Mechanical Engineering Problems

Hugh Jack

Table of contents :
1. TABLE OF CONTENTS……Page 2
2.2 TYPICAL MECHANICAL FUNCTIONS……Page 9
2.3 REFERENCES……Page 10
3. FORCES……Page 11
3.1 SOME BASIC CONCEPTS……Page 12
3.2 VECTOR AND SCALAR FORCES……Page 13
3.2.1 Cartesian Vector Notation……Page 17
3.2.2 Scalar Notation……Page 18
3.2.3 3D Vectors……Page 21
3.2.4 Dot (Scalar) Product……Page 29
3.2.6 Practice Problems……Page 34
4. EQUILLIBRIUM……Page 38
4.1 THE BASIC EQUATIONS OF STATICS……Page 39
4.1.1 Practice Problems……Page 42
4.2 FREE BODY DIAGRAMS (FBD)……Page 44
4.2.1 Pulleys and Springs……Page 46
4.3 PRACTICE PROBLEMS……Page 51
5. MOMENTS……Page 53
5.1 CALCULATING SCALAR AND VECTOR MOMENTS……Page 56
5.2 FORCE COUPLES TO MAKE CENTERLESS MOMENTS……Page 64
5.2.1 Moving Forces and Equivalent Force Moments……Page 67
5.2.2 Practice Problems……Page 71
6.1 REACTIONS AND SUPPORTS……Page 79
6.2 EQUILLIBRIUM OF FORCES AND MOMENTS……Page 82
6.3 SPECIAL CASES……Page 90
6.4 STATICALLY INDETERMINATE……Page 91
6.4.1 Summary……Page 92
7.1 WHAT ARE TRUSSES?……Page 93
7.2 STABILITY OF TRUSSES……Page 99
7.3 THE METHOD OF JOINTS……Page 102
7.3.1 Practice Problems……Page 109
7.4 THE METHOD OF SECTIONS……Page 114
7.4.1 Practice Problems……Page 118
7.5 METHOD OF MEMBERS……Page 122
7.5.1 Practice Problems……Page 125
8.1 THE BASIC PHYSICS OF FRICTION……Page 132
8.2 APPLICATIONS OF FRICTION……Page 137
8.2.1 Wedges……Page 138
8.2.1.1 – Practice Problems……Page 141
8.2.1.2 – References……Page 143
8.2.2 Belt Friction……Page 144
8.2.2.1 – Practice Problems……Page 148
8.2.2.2 – References……Page 149
9.1 INTRODUCTION……Page 150
9.2 SOME BASIC CONCEPTS……Page 151
9.3 VECTOR VS. SCALAR QUANTITIES……Page 152
9.4 MATH REVIEW……Page 153
9.5 RECTANGULAR FORM OF VECTORS……Page 158
9.6 POLAR FORM OF VECTORS……Page 159
9.6.1 Cartesian Vector Notation……Page 161
9.6.2 Scalar Notation……Page 163
9.6.3 Unit Vector Representation……Page 165
9.6.4 3D Vectors……Page 167
9.6.5 Dot (Scalar) Product……Page 176
9.6.7 Practice Problems……Page 183
10.2 THE BASIC EQUATIONS OF STATICS……Page 187
10.2.1 Practice Problems……Page 192
10.3 FREE BODY DIAGRAMS (FBD)……Page 196
10.3.1 Pulleys and Springs……Page 198
10.4 PRACTICE PROBLEMS……Page 205
10.4.1 References……Page 207
11.1 INTRODUCTION……Page 208
11.2 TYPES OF STRESS……Page 210
11.3 STRESS ANALYSIS……Page 214
11.4 STRAIN CAUSED BY AXIAL LOADS……Page 219
11.5 STRESS STRAIN CURVES……Page 220
11.6 ANALYSIS OF MEMBERS……Page 222
11.7 GENERALIZED STRESS……Page 225
11.9 STRESS ON OBLIQUE PLANES……Page 229
11.10 SHEAR STRAIN……Page 232
11.11 POISSON’S RATIO……Page 236
11.12 GENERALIZED HOOKES LAW……Page 238
11.13 REFERENCES……Page 239
12.1 INTRODUCTION……Page 240
12.2 CALCULATING SCALAR MOMENTS……Page 244
12.3 CALCULATING VECTOR MOMENTS……Page 247
12.4 MOMENTS ABOUT AN AXIS……Page 251
12.5 EQUILLIBRIUM OF MOMENTS……Page 256
12.6 FORCE COUPLES TO MAKE CENTERLESS MOMENTS……Page 260
12.6.1 Moving Forces and Equivalent Force Moments……Page 265
12.6.2 Practice Problems……Page 270
12.6.3 References……Page 275
13.1 INTRODUCTION……Page 276
13.2 THE RELATIONSHIP BETWEEN STRESS AND STRAIN IN TORSION……Page 280
13.2.1 Summary……Page 283
13.3 PRACTICE PROBLEMS……Page 285
13.4 REFERENCES……Page 286
14.1 INTRODUCTION……Page 287
14.2 CENTRE OF MASS……Page 290
14.3 CENTROIDS……Page 293
14.3.1 Finding Centroids Using Composite Shapes……Page 300
14.3.2 Pappus and Guldinus – Rotated Sections……Page 306
14.3.3 Practice Problems……Page 307
15.2 REACTIONS AND SUPPORTS……Page 311
15.3 EQUILLIBRIUM OF FORCES AND MOMENTS……Page 314
15.4 SPECIAL CASES……Page 323
15.5 STATICALLY INDETERMINATE……Page 324
15.5.1 Summary……Page 329
15.5.3 References……Page 330
16.2 WHAT ARE TRUSSES?……Page 331
16.3 STABILITY OF TRUSSES……Page 336
16.4 THE METHOD OF JOINTS……Page 339
16.4.1 Practice Problems……Page 348
16.5 THE METHOD OF SECTIONS……Page 352
16.5.1 Practice Problems……Page 358
16.6.2 Compound Trusses……Page 361
16.6.3 References……Page 362
17.1 FACTOR OF SAFETY……Page 363
17.1.1 Practice Problems……Page 366
17.2 REFERENCES……Page 367
18. STRAIN FAILURE……Page 368
18.1 POISSON’S RATIO……Page 369
18.2 GENERALIZED HOOKES LAW……Page 372
18.3 STRESS CONCENTRATIONS……Page 373
18.4 TORSION STRESS CONCENTRATIONS……Page 379
18.5 REFERENCES……Page 380
19.1 INTRODUCTION……Page 381
19.1.1 Practice Problems……Page 386
19.4 REFERENCES……Page 391
20.1 INTRODUCTION……Page 392
20.2 PRACTICE PROBLEMS……Page 405
20.3 REFERENCES……Page 409
21.2 STRESSES IN BEAMS……Page 410
21.3 MOMENT CURVATURE IN BEAMS……Page 412
21.4 EVALUATING THE SECOND MOMENT OF INERTIA……Page 414
21.5 MOMENTS OF INERTIAS B COMPOSITE AREAS……Page 415
21.6 POLAR MOMENT OF INERTIA……Page 419
21.7 REVIEW OF BASIC CALCUATIONS……Page 421
21.8 PRODUCT OF INERTIA……Page 433
21.9 REFERENCES……Page 435
22.1 INTRODUCTION……Page 436
22.2 TRANSVERSE SHEAR……Page 440
22.3 REVIEW OF TRANSVERSE LOADING……Page 445
22.4 REFERENCES……Page 447
23.2 THE BASIC PHYSICS OF FRICTION……Page 448
23.3 APPLICATIONS OF FRICTION……Page 454
23.3.1 Wedges……Page 455
23.3.1.1 – Practice Problems……Page 460
23.3.2 Belt Friction……Page 462
23.3.2.2 – References……Page 467
24. MASS PROPERTIES……Page 468
24.1 CENTRE OF MASS……Page 471
24.2 CENTROIDS……Page 474
24.2.1 Finding Centroids Using Composite Shapes……Page 478
24.2.2 Practice Problems……Page 481
24.3 MOMENTS OF INERTIA……Page 486
24.4 PRODUCT OF INERTIA……Page 496
25. INTERNAL FORCES IN MEMBERS……Page 498
26. STRESS……Page 507
26.1 TYPES OF STRESS……Page 508
26.2 STRESS ANALYSIS……Page 512
26.3 STRESS ON OBLIQUE PLANES……Page 517
26.4 GENERALIZED STRESS……Page 519
26.5 FACTOR OF SAFETY……Page 522
26.5.1 Practice Problems……Page 525
27.1 STRAIN CAUSED BY AXIAL LOADS……Page 526
27.2 STRESS STRAIN CURVES……Page 527
27.3 ANALYSIS OF MEMBERS……Page 529
27.4 POISSON’S RATIO……Page 533
27.6 SHEAR STRAIN……Page 535
27.7 STRESS CONCENTRATIONS……Page 540
27.8 TORSION……Page 543
27.9 TORSION STRESS CONCENTRATIONS……Page 546
27.10 PURE BENDING……Page 548
27.11 TRANSVERSE LOADING……Page 551
28.1 INTRODUCTION……Page 555
28.2 PLANAR……Page 556
28.2.1 Measuring Mass Properties……Page 561
28.3 PRACTICE PROBLEMS……Page 563
28.4 REFERENCES……Page 564
29.1 VIBRATION MODELLING……Page 565
29.1.1 Differential Equations……Page 566
29.1.2 Modeling Mechanical Systems with Laplace Transforms……Page 567
29.1.3 Second Order Systems……Page 568
29.2.1 Vibration Control……Page 572
29.3 VIBRATION CONTROL……Page 577
29.3.1 Isolation……Page 579
29.3.2 Inertial……Page 582
29.3.3 Active……Page 583
29.4 VIBRATION MEASUREMENT……Page 584
29.5 VIBRATION SIGNALS……Page 586
29.6.1 Velocity Pickups……Page 587
29.6.2 Accelerometers……Page 588
29.6.4 Modal Analysis……Page 590
29.7.1 Sources……Page 591
29.9 PRACTICE QUESTIONS……Page 592
29.10 SOUND/VIBRATIONS TERMS……Page 598
29.11 REFERENCES……Page 599
30.1 POWER……Page 600
30.2 KINEMATICS AND DYNAMICS……Page 603
30.3 REFERENCES……Page 605
1.1 BASIC PROPERTIES OF SOUND……Page 606
1.1.1 Adding/Subtracting/Averaging dB Values……Page 610
1.2 SOUND MEASUREMENTS……Page 612
1.2.1 Measurement Techniques……Page 613
1.3 THE HUMAN EFFECTS OF SOUND……Page 614
1.3.1 Background Noise……Page 619
1.3.2 Weighting Sound Values to Compensate for the Human Ear……Page 622
1.3.3 Speech Interference……Page 624
1.4 NOISE CONTROL REGULATIONS……Page 626
1.4.1 Other Hearing Effects of Noise Exposure……Page 627
1.4.2 Measuring Noise Revisited……Page 628
1.5.1 Noisy machines……Page 632
1.5.2 A Room (or not)……Page 633
1.5.3 Equipment (Microphones)……Page 635
1.5.5 Equipment – Spectrum Analyzers……Page 637
1.6 EQUIPMENT GENERATED NOISE……Page 642
1.7 ROOM ACCOUSTICS……Page 646
1.7.1 Sound Source Power Measurements Revisited Again……Page 652
1.7.2 How Sound Propagates……Page 653
1.8 ENCLOSURES, BARRIERS AND WALLS……Page 654
1.8.1 Noise Reduction Through Walls……Page 658
1.8.2 Walls with composite layers……Page 660
1.8.3 Enclosures……Page 661
1.8.4 Acoustic Barriers……Page 662
1.9 MATERIALS……Page 663
1.10 MUFFLERS……Page 664
1.10.1 Lined Ducts and Baffles……Page 665
1.10.2 Bends……Page 666
1.10.3 Plenum……Page 668
1.10.4 Expansion Chamber……Page 669
1.10.5 Cavity Resonator (Helmholtz)……Page 670
1.13 PRACTICE QUESTIONS……Page 671
2. INTRODUCTION TO KINEMATICS OF MECHANISMS……Page 685
2.1.1 Locking/Engaging……Page 687
2.1.3 Four Bar Linkages……Page 689
2.1.4 Reciprocating……Page 691
2.1.5 Six Bar Linkages……Page 693
2.2 SKELETON DIAGRAMS FOR MECHANISMS……Page 694
2.3 DOF AND THE KUTZBACH/GRUEBLER CRITERION……Page 695
2.4 KINEMATIC/GEOMETRIC INVERSION……Page 698
2.5 GRASHOF’S LAW……Page 700
2.6 MECHANICAL ADVANTAGE……Page 702
2.7 PRACTICE PROBLEMS……Page 703
2.8 REFERENCES……Page 704
3.1 MATHEMATICAL TOOLS……Page 705
3.2 DEFINING POSITIONS AND DISPLACEMENTS……Page 706
3.4 SOLVING FOR POSITIONS……Page 710
3.4.1 Trigonometry……Page 711
3.4.2 Complex Numbers……Page 713
3.4.3 Numerical Solutions……Page 715
3.5 GRAPHING OF POSITIONS……Page 716
3.6 DISPLACEMENT, TRANSLATION AND ROTATION……Page 718
3.8 REFERENCES……Page 721
4.1 THE BASIC RELATIONSHIPS……Page 723
4.2.1 Complex Algebra……Page 725
4.2.2 Analytic Methods……Page 729
4.3 INSTANTANEOUS CENTERS OF ROTATION……Page 730
4.3.1 Aronhold-Kennedy Theorem of Three Centers……Page 733
4.3.2 Some Examples of Instantaneous Centers……Page 734
4.3.3 The Angular-Velocity-Ratio Theorem……Page 736
4.3.4 Mechanical Advantage……Page 737
4.3.5 Freudenstein’s Theorem……Page 738
4.3.6 Centrodes……Page 739
4.4 PRACTICE PROBLEMS……Page 740
4.5 REFERENCES……Page 742
5.1 THE BASIC DEFINITION……Page 743
5.4 REFERENCES……Page 751
6.1 SELECTION OF MECHANISM TYPES……Page 752
6.2.1 Mechanism Typing……Page 753
6.3.1 Two Position Design……Page 755
6.4 REFERENCES……Page 761
7.1 BASICS……Page 762
7.1.1 Degrees of Freedom……Page 763
7.2 HOMOGENEOUS MATRICES……Page 764
7.2.1 Denavit-Hartenberg Transformation (D-H)……Page 769
7.2.2 Orientation……Page 771
7.2.3 Inverse Kinematics……Page 773
7.2.4 The Jacobian……Page 774
7.3.1 Moments of Inertia About Arbitrary Axes……Page 776
7.3.2 Euler’s Equations of Motion……Page 779
7.3.3.1 – Linear Momentum……Page 780
7.4 DYNAMICS FOR KINEMATICS CHAINS……Page 781
7.4.1 Euler-Lagrange……Page 782
7.5 REFERENCES……Page 784
7.6 PRACTICE PROBLEMS……Page 785
8. MECHANICAL COMPONENTS……Page 798
9.1 CAM TYPES……Page 799
9.2 CAM MOTION……Page 801
9.4 PRACTICE PROBLEMS……Page 812
9.5 REFERENCES……Page 814
10.1 SPUR GEARS……Page 815
10.1.1 Involute Profiles……Page 819
10.1.2 Design of Gears……Page 821
10.1.3.1 – Undercutting and Contact Ratios……Page 824
10.2 HELICAL GEARS……Page 826
10.2.1 Design of Helical Gears……Page 827
10.2.2 Perpendicular Helical Gears……Page 830
10.2.3 Practice Problems……Page 831
10.3 BEVEL GEARS……Page 832
10.3.1 Design of Bevel Gears……Page 833
10.4 WORM GEARS……Page 834
10.4.2 Practice Problems……Page 837
11.1 SIMPLE GEAR TRAINS……Page 838
11.1.1 Examples – Fixed Axis Gears……Page 839
11.1.2.1 – Epicyclic Gear Trains……Page 843
11.1.2.2 – Differentials……Page 846
12.1 INTRODUCTION……Page 848
12.2 ANALYSIS OF GEARS……Page 849
12.4 REFERENCES……Page 850
13.1 TRANFORMING DEVICES……Page 851
13.2 REFERENCES……Page 852
14.1.1.1.2 – Split Bearings……Page 853
14.1.1.2.1 – Ball Bearings……Page 854
14.1.1.2.2 – Roller Bearings……Page 855
14.1.1.2.3 – Thrust Bearings……Page 856
14.1.2 Threads……Page 857
14.1.2.1 – Metric……Page 859
14.1.2.2 – American National Standard……Page 860
14.1.2.3 – British Standard Whitworth (BSW)……Page 861
14.1.2.4 – The Unified Thread……Page 862
14.1.2.5 – American National ACME Thread……Page 863
14.1.2.6 – Brown and Sharpe Worm Thread……Page 864
14.1.3 Tapers……Page 865
14.2.1.1 – Spur Gears……Page 866
14.2.1.3 – Helical Gears……Page 867
14.2.1.4 – Herringbone Gears……Page 868
14.2.1.6 – Hypoid Gears……Page 869
14.2.1.8 – Rack and Pinion……Page 870
14.3.1 Hydraulics……Page 871