Mechanical Wave Vibrations – Analysis and Control

Mechanical Wave Vibrations – Analysis and Control
اسم المؤلف
Chunhui Mei
التاريخ
8 أكتوبر 2023
المشاهدات
321
التقييم
(لا توجد تقييمات)
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Mechanical Wave Vibrations – Analysis and Control
Chunhui Mei
University of Michigan-Dearborn
Dearborn, MI, USA
Contents
Preface xi
Acknowledgement xiii
About the Companion Website xv
1 Sign Conventions and Equations of Motion Derivations 1
1.1 Derivation of the Bending Equations of Motion by Various Sign Conventions 1
1.1.1 According to Euler–Bernoulli Bending Vibration Theory 2
1.1.2 According to Timoshenko Bending Vibration Theory 7
1.2 Derivation of the Elementary Longitudinal Equation of Motion by Various Sign Conventions 10
1.3 Derivation of the Elementary Torsional Equation of Motion by Various Sign Conventions 12
2 Longitudinal Waves in Beams 15
2.1 The Governing Equation and the Propagation Relationships 15
2.2 Wave Reflection at Classical and Non-Classical Boundaries 16
2.3 Free Vibration Analysis in Finite Beams – Natural Frequencies and Modeshapes 20
2.4 Force Generated Waves and Forced Vibration Analysis of Finite Beams 24
2.5 Numerical Examples and Experimental Studies 27
2.6 MATLAB Scripts 32
3 Bending Waves in Beams 39
3.1 The Governing Equation and the Propagation Relationships 39
3.2 Wave Reflection at Classical and Non-Classical Boundaries 40
3.3 Free Vibration Analysis in Finite Beams – Natural Frequencies and Modeshapes 46
3.4 Force Generated Waves and Forced Vibration Analysis of Finite Beams 50
3.5 Numerical Examples and Experimental Studies 55
3.6 MATLAB Scripts 59
4 Waves in Beams on a Winkler Elastic Foundation 69
4.1 Longitudinal Waves in Beams 69
4.1.1 The Governing Equation and the Propagation Relationships 69
4.1.2 Wave Reflection at Boundaries 70
4.1.3 Free Wave Vibration Analysis 71
4.1.4 Force Generated Waves and Forced Vibration Analysis of Finite Beams 72
4.1.5 Numerical Examples 76
4.2 Bending Waves in Beams 79
4.2.1 The Governing Equation and the Propagation Relationships 79
4.2.2 Wave Reflection at Classical Boundaries 82
4.2.3 Free Wave Vibration Analysis 84
4.2.4 Force Generated Waves and Forced Wave Vibration Analysis 84
4.2.5 Numerical Examples 89viii Contents
5 Coupled Waves in Composite Beams 97
5.1 The Governing Equations and the Propagation Relationships 97
5.2 Wave Reflection at Classical and Non-Classical Boundaries 100
5.3 Wave Reflection and Transmission at a Point Attachment 102
5.4 Free Vibration Analysis in Finite Beams – Natural Frequencies and Modeshapes 104
5.5 Force Generated Waves and Forced Vibration Analysis of Finite Beams 105
5.6 Numerical Examples 108
5.7 MATLAB Script 114
6 Coupled Waves in Curved Beams 119
6.1 The Governing Equations and the Propagation Relationships 119
6.2 Wave Reflection at Classical and Non-Classical Boundaries 121
6.3 Free Vibration Analysis in a Finite Curved Beam – Natural Frequencies and Modeshapes 127
6.4 Force Generated Waves and Forced Vibration Analysis of Finite Curved Beams 128
6.5 Numerical Examples 134
6.6 MATLAB Scripts 143
7 Flexural/Bending Vibration of Rectangular Isotropic Thin Plates with Two Opposite Edges
Simply-supported 151
7.1 The Governing Equations of Motion 151
7.2 Closed-form Solutions 152
7.3 Wave Reflection, Propagation, and Wave Vibration Analysis along the Simply-supported x Direction 154
7.4 Wave Reflection, Propagation, and Wave Vibration Analysis Along the y Direction 156
7.4.1 Wave Reflection at a Classical Boundary along the y Direction 157
7.4.2 Wave Propagation and Wave Vibration Analysis along the y Direction 159
7.5 Numerical Examples 159
8 In-Plane Vibration of Rectangular Isotropic Thin Plates with Two Opposite Edges Simply-supported 189
8.1 The Governing Equations of Motion 189
8.2 Closed-form Solutions 190
8.3 Wave Reflection, Propagation, and Wave Vibration Analysis along the
Simply-supported x Direction 192
8.3.1 Wave Reflection at a Simply-supported Boundary along the x Direction 192
8.3.2 Wave Propagation and Wave Vibration Analysis along the x Direction 195
8.4 Wave Reflection, Propagation, and Wave Vibration Analysis along the y Direction 197
8.4.1 Wave Reflection at a Classical Boundary along the y Direction 198
8.4.2 Wave Propagation and Wave Vibration Analysis along the y Direction 201
8.5 Special Situation of k0 = 0: Wave Reflection, Propagation, and Wave Vibration
Analysis along the y Direction 201
8.5.1 Wave Reflection at a Classical Boundary along the y Direction Corresponding to a Pair of Type I Simple Supports
along the x Direction When k0 = 0 202
8.5.2 Wave Reflection at a Classical Boundary along the y Direction Corresponding to a Pair of Type II Simple
Supports along the x Direction When k0 = 0 203
8.5.3 Wave Propagation and Wave Vibration Analysis along the y Direction When k0 = 0 205
8.6 Wave Reflection, Propagation, and Wave Vibration Analysis with a Pair of Simply-supported
Boundaries along the y Direction When k0 ≠ 0 207
8.6.1 Wave Reflection, Propagation, and Wave Vibration Analysis with a Pair of Simply-supported
Boundaries along the y Direction When k0 ≠ 0, k1 ≠ 0, and k2 ≠ 0 207
8.6.2 Wave Reflection, Propagation, and Wave Vibration Analysis with a Pair of Simply-supported Boundaries along
the y Direction When k0 = 0, and either k1 = 0 or k2 = 0 209
8.7 Numerical Examples 212Contents ix
8.7.1 Example 1: Two Pairs of the Same Type of Simple Supports along the x and y Directions 212
8.7.2 Example 2: One Pair of the Same Type Simple Supports along the x Direction, Various Combinations
of Classical Boundaries on Opposite Edges along the y Direction 217
8.7.3 Example 3: One Pair of Mixed Type Simple Supports along the x Direction, Various
Combinations of Classical Boundaries on Opposite Edges along the y Direction 223
9 Bending Waves in Beams Based on Advanced Vibration Theories 227
9.1 The Governing Equations and the Propagation Relationships 227
9.1.1 Rayleigh Bending Vibration Theory 227
9.1.2 Shear Bending Vibration Theory 228
9.1.3 Timoshenko Bending Vibration Theory 230
9.2 Wave Reflection at Classical and Non-Classical Boundaries 232
9.2.1 Rayleigh Bending Vibration Theory 232
9.2.2 Shear and Timoshenko Bending Vibration Theories 238
9.3 Waves Generated by Externally Applied Point Force and Moment on the Span 244
9.3.1 Rayleigh Bending Vibration Theory 245
9.3.2 Shear and Timoshenko Bending Vibration Theories 246
9.4 Waves Generated by Externally Applied Point Force and Moment at a Free End 247
9.4.1 Rayleigh Bending Vibration Theory 248
9.4.2 Shear and Timoshenko Bending Vibration Theories 249
9.5 Free and Forced Vibration Analysis 250
9.5.1 Free Vibration Analysis 250
9.5.2 Forced Vibration Analysis 250
9.6 Numerical Examples and Experimental Studies 252
9.7 MATLAB Scripts 257
10 Longitudinal Waves in Beams Based on Various Vibration Theories 263
10.1 The Governing Equations and the Propagation Relationships 263
10.1.1 Love Longitudinal Vibration Theory 263
10.1.2 Mindlin–Herrmann Longitudinal Vibration Theory 264
10.1.3 Three-mode Longitudinal Vibration Theory 265
10.2 Wave Reflection at Classical Boundaries 267
10.2.1 Love Longitudinal Vibration Theory 267
10.2.2 Mindlin–Herrmann Longitudinal Vibration Theory 268
10.2.3 Three-mode Longitudinal Vibration Theory 269
10.3 Waves Generated by External Excitations on the Span 271
10.3.1 Love Longitudinal Vibration Theory 271
10.3.2 Mindlin–Herrmann Longitudinal Vibration Theory 272
10.3.3 Three-mode Longitudinal Vibration Theory 273
10.4 Waves Generated by External Excitations at a Free End 275
10.4.1 Love Longitudinal Vibration Theory 275
10.4.2 Mindlin–Herrmann Longitudinal Vibration Theory 276
10.4.3 Three-mode Longitudinal Vibration Theory 276
10.5 Free and Forced Vibration Analysis 277
10.5.1 Free Vibration Analysis 278
10.5.2 Forced Vibration Analysis 278
10.6 Numerical Examples and Experimental Studies 281
11 Bending and Longitudinal Waves in Built-up Planar Frames 287
11.1 The Governing Equations and the Propagation Relationships 287
11.2 Wave Reflection at Classical Boundaries 289
11.3 Force Generated Waves 291x Contents
11.4 Free and Forced Vibration Analysis of a Multi-story Multi-bay Planar Frame 292
11.5 Reflection and Transmission of Waves in a Multi-story Multi-bay Planar Frame 304
11.5.1 Wave Reflection and Transmission at an L-shaped Joint 304
11.5.2 Wave Reflection and Transmission at a T-shaped Joint 308
11.5.3 Wave Reflection and Transmission at a Cross Joint 315
12 Bending, Longitudinal, and Torsional Waves in Built-up Space Frames 329
12.1 The Governing Equations and the Propagation Relationships 329
12.2 Wave Reflection at Classical Boundaries 333
12.3 Force Generated Waves 336
12.4 Free and Forced Vibration Analysis of a Multi-story Space Frame 338
12.5 Reflection and Transmission of Waves in a Multi-story Space Frame 341
12.5.1 Wave Reflection and Transmission at a Y-shaped Spatial Joint 343
12.5.2 Wave Reflection and Transmission at a K-shaped Spatial Joint 353
13 Passive Wave Vibration Control 369
13.1 Change in Cross Section or Material 369
13.1.1 Wave Reflection and Transmission at a Step Change by Euler–Bernoulli Bending Vibration Theory 371
13.1.2 Wave Reflection and Transmission at a Step Change by Timoshenko Bending Vibration Theory 372
13.2 Point Attachment 373
13.2.1 Wave Reflection and Transmission at a Point Attachment by Euler–Bernoulli Bending Vibration Theory 374
13.2.2 Wave Reflection and Transmission at a Point Attachment by Timoshenko Bending Vibration Theory 375
13.3 Beam with a Single Degree of Freedom Attachment 376
13.4 Beam with a Two Degrees of Freedom Attachment 378
13.5 Vibration Analysis of a Beam with Intermediate Discontinuities 380
13.6 Numerical Examples 381
13.7 MATLAB Scripts 390
14 Active Wave Vibration Control 401
14.1 Wave Control of Longitudinal Vibrations 401
14.1.1 Feedback Longitudinal Wave Control on the Span 401
14.1.2 Feedback Longitudinal Wave Control at a Free Boundary 405
14.2 Wave Control of Bending Vibrations 407
14.2.1 Feedback Bending Wave Control on the Span 407
14.2.2 Feedback Bending Wave Control at a Free Boundary 410
14.3 Numerical Examples 413
14.4 MATLAB Scripts 416
Index 421
Index
a
acceleration/force 32, 58–59, 257
Acoustics 116, 149, 260, 326, 367, 399, 419
active discontinuity 401
allowable deflection 2
amplitudes 15, 28, 34, 39, 55, 64, 70, 79, 97, 112, 119, 134, 153, 190,
227, 229–230, 251, 260, 263–264, 266
analysis 1, 7, 15, 28, 31–32, 39–40, 56, 58, 69, 97, 119, 151, 189, 197,
201, 205, 209–210, 212, 227, 232, 263, 267, 277, 280, 285, 287,
289, 326, 329, 333, 369, 401
analytical predictions 285
analytical results 31–32, 58–59, 252, 257, 281, 286
angle joint 287, 329
angular distortion 2, 7
anti-symmetrical mode 160, 187
applied forces 24, 26, 50, 53, 55, 72–74, 76, 84–86, 87, 89, 105,
107, 128–133, 244, 247, 271–273, 275–276, 291–292, 294,
297, 336, 341
area moment of inertia 5, 7, 39, 79, 97, 119, 227, 287, 330, 369
aspect ratio 159–160, 164–165, 169, 173–174, 178, 182, 186–187, 217
attached boundary 18–19, 42–43, 45, 124, 126, 143–144, 235–238,
243, 405, 411
attached spring 388, 401–402, 405, 408, 411, 420
attachment 117, 369, 381, 383, 385–387, 400
single DOF 385
audio frequency applications 231, 238, 243, 247, 250, 254, 288,
331, 369
axial deflection 10, 12, 69, 289, 402, 405
axial displacement 263, 266
axial force 1, 10, 24 –27, 30, 271, 292, 294, 341, 344, 353
applied 24, 26, 72–74, 291–292
external 24–25, 27, 30, 73–75
axial loading 1, 3
axial stiffness 69
b
beam 1–3, 5–8, 10–11, 13–79, 81–97, 100, 102, 105, 108, 117, 119,
125, 130–131, 133, 227, 229–231, 236–237, 241–242, 247,
250–254, 256–257, 261, 263–264, 266–267, 275, 278–281,
284–285, 287–288, 296–297, 303, 305–327, 330–331, 333, 342,
344–351, 353, 355–369, 373, 376–383, 385–389, 400, 402–403,
408, 411, 413–414, 419–420
cantilever 108, 115, 117
fixed-fixed 21, 76–77
fixed-free 76, 78
free-free 34, 37, 62
metal 31, 58, 281
stepped Timoshenko 382, 390
beam axis 15, 39, 69, 79, 97, 227, 263, 287, 330, 349, 360, 368–369
beam elements 2–3, 11–13, 292–295, 297, 303, 305–316, 318–325,
327, 338, 340–342, 344–345, 347–348, 350–351, 353–354,
357–358, 361, 363, 365, 368, 380
horizontal 292–293, 338, 342
vertical 292–293, 295, 338, 342, 353
bending decaying 99, 100, 115
bending deflection 39, 50, 85, 227, 229, 230, 232, 237–242, 244–246,
248–249, 287, 291, 306–307, 311, 313–314, 319, 321, 323–324,
348–351, 358, 360–361, 363, 369, 371, 374–375, 408
bending deformation 2, 4, 7
bending moment 1–4, 7, 8, 40, 41, 50–52, 79, 81, 84–88, 100–101,
105, 107, 111, 113, 121, 128, 141–142, 151–152, 232–233, 240,
244–248, 251–252, 289, 290–292, 294, 303, 333, 341, 344, 353,
369–371, 373, 408–409
bending slope 40, 44, 82, 101, 103–105, 125, 229–230, 232, 237–242,
245–246, 289–290, 336, 345, 349, 354, 360, 371, 374–375, 408
bending spring 101, 103
bending stiffness 101–102
bending-torsion coupling 97
bending vibration control 401, 410, 412, 416
bending vibration model 227, 231–232
bending vibration motions 108
bending vibrations 39, 58, 67, 89, 109, 227, 241, 261, 285, 287–288,
329–330, 369, 381, 407, 409, 411, 413–416, 420
forced 89
free 39, 79, 227–228, 230, 369
suppressing 413–414
bending vibration theory 6–8, 10, 39–42, 50–51, 53, 58–59, 82, 101,
227–228, 230–236, 238–258, 285, 287–290, 292, 307–308,
312–315, 319–320, 322–326, 329–331, 333–337, 349–353,
359–367, 369–372, 374–375, 381, 401, 408, 411, 416
bending wave control design 408, 411
bending wave components 288–289, 332–333, 369
bending wavenumber 39, 63, 65, 79, 227, 409, 411
bending waves 39–40, 46, 50, 81, 84, 228, 244, 254, 288, 290, 311, 319,
331, 348, 358, 360, 407, 410
injected 51, 86, 246–247
Bending Waves in Beams 39–66, 79, 83–93
Bending Waves in Beams Based on Advanced Vibration
Theories 227–254, 258, 260
boundaries 16–20, 23–27, 29, 36–37, 40–42, 44–46, 49, 51–56, 61, 63, 65,
70–76, 82–84, 86–90, 100, 102, 104–105, 107–108, 116–117,
121–127, 130–134, 150, 154–155, 157–159, 192–193, 195, 198–205,
223, 233, 235–236, 238–239, 241–242, 244, 250–252, 267–268,
278–280, 290, 292–293, 295–296, 326–327, 329, 334–336, 338–339,
341, 380–381, 393, 395, 405, 410–411, 413, 417–418
clamped-clamped 67
clamped-free 67
clamped-mass 67
clamped-pinned 67
corresponding 295
elastic 71, 125, 143, 233
fixed-free 21
free-fixed 38
free-free 20–22, 28, 38, 46–48, 55, 252, 281
free-spring 38
ideal 414, 416
left 381–382, 400, 413, 420
tuned 416
boundary conditions 16–17, 20, 23, 33–34, 40–42, 45–47, 71, 82–83,
121–123, 127, 150, 154, 157–158, 160, 190, 192–193, 198–200,
202–205, 209–211, 217, 223, 233–235, 240–241, 267–270, 290,
334, 392
boundary control 413, 417
boundary control forces 413, 420
boundary controller 406
boundary discontinuities 295, 341
boundary reflection coefficients 28, 160
boundary reflections 25–26, 51, 54, 73, 75, 130, 133, 250, 277
boundary reflection relationships 86, 88, 240
boundary with a mass attachment 19, 44, 125, 236, 242
boundary with spring attachments 42, 100, 123, 124, 235–236,
241, 411
Brüel & Kjær PULSE Unit 31, 58, 282, 368
Brüel & Kjær Type 4397 accelerometer 31, 58, 253, 282
Brüel & Kjær Type 8202 Impact hammer 31, 58, 253, 282, 368
c
cantilever 108–110
cantilever composite beam 108–111, 117
Cartesian coordinate system 1, 294, 341
causal controller 410
center of mass 44, 67, 125, 237, 261
centerline 2, 4, 7, 10, 12, 15, 39, 69–70, 79, 97, 105, 108, 119, 121, 125,
227, 229–230, 263–264, 266, 287, 330, 369
centroid 294, 330, 341
centroidal axis 5, 7, 294, 303, 341, 344
characteristic equation 20–23, 46–48, 72, 84, 105, 128, 150, 155–156,
159, 196, 197, 201, 205, 207–208, 250, 278
characteristic expressions 20, 32, 46, 60–61
characteristic matrix 32–33, 35–36, 61, 65, 116, 146, 260, 393,
398, 418
characteristic polynomials 28, 33, 37–38, 55, 67, 76–77, 80, 89–90, 93,
96, 109–110, 134, 137, 150, 160–161, 219, 220, 223–224, 254, 257,
261, 284–285, 327, 368, 381–383, 382–389, 389–390, 400
circular frequency 15–16, 32, 39, 69, 97, 99, 119, 152, 189, 227,
229–231, 263–264, 266, 288, 331, 369, 377, 381, 401
clamped 198, 202–204, 206, 217, 233–234, 236, 290–291
Clamped – Clamped 217
Clamped – Free 219
clamped boundary 41–43, 45, 48, 67, 83, 108, 117, 122–123, 125, 150,
157, 165–167, 169, 178–180, 182, 198–199, 202–204, 234, 236,
238, 240–243, 290–291, 334–335
clamped end 67, 115, 126, 150, 259, 261
Clamped-Clamped 47–48, 160
Clamped-Free 47, 160
Clamped-Pinned Boundary 47–48
Clamped–Simply-supported 160
Classical 16–17, 19, 40–41, 43, 45, 100–101, 121, 123, 125, 232–233,
235, 237, 239, 241, 243
classical boundary 15–17, 19, 46, 60, 67, 82–83, 102, 125, 127, 152,
157, 159, 190, 198, 201–203, 205, 207, 217, 219, 221, 223, 225,
236, 242, 267, 269, 289–290, 333, 335
closed-form solutions 151–153, 189–191
coefficient matrices 20–23, 26–28, 46–48, 52, 55, 72, 74, 76, 84,
87, 89, 98, 104–105, 108, 119, 128, 131, 133, 155–156, 159–160,
191, 195–197, 201, 205, 207–208, 219, 223, 229–230, 250, 252,
265–267, 273–280, 294, 341, 372–373, 375–376, 379, 381–382
coefficient matrix 20, 28, 46, 55, 72, 84, 98, 105, 119, 128, 155–156,
159–160, 191, 196–197, 201, 205, 207–208, 219, 223, 229–230,
250, 265–266, 278, 294, 341, 379, 382
coefficients 20, 238, 248, 330, 349, 360, 384
collocated feedback control 401
combinations 76, 196, 207, 217, 290, 292
classical boundaries 207, 219, 221, 225
classical boundaries on opposite edges 217, 223
composite beam 97, 104–105, 107–111, 117
uniform 97–98, 100, 104–105, 107, 114, 117, 390
concave 1, 4, 7, 294, 341
continuity and equilibrium 24, 50, 85, 128, 245–246, 271, 291, 336,
377, 402–403, 408
continuity 15, 24, 39, 50, 72, 85, 103–106, 128, 245–247, 271–274,
279, 291, 294, 299, 301, 303–305, 307–310, 312–318, 320,
322–325, 341, 344–347, 349–351, 353–357, 360, 362–367,
374–375, 377, 379, 402–403, 408
continuity conditions 50, 85, 103, 106, 128, 245, 247, 272–274, 299,
301, 305, 307–310, 312–318, 320, 322, 324–325, 336, 344–347,
349, 351, 353–357, 360, 362, 364–367
continuity equations 103, 273–274, 279, 305, 307–309, 312–315, 320,
322, 324–325, 345, 349, 351, 353–355, 360, 362, 364–367
group of 310, 316–318, 345–347, 355–357
multiple sets of equivalent 347, 357
scalar 307–308, 310, 312–315, 320, 322, 324–325, 347, 349, 351,
353, 357, 360, 362, 364–367
continuity in deflections 374, 379
control 1, 15, 39, 69, 97, 119, 151, 189, 227, 263, 286–287, 329, 369,
401, 404, 406, 410, 412–417, 420Index 423
control force 401, 404, 405–407, 409–410, 412–413, 420
wave 401
control gain(s) 404–407, 410, 412
controllers 403–406, 409–410, 412–414, 416, 420
ideal 410, 412, 416
convex 1, 4, 7
coordinate systems 1, 72, 84, 294, 297–299, 301, 303, 341–342
local 294, 297, 303, 342
rotating local 299–301, 303
varying local 298–299, 303
coupled bending 97, 104, 390
coupled torsional 108
coupled vibration 119, 390
coupled waves 97, 119
in composite beams 97–116
in curved beams 119–150
coupling coefficient 97–98, 108, 117
critical frequency 264, 282, 285
cross (“+”) joint(s) 293–295, 297, 301, 303–304, 320, 323–326
cross-sectional area 10, 12, 15, 39, 69, 79, 97, 119, 227, 263, 287, 330,
369, 403
curvature 2–4, 7, 119–120, 134, 144, 150
curved beam 119–121, 125, 127–128, 130–134, 138, 141–142,
144, 150
clamped uniform 138
uniform 119, 127, 130–131, 133–134
curved beam theory 121
cut-off circular frequency 156
cut-off frequency 69, 70, 76, 81, 89, 93, 96, 119–120, 134, 137, 150,
160, 187, 198, 223, 231–232, 244, 247, 250, 254, 265–266, 282,
284, 288, 331, 369, 371
d
D control/controller 404–406, 409–410, 412–416, 420
damper 16, 32, 386, 388, 408, 411
damper attachment 387
damping 19, 228, 229, 231, 241, 254, 257, 369, 389
damping constant 387, 405, 407, 410
damping effect, viscous 19, 42, 236, 386, 408, 411
dB Magnitude 37, 62, 66, 76, 89, 110, 116, 134, 137, 148–149, 254,
260, 327, 368, 395, 398, 419
decaying 40, 156, 228, 288, 331, 407–408, 410
decaying bending wave component 408
decaying bending waves 40, 228, 288, 331, 407, 410
decaying waves 46, 70, 81, 134, 156–157, 160, 198, 229, 231, 244, 247,
250, 254, 282, 284–285, 288, 408
negative-going 70, 198, 229, 254
single pair of 157, 284
decoupled governing equations of motion 232
deep composite beams 111
deflections 2, 6, 10, 12, 14, 24, 26–27, 49–50, 52–53, 55, 69, 72,
74, 76, 85, 87, 89, 108, 119, 128, 131, 133, 251–252,
267–269, 275–276, 279–280, 371–372, 374, 376–379, 401, 404,
413, 420
bending 39, 50, 85, 227, 229–230, 232, 237–242, 244–246, 248–249,
287, 291, 306–307, 311, 313–314, 319, 321, 323–324, 348–351,
358, 360–361, 363, 365, 369, 371, 374–375, 408
deformation 1–4, 7–9
bending 2, 4, 7
convex 4, 7
determinant 20–22, 33, 46–48, 72, 84, 98, 105, 119, 128, 155–156, 159,
191, 196–197, 201, 205, 207–208, 219, 223, 229–230, 250,
265–266, 278, 294, 341, 381–382
determinant of the coefficient matrix 20, 46, 72, 84, 98, 105, 119, 128,
155–156, 159, 191, 196–197, 201, 205, 207–208, 219, 223,
229–230, 250, 265–266, 278, 294, 341, 382
differential equation 15, 39, 69, 229–230
digital finite impulse response (FIR) filter 410
direction 1, 12, 151–186, 189–217, 219–225, 294, 303, 336,
341–342, 344
axial 402
given 4, 401
longitudinal 16
positive 294, 297, 303, 341–342
tangential 119, 121, 125
upward 297
y-axis 5, 8
disappearance of resonant peaks 29, 58, 92, 135, 137
discontinuity 15, 103–104, 117, 294–295, 341, 369, 371–373, 375–376,
379, 381–382, 384, 386, 394, 397–398, 400–402, 408, 411,
417–418
active 401
attached intermediate 117
attaching multiple identical 388
intermediate spring 398
joint 295
discontinuity component, single 382, 400
discriminant polynomial 282
dispersion equation 15, 39, 70, 79, 98, 119–120, 134, 148, 153, 191,
227, 229, 230, 263, 265–266, 282
dispersion relationships 119, 134–136
dispersive 39, 227, 229, 231, 254
dispersive longitudinal waves 282
displacements 19, 121, 125
in-plane plate 189–190
longitudinal 263
distance 4, 7, 16, 20, 24–27, 29, 40, 46, 49, 52–56, 70, 72–76, 81,
86–87, 89, 100, 104, 108, 121, 127, 130, 133, 154–155, 159, 195,
201, 228, 250–252, 264, 278–280, 289, 293, 332–333, 378, 381,
408, 411, 413, 420
dynamic equilibrium 305, 308, 315, 344, 353
dynamic spring stiffness 19, 42, 236, 386, 401, 408, 411
dynamic stiffness(es) 241, 373–374, 405
e
edge(s) 152, 160, 190, 196, 207–208, 212, 217–218
elastic foundation 69–72, 76, 79, 81, 84, 89–90, 93, 96
elementary longitudinal equation of motion 10–11
elementary longitudinal vibration theory 15, 263, 268, 272, 276, 282,
284–286, 287–290, 292, 329, 331, 401–402, 416
elementary theory 15, 287–288
elementary torsional equation of motion 12–13
end mass 44, 66, 125–126, 143–144, 236, 261
Endevco 2302 impact hammer 327
energy, optimal 404–406, 409–410, 412, 420
energy absorption, predicted 413–414424 Index
external force(s) 15, 24–27, 29, 35, 39, 50–54, 56–58, 72–76, 84–86,
87–92, 128–130, 132, 134–135, 228, 230, 245–249, 251–252,
258, 271–272, 275–276, 279–280, 291, 329, 336, 376–377, 381,
400, 402, 405, 408, 411
externally applied axial force 24, 26, 72–74, 291–292
externally applied end force 26, 74, 276–277
externally applied point force and moment 244–245, 247, 249
external transverse force 52, 86–87, 245, 248, 251–252
f
feedback bending wave control 407–408, 410–411
feedback bending wave controller 408, 411
feedback controller 401, 403–408
feedback control of flexural waves in beams 419
feedback longitudinal wave control 401, 405
feedback longitudinal wave controller 403
feedback PD wave controller 413–414
feedback wave control 401–403, 406–407, 416
feedback wave controllers 401, 403, 413–414, 416
Finite Impulse Response (FIR) filter 410, 412
fixed boundary 17–19, 21, 71, 76, 268–271, 280
fixed-damper boundary 22–23
fixed-fixed boundary 21, 23, 76–77
fixed-free boundary 21, 23, 76, 78
fixed-mass boundary 22–23
fixed-spring boundary 22–23
flexural/bending vibration of rectangular isotropic thin
plates 151–188
force 12, 24–26, 29, 35–37, 50–51, 53–54, 57, 65, 72–76,
86, 88, 91, 107, 128, 130, 132–133, 147, 241, 244, 246,
248, 250, 260, 276–277, 285, 291, 374, 379, 381, 393,
417–418
impact 31, 58, 253, 281
radial 121, 128, 141
flexural vibrations, see Euler–Bernoulli
forced in-plane bending vibrations 67, 261
forced out-of-plane vibrations in planar frames 326
forced response 36, 66, 147, 394–395, 418–419
forced vibration(s) 25–26, 29, 31, 34, 50, 51, 54, 73, 75–76, 86, 88,
107–108, 119, 130, 133–134, 150, 227, 251–252, 263, 281, 326,
367, 382, 400
forced vibration analysis 24–25, 28–29, 31, 34, 50–51, 53, 55–56, 64,
72, 76, 90, 105, 107, 135, 144, 250–251, 253, 258, 277–279,
292–293, 295, 297, 299, 301, 303, 338–339, 380
curved beam 128–129, 131, 133–144
forced wave vibration analysis 28, 56, 84, 116, 382,
390, 399
force generated waves 24–26, 35, 50–51, 53–54, 65, 72–73, 75, 84, 86,
88, 105, 107, 128–131, 133, 147, 248, 250, 260, 277, 291, 336,
337, 393, 417
frame 287, 293–295, 303, 327, 399
free bending vibrations 227, 369
free body diagram 3, 10–13, 16–19, 24, 26, 41–42, 44, 50, 53,
71–75, 84–85, 87–88, 101, 103, 105–106, 124–125, 128–129,
132, 233, 235–236, 241, 245, 247–248, 291, 294, 297-301,
304–305, 308–309, 315, 317, 341, 344, 374, 376, 378, 402,
405, 407, 408, 411
free boundaries 16, 21, 38, 47, 102, 160–161, 169–170, 172–176, 178,
182–183, 185–186
equation(s) of motion
“+” joint 315
cross joint 315
bending waves in beams on Winkler elastic foundation 79
coupled bending-torsion vibration in slender composite beams 97
elementary longitudinal vibration theory 11–12, 15, 329–330
elementary torsional vibration theory 13–14
Euler-Bernoulli bending vibration theory 6, 39, 232, 287, 329
in-plane vibration of thin plate 189
L joint 304-305
longitudinal waves in beams on Winkler elastic foundation 69
Love’s in-plane vibration theory in curved beams 119
Love longitudinal vibration theory 263
Mindlin-Herrmann longitudinal vibration theory 264
out-of-plane vibration of thin plate 151
Rayleigh bending vibration theory 227, 232
Shear bending vibration theory 228–229, 232
single DOF system 376
spatial K joint 353
spatial Y joint 343
T joint 308
two DOF system 378
Three-mode longitudinal vibration theory 265–266
Timoshenko bending vibration theory 9–10, 230, 232, 287, 329–330
equilibrium 15, 39, 72, 299, 301, 305–309, 311–315, 318, 320–321, 323–324,
326, 345, 347, 350–351, 353–354, 357, 360–361, 363, 365, 367
vector equation on 307–308, 313–315, 320, 323–324, 326, 350–351,
353, 361, 363, 365, 367
equilibrium conditions 17, 19, 24, 41, 44, 50–51, 53, 82, 85, 88, 101,
103, 106, 121, 125, 128–129, 233, 236, 240, 242, 245–247,
268–269, 271–274, 290–291, 334, 336, 374, 377, 402–403, 405,
408, 411
equilibrium equations 53, 87, 102, 104, 106, 132, 241–242, 247, 273,
275–277, 279–280, 303–304, 306–308, 312, 314–315, 318, 320,
323–326, 341, 347, 357, 361, 374
equivalent 306, 311, 318, 347, 357
scalar 306–308, 311–312, 314–315, 318, 320, 323–324, 326, 347,
350–351, 353, 357, 360–361, 363, 365, 367
equilibrium relationships 105, 294
equivalent continuity equations 305, 310, 316–318, 347, 357
Euler–Bernoulli 2, 6–8, 39–42, 50–51, 53, 58–59, 63, 65,
82–84, 97, 101, 116, 227, 231–236, 238, 246, 248, 253–257, 261,
285, 287–290, 292, 307–308, 312–315, 319–320, 322–326,
329–330, 333–335, 337, 349–353, 359–367, 369–372, 374–375,
377, 379–381, 396, 400–401, 408, 411, 416
experimental results 15, 31, 39, 58, 252, 257, 281, 286
experimental setup 31
experimental studies 27, 29, 31–32, 55, 57, 59, 252–253, 256–258,
281, 283, 285, 327, 367
experimental equipment
Brüel & Kjær PULSE Unit 31, 58, 282, 368
Brüel & Kjær Type 4397 accelerometer 31, 58, 253, 282
Brüel & Kjær Type 8202 Impact hammer 31, 58, 253, 282, 368
Endevco 2302 impact hammer 327
PCB 353 B12 Accelerometer 327
external excitation(s) 15, 24–29, 31, 38–39, 50–58, 67, 72–76, 84–92,
105–108, 111, 128–135, 137, 139, 146, 150, 228, 230, 244–249,
251–252, 258, 261, 271–280, 286, 291–292, 294, 297, 326, 336,
341, 367, 369, 376–377, 380–381, 400, 402, 405, 408, 411, 413, 420Index 425
Hz 29, 31, 33, 37, 56, 58, 62, 66, 110, 116, 134–135, 138, 145, 148–149,
258–260, 327, 368, 391, 395, 398, 414, 417, 419
i
identity matrix 16, 40–41, 46, 52, 55, 71, 82–84, 87, 89, 100–103, 105,
108, 121–123, 127, 131, 154, 157–159, 192, 195, 199–205, 207,
233–234, 250, 267, 278, 280, 290, 294, 297, 303, 306–308,
311–315, 319–326, 334, 340, 347–351, 353, 357–358, 360–367,
369–370, 372–376, 401, 405, 407–408, 410
imaginary part 116, 136, 253, 255, 260, 281, 283
imaginary unit 15, 39, 69–70, 79, 97, 119, 190, 227, 229–230,
263–264, 266, 288–289, 291, 307–308, 312–314, 320, 322–323,
325, 332, 370–371, 403
incident waves 16, 40–41, 71, 82–83, 100–103, 121–123, 154,
157–159, 192, 199–200, 202–205, 233–234, 267, 290, 294,
297, 303, 306–308, 311–315, 319–326, 334, 340, 347–351,
353, 357–358, 360–367, 369–370, 372–376, 401, 405,
407–408, 410
incoming longitudinal vibration energy 404, 406
inertance frequency response(s), see frequency response(s)
inertia 5, 7, 12, 39, 44, 67, 79, 97, 119, 125, 227, 237, 261, 263–264,
287, 303, 330, 344, 369, 378, 382
infinite shear rigidity 229, 231–233, 245
injected waves 130, 292, 294, 297, 326, 337, 341, 367, 377, 379
in-plane 67, 119, 138, 149, 189–190, 193–194, 196, 201–202, 207, 226,
261, 287, 326–327, 330
in-plane bending wavenumbers 330
in-plane plate vibrations 193–194, 196, 201–202
one-dimensional 207
In-Plane Vibration of Rectangular Isotropic Thin Plates 189–226
integer multiples, odd 223–224
integer number of half waves 160
intermediate spring support 390
internal resistant axial force 10, 69, 289, 402, 405
internal resistant bending moment 4, 233, 408
internal resistant force(s) or/and moment(s) 1, 10, 12, 16, 24, 26, 50,
53, 72, 74, 85, 87, 128, 132, 245, 247, 267–269, 275, 291, 294,
307–308, 312, 314–315, 320, 323–326, 371, 374, 402
internal resistant shear force (and bending moment) 1–2, 8, 40–41,
81, 100, 232–233, 240, 246, 289–290, 303, 333, 344, 369–371,
373, 409
isotropic 151, 159, 189, 198, 207, 212, 226
j
joint parameters 350–351, 353, 360, 363, 365, 367
joints 293–297, 299, 303–304, 315, 317, 329, 338, 340–342
spatial angle 329
joints and boundaries 329, 338
k
K joint(s) 338, 341–343, 353–354, 357–367
kinetic energy 263, 272, 276
K-shaped 329, 353
l
L joint(s) 293, 296–297, 299, 301–308
load vectors 51, 86–87, 130–131, 246–249, 251–252, 273, 275–277,
279–280, 292, 337, 381
free boundary 16–17, 20–21, 26, 41, 43, 45–47, 54, 67, 82–83, 88, 102,
121, 125, 132–133, 150, 157–158, 160–161, 169–170, 172–176,
178, 182–183, 185–186, 199, 202–204, 233–234, 236, 238, 240,
242–243, 248–250, 268–270, 290, 334–235, 401–402, 405, 410,
413, 416
free end 18–19, 26–27, 31, 33, 38, 53–54, 58, 67, 74–75, 87, 89, 115,
125, 132–134, 247, 249, 252–253, 259, 261, 275, 280, 286
free flexural vibrations 151, 189, 226, 327
free longitudinal vibrations 96, 263, 264–265
free vibration 15, 28, 39, 69, 97, 151–152, 188–189, 232, 250, 287, 294,
297, 340, 368, 370, 399
free vibration analysis 20–21, 23, 28–29, 32–33, 46–47, 49, 55–57,
61–62, 71, 76, 84, 90, 104, 109, 117, 127, 135, 149, 250, 278,
327, 368
Planar Curved Beams 149
free-free boundary 20–23, 28, 31, 33, 38, 46, 47–48, 55, 160, 219, 252,
281
free vibration responses 250, 294, 297, 340
frequency domain 401–402, 404, 408
frequency resolution 31, 58, 134, 253, 282
frequency response(s) 29, 31, 57–58, 92–93, 111, 113, 135, 137, 282
inertance 31–32, 58–59, 253, 255–257, 285–286
receptance 29–30, 38, 56–57, 67, 76, 80, 90, 92–94, 135, 141–142,
150, 261, 382–388, 400, 413–416, 420
steady state 15, 39, 119
frequency span 31, 58, 253, 281
functions
exponential 153
parabolic distribution 266
g
generated wave(s) 24–26, 35, 51, 54, 65, 73–75, 86, 88, 107, 130, 133,
246, 248–250, 260, 271–273, 275–277, 393, 417
geometrical parameters 32–33, 61–62, 144
geometrical properties, see material (and geometrical) properties
governing equation(s), see also equation(s) of motion
governing equation of bending vibrations 39
governing equation of longitudinal vibrations 15
governing equation of motion for free bending vibrations 79, 227
governing equations 15, 39, 69, 79, 97, 99, 119, 191, 227, 229, 231,
263, 265, 287, 294, 329, 331, 341
decoupled 232
higher fourth-order partial differential 39
governing equation(s) of motion 1, 7, 39, 69, 79, 97, 119, 151–152,
189, 227–228, 230–232, 263, 287, 369
governing equations of motion for free bending vibration 228, 230
governing equations of motion for free in-plane vibration 119
governing equations of motion for free vibration 39, 69, 97, 152, 287
gravity 376, 378, 382
h
half harmonic waves 156, 196
half waves 160
integer multiples of 159–161, 219–220, 223
Homework Project 38, 67, 96, 117, 150, 261, 327, 368, 400, 420
Hooke’s law 8, 17, 71
horizontal beam elements 292–293, 338, 342
H-shaped frame 287, 327426 Index
mechanics 10, 12–14
Mei 99, 101, 103, 111, 116, 231, 260, 263, 286, 305, 309, 326, 339,
342–343, 354, 359, 367, 378–379, 399, 405, 407, 410, 416, 419
Mindlin-Herrmann 263–266, 268, 269, 272, 276–277, 280, 282,
284–286
minimum, local 76, 134
miscounting of natural frequencies 58, 137
modal approach 329
mode number 77–78, 110, 134, 139
modes 28–29, 35, 55–56, 58–59, 64, 76, 79, 89–93, 111–113, 134–140,
150, 160, 164–165, 169, 173–174, 178, 182, 186, 269–271
anti-symmetrical 160, 187
body 55–56, 64
corresponding 76, 93
one-dimensional 196, 201–202
rigid body 21–22, 28, 34, 47–48, 62, 64
symmetrical 160, 187
modeshape amplitudes 24, 49
modeshape curves 29, 57, 89
modeshape 15, 20–21, 23–24, 28–29, 33–34, 38–39, 46-49, 55-57, 62,
64, 67, 72, 76, 78–79, 84, 89–91, 104–105, 111, 119, 127–128,
134–135, 138–139
corresponding 23, 84
normalized 34, 64, 76, 89
overlaid 76, 79, 91, 134, 138
modeshapes by mode number 139
modes of vibrations for flexural motion 134
mode transition point 282
moment 1, 6, 10, 14, 50, 52–56, 65, 67, 84–90, 92, 105, 128–134, 141,
151–152, 241, 244–247, 249–252, 260, 291–292, 294, 297,
307–308, 312, 314–315, 320, 323–326, 329, 333, 336, 341, 344,
371, 373, 393
motion 1–15, 23, 32, 39, 48, 69, 79, 97, 119, 128, 151–152, 156, 189–190,
227–232, 263–265, 287, 294, 329–331, 341, 369, 376, 378
flexural 134–135, 137–138
mounting 382, 384, 388
Multi-story Multi-bay Planar Frame 292–293, 295–325
Multi-story Space Frame 338–365
n
natural frequency 15, 20–24, 28–29, 31–34, 37–39, 46–49, 55, 57–59,
61–62, 67, 72, 76–78, 84, 89–91, 93, 96, 104–105, 109–110, 117,
119, 127, 128, 134–135, 137–138, 150, 155–156, 159–160,
162–167, 169–170, 172–180, 182–183, 185–187, 196–197, 201,
206–209, 211–219, 221–223, 225, 250, 254, 257, 261, 278,
284–285, 294, 327, 341, 368–369, 381–385, 387–389
measured 327, 368
non-dimensional 21, 90, 160, 164, 169, 173, 178, 182, 186
Natural Frequencies and Modeshapes 20–21, 23, 46–47, 49, 104, 127
natural frequencies of a structure 20, 46, 58, 72, 137, 257, 278, 369
natural frequencies of the beam 28, 46, 55, 72, 84, 90, 96,
250, 381, 388, 389
natural frequencies of the two DOF spring-mass system 382
Natural frequency expressions, non-dimensional 23
natural frequency values 24, 48–49, 93
nearfield wave 156, 228, 408, 411
negative directions 1, 16, 153, 157, 198, 227, 229, 231, 253
negative-going waves 70, 100, 120, 154, 195, 198, 201, 248–249, 264,
273–277, 279–281, 288–289, 292
local coordinates 294, 297, 303–304, 315, 341, 343
local coordinate systems 294, 297–300, 303, 306–307, 311, 313–314,
319, 321, 323–324, 341–342, 348, 350–351, 358, 361, 363
local minima 76, 89, 93, 109, 134, 160, 219, 223, 254, 285, 382–383,
385, 387, 389
local minimum 76, 134
location 29, 31, 37, 49, 56–59, 66, 76, 79, 91–93, 135, 137, 139–140,
154–155, 157–159, 192–193, 195, 198–200, 202–204, 292–294,
296–297, 339, 341, 389, 401
ideal 31, 58, 137
longitudinal axis 1, 59, 253, 257, 294, 341
longitudinal and torsional wave components 333, 358
longitudinal deflection 10, 15, 17–19, 24, 69–70, 72, 263–264, 266,
287, 330, 334
longitudinal force 15, 17, 303, 333–334, 344
longitudinal motion 134–135, 137–138
longitudinal propagation coefficient 16, 20, 25, 27, 70, 72–73, 75
longitudinal vibration 11–12, 15, 38–39, 96, 263–273, 275–277,
280–282, 284–285, 287, 401–404, 406–407, 413, 416, 419
free 15, 96, 263–265
longitudinal vibration control 395
longitudinal vibration theory 17, 263, 265–271, 276–277, 280–282,
284–285, 289–290, 292, 401–402
longitudinal wave control design 408, 411
longitudinal waves 15–16, 70, 120, 281–282, 287–326, 401–402, 405
longitudinal wavenumber 15, 16, 70, 263, 281, 283, 288, 331
longitudinal waves in beams 15–38, 69, 71, 73, 75, 77
longitudinally vibrating beams 32, 263
Love longitudinal vibration theory 119, 121, 138, 263, 265, 267–268,
271, 275–277, 280–282, 284–286
low frequency mechanical vibrations 389
L-shaped Joint 287, 304, 326
lumped mass boundaries 22
m
magnitude plots 96, 382, 389
dB 76, 109, 134, 160, 219, 223
magnitudes 28, 32–33, 37–38, 47–48, 55, 58, 62, 66–67, 76–77, 80,
89–90, 93, 96, 109, 116, 153, 161, 220, 224, 253–257, 261,
281–285, 349, 360, 382–390, 395, 398
mass 19–20, 32, 44–45, 67, 125–126, 143–144, 236–238, 241–243, 261,
303, 344, 368, 376, 378, 382
mass attached 19, 44–45, 67, 125–126, 143–144, 238, 243, 261
mass attachments 16, 19, 40, 44, 121, 125–126, 233, 236, 242
mass block 19, 44, 125–226, 237–238, 243, 382
mass block attachment 236, 238
mass density 27, 38, 55, 67, 108, 189, 212, 252, 261, 327, 368, 381,
400, 413, 420
mass moment of inertia 44, 67, 125, 237, 261, 378, 382
material coupling 97, 109, 116
material (and geometrical) properties 27, 38, 55, 67, 99, 108, 117,
134, 150, 159, 212, 223, 227, 229, 231, 252, 258, 261, 264, 281,
327, 354, 359, 368–369, 375, 381, 389, 400, 413, 420
materials 10, 12–14, 20, 32–33, 46, 61–62, 99, 108, 117, 134, 144, 150,
198, 227, 229, 231, 252, 264, 369–371, 382, 391, 408
MATLAB 15, 20–22, 28, 31–32, 39, 45–48, 56, 59, 96, 109, 114, 117,
119, 125–126, 135, 143, 145, 147, 227, 244, 253, 257–259, 382,
389–391, 393, 395, 397, 400, 407, 416–417, 420
matrix equation 45, 236, 238, 242–243, 293, 338, 377Index 427
bending wavenumbers 330
forced 326
free vibrations 159
plate vibrations 154
vibration 151, 152
over-predicts 58–59, 257
p
pair of decaying waves 134, 157, 198, 231, 244, 247, 250, 282, 284, 288
pair of propagating waves 134, 153, 157, 198, 231, 244, 247, 250, 254,
282, 284, 288
pair of simply-supported boundaries 153, 201, 207, 209, 211
pair of type I 202–203, 205–207, 209–214, 216–217, 219
pair of Type II 203–207, 210–212, 214–217, 219, 222–223
parabolic distribution 263, 266
parameters 1, 4, 7, 10, 12, 150, 264, 266, 303, 305, 309–310, 315, 318,
344–345, 347, 350–351, 353–354, 357, 360, 363, 365, 367, 369,
403, 408, 411
passing band 388–389
passive vibration control 388–389
Passive Wave Vibration Control 369–400
passively implemented control 405, 407
PCB 353 B12 Accelerometer 327
PD control/controller 404–406, 409–410, 412–415, 420
Periodical structures 388–389
phase speeds 198, 207
phase velocity 16, 39, 227, 229, 231, 254, 282
pinned boundary 42–43, 45–46, 48, 67, 233–236, 238, 241–243,
290–291
Pinned / Simply Supported Boundary 40, 45–46
Pinned-Pinned Boundary 48
planar frame 287, 292–295, 297, 299, 303–304, 326–327, 342
general 292
plane 1–2, 5, 7, 44, 67, 125, 151, 189, 237, 253, 261, 294, 303, 330–332,
336, 344–345, 348–349, 354, 358, 360
plate 151–153, 156–157, 159–161, 189–190, 198, 207, 212, 219, 223
uniform 154–155, 159, 195–196, 201, 205
plate flexural rigidity 151
Point Attachment 102–3, 117, 373–375, 381
point axial force 24, 72, 273
point control force 401
point discontinuity 402, 408, 411
point downwards 342
point force 27, 75, 244, 250
point spring 401–402, 405, 407
point support 102–104, 373–375
point transverse force 50, 84, 105, 107, 111, 113, 244, 377
Poisson effect 263
Poisson’s ratio 151, 159, 160, 187, 189, 198, 212, 253, 261, 263–264,
330, 368, 381, 400
polar moment of inertia 12, 97, 263, 330
polynomial equation, cubic 98, 282
positive direction of angle 1, 294
positive shear force 81, 101
positive sign directions 1–2, 10, 12, 79, 333
power 153
principle of d’Alembert 305, 308, 315, 344, 353
propagating waves 46, 70, 134, 153, 157, 198, 228–229, 231, 247, 250,
254, 282, 284, 288, 408, 410–412
negative wavenumber values 253, 281
neutral axis 1–4, 7
neutral line 7
Newton’s second law 10–12, 14, 19, 44, 125, 236, 301, 304, 308, 315,
343, 353, 376, 378
nodal point 29–31, 35, 56–58, 64, 76, 90–92, 135, 137, 139
nodal point of modes 29, 35, 56, 58, 64, 135, 137, 139
nodal point of odd modes 29, 35
non-causal 410, 412
non-classical boundary 15–17, 19, 40–41, 43, 45, 100–101, 119, 121,
123, 125, 232–233, 235, 237, 239, 241, 243
non-dimensional coordinate 71–72
non-dimensional displacement 70, 82
non-dimensional flexibility coefficient 71, 76, 96
non-dimensional frequency 21–23, 46, 69, 81, 96
non-dimensional natural frequency 21–23, 47–48, 76–77, 160, 164,
169, 173, 178, 182, 186, 217–218, 222–223
non-dimensional stiffness 69, 76, 81, 89, 96
non-dispersive 16, 282
nonlinear equation 22–23
non-nodal point 29–30, 35, 56–58, 64, 76, 90–92, 135, 139
non-rigid body modes 55–56, 64
non-trivial solution 20, 46, 72, 105, 250, 278
normal strain 4, 7
normal stress 4, 5, 190
normalized modeshape 34, 64, 76, 89, 90
n-story m-bay 293
numerical examples 27, 55, 76, 89, 108–109, 111, 113, 134–135, 137,
159–160, 163–187, 212–213, 252, 281, 326, 367, 381, 383, 385,
387, 413, 415
Test A 135, 137, 139–141
Test B 135, 137, 139–140, 142
Numerical Examples and Experimental Studies 27, 29, 31, 55, 57,
252–253, 281, 283, 285
o
observation location 29, 58, 91–93, 135, 137, 139, 150
observation point 29, 58, 92
observations 139, 150, 160, 257, 413–414
occurrence of natural frequencies 160, 187
occurrence of resonant peaks 29, 57–59, 91, 93, 135, 140
odd modes 29, 35
on span control 413, 417, 420
opposite edges 151, 153, 159–160, 189, 191, 196, 201, 205, 207–209,
217, 219, 223
Opposite Edges Simply-supported 151–226
optimal D controller 404, 406, 410, 412
optimal energy absorbing D control gain 406
optimal energy absorbing PD control gains 404, 406, 409, 412
optimal energy absorbing PD controller 404–406, 409–410, 412
optimal PD controller 406
optimal performance at a frequency 410, 412
origin 17–19, 24, 26, 41–42, 44, 50–51, 53, 71–72, 74, 82–83, 85, 88,
102–104, 106, 122–124, 126, 128–129, 132, 154, 157–158,
192–193, 198–200, 202–204, 233–235, 237, 240–243, 245–249,
268–274, 276–277, 307–308, 312–315, 320, 322–326, 334–336,
349–351, 353, 360, 362–367, 372–375, 403, 405
out-of-plane 159, 329–330
out-of-plane vibrations 151–152, 326428 Index
reflection matrix 27, 39–43, 45–48, 52, 54–55, 59,
82–84, 86, 88, 102, 104, 107, 115, 117, 121, 123–127, 130,
132–133, 143, 155, 157–159, 192–195, 199–201, 207, 233–244,
249–252, 257, 259, 267, 269–271, 278–280, 290, 293, 295–296,
306, 311, 319, 334–336, 339, 347–348, 357, 369, 374, 381, 393,
407–408, 410–411
reflection relationships 72, 83–84, 250, 293, 338–339, 420
resonances 387–388
resonant peaks 29, 57–59, 76, 91–93, 135–137, 140
response observation 29, 57–58, 76, 90–91, 93, 135, 137, 139
locations 29, 57, 91, 93, 135
points 76, 90–91, 135, 139
responses 24, 26–27, 29, 31, 38, 49, 52, 55–56, 58, 64, 67, 72, 74, 76,
87, 92, 108, 131, 133, 135, 137, 139, 150, 241, 252–253,
257–258, 261, 279–280, 285, 373–374, 381, 389, 391, 400, 417
dB magnitude 37, 284
forced 37, 76, 108, 261, 294, 297, 341
forced vibration 75, 135
imaginary 134, 137
right hand rule 12, 294, 341
right side 1, 10, 12, 24, 48, 50, 72, 85, 103–105, 128, 244–245, 251,
271–273, 291, 293, 296, 299, 300–304, 336, 338, 341, 371–376,
391, 402, 408
rigid body 19, 21–22, 28, 34, 44, 47–48, 62, 64, 67, 126, 237, 261, 303,
308, 344–345, 347, 354, 357, 378
joint related parameters 305, 309–310, 315, 318, 345, 347, 354, 357
mass attachment 126
motion 207
rigid mass block 45, 59, 67, 239, 242, 244, 257, 261, 376, 378, 382
ring frequency 120, 134, 150
rod theories 286
roots 20, 28, 46, 55, 72, 84, 99, 105, 120, 128, 134, 145, 148, 150, 153,
160, 191, 219, 223, 278, 282, 284, 381
rotary inertia 6, 97, 111, 138, 227, 230–233, 235, 257, 269
rotating coordinate systems 297, 303, 342
rotation 1–2, 7, 44, 121, 125, 128, 135, 138, 237, 241–242, 303, 344,
373–374
angular 378
rotation angle 294, 341
rotational motion 12, 14, 42, 236
rotational relationships 297–299, 303, 342
rotational stiffness 42, 235–236
rotationally asymmetric cross section 13, 330
rotationally symmetric cross section 330–331
s
scalars 277, 280, 307, 360, 372
scalar equation 305–306, 311, 318
scalar equations of continuity, six 347, 349, 351, 353, 357, 360, 362,
364–367
scalar equations of continuity, three 307–308, 310, 312–317, 320,
322, 324–325
scalar equations of equilibrium 306–308, 311–312, 314–315,
318, 320, 323–324, 326, 347, 350–351, 353, 357, 360–361, 363,
365, 367
sensor 35, 64, 258, 391, 417
separation of variables 15, 39, 69, 79, 97, 119, 152–153, 190–191, 227,
229–230, 263–264, 266, 288, 330
sequence 145, 164–165, 169, 173–174, 178, 182, 186
bending 99–100, 115, 331
propagating decaying waves 134
propagating wave component 408, 411
propagating waves 39, 46, 134, 153, 157, 198, 229, 231, 244, 247, 250,
254, 284, 288
reflected 408, 411–412
transmitted 410
propagation 15–16, 28, 39–40, 46, 55, 62–63, 65, 72, 84, 86, 88, 104,
107, 116, 120, 147, 154, 160, 195, 201–211, 250–252, 264, 289,
293, 338, 381, 393, 401, 418
propagation coefficient 15, 155, 205, 264, 280
propagation matrices 107, 117, 251, 259, 279, 339
propagation matrix 39–40, 46, 49, 52, 54, 81, 86, 89, 100, 104, 121,
127, 130, 133, 159, 195, 201, 228, 250, 252, 265, 267, 278, 280,
289, 293, 333, 381
bending 40, 46, 52, 54, 81, 228, 381
propagation relationships 15, 20, 25–27, 33–34, 39, 46, 51–52, 54,
69–70, 73, 75, 79, 86, 89, 97, 99–100, 104, 107, 127, 130, 133,
155, 159, 195, 201, 205, 227–229, 231, 250, 252, 263, 265, 267,
277–280, 287, 293–295, 297, 329, 331–332, 338, 341, 380
Proportional Derivative (PD) control 404–406, 409, 410, 412–415, 420
proximity, close 134
pseudo cut-off frequency 282, 284
r
Radial direction 119–121, 125, 128, 142
radial force 121, 128, 141–142
radial stiffness 123
radius of curvature 2–4, 7, 119–120, 134, 144, 150
Rayleigh 227, 231–235, 238–239, 245, 248, 251–257, 285
receptance frequency response(s), see frequency response(s)
rectangular (thin) plate(s) 151, 155, 159, 188–189, 195, 201, 205,
215–218, 226
References 4, 14, 37, 66, 95, 116, 149, 188, 226, 260, 286, 326, 367,
399, 419
reflected and transmitted propagating wave components 408, 410
reflected and transmitted waves 15, 39, 104, 292, 295, 305, 309, 315,
329, 338, 341, 345, 354, 369, 380, 401, 408, 410
reflected bending vibration energy 411
reflected bending waves 407, 410
reflected longitudinal vibration waves 401
reflected propagating waves 408, 411–412
reflected vibration energy 403, 406
reflected vibrations 401
reflected wave 16, 40, 41, 71, 82–84, 100, 102, 104, 121–123, 154,
157–159, 192–194, 199–200, 202–205, 233–234, 250, 267, 290,
306, 311, 318–319, 334, 347–348, 357, 369, 374, 401, 405
reflection 15, 20, 33, 46, 49, 71, 104, 107, 115, 127, 153, 155, 192, 195,
197, 250–252, 278, 293, 296, 304–307, 309–325, 338, 340–341,
343–365, 381, 390, 401–403, 405, 407
Reflection and Transmission in Curved Beams 149
reflection and transmission matrices 39, 104, 107, 294–295, 297,
306–308, 311, 313–315, 319, 321, 323–324, 326, 338, 340, 348,
350–351, 353, 358, 361, 363, 367, 371–373, 375–376, 390, 408
reflection and transmission relationships 15, 293–296, 303–306, 309,
311, 315, 318, 338–342, 345, 347, 354, 357
reflection coefficient 15–22, 25–26, 28, 46, 71–73, 75, 82, 154–155,
160, 197, 202–207, 210–211, 219, 267–268, 280, 401, 403,
405–406Index 429
spring attachments 16–17, 19, 41–42, 100, 123–124, 233, 235–236,
241, 379, 388–390, 395, 408, 411
discrete 389
intermediate 117, 395–397, 400
spaced 389
spring boundaries 22
spring forces 379
spring-mass 369, 376
spring-mass system 376, 378
single DOF 376, 381–382, 384
spring stiffness 19, 42, 236, 386, 407
steady state frequency responses, see frequency response(s)
steel beam 27, 31–33, 35, 38, 55, 61, 62, 67, 96, 134, 144, 150, 252,
257–258, 261, 281–282, 284, 368, 381–382, 384, 386, 390, 395,
400, 413, 416, 420
curved 134, 150
step change 382, 391, 394
step size 33, 35, 61, 64, 114, 145, 259, 391, 396, 417
stiffness 17, 38, 71, 76, 79, 90, 93, 97, 241, 387, 389–390, 410–411
dynamic 241, 374, 401
dynamic spring 19, 42, 236, 386, 401
normalized spring 242
stopping bands 388, 389
strain 3, 4, 8
strain energy 263
structural discontinuity 15, 39, 292, 294–295, 329, 338, 341, 369, 380,
381, 388, 390, 408, 411
adjacent 408
intermediate 295, 341, 380
structure 15, 20, 23–24, 29, 31, 37, 46, 48–50, 56, 58, 72, 84, 90, 95, 97,
99, 105, 107, 128, 135, 137, 150–151, 227, 229, 231, 244, 250,
257, 261, 271, 278, 286, 291, 294, 327, 329, 336, 338, 341, 369,
380, 388, 390, 401
structural elements 329, 332, 338, 380, 401
symmetrical quadratic equation 191
t
T joint(s) 293, 296–297, 299, 301–304, 308–309, 311–315
tangential deflections 119, 121, 125, 128
tangential force 121, 128, 141–142
tangential vibrations 120
Test 29, 31, 56, 58–59, 90–93, 135, 137, 139–140, 253, 281, 368
theoretical natural frequency values 34, 37, 62
theoretical values of natural frequencies 28, 55
theoretical values of natural frequency 33
thickness 32–33, 55, 58–59, 61–62, 64, 67, 108, 117, 151, 159, 189,
253–257, 261, 266, 303, 305, 308, 315, 327, 344, 368, 382, 400,
413, 420
thickness ratio 382–383, 391
thin plate 151, 162–186, 212, 217–218, 223
thin plate bending theory 151
Three-mode Theory 263, 265–266, 269, 270–271, 273, 276–277, 280,
282, 284–286
three-story two-bay 294–295, 297–299
time dependence 16, 40, 82, 99, 120, 228–229, 231, 264–266,
288, 332
time domain 404–407, 409–410, 412
time harmonic motion 15, 18–19, 39, 69, 79, 97, 119, 152, 190, 227,
229–230, 263–264, 266, 288, 330, 377–378, 384
shaft 12–13
sharpness 387–388
shear 2, 7, 189, 197–198, 207, 227–228, 231–232, 238–242, 244, 246,
249, 251–257, 285
shear angle 229, 287, 330, 369
Shear bending vibration theory 228, 231, 238
shear coefficient 8, 14, 229, 252, 261, 287, 330, 369
shear deformation 7–8, 97, 111, 138, 190, 230–231, 233, 245,
249, 257, 287
transverse 227
shear distortion 227
shear force 1, 7, 8, 79, 81, 100–101, 151–152, 289, 294, 303, 333, 341,
344, 353, 371, 408
shear modulus 8, 97, 108, 117, 189, 212, 229, 252, 261, 287, 330,
368–369, 381, 400
shear rigidity 229, 231–233, 245, 264, 269
shear stresses 8, 189–190
sign convention 1–3, 6, 8, 10–14, 16, 40, 69, 121, 151, 232, 240, 246,
248–249, 267–269, 289, 294, 333, 341, 369, 374, 401–402, 405,
408, 411
simple support 158, 178–180, 182–183, 185–186, 190, 192–194,
196–197, 200–204, 206–208, 217, 219
Simple Support – Type I 192
Simple Support – Type II 193
Simply-supported boundary 40, 82–83, 89, 102, 115, 121, 123, 125,
127, 151–158, 160, 189–194, 197–198, 200–201, 203–205, 207,
209, 211–212, 217, 334–336
Simply-supported–Free 160
Simply-supported–Simply-Supported 160
single degree-of-freedom (DOF) spring-mass 369
single DOF spring-mass system 376, 381–382, 384–385
slender composite beams 111
slopes, bending 40, 44, 82, 101, 103–105, 125, 229–230, 232,
237–242, 245–246, 289–290, 307–308, 312–314, 319–320,
322–323, 325, 333, 336, 345, 349–352, 354, 359–362, 364, 366,
371, 374–375, 408
Space Frames 329, 338–339, 341–342, 367–368
n-story 338–339
two-story 341–342
three-story two-bay 294–295, 297, 299
span 24–25, 28, 52, 56, 58, 73–74, 84, 87, 117, 244–245, 251, 271, 273,
278, 280, 401–402, 406–407, 413–417, 420
span angle 120, 134, 144, 150
spatial 338–344, 347–351, 353, 357–358, 360–368
spatial angle joint 329
special situation 18–19, 43, 45, 67, 71, 97, 120, 125–126, 132, 189,
196, 198, 201–203, 205, 209–212, 214, 216–217, 219, 223, 227,
231, 236, 238, 242–243, 248–249, 382, 384
spring 16–20, 32, 38, 40, 42–43, 45, 71, 76, 96, 101–102, 117, 121,
124–125, 143, 233, 235–236, 239, 241, 244, 369, 374–375,
377–379, 382, 386–388, 390, 395, 400, 403, 405, 408, 411
attached 388, 401–402, 405, 408, 411, 420
attached boundary 18, 42–43, 124, 143, 235, 236, 405, 411
attached end 18, 43, 71, 125, 236
attachment 16–17, 19, 117, 386, 388, 395–397, 400, 408
constant 376, 378, 382
stiffness 19, 42, 102, 242, 386, 374, 402, 405, 407–408, 411
spring and mass attachments 40, 121, 233
spring and viscous damper 402–403, 405, 407–408, 410430 Index
tuning 412
Two DOF attachment 384
two DOF spring-mass system 369, 378–379, 381–383
two-story space frame in figure 342
Type I 196, 202–203, 205–212, 214, 216–217, 219
Type II 189–191, 193–194, 196, 200–215, 216–219, 222–223
two wave mode transitions 282
two-dimensional rectangular plates 151, 189
Type I – Type I support 212, 217
Type I simple support 190, 192–193, 196, 200–201, 203–204, 206–207,
208, 217, 219
Type I simple support – Clamped 217
Type I simple support – Free 219
Type II – Type II support 212, 217
Type II simple support 190, 193–194, 196, 200–208, 210–219, 222–223
Type II simple support – Clamped 217
Type II simple support – Free 219
u
uncoupled bending 98
uniform beam 15–16, 20, 24–25, 27–28, 38–40, 46, 52, 54–56, 69–72,
74–76, 79, 81, 84, 87, 89, 96, 104, 107, 111, 113, 117, 227–228,
230, 250–252, 263–265, 278, 281–282, 287, 289, 293–295, 327,
329, 332, 338, 341, 369, 380, 383–391, 394–395, 400–401,
407–408
uniform cantilever beam 67, 261
uniform cantilever composite beam 117
uniform composite beam 97, 98, 100, 104–105, 107, 114, 117, 390
uniform plate 154–155, 159, 195–196, 201, 205
uniform steel beam 27, 38, 55, 67, 96, 252, 261, 281, 381–382, 384,
386, 389, 400, 413
uniform waveguide 154, 159, 326, 367
v
vector equation 307–308, 312–315, 320, 322–326, 349–351, 353,
360–367, 374
continuity 307–308, 312–313, 315, 320, 322, 324–325, 349, 351,
353, 360, 362–364, 365–367
equilibrium 307–308, 313–315, 320, 323–324, 326, 350–351, 353,
361, 363, 365, 367
velocity
phase 39, 227, 229, 231, 254, 281–282
transverse 263
vertical beam elements 292–293, 295, 338, 342, 353
vibrating 20, 46, 69, 71, 79, 84, 96, 100
vibrating shafts 32
vibrating string 32
Vibrational Power Transmission in Curved Beams 149
vibration analysis 1, 15, 28, 55, 90, 285, 329, 338, 380
vibration characteristics 382–383, 385–386, 388, 390
vibration control, bending 401, 410, 412, 416
vibration energy 408, 410, 412
bending 410, 412
incoming longitudinal 404, 406
reflected 403, 406
total 404
vibration isolation standpoint 31, 58, 137
Vibration mode 23, 93, 134, 221–223, 225–226
Timoshenko 7, 10, 227, 230–232, 238–244, 246–247, 249–261, 285,
287–288, 290, 292, 307–308, 312–315, 320, 322–326, 329, 331,
334–337, 349–353, 360–367, 369–372, 374–375, 377–382, 390,
396, 399, 400
bending vibration theory 7, 10, 230–232, 238, 243, 247, 250,
253–254, 257–258, 260, 285, 287–290, 292, 307–308, 312–315,
320, 322–326, 329, 331, 334–337, 349–353, 360–367, 369–372,
375, 381
Timoshenko and Shear 238, 240–242, 244, 246, 257
torque 12, 14, 100–101, 105, 107, 113, 151–152, 333–334, 336, 341,
344, 353
internal resistant 12–14
torsional deflection 12–14, 330, 334, 336, 341
torsional propagating 99–100
torsional rigidity 13, 330
torsional rotation 97, 100, 103–105, 108, 111–112
torsional theories 337
torsional vibration(s) 1, 13, 14, 97, 104, 108, 326, 329, 331,
336, 349, 390
torsional vibrations in composite beams 97
torsional vibration waves 326, 329, 336
torsional wavenumber 109, 331
torsional waves 98, 329–330, 332, 348, 358
torsionally vibrating shafts 32
transition of wave mode 156, 244, 247, 250, see also wave mode
transition(s)
translation 2, 7, 241, 373–374
translational constraints 40, 102, 233, 235, 241, 303, 373, 390
translational motion 42, 236, 344
translational spring 101, 103, 117, 386, 400, 410
spring stiffness 42, 101, 123, 236, 241, 373, 386, 389, 400, 407, 410
transmission 37, 66, 102–103, 117, 304–305, 307–326, 338, 341–365,
367, 369–372, 374–375, 381, 390, 393, 401–403, 405, 407–408, 420
transmission and reflection coefficients 401–403
transmission and reflection matrices 102, 117, 369, 374, 381, 393,
407–408
transmission matrix 39, 104, 107, 294–295, 297, 303, 306–308, 311,
313–315, 319, 321, 323–324, 326, 338, 340, 347–348, 350–351,
353, 357–358, 361, 363, 365, 367, 371–373, 375–376, 390, 408
transmission relationships 15, 293–296, 303–306, 309, 311, 315, 318,
338–342, 345, 347, 354, 357
transmitted and reflected bending waves 407
transmitted and reflected nearfield wave components 408
transmitted and reflected propagating waves 408
transmitted propagating bending wave component 408
transmitted waves 103, 305–306, 309, 311, 315, 318–319, 345,
347–348, 354, 357, 369–370, 401
transverse contraction 263–264, 266
transverse deflections 1–2, 4, 7, 40, 44, 79, 82, 97, 100, 103–105, 108,
151, 263, 289–290, 330, 333, 345, 354, 376, 378
transverse displacement 44, 237, 242
transverse force 39, 50–52, 84–88, 105, 107, 111, 113, 244–248,
251–252, 291–292, 377, 379
applied 39, 50–51, 85–86, 88, 244–247, 379
transverse load 1
transverse shear deformation 227
T-shaped Frame 287
T-shaped Joint 308Index 431
wave propagation 16, 20, 37, 40, 46, 71, 100, 120, 127, 154–155, 159,
195, 201, 205, 228, 250–251, 264, 278, 286, 289, 294, 326, 329,
332, 342, 367, 399
Wave Reflection 16–17, 19, 40–41, 43, 45, 82, 100–103, 121, 123, 125,
154–158, 192–193, 195, 197–205, 207, 209, 211, 232–233, 235,
237, 239, 241, 243, 267–269, 289–290, 304, 307–308, 312, 315,
321, 326, 333, 335, 367, 370–375
wave relationships 117, 127, 131, 133, 195, 201, 205, 252, 278, 280,
294, 341, 380–381
waves 15–16, 24–27, 40, 49–50, 52–54, 72–75, 84–85, 87–89, 100,
105–107, 119–120, 128–129, 131–132, 134, 153–154, 156–157,
159–160, 195, 197–198, 227–229, 231, 244–245, 248, 251–253,
257, 264, 271, 275, 278, 280, 284, 289, 291–292, 295, 304–305,
307–325, 332, 336, 338–339, 341–365, 377, 379–381, 399, 401
axial 71, 271
coefficients of positive- and negative-going 249, 279
extensional 197–198
extensional in-plane 207
half harmonic 156, 196
incoming 381, 401
injected 326, 367, 377, 379
injecting 24, 50, 72, 105, 128, 244, 271, 291, 336, 380
nearfield 156, 228, 408
outgoing 250–252, 293, 338
positive-going 102, 156, 196–197, 208–209
quarter 201, 223–224
reflected 16, 40–41, 71, 82–84, 100, 102, 104, 121–123, 154,
157–159, 192–194, 199–200, 202–205, 233–234, 250, 267, 290,
306, 311, 318–319, 334, 347–348, 357, 369, 374, 401, 405
Waves generated by external excitation(s) 50, 84, 106, 129, 244–245,
247–249, 271, 273, 275, 291, 336
waves generated by spring forces 379
Waves in Beams on a Winkler Elastic Foundation 69–96
wave standpoint 15, 20, 39, 46, 72, 84, 127, 227, 292, 338
wave vectors 26–27, 40, 46, 52, 55, 74, 76, 81, 87, 89, 105, 108, 121,
128, 131, 133, 155, 159, 195, 201, 228, 278–279, 289, 294, 306,
311, 319, 333, 341, 348, 358, 372–373, 375–376, 381
Wave Vibration Analysis 117, 151, 154–57, 159, 192–193, 195–211,
326, 342, 367
wave vibration standpoint 104, 119, 151, 189, 263, 377, 379
wavenumber 15–16, 33–34, 39–40, 63, 65, 70, 79, 81, 83, 97–100,
108–109, 119–120, 134, 145, 153–154, 156, 159–160, 189–190,
192, 195–198, 201, 205–209, 211, 227–232, 238, 240, 246, 253,
254, 255, 257, 263–266, 281–284, 288–289, 307–308, 312–314,
320, 322–323, 325, 330–332, 349, 360, 369, 370, 403, 409, 411
width 32–33, 44, 55, 58–59, 61–62, 64, 67, 108, 117, 125, 151, 237,
253–257, 261, 327, 344, 368, 391, 400, 413, 420
Winkler Elastic Foundation 70–96, 76, 79, 81, 84, 89–90, 93, 96
x
x-axis 1, 3, 10, 12, 97, 294–295, 297, 303, 329–330, 336,
341–342
y
y anti-symmetrical mode 160, 187
Y joint(s) 338–339, 341, 343–344, 348–351, 353
y symmetrical mode 160, 187
vibration motions 134, 139
bending 108
coupled 119, 390
vibrations 15, 23, 27, 31, 37, 55, 58, 66, 72, 84, 95, 97, 104, 116, 134,
149, 226, 253, 260, 281, 286–287, 292, 326, 329, 338, 367, 369,
380, 387, 389, 399, 413, 416, 419
vibration tests 31, 58
vibration theories 234, 244, 250, 254, 263–286, 380
advanced bending 59, 227
elementary 327, 329, 331, 368
engineering bending 231–232
vibration waves 117, 154, 159, 293, 296, 305, 309, 315, 332, 338, 345,
354, 377, 379, 401
applied force injects 377
bending 69, 401
incoming 295, 341
injecting 84
longitudinal 15, 282
reflected 381
reflected longitudinal 401
viscous damper 18–20, 369, 386–387, 402–403, 405, 407–408,
410–411, 420
attachment 18–19, 401–403, 405, 407–408, 410–411
boundaries 22
viscous damping 18–19, 42, 236, 374, 386, 387
constant 18, 402, 408, 411
effect 408
volume mass density 5, 10, 12, 15, 32, 33, 39, 69, 79, 97, 108, 117,
119, 134, 150–151, 159, 189, 227, 263, 281, 287, 330, 369, 403
w
wave amplitudes 99, 120, 197, 230–231, 265, 267
Wave Analysis 38, 67, 96, 260–261, 326–327, 368, 400
wave component(s) 20, 23–24, 26–27, 33–34, 46, 48–49, 52, 55, 62,
65, 70, 72, 74, 76, 84, 87, 89, 100, 105, 108, 116, 120, 128, 131,
133, 153, 155, 159, 191, 195, 201, 231, 238, 244, 247, 250–252,
260, 265, 267, 279, 280, 282, 288–289, 292, 294, 307–308,
312–315, 320, 322–326, 332–333, 338, 341, 349, 351, 353, 358,
360, 362, 364–367, 369, 377, 379, 381, 402, 405, 408, 411
wave control of bending vibrations 407, 409, 411, 420
wave control of longitudinal vibrations 401, 403, 405
wave control force 401
waveguides 15, 294–295, 341
wave modes 156, 265–266
wave mode conversion 287, 329
wave mode transition(s) 11, 69–70, 81, 120, 134, 160, 198, 223,
231–232, 254, 282, 284, 288, 331, 369, see also transition of
wave mode
wavenumbers 15–16, 33–34, 39–40, 70, 79, 83, 97–100, 108, 119–120,
134, 153–154, 156, 159–160, 189–190, 192, 195–198, 201,
205–209, 211, 227–232, 238, 240, 246, 253–255, 257, 263–266,
282, 284, 288–289, 307–308, 312–314, 320, 322, 325, 330, 332,
369–370
bending 39, 79, 81, 109, 227, 229, 288, 330–331, 409, 411
imaginary 198, 254
longitudinal 15–16, 70, 263, 281, 283, 288, 331
wavenumber sequence 307–308, 312–314, 320, 322–323, 325432 Index
z
z-axis 97, 330, 341, 344, 354
zero matrix/matrices 46, 52, 55, 84, 87, 89, 105, 108, 127, 131, 159,
195, 201, 278, 280
zero shear deformation 229, 231
y-axis 1, 294, 297, 330
Young’s modulus 4, 15, 27, 32–33, 38–39, 55, 67, 69, 79, 97, 108, 117,
119, 134, 150–151, 159, 189, 212, 227, 252, 261, 263, 281, 287,
327, 330, 368–369, 381, 400, 403, 413, 420
Y-shaped Space Frame 329, 343, 36Index 423
control force 401, 404, 405–407, 409–410, 412–413, 420
wave 401
control gain(s) 404–407, 410, 412
controllers 403–406, 409–410, 412–414, 416, 420
ideal 410, 412, 416
convex 1, 4, 7
coordinate systems 1, 72, 84, 294, 297–299, 301, 303, 341–342
local 294, 297, 303, 342
rotating local 299–301, 303
varying local 298–299, 303
coupled bending 97, 104, 390
coupled torsional 108
coupled vibration 119, 390
coupled waves 97, 119
in composite beams 97–116
in curved beams 119–150
coupling coefficient 97–98, 108, 117
critical frequency 264, 282, 285
cross (“+”) joint(s) 293–295, 297, 301, 303–304, 320, 323–326
cross-sectional area 10, 12, 15, 39, 69, 79, 97, 119, 227, 263, 287, 330,
369, 403
curvature 2–4, 7, 119–120, 134, 144, 150
curved beam 119–121, 125, 127–128, 130–134, 138, 141–142,
144, 150
clamped uniform 138
uniform 119, 127, 130–131, 133–134
curved beam theory 121
cut-off circular frequency 156
cut-off frequency 69, 70, 76, 81, 89, 93, 96, 119–120, 134, 137, 150,
160, 187, 198, 223, 231–232, 244, 247, 250, 254, 265–266, 282,
284, 288, 331, 369, 371
d
D control/controller 404–406, 409–410, 412–416, 420
damper 16, 32, 386, 388, 408, 411
damper attachment 387
damping 19, 228, 229, 231, 241, 254, 257, 369, 389
damping constant 387, 405, 407, 410
damping effect, viscous 19, 42, 236, 386, 408, 411
dB Magnitude 37, 62, 66, 76, 89, 110, 116, 134, 137, 148–149, 254,
260, 327, 368, 395, 398, 419
decaying 40, 156, 228, 288, 331, 407–408, 410
decaying bending wave component 408
decaying bending waves 40, 228, 288, 331, 407, 410
decaying waves 46, 70, 81, 134, 156–157, 160, 198, 229, 231, 244, 247,
250, 254, 282, 284–285, 288, 408
negative-going 70, 198, 229, 254
single pair of 157, 284
decoupled governing equations of motion 232
deep composite beams 111
deflections 2, 6, 10, 12, 14, 24, 26–27, 49–50, 52–53, 55, 69, 72,
74, 76, 85, 87, 89, 108, 119, 128, 131, 133, 251–252,
267–269, 275–276, 279–280, 371–372, 374, 376–379, 401, 404,
413, 420
bending 39, 50, 85, 227, 229–230, 232, 237–242, 244–246, 248–249,
287, 291, 306–307, 311, 313–314, 319, 321, 323–324, 348–351,
358, 360–361, 363, 365, 369, 371, 374–375, 408
deformation 1–4, 7–9
bending 2, 4, 7
convex 4, 7
determinant 20–22, 33, 46–48, 72, 84, 98, 105, 119, 128, 155–156, 159,
191, 196–197, 201, 205, 207–208, 219, 223, 229–230, 250,
265–266, 278, 294, 341, 381–382
determinant of the coefficient matrix 20, 46, 72, 84, 98, 105, 119, 128,
155–156, 159, 191, 196–197, 201, 205, 207–208, 219, 223,
229–230, 250, 265–266, 278, 294, 341, 382
differential equation 15, 39, 69, 229–230
digital finite impulse response (FIR) filter 410
direction 1, 12, 151–186, 189–217, 219–225, 294, 303, 336,
341–342, 344
axial 402
given 4, 401
longitudinal 16
positive 294, 297, 303, 341–342
tangential 119, 121, 125
upward 297
y-axis 5, 8
disappearance of resonant peaks 29, 58, 92, 135, 137
discontinuity 15, 103–104, 117, 294–295, 341, 369, 371–373, 375–376,
379, 381–382, 384, 386, 394, 397–398, 400–402, 408, 411,
417–418
active 401
attached intermediate 117
attaching multiple identical 388
intermediate spring 398
joint 295
discontinuity component, single 382, 400
discriminant polynomial 282
dispersion equation 15, 39, 70, 79, 98, 119–120, 134, 148, 153, 191,
227, 229, 230, 263, 265–266, 282
dispersion relationships 119, 134–136
dispersive 39, 227, 229, 231, 254
dispersive longitudinal waves 282
displacements 19, 121, 125
in-plane plate 189–190
longitudinal 263
distance 4, 7, 16, 20, 24–27, 29, 40, 46, 49, 52–56, 70, 72–76, 81,
86–87, 89, 100, 104, 108, 121, 127, 130, 133, 154–155, 159, 195,
201, 228, 250–252, 264, 278–280, 289, 293, 332–333, 378, 381,
408, 411, 413, 420
dynamic equilibrium 305, 308, 315, 344, 353
dynamic spring stiffness 19, 42, 236, 386, 401, 408, 411
dynamic stiffness(es) 241, 373–374, 405
e
edge(s) 152, 160, 190, 196, 207–208, 212, 217–218
elastic foundation 69–72, 76, 79, 81, 84, 89–90, 93, 96
elementary longitudinal equation of motion 10–11
elementary longitudinal vibration theory 15, 263, 268, 272, 276, 282,
284–286, 287–290, 292, 329, 331, 401–402, 416
elementary theory 15, 287–288
elementary torsional equation of motion 12–13
end mass 44, 66, 125–126, 143–144, 236, 261
Endevco 2302 impact hammer 327
energy, optimal 404–406, 409–410, 412, 420
energy absorption, predicted 413–414424 Index
external force(s) 15, 24–27, 29, 35, 39, 50–54, 56–58, 72–76, 84–86,
87–92, 128–130, 132, 134–135, 228, 230, 245–249, 251–252,
258, 271–272, 275–276, 279–280, 291, 329, 336, 376–377, 381,
400, 402, 405, 408, 411
externally applied axial force 24, 26, 72–74, 291–292
externally applied end force 26, 74, 276–277
externally applied point force and moment 244–245, 247, 249
external transverse force 52, 86–87, 245, 248, 251–252
f
feedback bending wave control 407–408, 410–411
feedback bending wave controller 408, 411
feedback controller 401, 403–408
feedback control of flexural waves in beams 419
feedback longitudinal wave control 401, 405
feedback longitudinal wave controller 403
feedback PD wave controller 413–414
feedback wave control 401–403, 406–407, 416
feedback wave controllers 401, 403, 413–414, 416
Finite Impulse Response (FIR) filter 410, 412
fixed boundary 17–19, 21, 71, 76, 268–271, 280
fixed-damper boundary 22–23
fixed-fixed boundary 21, 23, 76–77
fixed-free boundary 21, 23, 76, 78
fixed-mass boundary 22–23
fixed-spring boundary 22–23
flexural/bending vibration of rectangular isotropic thin
plates 151–188
force 12, 24–26, 29, 35–37, 50–51, 53–54, 57, 65, 72–76,
86, 88, 91, 107, 128, 130, 132–133, 147, 241, 244, 246,
248, 250, 260, 276–277, 285, 291, 374, 379, 381, 393,
417–418
impact 31, 58, 253, 281
radial 121, 128, 141
flexural vibrations, see Euler–Bernoulli
forced in-plane bending vibrations 67, 261
forced out-of-plane vibrations in planar frames 326
forced response 36, 66, 147, 394–395, 418–419
forced vibration(s) 25–26, 29, 31, 34, 50, 51, 54, 73, 75–76, 86, 88,
107–108, 119, 130, 133–134, 150, 227, 251–252, 263, 281, 326,
367, 382, 400
forced vibration analysis 24–25, 28–29, 31, 34, 50–51, 53, 55–56, 64,
72, 76, 90, 105, 107, 135, 144, 250–251, 253, 258, 277–279,
292–293, 295, 297, 299, 301, 303, 338–339, 380
curved beam 128–129, 131, 133–144
forced wave vibration analysis 28, 56, 84, 116, 382,
390, 399
force generated waves 24–26, 35, 50–51, 53–54, 65, 72–73, 75, 84, 86,
88, 105, 107, 128–131, 133, 147, 248, 250, 260, 277, 291, 336,
337, 393, 417
frame 287, 293–295, 303, 327, 399
free bending vibrations 227, 369
free body diagram 3, 10–13, 16–19, 24, 26, 41–42, 44, 50, 53,
71–75, 84–85, 87–88, 101, 103, 105–106, 124–125, 128–129,
132, 233, 235–236, 241, 245, 247–248, 291, 294, 297-301,
304–305, 308–309, 315, 317, 341, 344, 374, 376, 378, 402,
405, 407, 408, 411
free boundaries 16, 21, 38, 47, 102, 160–161, 169–170, 172–176, 178,
182–183, 185–186
equation(s) of motion
“+” joint 315
cross joint 315
bending waves in beams on Winkler elastic foundation 79
coupled bending-torsion vibration in slender composite beams 97
elementary longitudinal vibration theory 11–12, 15, 329–330
elementary torsional vibration theory 13–14
Euler-Bernoulli bending vibration theory 6, 39, 232, 287, 329
in-plane vibration of thin plate 189
L joint 304-305
longitudinal waves in beams on Winkler elastic foundation 69
Love’s in-plane vibration theory in curved beams 119
Love longitudinal vibration theory 263
Mindlin-Herrmann longitudinal vibration theory 264
out-of-plane vibration of thin plate 151
Rayleigh bending vibration theory 227, 232
Shear bending vibration theory 228–229, 232
single DOF system 376
spatial K joint 353
spatial Y joint 343
T joint 308
two DOF system 378
Three-mode longitudinal vibration theory 265–266
Timoshenko bending vibration theory 9–10, 230, 232, 287, 329–330
equilibrium 15, 39, 72, 299, 301, 305–309, 311–315, 318, 320–321, 323–324,
326, 345, 347, 350–351, 353–354, 357, 360–361, 363, 365, 367
vector equation on 307–308, 313–315, 320, 323–324, 326, 350–351,
353, 361, 363, 365, 367
equilibrium conditions 17, 19, 24, 41, 44, 50–51, 53, 82, 85, 88, 101,
103, 106, 121, 125, 128–129, 233, 236, 240, 242, 245–247,
268–269, 271–274, 290–291, 334, 336, 374, 377, 402–403, 405,
408, 411
equilibrium equations 53, 87, 102, 104, 106, 132, 241–242, 247, 273,
275–277, 279–280, 303–304, 306–308, 312, 314–315, 318, 320,
323–326, 341, 347, 357, 361, 374
equivalent 306, 311, 318, 347, 357
scalar 306–308, 311–312, 314–315, 318, 320, 323–324, 326, 347,
350–351, 353, 357, 360–361, 363, 365, 367
equilibrium relationships 105, 294
equivalent continuity equations 305, 310, 316–318, 347, 357
Euler–Bernoulli 2, 6–8, 39–42, 50–51, 53, 58–59, 63, 65,
82–84, 97, 101, 116, 227, 231–236, 238, 246, 248, 253–257, 261,
285, 287–290, 292, 307–308, 312–315, 319–320, 322–326,
329–330, 333–335, 337, 349–353, 359–367, 369–372, 374–375,
377, 379–381, 396, 400–401, 408, 411, 416
experimental results 15, 31, 39, 58, 252, 257, 281, 286
experimental setup 31
experimental studies 27, 29, 31–32, 55, 57, 59, 252–253, 256–258,
281, 283, 285, 327, 367
experimental equipment
Brüel & Kjær PULSE Unit 31, 58, 282, 368
Brüel & Kjær Type 4397 accelerometer 31, 58, 253, 282
Brüel & Kjær Type 8202 Impact hammer 31, 58, 253, 282, 368
Endevco 2302 impact hammer 327
PCB 353 B12 Accelerometer 327
external excitation(s) 15, 24–29, 31, 38–39, 50–58, 67, 72–76, 84–92,
105–108, 111, 128–135, 137, 139, 146, 150, 228, 230, 244–249,
251–252, 258, 261, 271–280, 286, 291–292, 294, 297, 326, 336,
341, 367, 369, 376–377, 380–381, 400, 402, 405, 408, 411, 413, 420Index 425
Hz 29, 31, 33, 37, 56, 58, 62, 66, 110, 116, 134–135, 138, 145, 148–149,
258–260, 327, 368, 391, 395, 398, 414, 417, 419
i
identity matrix 16, 40–41, 46, 52, 55, 71, 82–84, 87, 89, 100–103, 105,
108, 121–123, 127, 131, 154, 157–159, 192, 195, 199–205, 207,
233–234, 250, 267, 278, 280, 290, 294, 297, 303, 306–308,
311–315, 319–326, 334, 340, 347–351, 353, 357–358, 360–367,
369–370, 372–376, 401, 405, 407–408, 410
imaginary part 116, 136, 253, 255, 260, 281, 283
imaginary unit 15, 39, 69–70, 79, 97, 119, 190, 227, 229–230,
263–264, 266, 288–289, 291, 307–308, 312–314, 320, 322–323,
325, 332, 370–371, 403
incident waves 16, 40–41, 71, 82–83, 100–103, 121–123, 154,
157–159, 192, 199–200, 202–205, 233–234, 267, 290, 294,
297, 303, 306–308, 311–315, 319–326, 334, 340, 347–351,
353, 357–358, 360–367, 369–370, 372–376, 401, 405,
407–408, 410
incoming longitudinal vibration energy 404, 406
inertance frequency response(s), see frequency response(s)
inertia 5, 7, 12, 39, 44, 67, 79, 97, 119, 125, 227, 237, 261, 263–264,
287, 303, 330, 344, 369, 378, 382
infinite shear rigidity 229, 231–233, 245
injected waves 130, 292, 294, 297, 326, 337, 341, 367, 377, 379
in-plane 67, 119, 138, 149, 189–190, 193–194, 196, 201–202, 207, 226,
261, 287, 326–327, 330
in-plane bending wavenumbers 330
in-plane plate vibrations 193–194, 196, 201–202
one-dimensional 207
In-Plane Vibration of Rectangular Isotropic Thin Plates 189–226
integer multiples, odd 223–224
integer number of half waves 160
intermediate spring support 390
internal resistant axial force 10, 69, 289, 402, 405
internal resistant bending moment 4, 233, 408
internal resistant force(s) or/and moment(s) 1, 10, 12, 16, 24, 26, 50,
53, 72, 74, 85, 87, 128, 132, 245, 247, 267–269, 275, 291, 294,
307–308, 312, 314–315, 320, 323–326, 371, 374, 402
internal resistant shear force (and bending moment) 1–2, 8, 40–41,
81, 100, 232–233, 240, 246, 289–290, 303, 333, 344, 369–371,
373, 409
isotropic 151, 159, 189, 198, 207, 212, 226
j
joint parameters 350–351, 353, 360, 363, 365, 367
joints 293–297, 299, 303–304, 315, 317, 329, 338, 340–342
spatial angle 329
joints and boundaries 329, 338
k
K joint(s) 338, 341–343, 353–354, 357–367
kinetic energy 263, 272, 276
K-shaped 329, 353
l
L joint(s) 293, 296–297, 299, 301–308
load vectors 51, 86–87, 130–131, 246–249, 251–252, 273, 275–277,
279–280, 292, 337, 381
free boundary 16–17, 20–21, 26, 41, 43, 45–47, 54, 67, 82–83, 88, 102,
121, 125, 132–133, 150, 157–158, 160–161, 169–170, 172–176,
178, 182–183, 185–186, 199, 202–204, 233–234, 236, 238, 240,
242–243, 248–250, 268–270, 290, 334–235, 401–402, 405, 410,
413, 416
free end 18–19, 26–27, 31, 33, 38, 53–54, 58, 67, 74–75, 87, 89, 115,
125, 132–134, 247, 249, 252–253, 259, 261, 275, 280, 286
free flexural vibrations 151, 189, 226, 327
free longitudinal vibrations 96, 263, 264–265
free vibration 15, 28, 39, 69, 97, 151–152, 188–189, 232, 250, 287, 294,
297, 340, 368, 370, 399
free vibration analysis 20–21, 23, 28–29, 32–33, 46–47, 49, 55–57,
61–62, 71, 76, 84, 90, 104, 109, 117, 127, 135, 149, 250, 278,
327, 368
Planar Curved Beams 149
free-free boundary 20–23, 28, 31, 33, 38, 46, 47–48, 55, 160, 219, 252,
281
free vibration responses 250, 294, 297, 340
frequency domain 401–402, 404, 408
frequency resolution 31, 58, 134, 253, 282
frequency response(s) 29, 31, 57–58, 92–93, 111, 113, 135, 137, 282
inertance 31–32, 58–59, 253, 255–257, 285–286
receptance 29–30, 38, 56–57, 67, 76, 80, 90, 92–94, 135, 141–142,
150, 261, 382–388, 400, 413–416, 420
steady state 15, 39, 119
frequency span 31, 58, 253, 281
functions
exponential 153
parabolic distribution 266
g
generated wave(s) 24–26, 35, 51, 54, 65, 73–75, 86, 88, 107, 130, 133,
246, 248–250, 260, 271–273, 275–277, 393, 417
geometrical parameters 32–33, 61–62, 144
geometrical properties, see material (and geometrical) properties
governing equation(s), see also equation(s) of motion
governing equation of bending vibrations 39
governing equation of longitudinal vibrations 15
governing equation of motion for free bending vibrations 79, 227
governing equations 15, 39, 69, 79, 97, 99, 119, 191, 227, 229, 231,
263, 265, 287, 294, 329, 331, 341
decoupled 232
higher fourth-order partial differential 39
governing equation(s) of motion 1, 7, 39, 69, 79, 97, 119, 151–152,
189, 227–228, 230–232, 263, 287, 369
governing equations of motion for free bending vibration 228, 230
governing equations of motion for free in-plane vibration 119
governing equations of motion for free vibration 39, 69, 97, 152, 287
gravity 376, 378, 382
h
half harmonic waves 156, 196
half waves 160
integer multiples of 159–161, 219–220, 223
Homework Project 38, 67, 96, 117, 150, 261, 327, 368, 400, 420
Hooke’s law 8, 17, 71
horizontal beam elements 292–293, 338, 342
H-shaped frame 287, 327426 Index
mechanics 10, 12–14
Mei 99, 101, 103, 111, 116, 231, 260, 263, 286, 305, 309, 326, 339,
342–343, 354, 359, 367, 378–379, 399, 405, 407, 410, 416, 419
Mindlin-Herrmann 263–266, 268, 269, 272, 276–277, 280, 282,
284–286
minimum, local 76, 134
miscounting of natural frequencies 58, 137
modal approach 329
mode number 77–78, 110, 134, 139
modes 28–29, 35, 55–56, 58–59, 64, 76, 79, 89–93, 111–113, 134–140,
150, 160, 164–165, 169, 173–174, 178, 182, 186, 269–271
anti-symmetrical 160, 187
body 55–56, 64
corresponding 76, 93
one-dimensional 196, 201–202
rigid body 21–22, 28, 34, 47–48, 62, 64
symmetrical 160, 187
modeshape amplitudes 24, 49
modeshape curves 29, 57, 89
modeshape 15, 20–21, 23–24, 28–29, 33–34, 38–39, 46-49, 55-57, 62,
64, 67, 72, 76, 78–79, 84, 89–91, 104–105, 111, 119, 127–128,
134–135, 138–139
corresponding 23, 84
normalized 34, 64, 76, 89
overlaid 76, 79, 91, 134, 138
modeshapes by mode number 139
modes of vibrations for flexural motion 134
mode transition point 282
moment 1, 6, 10, 14, 50, 52–56, 65, 67, 84–90, 92, 105, 128–134, 141,
151–152, 241, 244–247, 249–252, 260, 291–292, 294, 297,
307–308, 312, 314–315, 320, 323–326, 329, 333, 336, 341, 344,
371, 373, 393
motion 1–15, 23, 32, 39, 48, 69, 79, 97, 119, 128, 151–152, 156, 189–190,
227–232, 263–265, 287, 294, 329–331, 341, 369, 376, 378
flexural 134–135, 137–138
mounting 382, 384, 388
Multi-story Multi-bay Planar Frame 292–293, 295–325
Multi-story Space Frame 338–365
n
natural frequency 15, 20–24, 28–29, 31–34, 37–39, 46–49, 55, 57–59,
61–62, 67, 72, 76–78, 84, 89–91, 93, 96, 104–105, 109–110, 117,
119, 127, 128, 134–135, 137–138, 150, 155–156, 159–160,
162–167, 169–170, 172–180, 182–183, 185–187, 196–197, 201,
206–209, 211–219, 221–223, 225, 250, 254, 257, 261, 278,
284–285, 294, 327, 341, 368–369, 381–385, 387–389
measured 327, 368
non-dimensional 21, 90, 160, 164, 169, 173, 178, 182, 186
Natural Frequencies and Modeshapes 20–21, 23, 46–47, 49, 104, 127
natural frequencies of a structure 20, 46, 58, 72, 137, 257, 278, 369
natural frequencies of the beam 28, 46, 55, 72, 84, 90, 96,
250, 381, 388, 389
natural frequencies of the two DOF spring-mass system 382
Natural frequency expressions, non-dimensional 23
natural frequency values 24, 48–49, 93
nearfield wave 156, 228, 408, 411
negative directions 1, 16, 153, 157, 198, 227, 229, 231, 253
negative-going waves 70, 100, 120, 154, 195, 198, 201, 248–249, 264,
273–277, 279–281, 288–289, 292
local coordinates 294, 297, 303–304, 315, 341, 343
local coordinate systems 294, 297–300, 303, 306–307, 311, 313–314,
319, 321, 323–324, 341–342, 348, 350–351, 358, 361, 363
local minima 76, 89, 93, 109, 134, 160, 219, 223, 254, 285, 382–383,
385, 387, 389
local minimum 76, 134
location 29, 31, 37, 49, 56–59, 66, 76, 79, 91–93, 135, 137, 139–140,
154–155, 157–159, 192–193, 195, 198–200, 202–204, 292–294,
296–297, 339, 341, 389, 401
ideal 31, 58, 137
longitudinal axis 1, 59, 253, 257, 294, 341
longitudinal and torsional wave components 333, 358
longitudinal deflection 10, 15, 17–19, 24, 69–70, 72, 263–264, 266,
287, 330, 334
longitudinal force 15, 17, 303, 333–334, 344
longitudinal motion 134–135, 137–138
longitudinal propagation coefficient 16, 20, 25, 27, 70, 72–73, 75
longitudinal vibration 11–12, 15, 38–39, 96, 263–273, 275–277,
280–282, 284–285, 287, 401–404, 406–407, 413, 416, 419
free 15, 96, 263–265
longitudinal vibration control 395
longitudinal vibration theory 17, 263, 265–271, 276–277, 280–282,
284–285, 289–290, 292, 401–402
longitudinal wave control design 408, 411
longitudinal waves 15–16, 70, 120, 281–282, 287–326, 401–402, 405
longitudinal wavenumber 15, 16, 70, 263, 281, 283, 288, 331
longitudinal waves in beams 15–38, 69, 71, 73, 75, 77
longitudinally vibrating beams 32, 263
Love longitudinal vibration theory 119, 121, 138, 263, 265, 267–268,
271, 275–277, 280–282, 284–286
low frequency mechanical vibrations 389
L-shaped Joint 287, 304, 326
lumped mass boundaries 22
m
magnitude plots 96, 382, 389
dB 76, 109, 134, 160, 219, 223
magnitudes 28, 32–33, 37–38, 47–48, 55, 58, 62, 66–67, 76–77, 80,
89–90, 93, 96, 109, 116, 153, 161, 220, 224, 253–257, 261,
281–285, 349, 360, 382–390, 395, 398
mass 19–20, 32, 44–45, 67, 125–126, 143–144, 236–238, 241–243, 261,
303, 344, 368, 376, 378, 382
mass attached 19, 44–45, 67, 125–126, 143–144, 238, 243, 261
mass attachments 16, 19, 40, 44, 121, 125–126, 233, 236, 242
mass block 19, 44, 125–226, 237–238, 243, 382
mass block attachment 236, 238
mass density 27, 38, 55, 67, 108, 189, 212, 252, 261, 327, 368, 381,
400, 413, 420
mass moment of inertia 44, 67, 125, 237, 261, 378, 382
material coupling 97, 109, 116
material (and geometrical) properties 27, 38, 55, 67, 99, 108, 117,
134, 150, 159, 212, 223, 227, 229, 231, 252, 258, 261, 264, 281,
327, 354, 359, 368–369, 375, 381, 389, 400, 413, 420
materials 10, 12–14, 20, 32–33, 46, 61–62, 99, 108, 117, 134, 144, 150,
198, 227, 229, 231, 252, 264, 369–371, 382, 391, 408
MATLAB 15, 20–22, 28, 31–32, 39, 45–48, 56, 59, 96, 109, 114, 117,
119, 125–126, 135, 143, 145, 147, 227, 244, 253, 257–259, 382,
389–391, 393, 395, 397, 400, 407, 416–417, 420
matrix equation 45, 236, 238, 242–243, 293, 338, 377Index 427
bending wavenumbers 330
forced 326
free vibrations 159
plate vibrations 154
vibration 151, 152
over-predicts 58–59, 257
p
pair of decaying waves 134, 157, 198, 231, 244, 247, 250, 282, 284, 288
pair of propagating waves 134, 153, 157, 198, 231, 244, 247, 250, 254,
282, 284, 288
pair of simply-supported boundaries 153, 201, 207, 209, 211
pair of type I 202–203, 205–207, 209–214, 216–217, 219
pair of Type II 203–207, 210–212, 214–217, 219, 222–223
parabolic distribution 263, 266
parameters 1, 4, 7, 10, 12, 150, 264, 266, 303, 305, 309–310, 315, 318,
344–345, 347, 350–351, 353–354, 357, 360, 363, 365, 367, 369,
403, 408, 411
passing band 388–389
passive vibration control 388–389
Passive Wave Vibration Control 369–400
passively implemented control 405, 407
PCB 353 B12 Accelerometer 327
PD control/controller 404–406, 409–410, 412–415, 420
Periodical structures 388–389
phase speeds 198, 207
phase velocity 16, 39, 227, 229, 231, 254, 282
pinned boundary 42–43, 45–46, 48, 67, 233–236, 238, 241–243,
290–291
Pinned / Simply Supported Boundary 40, 45–46
Pinned-Pinned Boundary 48
planar frame 287, 292–295, 297, 299, 303–304, 326–327, 342
general 292
plane 1–2, 5, 7, 44, 67, 125, 151, 189, 237, 253, 261, 294, 303, 330–332,
336, 344–345, 348–349, 354, 358, 360
plate 151–153, 156–157, 159–161, 189–190, 198, 207, 212, 219, 223
uniform 154–155, 159, 195–196, 201, 205
plate flexural rigidity 151
Point Attachment 102–3, 117, 373–375, 381
point axial force 24, 72, 273
point control force 401
point discontinuity 402, 408, 411
point downwards 342
point force 27, 75, 244, 250
point spring 401–402, 405, 407
point support 102–104, 373–375
point transverse force 50, 84, 105, 107, 111, 113, 244, 377
Poisson effect 263
Poisson’s ratio 151, 159, 160, 187, 189, 198, 212, 253, 261, 263–264,
330, 368, 381, 400
polar moment of inertia 12, 97, 263, 330
polynomial equation, cubic 98, 282
positive direction of angle 1, 294
positive shear force 81, 101
positive sign directions 1–2, 10, 12, 79, 333
power 153
principle of d’Alembert 305, 308, 315, 344, 353
propagating waves 46, 70, 134, 153, 157, 198, 228–229, 231, 247, 250,
254, 282, 284, 288, 408, 410–412
negative wavenumber values 253, 281
neutral axis 1–4, 7
neutral line 7
Newton’s second law 10–12, 14, 19, 44, 125, 236, 301, 304, 308, 315,
343, 353, 376, 378
nodal point 29–31, 35, 56–58, 64, 76, 90–92, 135, 137, 139
nodal point of modes 29, 35, 56, 58, 64, 135, 137, 139
nodal point of odd modes 29, 35
non-causal 410, 412
non-classical boundary 15–17, 19, 40–41, 43, 45, 100–101, 119, 121,
123, 125, 232–233, 235, 237, 239, 241, 243
non-dimensional coordinate 71–72
non-dimensional displacement 70, 82
non-dimensional flexibility coefficient 71, 76, 96
non-dimensional frequency 21–23, 46, 69, 81, 96
non-dimensional natural frequency 21–23, 47–48, 76–77, 160, 164,
169, 173, 178, 182, 186, 217–218, 222–223
non-dimensional stiffness 69, 76, 81, 89, 96
non-dispersive 16, 282
nonlinear equation 22–23
non-nodal point 29–30, 35, 56–58, 64, 76, 90–92, 135, 139
non-rigid body modes 55–56, 64
non-trivial solution 20, 46, 72, 105, 250, 278
normal strain 4, 7
normal stress 4, 5, 190
normalized modeshape 34, 64, 76, 89, 90
n-story m-bay 293
numerical examples 27, 55, 76, 89, 108–109, 111, 113, 134–135, 137,
159–160, 163–187, 212–213, 252, 281, 326, 367, 381, 383, 385,
387, 413, 415
Test A 135, 137, 139–141
Test B 135, 137, 139–140, 142
Numerical Examples and Experimental Studies 27, 29, 31, 55, 57,
252–253, 281, 283, 285
o
observation location 29, 58, 91–93, 135, 137, 139, 150
observation point 29, 58, 92
observations 139, 150, 160, 257, 413–414
occurrence of natural frequencies 160, 187
occurrence of resonant peaks 29, 57–59, 91, 93, 135, 140
odd modes 29, 35
on span control 413, 417, 420
opposite edges 151, 153, 159–160, 189, 191, 196, 201, 205, 207–209,
217, 219, 223
Opposite Edges Simply-supported 151–226
optimal D controller 404, 406, 410, 412
optimal energy absorbing D control gain 406
optimal energy absorbing PD control gains 404, 406, 409, 412
optimal energy absorbing PD controller 404–406, 409–410, 412
optimal PD controller 406
optimal performance at a frequency 410, 412
origin 17–19, 24, 26, 41–42, 44, 50–51, 53, 71–72, 74, 82–83, 85, 88,
102–104, 106, 122–124, 126, 128–129, 132, 154, 157–158,
192–193, 198–200, 202–204, 233–235, 237, 240–243, 245–249,
268–274, 276–277, 307–308, 312–315, 320, 322–326, 334–336,
349–351, 353, 360, 362–367, 372–375, 403, 405
out-of-plane 159, 329–330
out-of-plane vibrations 151–152, 326428 Index
reflection matrix 27, 39–43, 45–48, 52, 54–55, 59,
82–84, 86, 88, 102, 104, 107, 115, 117, 121, 123–127, 130,
132–133, 143, 155, 157–159, 192–195, 199–201, 207, 233–244,
249–252, 257, 259, 267, 269–271, 278–280, 290, 293, 295–296,
306, 311, 319, 334–336, 339, 347–348, 357, 369, 374, 381, 393,
407–408, 410–411
reflection relationships 72, 83–84, 250, 293, 338–339, 420
resonances 387–388
resonant peaks 29, 57–59, 76, 91–93, 135–137, 140
response observation 29, 57–58, 76, 90–91, 93, 135, 137, 139
locations 29, 57, 91, 93, 135
points 76, 90–91, 135, 139
responses 24, 26–27, 29, 31, 38, 49, 52, 55–56, 58, 64, 67, 72, 74, 76,
87, 92, 108, 131, 133, 135, 137, 139, 150, 241, 252–253,
257–258, 261, 279–280, 285, 373–374, 381, 389, 391, 400, 417
dB magnitude 37, 284
forced 37, 76, 108, 261, 294, 297, 341
forced vibration 75, 135
imaginary 134, 137
right hand rule 12, 294, 341
right side 1, 10, 12, 24, 48, 50, 72, 85, 103–105, 128, 244–245, 251,
271–273, 291, 293, 296, 299, 300–304, 336, 338, 341, 371–376,
391, 402, 408
rigid body 19, 21–22, 28, 34, 44, 47–48, 62, 64, 67, 126, 237, 261, 303,
308, 344–345, 347, 354, 357, 378
joint related parameters 305, 309–310, 315, 318, 345, 347, 354, 357
mass attachment 126
motion 207
rigid mass block 45, 59, 67, 239, 242, 244, 257, 261, 376, 378, 382
ring frequency 120, 134, 150
rod theories 286
roots 20, 28, 46, 55, 72, 84, 99, 105, 120, 128, 134, 145, 148, 150, 153,
160, 191, 219, 223, 278, 282, 284, 381
rotary inertia 6, 97, 111, 138, 227, 230–233, 235, 257, 269
rotating coordinate systems 297, 303, 342
rotation 1–2, 7, 44, 121, 125, 128, 135, 138, 237, 241–242, 303, 344,
373–374
angular 378
rotation angle 294, 341
rotational motion 12, 14, 42, 236
rotational relationships 297–299, 303, 342
rotational stiffness 42, 235–236
rotationally asymmetric cross section 13, 330
rotationally symmetric cross section 330–331
s
scalars 277, 280, 307, 360, 372
scalar equation 305–306, 311, 318
scalar equations of continuity, six 347, 349, 351, 353, 357, 360, 362,
364–367
scalar equations of continuity, three 307–308, 310, 312–317, 320,
322, 324–325
scalar equations of equilibrium 306–308, 311–312, 314–315,
318, 320, 323–324, 326, 347, 350–351, 353, 357, 360–361, 363,
365, 367
sensor 35, 64, 258, 391, 417
separation of variables 15, 39, 69, 79, 97, 119, 152–153, 190–191, 227,
229–230, 263–264, 266, 288, 330
sequence 145, 164–165, 169, 173–174, 178, 182, 186
bending 99–100, 115, 331
propagating decaying waves 134
propagating wave component 408, 411
propagating waves 39, 46, 134, 153, 157, 198, 229, 231, 244, 247, 250,
254, 284, 288
reflected 408, 411–412
transmitted 410
propagation 15–16, 28, 39–40, 46, 55, 62–63, 65, 72, 84, 86, 88, 104,
107, 116, 120, 147, 154, 160, 195, 201–211, 250–252, 264, 289,
293, 338, 381, 393, 401, 418
propagation coefficient 15, 155, 205, 264, 280
propagation matrices 107, 117, 251, 259, 279, 339
propagation matrix 39–40, 46, 49, 52, 54, 81, 86, 89, 100, 104, 121,
127, 130, 133, 159, 195, 201, 228, 250, 252, 265, 267, 278, 280,
289, 293, 333, 381
bending 40, 46, 52, 54, 81, 228, 381
propagation relationships 15, 20, 25–27, 33–34, 39, 46, 51–52, 54,
69–70, 73, 75, 79, 86, 89, 97, 99–100, 104, 107, 127, 130, 133,
155, 159, 195, 201, 205, 227–229, 231, 250, 252, 263, 265, 267,
277–280, 287, 293–295, 297, 329, 331–332, 338, 341, 380
Proportional Derivative (PD) control 404–406, 409, 410, 412–415, 420
proximity, close 134
pseudo cut-off frequency 282, 284
r
Radial direction 119–121, 125, 128, 142
radial force 121, 128, 141–142
radial stiffness 123
radius of curvature 2–4, 7, 119–120, 134, 144, 150
Rayleigh 227, 231–235, 238–239, 245, 248, 251–257, 285
receptance frequency response(s), see frequency response(s)
rectangular (thin) plate(s) 151, 155, 159, 188–189, 195, 201, 205,
215–218, 226
References 4, 14, 37, 66, 95, 116, 149, 188, 226, 260, 286, 326, 367,
399, 419
reflected and transmitted propagating wave components 408, 410
reflected and transmitted waves 15, 39, 104, 292, 295, 305, 309, 315,
329, 338, 341, 345, 354, 369, 380, 401, 408, 410
reflected bending vibration energy 411
reflected bending waves 407, 410
reflected longitudinal vibration waves 401
reflected propagating waves 408, 411–412
reflected vibration energy 403, 406
reflected vibrations 401
reflected wave 16, 40, 41, 71, 82–84, 100, 102, 104, 121–123, 154,
157–159, 192–194, 199–200, 202–205, 233–234, 250, 267, 290,
306, 311, 318–319, 334, 347–348, 357, 369, 374, 401, 405
reflection 15, 20, 33, 46, 49, 71, 104, 107, 115, 127, 153, 155, 192, 195,
197, 250–252, 278, 293, 296, 304–307, 309–325, 338, 340–341,
343–365, 381, 390, 401–403, 405, 407
Reflection and Transmission in Curved Beams 149
reflection and transmission matrices 39, 104, 107, 294–295, 297,
306–308, 311, 313–315, 319, 321, 323–324, 326, 338, 340, 348,
350–351, 353, 358, 361, 363, 367, 371–373, 375–376, 390, 408
reflection and transmission relationships 15, 293–296, 303–306, 309,
311, 315, 318, 338–342, 345, 347, 354, 357
reflection coefficient 15–22, 25–26, 28, 46, 71–73, 75, 82, 154–155,
160, 197, 202–207, 210–211, 219, 267–268, 280, 401, 403,
405–406Index 429
spring attachments 16–17, 19, 41–42, 100, 123–124, 233, 235–236,
241, 379, 388–390, 395, 408, 411
discrete 389
intermediate 117, 395–397, 400
spaced 389
spring boundaries 22
spring forces 379
spring-mass 369, 376
spring-mass system 376, 378
single DOF 376, 381–382, 384
spring stiffness 19, 42, 236, 386, 407
steady state frequency responses, see frequency response(s)
steel beam 27, 31–33, 35, 38, 55, 61, 62, 67, 96, 134, 144, 150, 252,
257–258, 261, 281–282, 284, 368, 381–382, 384, 386, 390, 395,
400, 413, 416, 420
curved 134, 150
step change 382, 391, 394
step size 33, 35, 61, 64, 114, 145, 259, 391, 396, 417
stiffness 17, 38, 71, 76, 79, 90, 93, 97, 241, 387, 389–390, 410–411
dynamic 241, 374, 401
dynamic spring 19, 42, 236, 386, 401
normalized spring 242
stopping bands 388, 389
strain 3, 4, 8
strain energy 263
structural discontinuity 15, 39, 292, 294–295, 329, 338, 341, 369, 380,
381, 388, 390, 408, 411
adjacent 408
intermediate 295, 341, 380
structure 15, 20, 23–24, 29, 31, 37, 46, 48–50, 56, 58, 72, 84, 90, 95, 97,
99, 105, 107, 128, 135, 137, 150–151, 227, 229, 231, 244, 250,
257, 261, 271, 278, 286, 291, 294, 327, 329, 336, 338, 341, 369,
380, 388, 390, 401
structural elements 329, 332, 338, 380, 401
symmetrical quadratic equation 191
t
T joint(s) 293, 296–297, 299, 301–304, 308–309, 311–315
tangential deflections 119, 121, 125, 128
tangential force 121, 128, 141–142
tangential vibrations 120
Test 29, 31, 56, 58–59, 90–93, 135, 137, 139–140, 253, 281, 368
theoretical natural frequency values 34, 37, 62
theoretical values of natural frequencies 28, 55
theoretical values of natural frequency 33
thickness 32–33, 55, 58–59, 61–62, 64, 67, 108, 117, 151, 159, 189,
253–257, 261, 266, 303, 305, 308, 315, 327, 344, 368, 382, 400,
413, 420
thickness ratio 382–383, 391
thin plate 151, 162–186, 212, 217–218, 223
thin plate bending theory 151
Three-mode Theory 263, 265–266, 269, 270–271, 273, 276–277, 280,
282, 284–286
three-story two-bay 294–295, 297–299
time dependence 16, 40, 82, 99, 120, 228–229, 231, 264–266,
288, 332
time domain 404–407, 409–410, 412
time harmonic motion 15, 18–19, 39, 69, 79, 97, 119, 152, 190, 227,
229–230, 263–264, 266, 288, 330, 377–378, 384
shaft 12–13
sharpness 387–388
shear 2, 7, 189, 197–198, 207, 227–228, 231–232, 238–242, 244, 246,
249, 251–257, 285
shear angle 229, 287, 330, 369
Shear bending vibration theory 228, 231, 238
shear coefficient 8, 14, 229, 252, 261, 287, 330, 369
shear deformation 7–8, 97, 111, 138, 190, 230–231, 233, 245,
249, 257, 287
transverse 227
shear distortion 227
shear force 1, 7, 8, 79, 81, 100–101, 151–152, 289, 294, 303, 333, 341,
344, 353, 371, 408
shear modulus 8, 97, 108, 117, 189, 212, 229, 252, 261, 287, 330,
368–369, 381, 400
shear rigidity 229, 231–233, 245, 264, 269
shear stresses 8, 189–190
sign convention 1–3, 6, 8, 10–14, 16, 40, 69, 121, 151, 232, 240, 246,
248–249, 267–269, 289, 294, 333, 341, 369, 374, 401–402, 405,
408, 411
simple support 158, 178–180, 182–183, 185–186, 190, 192–194,
196–197, 200–204, 206–208, 217, 219
Simple Support – Type I 192
Simple Support – Type II 193
Simply-supported boundary 40, 82–83, 89, 102, 115, 121, 123, 125,
127, 151–158, 160, 189–194, 197–198, 200–201, 203–205, 207,
209, 211–212, 217, 334–336
Simply-supported–Free 160
Simply-supported–Simply-Supported 160
single degree-of-freedom (DOF) spring-mass 369
single DOF spring-mass system 376, 381–382, 384–385
slender composite beams 111
slopes, bending 40, 44, 82, 101, 103–105, 125, 229–230, 232,
237–242, 245–246, 289–290, 307–308, 312–314, 319–320,
322–323, 325, 333, 336, 345, 349–352, 354, 359–362, 364, 366,
371, 374–375, 408
Space Frames 329, 338–339, 341–342, 367–368
n-story 338–339
two-story 341–342
three-story two-bay 294–295, 297, 299
span 24–25, 28, 52, 56, 58, 73–74, 84, 87, 117, 244–245, 251, 271, 273,
278, 280, 401–402, 406–407, 413–417, 420
span angle 120, 134, 144, 150
spatial 338–344, 347–351, 353, 357–358, 360–368
spatial angle joint 329
special situation 18–19, 43, 45, 67, 71, 97, 120, 125–126, 132, 189,
196, 198, 201–203, 205, 209–212, 214, 216–217, 219, 223, 227,
231, 236, 238, 242–243, 248–249, 382, 384
spring 16–20, 32, 38, 40, 42–43, 45, 71, 76, 96, 101–102, 117, 121,
124–125, 143, 233, 235–236, 239, 241, 244, 369, 374–375,
377–379, 382, 386–388, 390, 395, 400, 403, 405, 408, 411
attached 388, 401–402, 405, 408, 411, 420
attached boundary 18, 42–43, 124, 143, 235, 236, 405, 411
attached end 18, 43, 71, 125, 236
attachment 16–17, 19, 117, 386, 388, 395–397, 400, 408
constant 376, 378, 382
stiffness 19, 42, 102, 242, 386, 374, 402, 405, 407–408, 411
spring and mass attachments 40, 121, 233
spring and viscous damper 402–403, 405, 407–408, 410430 Index
tuning 412
Two DOF attachment 384
two DOF spring-mass system 369, 378–379, 381–383
two-story space frame in figure 342
Type I 196, 202–203, 205–212, 214, 216–217, 219
Type II 189–191, 193–194, 196, 200–215, 216–219, 222–223
two wave mode transitions 282
two-dimensional rectangular plates 151, 189
Type I – Type I support 212, 217
Type I simple support 190, 192–193, 196, 200–201, 203–204, 206–207,
208, 217, 219
Type I simple support – Clamped 217
Type I simple support – Free 219
Type II – Type II support 212, 217
Type II simple support 190, 193–194, 196, 200–208, 210–219, 222–223
Type II simple support – Clamped 217
Type II simple support – Free 219
u
uncoupled bending 98
uniform beam 15–16, 20, 24–25, 27–28, 38–40, 46, 52, 54–56, 69–72,
74–76, 79, 81, 84, 87, 89, 96, 104, 107, 111, 113, 117, 227–228,
230, 250–252, 263–265, 278, 281–282, 287, 289, 293–295, 327,
329, 332, 338, 341, 369, 380, 383–391, 394–395, 400–401,
407–408
uniform cantilever beam 67, 261
uniform cantilever composite beam 117
uniform composite beam 97, 98, 100, 104–105, 107, 114, 117, 390
uniform plate 154–155, 159, 195–196, 201, 205
uniform steel beam 27, 38, 55, 67, 96, 252, 261, 281, 381–382, 384,
386, 389, 400, 413
uniform waveguide 154, 159, 326, 367
v
vector equation 307–308, 312–315, 320, 322–326, 349–351, 353,
360–367, 374
continuity 307–308, 312–313, 315, 320, 322, 324–325, 349, 351,
353, 360, 362–364, 365–367
equilibrium 307–308, 313–315, 320, 323–324, 326, 350–351, 353,
361, 363, 365, 367
velocity
phase 39, 227, 229, 231, 254, 281–282
transverse 263
vertical beam elements 292–293, 295, 338, 342, 353
vibrating 20, 46, 69, 71, 79, 84, 96, 100
vibrating shafts 32
vibrating string 32
Vibrational Power Transmission in Curved Beams 149
vibration analysis 1, 15, 28, 55, 90, 285, 329, 338, 380
vibration characteristics 382–383, 385–386, 388, 390
vibration control, bending 401, 410, 412, 416
vibration energy 408, 410, 412
bending 410, 412
incoming longitudinal 404, 406
reflected 403, 406
total 404
vibration isolation standpoint 31, 58, 137
Vibration mode 23, 93, 134, 221–223, 225–226
Timoshenko 7, 10, 227, 230–232, 238–244, 246–247, 249–261, 285,
287–288, 290, 292, 307–308, 312–315, 320, 322–326, 329, 331,
334–337, 349–353, 360–367, 369–372, 374–375, 377–382, 390,
396, 399, 400
bending vibration theory 7, 10, 230–232, 238, 243, 247, 250,
253–254, 257–258, 260, 285, 287–290, 292, 307–308, 312–315,
320, 322–326, 329, 331, 334–337, 349–353, 360–367, 369–372,
375, 381
Timoshenko and Shear 238, 240–242, 244, 246, 257
torque 12, 14, 100–101, 105, 107, 113, 151–152, 333–334, 336, 341,
344, 353
internal resistant 12–14
torsional deflection 12–14, 330, 334, 336, 341
torsional propagating 99–100
torsional rigidity 13, 330
torsional rotation 97, 100, 103–105, 108, 111–112
torsional theories 337
torsional vibration(s) 1, 13, 14, 97, 104, 108, 326, 329, 331,
336, 349, 390
torsional vibrations in composite beams 97
torsional vibration waves 326, 329, 336
torsional wavenumber 109, 331
torsional waves 98, 329–330, 332, 348, 358
torsionally vibrating shafts 32
transition of wave mode 156, 244, 247, 250, see also wave mode
transition(s)
translation 2, 7, 241, 373–374
translational constraints 40, 102, 233, 235, 241, 303, 373, 390
translational motion 42, 236, 344
translational spring 101, 103, 117, 386, 400, 410
spring stiffness 42, 101, 123, 236, 241, 373, 386, 389, 400, 407, 410
transmission 37, 66, 102–103, 117, 304–305, 307–326, 338, 341–365,
367, 369–372, 374–375, 381, 390, 393, 401–403, 405, 407–408, 420
transmission and reflection coefficients 401–403
transmission and reflection matrices 102, 117, 369, 374, 381, 393,
407–408
transmission matrix 39, 104, 107, 294–295, 297, 303, 306–308, 311,
313–315, 319, 321, 323–324, 326, 338, 340, 347–348, 350–351,
353, 357–358, 361, 363, 365, 367, 371–373, 375–376, 390, 408
transmission relationships 15, 293–296, 303–306, 309, 311, 315, 318,
338–342, 345, 347, 354, 357
transmitted and reflected bending waves 407
transmitted and reflected nearfield wave components 408
transmitted and reflected propagating waves 408
transmitted propagating bending wave component 408
transmitted waves 103, 305–306, 309, 311, 315, 318–319, 345,
347–348, 354, 357, 369–370, 401
transverse contraction 263–264, 266
transverse deflections 1–2, 4, 7, 40, 44, 79, 82, 97, 100, 103–105, 108,
151, 263, 289–290, 330, 333, 345, 354, 376, 378
transverse displacement 44, 237, 242
transverse force 39, 50–52, 84–88, 105, 107, 111, 113, 244–248,
251–252, 291–292, 377, 379
applied 39, 50–51, 85–86, 88, 244–247, 379
transverse load 1
transverse shear deformation 227
T-shaped Frame 287
T-shaped Joint 308Index 431
wave propagation 16, 20, 37, 40, 46, 71, 100, 120, 127, 154–155, 159,
195, 201, 205, 228, 250–251, 264, 278, 286, 289, 294, 326, 329,
332, 342, 367, 399
Wave Reflection 16–17, 19, 40–41, 43, 45, 82, 100–103, 121, 123, 125,
154–158, 192–193, 195, 197–205, 207, 209, 211, 232–233, 235,
237, 239, 241, 243, 267–269, 289–290, 304, 307–308, 312, 315,
321, 326, 333, 335, 367, 370–375
wave relationships 117, 127, 131, 133, 195, 201, 205, 252, 278, 280,
294, 341, 380–381
waves 15–16, 24–27, 40, 49–50, 52–54, 72–75, 84–85, 87–89, 100,
105–107, 119–120, 128–129, 131–132, 134, 153–154, 156–157,
159–160, 195, 197–198, 227–229, 231, 244–245, 248, 251–253,
257, 264, 271, 275, 278, 280, 284, 289, 291–292, 295, 304–305,
307–325, 332, 336, 338–339, 341–365, 377, 379–381, 399, 401
axial 71, 271
coefficients of positive- and negative-going 249, 279
extensional 197–198
extensional in-plane 207
half harmonic 156, 196
incoming 381, 401
injected 326, 367, 377, 379
injecting 24, 50, 72, 105, 128, 244, 271, 291, 336, 380
nearfield 156, 228, 408
outgoing 250–252, 293, 338
positive-going 102, 156, 196–197, 208–209
quarter 201, 223–224
reflected 16, 40–41, 71, 82–84, 100, 102, 104, 121–123, 154,
157–159, 192–194, 199–200, 202–205, 233–234, 250, 267, 290,
306, 311, 318–319, 334, 347–348, 357, 369, 374, 401, 405
Waves generated by external excitation(s) 50, 84, 106, 129, 244–245,
247–249, 271, 273, 275, 291, 336
waves generated by spring forces 379
Waves in Beams on a Winkler Elastic Foundation 69–96
wave standpoint 15, 20, 39, 46, 72, 84, 127, 227, 292, 338
wave vectors 26–27, 40, 46, 52, 55, 74, 76, 81, 87, 89, 105, 108, 121,
128, 131, 133, 155, 159, 195, 201, 228, 278–279, 289, 294, 306,
311, 319, 333, 341, 348, 358, 372–373, 375–376, 381
Wave Vibration Analysis 117, 151, 154–57, 159, 192–193, 195–211,
326, 342, 367
wave vibration standpoint 104, 119, 151, 189, 263, 377, 379
wavenumber 15–16, 33–34, 39–40, 63, 65, 70, 79, 81, 83, 97–100,
108–109, 119–120, 134, 145, 153–154, 156, 159–160, 189–190,
192, 195–198, 201, 205–209, 211, 227–232, 238, 240, 246, 253,
254, 255, 257, 263–266, 281–284, 288–289, 307–308, 312–314,
320, 322–323, 325, 330–332, 349, 360, 369, 370, 403, 409, 411
width 32–33, 44, 55, 58–59, 61–62, 64, 67, 108, 117, 125, 151, 237,
253–257, 261, 327, 344, 368, 391, 400, 413, 420
Winkler Elastic Foundation 70–96, 76, 79, 81, 84, 89–90, 93, 96
x
x-axis 1, 3, 10, 12, 97, 294–295, 297, 303, 329–330, 336,
341–342
y
y anti-symmetrical mode 160, 187
Y joint(s) 338–339, 341, 343–344, 348–351, 353
y symmetrical mode 160, 187
vibration motions 134, 139
bending 108
coupled 119, 390
vibrations 15, 23, 27, 31, 37, 55, 58, 66, 72, 84, 95, 97, 104, 116, 134,
149, 226, 253, 260, 281, 286–287, 292, 326, 329, 338, 367, 369,
380, 387, 389, 399, 413, 416, 419
vibration tests 31, 58
vibration theories 234, 244, 250, 254, 263–286, 380
advanced bending 59, 227
elementary 327, 329, 331, 368
engineering bending 231–232
vibration waves 117, 154, 159, 293, 296, 305, 309, 315, 332, 338, 345,
354, 377, 379, 401
applied force injects 377
bending 69, 401
incoming 295, 341
injecting 84
longitudinal 15, 282
reflected 381
reflected longitudinal 401
viscous damper 18–20, 369, 386–387, 402–403, 405, 407–408,
410–411, 420
attachment 18–19, 401–403, 405, 407–408, 410–411
boundaries 22
viscous damping 18–19, 42, 236, 374, 386, 387
constant 18, 402, 408, 411
effect 408
volume mass density 5, 10, 12, 15, 32, 33, 39, 69, 79, 97, 108, 117,
119, 134, 150–151, 159, 189, 227, 263, 281, 287, 330, 369, 403
w
wave amplitudes 99, 120, 197, 230–231, 265, 267
Wave Analysis 38, 67, 96, 260–261, 326–327, 368, 400
wave component(s) 20, 23–24, 26–27, 33–34, 46, 48–49, 52, 55, 62,
65, 70, 72, 74, 76, 84, 87, 89, 100, 105, 108, 116, 120, 128, 131,
133, 153, 155, 159, 191, 195, 201, 231, 238, 244, 247, 250–252,
260, 265, 267, 279, 280, 282, 288–289, 292, 294, 307–308,
312–315, 320, 322–326, 332–333, 338, 341, 349, 351, 353, 358,
360, 362, 364–367, 369, 377, 379, 381, 402, 405, 408, 411
wave control of bending vibrations 407, 409, 411, 420
wave control of longitudinal vibrations 401, 403, 405
wave control force 401
waveguides 15, 294–295, 341
wave modes 156, 265–266
wave mode conversion 287, 329
wave mode transition(s) 11, 69–70, 81, 120, 134, 160, 198, 223,
231–232, 254, 282, 284, 288, 331, 369, see also transition of
wave mode
wavenumbers 15–16, 33–34, 39–40, 70, 79, 83, 97–100, 108, 119–120,
134, 153–154, 156, 159–160, 189–190, 192, 195–198, 201,
205–209, 211, 227–232, 238, 240, 246, 253–255, 257, 263–266,
282, 284, 288–289, 307–308, 312–314, 320, 322, 325, 330, 332,
369–370
bending 39, 79, 81, 109, 227, 229, 288, 330–331, 409, 411
imaginary 198, 254
longitudinal 15–16, 70, 263, 281, 283, 288, 331
wavenumber sequence 307–308, 312–314, 320, 322–323, 325432 Index
z
z-axis 97, 330, 341, 344, 354
zero matrix/matrices 46, 52, 55, 84, 87, 89, 105, 108, 127, 131, 159,
195, 201, 278, 280
zero shear deformation 229, 231
y-axis 1, 294, 297, 330
Young’s modulus 4, 15, 27, 32–33, 38–39, 55, 67, 69, 79, 97, 108, 117,
119, 134, 150–151, 159, 189, 212, 227, 252, 261, 263, 281, 287,
327, 330, 368–369, 381, 400, 403, 413, 420
Y-shaped Space Frame 329, 343, 364

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