θ) calculated in step 2 above the horizontal. The velocity (vH ) is to the right at the magnitude calculated in step 1 at the angle (θ) calculated in step 2 above the horizontal. 10.4.2 Pulley Systems The simplest pulley system is shown in Fig. 10.24, where a single pulley transfers a downward force (P) into an upward force (P) to lift the load (W). vP Cable P W (load) vW FIGURE 10.24 Simplest pulley system. The downward velocity (vP ) of the force (P) is equal to the upward velocity (vW ) of the load (W) given by Eq. (10.57) as vW = vP (10.57) Therefore, for this simplest of pulley systems there is no mechanical advantage, meaning the force (P) is the same magnitude as the load (W), and the velocities (vP ) and (vW ) are equal. Consider the two pulley system shown in Fig. 10.25 where the upper pulley (1) is twice the diameter of the lower pulley (2).
436 APPLICATION TO MACHINES vP Cable P W (load) vW 1 2 FIGURE 10.25 Two pulley system. As the tension in the cable on each side of the lower pulley (2) is equal to the force (P), the mechanical advantage is (2:1), meaning the force (P) is half the magnitude of the load (W). Also, as the force (P) moves downward the lower pulley (2) rolls like a wheel up the cable that is attached to the center of the upper pulley (1). From the last section on rolling wheels, the velocity of the center of a wheel is half the velocity at a point at the top of the wheel. Therefore, the velocity (vW ) of the load (W), which is equal to the velocity of the center of the lower pulley (2), is half the velocity (vP ) of the force (P) and given by Eq. (10.58) as vW = 1 2 vP (10.58) Finally, consider the complex pulley system shown in Fig. 10.26 where pulleys (3) and (4) are connected to pulleys (1) and (2), respectively, by rigid links. Pulleys (1) and (2) have the same diameter, and pulleys (3) and (4) have the same diameter. As there is only one active cable, the mechanical advantage is (4:1), meaning the force (P) is one-fourth the magnitude of the load (W). Also, depending on the relative diameters of the large pulleys (1) and (2) as compared to the small pulleys (3) and (4), the upward velocity (vW ) will be some fraction of the velocity (vP ) of the force (P) as it moves downward. From the configuration of the pulleys in Fig. 10.26, the lower pulley (2) will again roll like a wheel up the cable that passes around pulley (3), even though this cable is not perfectly vertical. As the separation distance between the centers of pulleys (3) and (4) would be much larger than that shown in Fig. 10.26, the angle by which this cable and the cable that passes around pulley (4) and goes up to the center of pulley (3) is off from the vertical will be small.
MACHINE MOTION 437 vP Cable P W (load) vW 1 2 3 4 FIGURE 10.26 Complex pulley system. Therefore, the velocity (vW ) of the load (W), which is equal to the velocity of the center of the lower pulley (2), is the fraction (1/x) of the velocity (vP ) of the force (P) and given by Eq. (10.59) as vW = 1 x vP (10.59) where (x) is determined for specific diameters of all four pulleys and for a particular distance between pulleys (3) and (4). This completes the chapter on machine motion and thus we come to “The End” of the ten chapters of this first edition of Marks’ Calculations for Machine Design. It has been a pleasure uncovering the mystery of the formulas in machine design that are so important to bring about a safe and operationally sound design. Having said so, it must also be mentioned that for you it is just the beginning of a creative odyssey. The machines you would design based on the knowledge gained from this book, and the new ideas, theories, equations, and techniques you would come up with would go to increase the volume of books like this considerably in each subsequent edition. Best wishes for your designs.
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BIBLIOGRAPHY American Institute of Steel Construction (AISC), Manual of Steel Construction, 8th ed., American Institute of Steel Construction, Chicago, IL, 1980. Design Handbook, Associated Spring-Barnes Group, Bristol, CT, 1981. Walton, C. F. (ed.), Iron Casting Handbook, 3d ed., Iron Founder’s Society, Rocky River, OH, 1981. E. A. Avallone and T. Baumeister, III, Eds. Marks’ Standard Handbook for Mechanical Engineers, 10th ed., McGraw-Hill, New York, 1996. Marin, J., Mechanical Behavior of Engineering Materials, Prentice-Hall, Englewood Cliffs, NJ, 1962. Shigley, J. E., and C. R. Mischke, Mechanical Engineering Design, 6th ed., McGraw-Hill, Boston, MA, 2001. Wahl, A. M., Mechanical Springs, 2nd ed., McGraw-Hill, New York, 1963. Gere, J. M., and S. P. Timoshenko, Mechanics of Materials, 4th ed., PWS-Kent Publishing, Boston, MA, 1997. American National Standard Institute, Preferred Limits and Fits for Cylindrical Parts, ANSI B4.1–1967, Washington, DC (Revised 1999). American National Standard Institute, Preferred Metric Limits and Fits for Cylindrical Parts, ANSI B4.2–1978, Washington, DC (Revised 1999). Shigley, J. E., C. R. Mischke, and T. H. Brown, Jr. (eds.), Standard Handbook of Machine Design, 3d ed., McGraw-Hill, New York, 2004. Peterson, R. E., Stress Concentration Factors, Wiley, New York, 1974. 439 Copyright © 2005 by The McGraw-Hill Companies, Inc. Click here for terms of use.
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INDEX Acceleration analysis, 416–419 Advanced loadings, 127 American Welding Society (AWS), 348 Amplitude stress, 285 Angle of twist, 19 Angular rotation, 392 Area: tensile-stress, 324 Average stress, 196 Axial loading, 4, 156, 159, 164, 172 Axial strain, 5 Axial stress, 4 in cylinders: thin-walled, 129, 191 thick-walled, 133–134 prismatic, 4 Beams, 33 cantilevered, 33–34, 97 double overhanging, 35 simply-supported, 33, 35 single overhanging, 35 Beam loadings: concentrated couple, 48, 110 concentrated force at free end(s), 73, 86, 98 concentrated force at intermediate point, 41, 104 concentrated force at midpoint, 36, 159 triangular load, 60, 120 twin concentrated forces, 67 uniform load, 55, 79, 92, 115 Beam supports: cantilever, 35 pin, 34 roller, 34 Bending, 24, 158, 167, 184 Bergstrasser factor, 370 Biaxial stress element, 148 Bolt: length, 323 strength, 331 Bolted connections, 321 Butt welds, 348 Cantilevered beams, 33–34, 97, 98, 104, 110, 115, 120 Change in length: prismatic bar, 8 Classic mechanism designs, 410 Coefficient of expansion, 10 Coefficient of speed fluctuation, 393, 397 Column buckling, 260 Euler formula, 261 Parabolic formula, 263 Secant formula, 266 Short columns, 270 Column end types, 262 Combined loadings: dynamic, 311 static: axial and bending, 159 axial and pressure, 172 axial and thermal, 164 axial and torsion, 156 bending and pressure, 184 torsion and bending, 167 torsion and pressure, 175 Complex planetary gear trains, 430 Complex pulley systems, 437 Composite flywheels, 401–403 Compression springs, 380–382 Concentrated couple loading, 48, 110 Cone angle in bolted connections, 327 Contact loading: cylinders in contact, 143 spheres in contact, 139 441 Copyright © 2005 by The McGraw-Hill Companies, Inc. Click here for terms of use.
442 INDEX Contact pressure, maximum: between cylinders, 144 between spheres, 140 Corrosion effect, 283 Coulomb-Mohr theory, 248 graphical representation, 248 Critical frequency, 383–384 Cycle frequency effect, 283 Cyclic loading, 276 Cyclic motion, 421–424 Cylinders: thin-walled, 129 thick-walled, 130 Design: dynamic, 273 static, 233 Direct shear loading, 3, 11 Disassemblable joint bolt preload, 332 Distortion-energy theory, 237 graphical representation, 237 Double overhanging beams, 35, 86, 92 Elastic limit, 6, 14 Eccentricity ratio, 267 graphical representation, 267 Effective diameter, 281 Electrolytic plating effect, 283 Elongation method for bolt preload, 332 Endurance limit, 276 Energy, 375–376 Extension springs, 379–380 hook geometry, 379 External load on bolted joints, 332–334 Euler formula, 261 Factors, Marin equation: load type, 282 miscellaneous effects, 282–283 size, 280–281 surface finish, 280 Factors of safety: against bolted joint separation, 335–336 dynamic design: fluctuating loading: Gerber theory, 288–289 Goodman theory, 288–292 Modified Goodman theory, 288–289 stress-concentration factor, 304 torsional loading, 305 static design: brittle materials: Coulomb-Mohr theory, 249 Maximum-normal-stress theory, 248 Modified Coulomb-Mohr theory, 250 ductile materials: Distortion-energy theory, 238 Maximum-normal-stress theory, 235 Maximum-shear-stress theory, 237 yielding of a bolted connection, 339 Fastener assembly types, 321 Fatigue, 273 Fatigue loading of: bolted connections, 337–340 helical spring, 385–386 welded connections, 365–366 Fillet weld geometry for: bending, 357 torsion, 353 Fillet welds, 348, 350–358 Fillet welds treated as lines, 360–363 Finite life, 276, 277 First moment of area: definition, 27 formula for the maximum value: circular cross section, 31 rectangular cross section, 28 Fit standards, 137 Fluctuating design criteria, 287 Flywheels, 388 Four-bar linkage, 410 Four-stroke engine, 392 Fracture point, 6, 14 Free Body Diagram (FBD) of a helical spring, 368 Frettage effect, 283 Frustum in bolted connections, 326–327 Fundamental loadings, 3 summary table of formulas, 153 Gaskets in bolted connections, 326 Gear trains, 424 Geometry of: fillet welds as lines, 361 helical springs, 368 slider-crank linkage, 414 Gerber theory, 288–289 Goodman Diagram for: bolted connections, 338 fluctuating torsional loading, 386 welded connections, 366 Goodman theory, 288–289
INDEX 443 Grip, of a bolted connection, 323 Groove welds, 348 Helical spring deflection, 371 Helical springs, 367 Hole punching, 15 Hooke’s law: axial, 6, 9, 10 shear, 14 Hoop stress in cylinders, 129, 191 Infinite life, 276 Instantaneous contact point, 432 Interface pressure, 135 Internal combustion engines, 392–394 Joint contact, 334 Lap joint, 350 Lateral strain, 7 Linkages, 410 Load factor on bolted connections, 335 Load type factor for Marin equation, 282 Loadings: advanced, 127 axial, 4, 156, 159, 164 bending, 24, 159, 167, 184 combined, 153 contact, 139 direct shear, 11 fluctuating, 285 fundamental, 3 pressure, 127, 172, 175, 184 reversed, 274 rotational, 147 summary table of formulas, 153, 154 thermal, 10, 164 torsion, 16, 156, 167, 175 triangular, 60, 120 uniform, 55, 79, 92, 115 Machine: assembly, 321 energy, 367 motion, 409 Marin equation, 279 Maximum-normal-stress theory: brittle materials, 247 graphical representation, 247 ductile materials, 234 graphical representation, 247 Maximum-shear-stress theory, 235 graphical representation, 236 Maximum shear stress, 196 Mean stress, 285 Mechanical advantage, 436 Members, in a bolted connection, 326 Minimum shear stress, 196 Miscellaneous effects factors for Marin equation, 282–283 Modified Coulomb-Mohr theory, 249 graphical representation, 249 Modified Goodman theory, 288–289 Modulus of elasticity: axial, 6, 15 shear, 14, 15 Moment of inertia: circular beam, 31 flywheel, 389 rectangular beam, 25 Mohr’s Circle, 205 graphical process, 208 triaxial stress, 231 Neutral axis, 24 Notch sensitivity, 260, 283 Number of active coils in a helical spring, 372 Parabolic formula, 263 Parallel arrangement of springs, 377–378 Permanent joint bolt preload, 332 Pin supports, 34 Pitch of a helical spring, 380 Plane stress element, 153–154, 189 Planetary gears, 428–431 Poisson’s ratio, 7, 15 Polar moment of inertia: hollow shaft, 17 solid shaft, 17 thin-walled rectangular tube, 22 welded connection, 354 Potential energy of a spring, 376 Preload, bolt, 331–332 techniques to verify, 332 Press fits, 134, 175 Pressure: contact: between cylinders, 144 between spheres, 140 interface, 135 internal, 128
444 INDEX Pressure loadings, 127, 172, 175, 184 summary table of formulas, 154 Pressure vessels: thin-walled: cylindrical, 129, 191 spherical, 128 Pressurized tank, 184, 190 Principal stresses, 190, 195 Prismatic bar, 4, 156, 258, 322 Proof strength, 332 Proportional limit, 6, 14 Pulley systems, 435–437 Punch press: cycle, 396 flywheels, 395–398 Punching time, 397 Pure motions: rotation, 412 translation, 412 Pure shear element, 156 Quick-return linkage, 411 Radial interference, 135 Radial stress, 131–132, 148 Radius of gyration: 260 circular cross section, 265 rectangular cross section, 261 Rated torque, 395 Recovery time, punch presses, 397 Relative motion, 412–415, 416–419 Riveted joint, 11, 30 Roller supports, 34 Rolling wheels, 432–434 Rotated plane stress element, 190, 205 Rotating disk, 148 R. R. Moore rotating-beam machine, 275 Secant formula, 266 Section modulus, 26 Series arrangement of springs, 377–378 Shear: direct, 11 strain, 13, 19 stress, 12, 16, 22, 27 maximum, 196 minimum, 196 Shear-stress correction factor, 370 Shrink fits, 134, 175 Simply-supported beams, 33, 35, 36, 41, 48, 55, 60, 67, 73, 79, 86, 92 Single overhanging beams, 35, 73, 79 Size factor for Marin equation, 280–281 Slenderness ratio, 260, 261, 263, 266, 271 Slider-crank linkage, 411 S-N Diagram, 275–276 Solid disk flywheel, 388–389 Spheres: thin-walled, 128 Spring deflection, 371 Spring index, 369 Spring rate: bolt, 322–323 capscrew, 322–323 frustum of a cone, 327 helical springs, 371–372 members in a bolted connection, 326–327 Spur gears, 425–427 Static design: coordinate system, 234 brittle materials, 246 comparison with experimental data, 250 recommendations, 251–252 ductile materials, 234 comparison with experimental data, 238 recommendations, 238–239 theories: Coulomb-Mohr, 248 Distortion-energy, 237 Maximum-normal-stress, 234, 247 Maximum-shear-stress, 235 Modified Coulomb-Mohr, 249 Stability of helical springs, 381–382 Static loading of bolted connections, 335–336 Stiffness: bolt, 322–323 capscrew, 322–323 frustum of a cone, 327 members in a bolted connection, 326–327 Strain: axial, 5 lateral, 7 shear, 13, 19 thermal, 10 Strength, proof, 332 Stress: alternating, 285 average, 196 axial, 4, 129, 133–134 bending, 24 contact, 140, 144 critical, 261, 264, 266, 271
INDEX 445 direct shear, 12 hoop, 129, 191 mean, 285 normal, in spheres, 128 principal, 190 radial, 131–132, 148 shear, 16, 22, 27 tangential, 131, 148 thermal, 10 triaxial, 230 Stress-concentration factors, 258, 283, 304 Stress elements: biaxial, 148, 219, 223 maximum, 195 plane, 153, 205 pure shear, 156, 157, 219, 227 uniaxial, 155, 219 Stress-strain diagrams: axial loading: brittle materials, 7 ductile materials, 6 high-strength bolt or capscrew, 332 shear loading: brittle materials, 14 ductile materials, 14 Supports: cantilever, 33–34 pin, 34 roller, 34 Surface finish factor for Marin equation, 280 Synchronous angular velocity, 395 Tangential stress, 130, 148 Tee joint, 351 Temperature factor in Marin equation, 282 Tensile-stress area, 324, 331 Thermal: loading, 164 strain, 10 stress, 10 Thick-walled cylinders, 130 Thin rotating disks, 148 Thin-walled: tubes, 22 vessels, 128 cylindrical, 129 spherical, 128 Thread length, 323–324 Torque as a function of: angular velocity, 395 rotation angle, 392 Torque wrench method for bolt preload, 332 Torsion, 16, 156, 167, 175 Torsional loading: fluctuating, 304–305 welded connections, 352–354 Transformation equations, 190 Transverse joint, 351, 352 Triangular loading, 60, 120 Triaxial stress, 230 Turn-of-the-nut method for bolt preload, 332 Two-stroke engine, 392 Ultimate strength, 6, 14 Uniaxial stress element, 155 Uniform loading, 55, 79, 92, 115 Velocity analysis, 412–415 Vessels: thin-walled: cylinders, 129 spheres, 128 Wahl factor, 370 Welded connections, 348 Wheels and pulleys, 431 Work and energy, 375–376 Yield point, 6, 14
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ABOUT THE AUTHOR Thomas H. Brown, Jr., Ph.D., P.E., manages the Fundamentals of Engineering Review Program and the Civil Engineering Professional Engineering Review Program at the Institute for Transportation Research and Education at North Carolina State University. Dr. Brown has taught review courses for the Mechanical Engineering Professional Engineering Review Program offered by the Industrial Extension Service and taught undergraduate machine design courses for almost a dozen years in the Mechanical and Aerospace Engineering Department, both in the College of Engineering at North Carolina State University. Copyright © 2005 by The McGraw-Hill Companies, Inc. Click here for terms of use.