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عدد المساهمات : 18893 التقييم : 35191 تاريخ التسجيل : 01/07/2009 الدولة : مصر العمل : مدير منتدى هندسة الإنتاج والتصميم الميكانيكى
 موضوع: كتاب Mechanical Wave Vibrations  Analysis and Control الأحد 08 أكتوبر 2023, 3:43 am  

أخواني في الله أحضرت لكم كتاب Mechanical Wave Vibrations  Analysis and Control Chunhui Mei University of MichiganDearborn Dearborn,MI,USA
و المحتوى كما يلي :
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 NonClassical 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 NonClassical 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 NonClassical 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 NonClassical 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 Simplysupported 151 7.1 The Governing Equations of Motion 151 7.2 Closedform Solutions 152 7.3 Wave Reflection,Propagation,and Wave Vibration Analysis along the Simplysupported 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 InPlane Vibration of Rectangular Isotropic Thin Plates with Two Opposite Edges Simplysupported 189 8.1 The Governing Equations of Motion 189 8.2 Closedform Solutions 190 8.3 Wave Reflection,Propagation,and Wave Vibration Analysis along the Simplysupported x Direction 192 8.3.1 Wave Reflection at a Simplysupported 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 Simplysupported Boundaries along the y Direction When k0 ≠ 0 207 8.6.1 Wave Reflection,Propagation,and Wave Vibration Analysis with a Pair of Simplysupported 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 Simplysupported 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 NonClassical 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 Threemode 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 Threemode 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 Threemode 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 Threemode 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 Builtup 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 Multistory Multibay Planar Frame 292 11.5 Reflection and Transmission of Waves in a Multistory Multibay Planar Frame 304 11.5.1 Wave Reflection and Transmission at an Lshaped Joint 304 11.5.2 Wave Reflection and Transmission at a Tshaped Joint 308 11.5.3 Wave Reflection and Transmission at a Cross Joint 315 12 Bending,Longitudinal,and Torsional Waves in Builtup 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 Multistory Space Frame 338 12.5 Reflection and Transmission of Waves in a Multistory Space Frame 341 12.5.1 Wave Reflection and Transmission at a Yshaped Spatial Joint 343 12.5.2 Wave Reflection and Transmission at a Kshaped 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 antisymmetrical 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 408,411,413–414,419–420 cantilever 108,115,117 fixedfixed 21,76–77 fixedfree 76,78 freefree 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 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 bendingtorsion 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 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 clampedclamped 67 clampedfree 67 clampedmass 67 clampedpinned 67 corresponding 295 elastic 71,125,143,233 fixedfree 21 freefixed 38 freefree 20–22,28,38,46–48,55,252,281 freespring 38 ideal 414,416 left 381–382,400,413,420 tuned 416 boundary conditions 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 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 ClampedClamped 47–48,160 ClampedFree 47,160 ClampedPinned Boundary 47–48 Clamped–Simplysupported 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 closedform solutions 151–153,189–191 coefficient matrices 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 crosssectional 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 cutoff circular frequency 156 cutoff 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 negativegoing 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 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 yaxis 5,8 disappearance of resonant peaks 29,58,92,135,137 discontinuity 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 inplane 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 fixeddamper boundary 22–23 fixedfixed boundary 21,23,76–77 fixedfree boundary 21,23,76,78 fixedmass boundary 22–23 fixedspring 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 inplane bending vibrations 67,261 forced outofplane 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 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 bendingtorsion vibration in slender composite beams 97 elementary longitudinal vibration theory 11–12,15,329–330 elementary torsional vibration theory 13–14 EulerBernoulli bending vibration theory 6,39,232,287,329 inplane vibration of thin plate 189 L joint 304305 longitudinal waves in beams on Winkler elastic foundation 69 Love’s inplane vibration theory in curved beams 119 Love longitudinal vibration theory 263 MindlinHerrmann longitudinal vibration theory 264 outofplane 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 Threemode 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 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 equilibrium relationships 105,294 equivalent continuity equations 305,310,316–318,347,357 Euler–Bernoulli 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) Index 425 Hz identity matrix imaginary part 116,136,253,255,260,281,283 imaginary unit incident waves 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 inplane 67,119,138,149,189–190,193–194,196,201–202,207,226, 261,287,326–327,330 inplane bending wavenumbers 330 inplane plate vibrations 193–194,196,201–202 onedimensional 207 InPlane 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 Kshaped 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 freefree 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 fourthorder 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 inplane 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 Hshaped 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 MindlinHerrmann 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 antisymmetrical 160,187 body 55–56,64 corresponding 76,93 onedimensional 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 15,20–21,23–24,28–29,33–34,38–39,4649,5557,62, 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 Multistory Multibay Planar Frame 292–293,295–325 Multistory 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 nondimensional 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 springmass system 382 Natural frequency expressions,nondimensional 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 negativegoing 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 Lshaped 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 overpredicts 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 simplysupported 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 PinnedPinned 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 noncausal 410,412 nonclassical boundary 15–17,19,40–41,43,45,100–101,119,121, 123,125,232–233,235,237,239,241,243 nondimensional coordinate 71–72 nondimensional displacement 70,82 nondimensional flexibility coefficient 71,76,96 nondimensional frequency 21–23,46,69,81,96 nondimensional natural frequency 21–23,47–48,76–77,160,164, 169,173,178,182,186,217–218,222–223 nondimensional stiffness 69,76,81,89,96 nondispersive 16,282 nonlinear equation 22–23 nonnodal point 29–30,35,56–58,64,76,90–92,135,139 nonrigid body modes 55–56,64 nontrivial solution 20,46,72,105,250,278 normal strain 4,7 normal stress 4,5,190 normalized modeshape 34,64,76,89,90 nstory mbay 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 Simplysupported 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 outofplane 159,329–330 outofplane 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 cutoff 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 springmass 369,376 springmass 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 Threemode Theory 263,265–266,269,270–271,273,276–277,280, 282,284–286 threestory twobay 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 Simplysupported 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 Simplysupported–Free 160 Simplysupported–SimplySupported 160 single degreeoffreedom (DOF) springmass 369 single DOF springmass 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 nstory 338–339 twostory 341–342 threestory twobay 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 springmass system 369,378–379,381–383 twostory 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 twodimensional 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 Tshaped Frame 287 Tshaped 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 negativegoing 249,279 extensional 197–198 extensional inplane 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 positivegoing 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 xaxis 1,3,10,12,97,294–295,297,303,329–330,336, 341–342 y y antisymmetrical 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 zaxis 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 yaxis 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 Yshaped 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 crosssectional 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 cutoff circular frequency 156 cutoff 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 negativegoing 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
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