DYNAMIC RESPONSE OF BEAMS WITH PASSIVE TUNED MASS DAMPERS A Thesis Submitted to the Faculty of Purdue University by Mustafa Kemal Ozkan In Partial Fulfillment of the Requirements for the Degree of Master of Science in Civil Engineering May 2010 Purdue University West Lafayette, Indiana
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DYNAMIC RESPONSE OF BEAMS WITH PASSIVE TUNED MASS DAMPERS
A Thesis
Submitted to the Faculty
of
Purdue University
by
Mustafa Kemal Ozkan
In Partial Fulfillment of the
Requirements for the Degree
of
Master of Science in Civil Engineering
May 2010
Purdue University
West Lafayette, Indiana
ii
To the memory of my grandfather, Ali Bicer.
To my parents for their endless love, support and encouragement.
iii
ACKNOWLEDGMENTS
I would here like to express my thanks to the people who have been very
helpful to me during the time it took me to write this thesis.
I would like to express the deepest appreciation to my advisor and mentor,
Professor Ayhan Irfanoglu for giving me the opportunity to work in a very
interesting area and for his support and guidance throughout my graduate
studies at Purdue University.
I also would like to thank the members of my graduate committee,
Professor Mete A. Sozen and Professor Michael E. Kreger, for their time and
suggestions on this thesis.
I wish also to thank Professor Robert J. Connor and Ryan J. Sherman for
kindly sharing data for the high-mast lighting towers.
I would like to express my sincere thanks to my friends and colleagues,
particularly Fabian Consuegra and Bismarck Luna, for providing a very enjoyable
working environment.
I thank to the faculty and staff of the Structural Engineering department,
especially to Molly Stetler, for their kindness and support.
The last, and the most, I want to thank my family for their love, support
and encouragement.
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To my sister Aysenur Ozkan, I appreciate that you are just my sister and
thank you for being there always. I owe so much thanks to my grandmother,
Sabire Bicer, who has always supported me since the start of my life.
I am greatly indebted to my mother, Nurten Ozkan, and my father, Taki
Ozkan, thank you for providing me with the opportunity to be where I am.
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TABLE OF CONTENTS
Page LIST OF TABLES ............................................................................................... viii LIST OF FIGURES ............................................................................................ xxii LIST OF SYMBOLS ......................................................................................... xxvii ABSTRACT ...................................................................................................... xxix CHAPTER 1. INTRODUCTION ............................................................................ 1
1.1. General ....................................................................................................... 1 1.1.1. Sources of Dynamic Excitation ............................................................. 1 1.1.2. Dynamic Loadings ................................................................................ 2 1.1.3. Consequences of Vibration ................................................................... 2 1.1.4. Vibration Control ................................................................................... 3
1.2. Object and Scope ....................................................................................... 4 1.3. Organization ............................................................................................... 5
CHAPTER 2. BACKGROUND AND PREVIOUS RESEARCH ............................. 6 2.1. Human-Structure Dynamic Interaction and Human Induced Vibration ....... 6 2.2. Vibration Criteria ....................................................................................... 12
2.2.1. ISO International Standard ................................................................. 12 2.2.2. Murray’s Criterion ............................................................................... 14 2.2.3. Other Recommendations and Criteria ................................................ 14 2.2.4. Recommended Criteria for Sensitive Laboratory and Healthcare
2.4. Tuned Mass Dampers (TMDs) Overview .................................................. 32 2.4.1. Introduction ......................................................................................... 32 2.4.2. An Introductory Example of a TMD for an Undamped SDOF System ..................................................................................... 37
CHAPTER 3. FREE AND FORCED VIBRATION OF BEAMS WITH ANY NUMBER OF ATTACHED SPRING MASS SYSTEMS SUBJECTED TO DIFFERENT TYPES OF DYNAMIC LOADS ........................................................................ 40
3.1. Introduction ............................................................................................... 40 3.2. Formulation of the Free Vibration Problem for Uniform Beams Carrying Spring-Mass Systems ................................................................. 42
3.2.1. Equations of Motion and Displacement Functions .............................. 42
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Page 3.2.2. Derivation of Eigenfunctions for the Constrained Beam ..................... 43
3.3. Formulation of the Free Vibration Problem for Non-Uniform Beams Carrying Spring-Mass Systems ................................................................. 51
3.3.1. Equations of Motion and Derivation of Eigenfunctions for the Constrained Beam .............................................................................. 51
3.3.2. Coefficient Matrix [Bv] for the v-th Attaching Point .............................. 54 3.3.3. Coefficient Matrix [BL] for the Left End of the Beam............................ 59 3.3.4. Coefficient Matrix [BR] for the Right End of the Beam ......................... 60
3.4. Formulation of the Free Vibration Problem for Uniform Multi-Span Beams Carrying Spring-Mass Systems ................................... 63
3.4.1. Equations of Motion and Displacement Function ................................ 63 3.4.2. Coefficient Matrices and Determination of Natural Frequencies and Mode Shapes ............................................................................... 65
3.5. Forced Vibration of Euler-Bernoulli Beams ............................................... 74 3.5.1. Introduction ......................................................................................... 74 3.5.2. Formulation of Forced Vibration for Beams ........................................ 74
CHAPTER 4. NUMERICAL RESULTS ............................................................... 81 4.1. Introduction ............................................................................................... 81 4.2. Free Vibration Analysis of Single Span Uniform Beam Carrying One, Two and Three Spring-Mass Systems.............................................. 82 4.3. Free Vibration Analysis of Single Span Non-Uniform Beam Carrying
Spring-Mass Systems ............................................................................... 94 4.4. Free Vibration Analysis of Uniform Multi-Span Beam Carrying Spring-Mass Systems ............................................................................. 102
4.4.1. Free Vibration Analysis of Two Span Beam Carrying One Spring-Mass System ................................................................. 102 4.4.2. Free Vibration Analysis of Two Span Beam Carrying Two Spring-Mass Systems ............................................................... 106 4.4.3. Free Vibration Analysis of Three Span Beam Carrying One Spring-Mass Systems ............................................................... 109 4.4.4. Free Vibration Analysis of Three Span Beam Carrying Two Spring-Mass Systems ............................................................... 112
4.5. Forced Vibration Analysis of Single Span Uniform Beam Carrying One, Two and Three Spring-Mass Systems ........................................... 120
4.6. Forced Vibration Analysis of High-Mast Lighting Tower under Wind Load ............................................................................................... 158
4.7. Forced Vibration Analysis of Multi Span Uniform Beams Carrying One and Two Spring-Mass Systems ....................................................... 162
4.7.1. Forced Vibration Analysis of Two Span Beam Carrying One Spring-Mass System ................................................................. 162
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Page 4.7.2. Forced Vibration Analysis of Two Span Beam Carrying Two Spring-Mass Systems ............................................................... 176 4.7.3. Forced Vibration Analysis of Three Span Beam Carrying One Spring-Mass Systems ............................................................... 189 4.7.4. Forced Vibration Analysis of Three Span Beam Carrying Two Spring-Mass Systems ............................................................... 207
CHAPTER 5. SUMMARY AND CONCLUSIONS ............................................. 215 5.1. Summary ................................................................................................ 215 5.2. Conclusion .............................................................................................. 217 5.3. Future Work ............................................................................................ 220
Appendix A. ................................................................................................... 228 Appendix B. ................................................................................................... 234 Appendix C. ................................................................................................... 238 Appendix D. ................................................................................................... 241
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LIST OF TABLES
Table Page 2.1 Recommended values for αi (Murray et al., 1997) ................................... 8 2.2 Recommended acceleration limits for vibration due to rhythmic activities (Allen, 1990) ............................................................................ 17 2.3 Suggested design parameters for rhythmic activities (Allen et al., 1985) .................................................................................. 18 2.4 Minimum recommended natural assembly floor frequencies, Hz (Allen et al., 1985) .................................................................................. 18 2.5 Application and Interpretation of Generic Vibration Criteria (Pan et al.2008) ...................................................................................... 21 4.1 The lowest seven natural frequencies of the uniform beam carrying one spring-mass system (m1/mb=0.01) ................................................... 83 4.2 The lowest seven natural frequencies of the uniform beam carrying one spring-mass system (m1/mb=0.02) ................................................... 83 4.3 The lowest seven natural frequencies of the uniform beam carrying one spring-mass system (m1/mb=0.05) ................................................... 84 4.4 The lowest seven natural frequencies of the uniform beam carrying one spring-mass system (m1/mb=0.1) ..................................................... 84 4.5 The lowest seven natural frequencies of the uniform beam carrying one spring-mass system (m1/mb=0.2) ..................................................... 84 4.6 The lowest seven natural frequencies of the uniform beam carrying two spring-mass systems (m1/mb=m2/mb=0.01) ...................................... 85 4.7 The lowest seven natural frequencies of the uniform beam carrying two spring-mass systems (m1/mb=m2/mb=0.02) ...................................... 85 4.8 The lowest seven natural frequencies of the uniform beam carrying two spring-mass systems (m1/mb=m2/mb=0.05) ...................................... 85 4.9 The lowest seven natural frequencies of the uniform beam carrying two spring-mass systems (m1/mb=m2/mb=0.1) ........................................ 86 4.10 The lowest seven natural frequencies of the uniform beam carrying two spring-mass systems (m1/mb=m2/mb=0.2) ........................................ 86 4.11 The lowest eight natural frequencies of the uniform beam carrying three spring-mass systems (m1/mb=m2/mb=m3/mb=0.01) ....................... 87 4.12 The lowest eight natural frequencies of the uniform beam carrying three spring-mass systems (m1/mb=m2/mb=m3/mb=0.02) ....................... 87 4.13 The lowest eight natural frequencies of the uniform beam carrying three spring-mass systems (m1/mb=m2/mb=m3/mb=0.05) ....................... 88
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Table Page 4.14 The lowest eight natural frequencies of the uniform beam carrying three spring-mass systems (m1/mb=m2/mb=m3/mb=0.1) ......................... 88 4.15 The lowest eight natural frequencies of the uniform beam carrying three spring-mass systems (m1/mb=m2/mb=m3/mb=0.2) ......................... 89 4.16 The lowest six natural frequencies of the bare uniform beam ................ 89 4.17 The lowest six natural frequencies of the non-uniform beam carrying one spring-mass system (m1/mb=0.01) ................................................... 95 4.18 The lowest six natural frequencies of the non-uniform beam carrying one spring-mass system (m1/mb=0.02) ................................................... 95 4.19 The lowest six natural frequencies of the non-uniform beam carrying one spring-mass system (m1/mb=0.05) ................................................... 95 4.20 The lowest six natural frequencies of the non-uniform beam carrying one spring-mass system (m1/mb=0.1) ..................................................... 96 4.21 The lowest six natural frequencies of the non-uniform beam carrying one spring-mass system (m1/mb=0.2) ..................................................... 96 4.22 The lowest five natural frequencies of the bare non-uniform beam ........ 96 4.23 The lowest seven natural frequencies of the high-mast lighting tower
carrying one spring-mass system at the free end ................................. 100 4.24 Comparison of the lowest four natural frequencies of the bare high-mast lighting tower ........................................................................ 100 4.25 The lowest six natural frequencies of the two-span beam carrying one spring-mass system at second span ............................................. 103 4.26 The lowest six natural frequencies of the two-span beam carrying one spring-mass system at first span ................................................... 104 4.27 The lowest six natural frequencies of the two-span beam carrying two spring-mass systems based on case 1 .......................................... 106 4.28 The lowest six natural frequencies of the two-span beam carrying two spring-mass systems based on case 2 .......................................... 107 4.29 The lowest six natural frequencies of the three-span beam carrying one spring-mass system based on case 1 ........................................... 109 4.30 The lowest six natural frequencies of the three-span beam carrying one spring-mass system based on case 2 ........................................... 110 4.31 The lowest six natural frequencies of the three-span beam carrying two spring-mass systems based on case 1 .......................................... 112 4.32 The lowest six natural frequencies of the three-span beam carrying two spring-mass systems based on case 2 .......................................... 113 4.33 The lowest six natural frequencies of the three-span beam carrying two spring-mass systems based on case 3 .......................................... 114 4.34 The lowest six natural frequencies of the three-span beam carrying two spring-mass systems based on case 4 .......................................... 115 4.35 The lowest six natural frequencies of the three-span beam carrying two spring-mass systems based on case 5 .......................................... 116 4.36 Maximum and RMS responses at x=0.5L for SS, CC, CS beams and at x=L for CF beam without any spring mass system-Case 1 ........ 120
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Table Page 4.37 Maximum and RMS responses at x=0.5L for SS, CC, CS beams and at x=L for CF beam without any spring mass system-Case 2 ........ 120 4.38 Maximum and RMS responses at x=0.5L for SS, CC, CS beams and at x=L for CF beam carrying one spring-mass system under impact loading - Case 3 (m1/mb=0.01) ........................................ 121 4.39 Maximum and RMS responses at x=0.5L for SS, CC, CS beams and at x=L for CF beam carrying one spring-mass system under impact loading - Case 3 (m1/mb=0.02) ........................................ 122 4.40 Maximum and RMS responses at x=0.5L for SS, CC, CS beams and at x=L for CF beam carrying one spring-mass system under impact loading - Case 3 (m1/mb=0.05) ........................................ 122 4.41 Maximum and RMS responses at x=0.5L for SS, CC, CS beams and at x=L for CF beam carrying one spring-mass system under impact loading - Case 3 (m1/mb=0.1) .......................................... 122 4.42 Maximum and RMS responses at x=0.5L for SS, CC, CS beams and at x=L for CF beam carrying one spring-mass system under impact loading - Case 3 (m1/mb=0.2) .......................................... 123 4.43 Maximum and RMS responses at x=0.5L for SS, CC, CS beams and at x=L for CF beam carrying one spring-mass system under impact loading - Case 4 (m1/mb=0.01) ........................................ 124 4.44 Maximum and RMS responses at x=0.5L for SS, CC, CS beams and at x=L for CF beam carrying one spring-mass system under impact loading - Case 4 (m1/mb=0.02) ........................................ 124 4.45 Maximum and RMS responses at x=0.5L for SS, CC, CS beams and at x=L for CF beam carrying one spring-mass system under impact loading - Case 4 (m1/mb=0.05) ........................................ 124 4.46 Maximum and RMS responses at x=0.5L for SS, CC, CS beams and at x=L for CF beam carrying one spring-mass system under impact loading - Case 4 (m1/mb=0.1) ......................................... 125 4.47 Maximum and RMS responses at x=0.5L for SS, CC, CS beams and at x=L for CF beam carrying one spring-mass system under impact loading - Case 4 (m1/mb=0.2) .......................................... 125 4.48 Maximum and RMS responses at x=0.5L for SS, CC, CS beams and at x=L for CF beam carrying two spring-mass systems under impact loading - Case 5 (m1/mb=m2/mb=0.01) ............................ 126 4.49 Maximum and RMS responses at x=0.5L for SS, CC, CS beams and at x=L for CF beam carrying two spring-mass systems under impact loading - Case 5 (m1/mb=m2/mb=0.02) ............................ 126 4.50 Maximum and RMS responses at x=0.5L for SS, CC, CS beams and at x=L for CF beam carrying two spring-mass systems under impact loading - Case 5 (m1/mb=m2/mb=0.05) ............................ 127 4.51 Maximum and RMS responses at x=0.5L for SS, CC, CS beams and at x=L for CF beam carrying two spring-mass systems under impact loading - Case 5 (m1/mb=m2/mb=0.1) .............................. 127
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Table Page 4.52 Maximum and RMS responses at x=0.5L for SS, CC, CS beams and at x=L for CF beam carrying two spring-mass systems under impact loading - Case 5 (m1/mb=m2/mb=0.2) .............................. 127 4.53 Maximum and RMS responses at x=0.5L for SS, CC, CS beams and at x=L for CF beam carrying two spring-mass systems under impact loading - Case 6 (m1/mb=m2/mb=0.01) ............................ 128 4.54 Maximum and RMS responses at x=0.5L for SS, CC, CS beams and at x=L for CF beam carrying two spring-mass systems under impact loading - Case 6 (m1/mb=m2/mb=0.02) ............................ 128 4.55 Maximum and RMS responses at x=0.5L for SS, CC, CS beams and at x=L for CF beam carrying two spring-mass systems under impact loading - Case 6 (m1/mb=m2/mb=0.05) ............................ 129 4.56 Maximum and RMS responses at x=0.5L for SS, CC, CS beams and at x=L for CF beam carrying two spring-mass systems under impact loading - Case 6 (m1/mb=m2/mb=0.1) .............................. 129 4.57 Maximum and RMS responses at x=0.5L for SS, CC, CS beams and at x=L for CF beam carrying two spring-mass systems under impact loading - Case 6 (m1/mb=m2/mb=0.2) .............................. 129 4.58 Maximum and RMS responses at x=0.5L for SS, CC, CS beams and at x=L for CF beam carrying three spring-mass systems under impact loading - Case 7 (m1/mb=m2/mb=m3/mb=0.01) ................ 130 4.59 Maximum and RMS responses at x=0.5L for SS, CC, CS beams and at x=L for CF beam carrying three spring-mass systems under impact loading - Case 7 (m1/mb=m2/mb=m3/mb=0.02) ................ 130 4.60 Maximum and RMS responses at x=0.5L for SS, CC, CS beams and at x=L for CF beam carrying three spring-mass systems under impact loading - Case 7 (m1/mb=m2/mb=m3/mb=0.05) ................ 131 4.61 Maximum and RMS responses at x=0.5L for SS, CC, CS beams and at x=L for CF beam carrying three spring-mass systems under impact loading - Case 7 (m1/mb=m2/mb=m3/mb=0.1) .................. 131 4.62 Maximum and RMS responses at x=0.5L for SS, CC, CS beams and at x=L for CF beam carrying three spring-mass systems under impact loading - Case 7 (m1/mb=m2/mb=m3/mb=0.2) .................. 131 4.63 Maximum and RMS responses at x=0.5L for SS, CC, CS beams and at x=L for CF beam carrying three spring-mass systems under impact loading - Case 8 (m1/mb=m2/mb=m3/mb=0.01) ................ 132 4.64 Maximum and RMS responses at x=0.5L for SS, CC, CS beams and at x=L for CF beam carrying three spring-mass systems under impact loading - Case 8 (m1/mb=m2/mb=m3/mb=0.02) ................ 132 4.65 Maximum and RMS responses at x=0.5L for SS, CC, CS beams and at x=L for CF beam carrying three spring-mass systems under impact loading - Case 8 (m1/mb=m2/mb=m3/mb=0.05) ................ 133
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Table Page 4.66 Maximum and RMS responses at x=0.5L for SS, CC, CS beams and at x=L for CF beam carrying three spring-mass systems under impact loading - Case 8 (m1/mb=m2/mb=m3/mb=0.1) .................. 133 4.67 Maximum and RMS responses at x=0.5L for SS, CC, CS beams and at x=L for CF beam carrying three spring-mass systems under impact loading - Case 8 (m1/mb=m2/mb=m3/mb=0.2) .................. 133 4.68 Maximum and RMS responses at x=0.5L for SS, CC, CS beams and at x=L for CF beam without any spring mass system-Case 1 ........ 134 4.69 Maximum and RMS responses at x=0.5L for SS, CC, CS beams and at x=L for CF beam without any spring mass system-Case 2 ........ 134 4.70 Maximum and RMS responses at x=0.5L for SS, CC, CS beams and at x=L for CF beam carrying one spring-mass system under
harmonic loading - Case 3 (m1/mb=0.01) .............................................. 135 4.71 Maximum and RMS responses at x=0.5L for SS, CC, CS beams and at x=L for CF beam carrying one spring-mass system under
harmonic loading - Case 3 (m1/mb=0.02) .............................................. 136 4.72 Maximum and RMS responses at x=0.5L for SS, CC, CS beams and at x=L for CF beam carrying one spring-mass system under
harmonic loading - Case 3 (m1/mb=0.05) .............................................. 136 4.73 Maximum and RMS responses at x=0.5L for SS, CC, CS beams and at x=L for CF beam carrying one spring-mass system under
harmonic loading - Case 3 (m1/mb=0.1) ................................................ 136 4.74 Maximum and RMS responses at x=0.5L for SS, CC, CS beams and at x=L for CF beam carrying one spring-mass system under
harmonic loading - Case 3 (m1/mb=0.2) ................................................ 137 4.75 Maximum and RMS responses at x=0.5L for SS, CC, CS beams and at x=L for CF beam carrying one spring-mass system under
harmonic loading - Case 4 (m1/mb=0.01) .............................................. 138 4.76 Maximum and RMS responses at x=0.5L for SS, CC, CS beams and at x=L for CF beam carrying one spring-mass system under
harmonic loading - Case 4 (m1/mb=0.02) .............................................. 138 4.77 Maximum and RMS responses at x=0.5L for SS, CC, CS beams and at x=L for CF beam carrying one spring-mass system under
harmonic loading - Case 4 (m1/mb=0.05) .............................................. 138 4.78 Maximum and RMS responses at x=0.5L for SS, CC, CS beams and at x=L for CF beam carrying one spring-mass system under
harmonic loading - Case 4 (m1/mb=0.1) ................................................ 139 4.79 Maximum and RMS responses at x=0.5L for SS, CC, CS beams and at x=L for CF beam carrying one spring-mass system under
harmonic loading - Case 4 (m1/mb=0.2) ................................................ 139 4.80 Maximum and RMS responses at x=0.5L for SS, CC, CS beams and at x=L for CF beam carrying two spring-mass systems under
harmonic loading - Case 5 (m1/mb=m2/mb=0.01) .................................. 140
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Table Page 4.81 Maximum and RMS responses at x=0.5L for SS, CC, CS beams and at x=L for CF beam carrying two spring-mass systems under
harmonic loading - Case 5 (m1/mb=m2/mb=0.02) .................................. 140 4.82 Maximum and RMS responses at x=0.5L for SS, CC, CS beams and at x=L for CF beam carrying two spring-mass systems under
harmonic loading - Case 5 (m1/mb=m2/mb=0.05) .................................. 141 4.83 Maximum and RMS responses at x=0.5L for SS, CC, CS beams and at x=L for CF beam carrying two spring-mass systems under
harmonic loading - Case 5 (m1/mb=m2/mb=0.1) .................................... 141 4.84 Maximum and RMS responses at x=0.5L for SS, CC, CS beams and at x=L for CF beam carrying two spring-mass systems under
harmonic loading - Case 5 (m1/mb=m2/mb=0.2) .................................... 141 4.85 Maximum and RMS responses at x=0.5L for SS, CC, CS beams and at x=L for CF beam carrying two spring-mass systems under
harmonic loading - Case 6 (m1/mb=m2/mb=0.01) .................................. 142 4.86 Maximum and RMS responses at x=0.5L for SS, CC, CS beams and at x=L for CF beam carrying two spring-mass systems under
harmonic loading - Case 6 (m1/mb=m2/mb=0.02) .................................. 142 4.87 Maximum and RMS responses at x=0.5L for SS, CC, CS beams and at x=L for CF beam carrying two spring-mass systems under
harmonic loading - Case 6 (m1/mb=m2/mb=0.05) .................................. 143 4.88 Maximum and RMS responses at x=0.5L for SS, CC, CS beams and at x=L for CF beam carrying two spring-mass systems under
harmonic loading - Case 6 (m1/mb=m2/mb=0.1) .................................... 143 4.89 Maximum and RMS responses at x=0.5L for SS, CC, CS beams and at x=L for CF beam carrying two spring-mass systems under
harmonic loading - Case 6 (m1/mb=m2/mb=0.2) .................................... 143 4.90 Maximum and RMS responses at x=0.5L for SS, CC, CS beams and at x=L for CF beam carrying three spring-mass systems under
harmonic loading - Case 7 (m1/mb=m2/mb=m3/mb=0.01) ...................... 144 4.91 Maximum and RMS responses at x=0.5L for SS, CC, CS beams and at x=L for CF beam carrying three spring-mass systems under
harmonic loading - Case 7 (m1/mb=m2/mb=m3/mb=0.02) ...................... 144 4.92 Maximum and RMS responses at x=0.5L for SS, CC, CS beams and at x=L for CF beam carrying three spring-mass systems under
harmonic loading - Case 7 (m1/mb=m2/mb=m3/mb=0.05) ...................... 145 4.93 Maximum and RMS responses at x=0.5L for SS, CC, CS beams and at x=L for CF beam carrying three spring-mass systems under
harmonic loading - Case 7 (m1/mb=m2/mb=m3/mb=0.1) ........................ 145 4.94 Maximum and RMS responses at x=0.5L for SS, CC, CS beams and at x=L for CF beam carrying three spring-mass systems under
harmonic loading - Case 7 (m1/mb=m2/mb=m3/mb=0.2) ........................ 145
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Table Page 4.95 Maximum and RMS responses at x=0.5L for SS, CC, CS beams and at x=L for CF beam carrying three spring-mass systems under
harmonic loading - Case 8 (m1/mb=m2/mb=m3/mb=0.01) ...................... 146 4.96 Maximum and RMS responses at x=0.5L for SS, CC, CS beams and at x=L for CF beam carrying three spring-mass systems under
harmonic loading - Case 8 (m1/mb=m2/mb=m3/mb=0.02) ...................... 146 4.97 Maximum and RMS responses at x=0.5L for SS, CC, CS beams and at x=L for CF beam carrying three spring-mass systems under
harmonic loading - Case 8 (m1/mb=m2/mb=m3/mb=0.05) ...................... 147 4.98 Maximum and RMS responses at x=0.5L for SS, CC, CS beams and at x=L for CF beam carrying three spring-mass systems under
harmonic loading - Case 8 (m1/mb=m2/mb=m3/mb=0.1) ........................ 147 4.99 Maximum and RMS responses at x=0.5L for SS, CC, CS beams and at x=L for CF beam carrying three spring-mass systems under
harmonic loading - Case 8 (m1/mb=m2/mb=m3/mb=0.2) ........................ 147 4.100 Maximum and RMS responses at x=0.5L for SS beam carrying one spring-mass system under harmonic loading - Case 9 .................. 148 4.101 Maximum and RMS responses at x=0.5L for SS, CC, CS beams and at x=L for CF beam without any spring mass system-Case 1 ........ 148 4.102 Maximum and RMS responses at x=0.5L for SS, CC, CS beams and at x=L for CF beam carrying one spring-mass system under moving load -Case 2 (m1/mb=0.01) ............................................ 149 4.103 Maximum and RMS responses at x=0.5L for SS, CC, CS beams and at x=L for CF beam carrying one spring-mass system under moving load–Case 2 (m1/mb=0.02) ............................................. 150 4.104 Maximum and RMS responses at x=0.5L for SS, CC, CS beams and at x=L for CF beam carrying one spring-mass system under moving load -Case 2 (m1/mb=0.05) ....................................................... 150 4.105 Maximum and RMS responses at x=0.5L for SS, CC, CS beams and at x=L for CF beam carrying one spring-mass system under moving load -Case 2 (m1/mb=0.1) ......................................................... 150 4.106 Maximum and RMS responses at x=0.5L for SS, CC, CS beams and at x=L for CF beam carrying one spring-mass system under moving load -Case 2 (m1/mb=0.2) ......................................................... 151 4.107 Maximum and RMS responses at x=0.5L for SS, CC, CS beams and at x=L for CF beam carrying two spring-mass systems under moving load-Case 3 (m1/mb=m2/mb=0.01) ............................................ 151 4.108 Maximum and RMS responses at x=0.5L for SS, CC, CS beams and at x=L for CF beam carrying two spring-mass systems under moving load-Case 3 (m1/mb=m2/mb=0.02) ............................................ 151 4.109 Maximum and RMS responses at x=0.5L for SS, CC, CS beams and at x=L for CF beam carrying two spring-mass systems under moving load-Case 3 (m1/mb=m2/mb=0.05) ............................................ 152
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Table Page 4.110 Maximum and RMS responses at x=0.5L for SS, CC, CS beams and at x=L for CF beam carrying two spring-mass systems under moving load-Case 3 (m1/mb=m2/mb=0.1) .............................................. 152 4.111 Maximum and RMS responses at x=0.5L for SS, CC, CS beams and at x=L for CF beam carrying two spring-mass systems under moving load-Case 3 (m1/mb=m2/mb=0.2) .............................................. 152 4.112 Maximum and RMS responses at x=0.5L for SS, CC, CS beams and at x=L for CF beam without any spring mass system-Case 1 ........ 153 4.113 Maximum and RMS responses at x=0.5L for SS, CC, CS beams and at x=L for CF beam carrying one spring-mass system under moving pulsating force - Case 2 (m1/mb=0.01) ..................................... 154 4.114 Maximum and RMS responses at x=0.5L for SS, CC, CS beams and at x=L for CF beam carrying one spring-mass system under moving pulsating force - Case 2 (m1/mb=0.02) ..................................... 154 4.115 Maximum and RMS responses at x=0.5L for SS, CC, CS beams and at x=L for CF beam carrying one spring-mass system under moving pulsating force - Case 2 (m1/mb=0.05) ..................................... 154 4.116 Maximum and RMS responses at x=0.5L for SS, CC, CS beams and at x=L for CF beam carrying one spring-mass system under moving pulsating force - Case 2 (m1/mb=0.1) ....................................... 155 4.117 Maximum and RMS responses at x=0.5L for SS, CC, CS beams and at x=L for CF beam carrying one spring-mass system under moving pulsating force - Case 2 (m1/mb=0.2) ....................................... 155 4.118 Maximum and RMS responses at x=0.5L for SS, CC, CS beams and at x=L for CF beam carrying two spring-mass systems under moving pulsating force - Case 3 (m1/mb=m2/mb=0.01) ......................... 155 4.119 Maximum and RMS responses at x=0.5L for SS, CC, CS beams and at x=L for CF beam carrying two spring-mass systems under moving pulsating force - Case 3 (m1/mb=m2/mb=0.02) ......................... 156 4.120 Maximum and RMS responses at x=0.5L for SS, CC, CS beams and at x=L for CF beam carrying two spring-mass systems under moving pulsating force - Case 3 (m1/mb=m2/mb=0.05) ......................... 156 4.121 Maximum and RMS responses at x=0.5L for SS, CC, CS beams and at x=L for CF beam carrying two spring-mass systems under moving pulsating force - Case 3 (m1/mb=m2/mb=0.1) ........................... 156 4.122 Maximum and RMS responses at x=0.5L for SS, CC, CS beams and at x=L for CF beam carrying two spring-mass systems under moving pulsating force - Case 3 (m1/mb=m2/mb=0.2) ........................... 157 4.123 Maximum and RMS responses at x=0.5L for SS beam carrying one spring-mass system under moving pulsating force - Case 4 ......... 157 4.124 Maximum and RMS responses at x=0.75L for two-span uniform beam carrying one spring-mass system under impact loading - Case 1 ........................................................................ 163
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Table Page 4.125 Maximum and RMS responses at x=0.25L for two-span uniform beam carrying one spring-mass system under impact loading - Case 1 ........................................................................ 163 4.126 Maximum and RMS responses at x=0.75L for two-span uniform beam carrying one spring-mass system under impact loading - Case 2 ........................................................................ 164 4.127 Maximum and RMS responses at x=0.25L for two-span uniform beam carrying one spring-mass system under impact loading - Case 2 ........................................................................ 165 4.128 Maximum and RMS responses at x=0.75L for two-span uniform beam carrying one spring-mass system under impact loading - Case 3 ........................................................................ 166 4.129 Maximum and RMS responses at x=0.25L for two-span uniform beam carrying one spring-mass system under impact loading - Case 3 ........................................................................ 166 4.130 Maximum and RMS responses at x=0.75L for two-span uniform beam carrying one spring-mass system under harmonic loading - Case 1 .................................................................... 167 4.131 Maximum and RMS responses at x=0.25L for two-span uniform beam carrying one spring-mass system under harmonic loading - Case 1 .................................................................... 168 4.132 Maximum and RMS responses at x=0.75L for two-span uniform beam carrying one spring-mass system under harmonic loading - Case 2 .................................................................... 169 4.133 Maximum and RMS responses at x=0.25L for two-span uniform beam carrying one spring-mass system under harmonic loading - Case 2 .................................................................... 169 4.134 Maximum and RMS responses at x=0.75L for two-span uniform beam carrying one spring-mass system under harmonic loading - Case 3 .................................................................... 170 4.135 Maximum and RMS responses at x=0.25L for two-span uniform beam carrying one spring-mass system under harmonic loading - Case 3 .................................................................... 171 4.136 Maximum and RMS responses at x=0.75L for two-span uniform beam carrying one spring-mass system under harmonic loading - Case 4 .................................................................... 171 4.137 Maximum and RMS responses at x=0.75L for two-span uniform beam carrying one spring-mass system under moving load-Case 1 .............................................................................. 172 4.138 Maximum and RMS responses at x=0.25L for two-span uniform beam carrying one spring-mass system under moving load-Case 1 .............................................................................. 173
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Table Page 4.139 Maximum and RMS responses at x=0.75L for two-span uniform beam carrying one spring-mass system under moving pulsating force-Case 1 ............................................................. 174 4.140 Maximum and RMS responses at x=0.25L for two-span uniform beam carrying one spring-mass system under moving pulsating force-Case 1 ............................................................. 174 4.141 Maximum and RMS responses at x=0.75L for two-span uniform beam carrying one spring-mass system under harmonic loading - Case 2 .................................................................... 175 4.142 Maximum and RMS responses at x=0.25L for two-span uniform beam carrying two spring-mass systems under impact loading - Case 1 ........................................................................ 176 4.143 Maximum and RMS responses at x=0.75L for two-span uniform beam carrying two spring-mass systems under impact loading - Case 1 ........................................................................ 177 4.144 Maximum and RMS responses at x=0.25L for two-span uniform beam carrying two spring-mass systems under impact loading - Case 2 ........................................................................ 177 4.145 Maximum and RMS responses at x=0.75L for two-span uniform beam carrying two spring-mass systems under impact loading - Case 2 ........................................................................ 178 4.146 Maximum and RMS responses at both x=0.25L and x=0.75L for two-span uniform beam carrying two spring-mass systems under impact loading - Case 3 ........................................................................ 179 4.147 Maximum and RMS responses at x=0.25L for two-span uniform beam carrying two spring-mass systems under impact loading - Case 4 ........................................................................ 179 4.148 Maximum and RMS responses at x=0.75L for two-span uniform beam carrying two spring-mass systems under impact loading - Case 4 ........................................................................ 180 4.149 Maximum and RMS responses at x=0.25L for two-span uniform beam carrying two spring-mass systems under harmonic loading - Case 1 .................................................................... 181 4.150 Maximum and RMS responses at x=0.75L for two-span uniform beam carrying two spring-mass systems under harmonic loading - Case 1 .................................................................... 181 4.151 Maximum and RMS responses at x=0.25L for two-span uniform beam carrying two spring-mass systems under harmonic loading - Case 2 .................................................................... 182 4.152 Maximum and RMS responses at x=0.75L for two-span uniform beam carrying two spring-mass systems under harmonic loading - Case 2 .................................................................... 182
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Table Page 4.153 Maximum and RMS responses at both x=0.25L and x=0.75L for two-span uniform beam carrying two spring-mass systems under harmonic loading - Case 3 .................................................................... 183 4.154 Maximum and RMS responses at x=0.25L for two-span uniform beam carrying two spring-mass systems under harmonic loading - Case 4 .................................................................... 184 4.155 Maximum and RMS responses at x=0.75L for two-span uniform beam carrying two spring-mass systems under harmonic loading - Case 4 .................................................................... 184 4.156 Maximum and RMS responses at x=0.25L and x=0.75L for two-span uniform beam carrying two spring-mass systems under moving load-Case 1 .............................................................................. 185 4.157 Maximum and RMS responses at x=0.75L for two-span uniform beam carrying two spring-mass systems under moving load-Case 2 .............................................................................. 186 4.158 Maximum and RMS responses at x=0.25L for two-span uniform beam carrying two spring-mass systems under moving load-Case 2 .............................................................................. 186 4.159 Maximum and RMS responses at x=0.25L and x=0.75L for two-span uniform beam carrying two spring-mass systems under moving pulsating force-Case 1 ............................................................. 187 4.160 Maximum and RMS responses at x=0.75L for two-span uniform beam carrying two spring-mass systems under moving pulsating force-Case 2 ............................................................. 188 4.161 Maximum and RMS responses at x=0.25L for two-span uniform beam carrying two spring-mass systems under moving pulsating force-Case 2 ............................................................. 188 4.162 Maximum and RMS responses at x=(1/6)L for three-span uniform beam carrying one spring-mass system under impact loading - Case 1 ........................................................................ 189 4.163 Maximum and RMS responses at x=(3/6)L for three-span uniform beam carrying one spring-mass system under impact loading - Case 1 ........................................................................ 190 4.164 Maximum and RMS responses at x=(5/6)L for three-span uniform beam carrying one spring-mass system under impact loading - Case 1 ........................................................................ 190 4.165 Maximum and RMS responses at x=(1/6)L for three-span uniform beam carrying one spring-mass system under impact loading - Case 2 ........................................................................ 191 4.166 Maximum and RMS responses at x=(3/6)L for three-span uniform beam carrying one spring-mass system under impact loading - Case 2 ........................................................................ 191
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Table Page 4.167 Maximum and RMS responses at x=(5/6)L for three-span uniform beam carrying one spring-mass system under impact loading - Case 2 ........................................................................ 192 4.168 Maximum and RMS responses at x=(1/6)L for three-span uniform beam carrying one spring-mass system under impact loading - Case 3 ........................................................................ 193 4.169 Maximum and RMS responses at x=(3/6)L for three-span uniform beam carrying one spring-mass system under impact loading - Case 3 ........................................................................ 193 4.170 Maximum and RMS responses at x=(5/6)L for three-span uniform beam carrying one spring-mass system under impact loading - Case 3 ........................................................................ 193 4.171 Maximum and RMS responses at x=(1/6)L and x=(5/6)L for three-span uniform beam carrying one spring-mass system under impact loading - Case 4 ........................................................................ 194 4.172 Maximum and RMS responses at x=(3/6)L for three-span uniform beam carrying one spring-mass system under impact loading - Case 4 ........................................................................ 194 4.173 Maximum and RMS responses at x=(1/6)L for three-span uniform beam carrying one spring-mass system under harmonic loading - Case 1 .................................................................... 195 4.174 Maximum and RMS responses at x=(3/6)L for three-span uniform beam carrying one spring-mass system under harmonic loading - Case 1 .................................................................... 196 4.175 Maximum and RMS responses at x=(5/6)L for three-span uniform beam carrying one spring-mass system under harmonic loading - Case 1 .................................................................... 196 4.176 Maximum and RMS responses at x=(1/6)L for three-span uniform beam carrying one spring-mass system under harmonic loading - Case 2 .................................................................... 197 4.177 Maximum and RMS responses at x=(3/6)L for three-span uniform beam carrying one spring-mass system under harmonic loading - Case 2 .................................................................... 197 4.178 Maximum and RMS responses at x=(5/6)L for three-span uniform beam carrying one spring-mass system under harmonic loading - Case 2 .................................................................... 198 4.179 Maximum and RMS responses at x=(1/6)L for three-span uniform beam carrying one spring-mass system under harmonic loading - Case 3 .................................................................... 199 4.180 Maximum and RMS responses at x=(3/6)L for three-span uniform beam carrying one spring-mass system under harmonic loading - Case 3 .................................................................... 199
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Table Page 4.181 Maximum and RMS responses at x=(5/6)L for three-span uniform beam carrying one spring-mass system under harmonic loading - Case 3 .................................................................... 199 4.182 Maximum and RMS responses at x=(1/6)L and x=(5/6)L for three-span uniform beam carrying one spring-mass system under harmonic loading - Case 4 .................................................................... 200 4.183 Maximum and RMS responses at x=(3/6)L for three-span uniform beam carrying one spring-mass system under harmonic loading - Case 4 .................................................................... 200 4.184 Maximum and RMS responses at x=(1/6)L for three-span uniform beam carrying one spring-mass system under moving load-Case 1 .............................................................................. 201 4.185 Maximum and RMS responses at x=(3/6)L for three-span uniform beam carrying one spring-mass system under moving load-Case 1 .............................................................................. 202 4.186 Maximum and RMS responses at x=(5/6)L for three-span uniform beam carrying one spring-mass system under moving load-Case 1 .............................................................................. 202 4.187 Maximum and RMS responses at x=(1/6)L and x=(5/6)L for three-span uniform beam carrying one spring-mass system under moving load-Case 2 .............................................................................. 203 4.188 Maximum and RMS responses at x=(3/6)L for three-span uniform beam carrying one spring-mass system under moving load-Case 2 .............................................................................. 203 4.189 Maximum and RMS responses at x=(1/6)L for three-span uniform beam carrying one spring-mass system under moving pulsating force -Case 1 ............................................................ 204 4.190 Maximum and RMS responses at x=(3/6)L for three-span uniform beam carrying one spring-mass system under moving pulsating force -Case 1 ............................................................ 205 4.191 Maximum and RMS responses at x=(5/6)L for three-span uniform beam carrying one spring-mass system under moving pulsating force -Case 1 ............................................................ 205 4.192 Maximum and RMS responses at x=(1/6)L and x=(5/6)L for three-span uniform beam carrying one spring-mass system under moving pulsating force -Case 2 ............................................................ 206 4.193 Maximum and RMS responses at x=(3/6)L for three-span uniform beam carrying one spring-mass system under moving pulsating force -Case 2 ............................................................ 206 4.194 Maximum and RMS responses at x=(1/6)L for three-span uniform beam carrying two spring-mass systems under impact loading - Case 1 ........................................................................ 208
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Table Page 4.195 Maximum and RMS responses at x=(3/6)L for three-span uniform beam carrying two spring-mass systems under impact loading - Case 1 ........................................................................ 208 4.196 Maximum and RMS responses at x=(5/6)L for three-span uniform beam carrying two spring-mass systems under impact loading - Case 1 ........................................................................ 208 4.197 Maximum and RMS responses at x=(1/6)L for three-span uniform beam carrying two spring-mass systems under harmonic loading - Case 1 .................................................................... 209 4.198 Maximum and RMS responses at x=(3/6)L for three-span uniform beam carrying two spring-mass systems under harmonic loading - Case 1 .................................................................... 210 4.199 Maximum and RMS responses at x=(5/6)L for three-span uniform beam carrying two spring-mass systems under harmonic loading - Case 1 .................................................................... 210 4.200 Maximum and RMS responses at x=(1/6)L for three-span uniform beam carrying two spring-mass systems under moving load-Case 1 .............................................................................. 211 4.201 Maximum and RMS responses at x=(3/6)L for three-span uniform beam carrying two spring-mass systems under moving load-Case 1 .............................................................................. 212 4.202 Maximum and RMS responses at x=(5/6)L for three-span uniform beam carrying two spring-mass systems under moving load-Case 1 .............................................................................. 212 4.203 Maximum and RMS responses at x=(1/6)L for three-span uniform beam carrying two spring-mass systems under moving pulsating force -Case 1 ............................................................ 213 4.204 Maximum and RMS responses at x=(3/6)L for three-span uniform beam carrying two spring-mass systems under moving pulsating force -Case 1 ............................................................ 214 4.205 Maximum and RMS responses at x=(5/6)L for three-span uniform beam carrying two spring-mass systems under moving pulsating force -Case 1 ............................................................ 214
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LIST OF FIGURES
Figure Page 1.1 Types of Dynamic Loadings (Murray et al.1997) ..................................... 2 2.1 Directions of coordinate systems for vibrations influencing humans
(Naeim, 1991). ...................................................................................... 10 2.2 Average plot of force versus time for heel impact (Naeim, 1991) .......... 11 2.3 Typical floor response to heel impact (Naeim, 1991) ............................ 11 2.4 Recommended peak acceleration for human comfort for vibrations (Allen and Murray, 1993; ISO 2631/2, 1989). ........................................ 13 2.5 Modified Reiher-Meister perceptibility chart (Naeim, 1991) ................... 15 2.6 CSA annoyance criteria chart for floor vibrations (Naeim, 1991)........... 16 2.7 Generic Vibration Criteria of Gordon (Pan et al.2008) ........................... 20 2.8 Perception criteria (Ungar, 2007) .......................................................... 21 2.9 Viscous damper fitted between chevron braces beneath the deck of the London Millennium bridge (Nyawako and Reynolds, 2007) ........ 26 2.10 Free-layer damping and constrained-layer damping systems (Nyawako and Reynolds, 2007) ............................................................ 27 2.11 Friction damper device components and principle of action (Nyawako and Reynolds, 2007) ............................................................ 27 2.12 Illustration of a tuned sloshing damper (Nyawako and Reynolds, 2007) ............................................................ 29 2.13 Operating principles of an active control system (Nyawako and Reynolds, 2007) ............................................................ 30 2.14 Active mass dampers (Nyawako and Reynolds, 2007) ......................... 30 2.15 Uncontrolled and actively controlled velocity response of an office floor (Nyawako and Reynolds, 2007) ...................................... 31 2.16 Semi-active TMD on a vibrating system ................................................ 32 2.17 Undamped and damped vibration absorbers ........................................ 34 2.18 Tuned Mass Dampers beneath the London Millennium Bridge (Nyawako and Reynolds, 2007) ............................................................ 35 2.19 Example of the effect of damping ratio ξ of the vibration absorber on the frequency response of a primary system (Bachmann et al., 1994) ........................................................................ 37 2.20 SDOF-TMD system (Connor, 2003) ...................................................... 37 3.1 A cantilever beam carrying n spring-mass systems (Wu and Chou, 1999) ............................................................................ 42 3.2 A non-uniform cantilever beam carrying n spring-mass systems .......... 51
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Figure Page 3.3 A uniform multi-span beam carrying S spring-mass systems and T pinned supports (Lin and Tsai, 2007) ................................................. 64 3.4 Two-span uniform beam with one intermediate support and one spring-mass system ....................................................................... 73 3.5 Simply-supported beam subjected to step-function force F0 ................. 76 3.6 Simply-supported beam subjected to harmonic force F0sin(Ωt) ............ 77 3.7 Simply-supported beam subjected to moving load ................................ 78 3.8 Simply-supported beam subjected to moving pulsating load ................ 78 3.9 Simply-supported beam subjected to moving pulsating load ................ 79 4.1 Mode Shapes of Uniform SS, CC, CS and CF Beams Carrying One Spring-Mass System ..................................................................... 90 4.2 Mode Shapes of Uniform SS, CC, CS and CF Beams Carrying Two Spring-Mass Systems.................................................................... 91 4.3 Mode Shapes of Uniform SS, CC, CS and CF Beams Carrying Three Spring-Mass Systems ................................................................. 92 4.4 Mode Shapes of Bare SS, CC, CS and CF Uniform Beams ................. 93 4.5 Mode Shapes of Non-Uniform SS, CC, SC and FC Beams Carrying One Spring-Mass System ....................................................... 97 4.6 Mode Shapes of Bare SS, CC, CS and CF Non-Uniform Beams .............................................................................. 98 4.7 Mode Shapes of High-Mast Lighting Tower Carrying One Spring-Mass System on the Top ................................................. 101 4.8 Mode Shapes of Bare High-Mast Lighting Tower ................................ 101 4.9 Two-span beam carrying one spring-mass system attached to second span ..................................................................... 103 4.10 Two-span beam carrying one spring-mass system attached to first span ........................................................................... 104 4.11 Mode shapes of two-span beam carrying one spring-mass system at second span (mtmd=0.01mb) ................................................ 105 4.12 Mode shapes of two-span beam carrying one spring-mass system at first span (mtmd=0.01mb) ...................................................... 105 4.13 Two-span beam carrying two spring-mass systems (Case 1) ............. 106 4.14 Two-span beam carrying two spring-mass systems (Case 2) ............. 107 4.15 Mode shapes of two-span beam carrying two spring-mass systems tuned based on Case 1(m1tmd= m2tmd=0.01mb) ...................... 108 4.16 Mode shapes of two-span beam carrying two spring-mass systems tuned based on Case 2(m1tmd= m2tmd=0.01mb) ...................... 108 4.17 Three-span beam carrying one spring-mass system attached to first span ........................................................................... 109 4.18 Three-span beam carrying one spring-mass system attached to second span ..................................................................... 110 4.19 Mode shapes of three-span beam carrying one spring-mass system at first span (m1tmd=0.01mb) .................................................... 111
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Figure Page 4.20 Mode shapes of three-span beam carrying one spring-mass system at second span (m1tmd=0.01mb) .............................................. 111 4.21 Three-span beam carrying two spring-mass systems attached to first and second span (Case 1) ....................................................... 112 4.22 Three-span beam carrying two spring-mass systems attached to first and second span (Case 2) ....................................................... 113 4.23 Three-span beam carrying two spring-mass systems attached to first and second span (Case 3) ....................................................... 114 4.24 Three-span beam carrying two spring-mass systems attached to first and third span (Case 4) ............................................................ 115 4.25 Three-span beam carrying two spring-mass systems attached to first and third span (Case 5) ............................................................ 116 4.26 Mode shapes of three-span beam carrying two spring-mass systems at first span and second span tuned based on Case 1
(m1tmd=m2tmd=0.01mb) .......................................................................... 117 4.27 Mode shapes of three-span beam carrying two spring-mass systems at first and second span tuned based on Case 2 (m1tmd=m2tmd=0.01mb) .......................................................................... 117 4.28 Mode shapes of three-span beam carrying two spring-mass systems at first span and second span tuned based on Case 3
(m1tmd=m2tmd=0.01mb) .......................................................................... 118 4.29 Mode shapes of three-span beam carrying two spring-mass systems at first and third span tuned based on Case 4 (m1tmd=m2tmd=0.01mb) .......................................................................... 118 4.30 Mode shapes of three-span beam carrying two spring-mass systems at first span and third span tuned based on Case 5 (m1tmd=m2tmd=0.01mb) .......................................................................... 119 4.31 Simply supported beam carrying one spring-mass system subjected to step-function force at x=0.25L ......................................... 121 4.32 Simply supported beam carrying one spring-mass system subjected to step-function force at x=0.5L ........................................... 123 4.33 Simply supported beam carrying one spring-mass system subjected to harmonic force at x=0.25L .............................................. 135 4.34 Simply supported beam carrying one spring-mass system subjected to harmonic force at x=0.5L ................................................ 137 4.35 Simply supported beam carrying one spring-mass system subjected to moving load .................................................................... 149 4.36 Simply supported beam carrying one spring-mass system subjected to moving pulsating force .................................................... 153 4.37 Force and wind velocity profile of HMLT ............................................. 159 4.38 Dynamic responses of bare HMLT under wind load ............................ 160 4.39 Dynamic responses of HMLT carrying one spring-mass system at the shallow end under wind load (m1/mb=0.01) ................... 161
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Figure Page 4.40 Two-span simply supported beam carrying one spring-mass system subjected to step-function force at x=0.25L ............................ 162 4.41 Two-span simply supported beam carrying one spring-mass system subjected to step-function force at x=0.75L ............................ 164 4.42 Two-span simply supported beam carrying one spring-mass system subjected to step-function force at x=0.25L and x=0.75L ........ 165 4.43 Two-span simply supported beam carrying one spring-mass system subjected to harmonic force at x=0.25L .................................. 167 4.44 Two-span simply supported beam carrying one spring-mass system subjected to harmonic force at x=0.75L .................................. 168 4.45 Two-span simply supported beam carrying one spring-mass system subjected to harmonic force at x=0.25L and x=0.75L ............. 170 4.46 Two-span simply supported beam carrying one spring-mass system subjected to moving load ........................................................ 172 4.47 Two-span simply supported beam carrying one spring-mass system subjected to moving pulsating force ........................................ 173 4.48 Two-span simply supported beam carrying two spring-mass systems subjected to step-function force at x=0.75L .......................... 176 4.49 Two-span simply supported beam carrying two spring-mass systems subjected to step-function force at x=0.25L and x=0.75L ...................................................................... 178 4.50 Two-span simply supported beam carrying two spring-mass systems subjected to harmonic force at x=0.75L ................................ 180 4.51 Two-span simply supported beam carrying two spring-mass systems subjected to harmonic force at x=0.25L and x=0.75L ........... 183 4.52 Two-span simply supported beam carrying two spring-mass systems subjected to moving load ...................................................... 185 4.53 Two-span simply supported beam carrying two spring-mass systems subjected to moving pulsating load ....................................... 187 4.54 Three-span simply supported beam carrying one spring-mass system subjected to step-function force at x= (1/6) L .......................... 189 4.55 Three-span simply supported beam carrying one spring-mass system subjected to step-function force at x= (3/6) L .......................... 192 4.56 Three-span simply supported beam carrying one spring-mass system subjected to harmonic force at x= (1/6) L ................................ 195 4.57 Three-span simply supported beam carrying one spring-mass system subjected to harmonic force at x= (3/6) L ................................ 198 4.58 Three-span simply supported beam carrying one spring-mass system subjected to moving load ........................................................ 201 4.59 Three-span simply supported beam carrying one spring-mass system subjected to moving pulsating force ........................................ 204 4.60 Three-span simply supported beam carrying one spring-mass system subjected to step-function force at x= (1/6) L .......................... 207
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Figure Page 4.61 Three-span simply supported beam carrying one spring-mass system subjected to harmonic force at x= (1/6) L ................................ 209 4.62 Three-span simply supported beam carrying one spring-mass system subjected to moving load ........................................................ 211 4.63 Three-span simply supported beam carrying one spring-mass system subjected to moving pulsating force ........................................ 213 Appendix Figure A.1 SDOF-TMD system ............................................................................. 228
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LIST OF SYMBOLS
ω Natural frequency of primary system
c Viscous damping coefficient of primary system
ξ Damping ratio of primary system
k Stiffness of primary system
ωd Natural frequency of tuned mass damper
cd Viscous damping coefficient of tuned mass damper
ξd Damping ratio of tuned mass damper
kd Stiffness of tuned mass damper
Ω Forcing frequency
µ Mass ratio
ξe Equivalent damping ratio
Displacement of SDOF system
Velocity of SDOF system
Acceleration of SDOF system
Displacement of tuned mass damper
Velocity of tuned mass damper
Acceleration of tuned mass damper
Φ Phase angle
E Modulus of elasticity
I Moment of inertia
Beam mass per unit length
mv Point mass of the vth spring-mass system
kv Spring constant of the vth spring-mass system
zv Instantaneous displacement of vth spring-mass system
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Acceleration of vth spring-mass system
yv Displacement of constrained beam at the vth attaching point
Slope of constrained beam at the vth attaching point
Curvature of constrained beam at the vth attaching point
Yv(x) Amplitude of yv
Zv Amplitude of zv
L Represents the left as superscript
R Represents the right as superscript
y(x,t) Instantaneous displacement of the beam
ωv Natural frequency of spring-mass system
[Bv] Coefficient matrix for the vth attaching point
[BL] Coefficient matrix for the left end of the beam
[BR] Coefficient matrix for the right end of the beam
Cvi Integration constants
n Number of spring-mass systems
[B] Overall coefficient matrix
ρ Mass per unit volume
A Cross sectional area
A0 Cross sectional area at x=0
I0 Moment of inertia at x=0
r Radius
α Taper ratio of the beam
t Thickness
ηi(t) Generalized coordinates
Yi(x) ith normal mode shape of a beam
Qi(t) Generalized force corresponding to ηi(t)
w(x,t) Transverse deflection of a beam
f(x,t) External force per unit length
δij Kronecker delta
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ABSTRACT
Ozkan, Mustafa Kemal. M.S.C.E., Purdue University, May 2010. Dynamic Response of Beams with Passive Tuned Mass Dampers. Major Professor: Ayhan Irfanoglu.
Passive tuned mass damper (TMD) is a stand-alone vibrating system
attached to a primary structure and designed to reduce vibration of the structure
at selected frequency. This study focuses on the application of single or multiple
TMDs on Euler-Bernoulli beams and examines their effectiveness based on free
and forced vibration characteristics of the beams, i.e., the primary structures.
There is a gap in the existing literature in terms of free and forced vibration
analysis of beams carrying any number of concentrated elements. There are
methods developed for the free vibration analysis but they are not practical due
to the complex mathematical expressions. Numerical assembly method (Wu and
Chou, 1999) is used to determine free vibration characteristics of beams in order
to get over the drawbacks of other approaches in the literature and forced
vibration response is obtained based on modal analysis approach and
orthogonality condition.
The free-vibration formulations for uniform, non-uniform single-span and
multi-span continuous beams carrying any number of elastically mounted
masses are derived for various boundary conditions. Numerical solutions for
dynamic responses of these beams subjected to impulsive, harmonic, moving
and moving pulsating loads are presented. A numerical eigenvalue solution is
used to obtain the modal properties of the entire beam at its fundamental and
lower normal modes. The modal analysis approach allows calculating
xxx
displacement, velocity, acceleration and jerk responses at any point on the
beam. The resultant dynamic responses of beams with and without TMDs are
compared with each other in order to observe the performance of TMDs.
Numerical examples are given to confirm the validity and efficiency of the
proposed method. Natural frequencies and mode shapes of several structures
studied in literature are calculated and compared with those in existing literature
to verify the accuracy of the developed algorithm. The illustrative forcing
functions are considered as human-induced dynamic loads for uniform single and
multiple span beams. The results demonstrate that passive TMDs are efficient in
reducing the dynamic responses of beams subjected to harmonic excitations.
However, passive TMDs do not show the same level of performance under non-
harmonic loads. Additionally, wind load analysis is performed for a sample high
mast lighting tower (HMLT) represented as a cantilever non-uniform beam in this
study and the efficiency of attached TMD is analyzed. Experimental wind velocity
data is used to generate the wind induced dynamic load on the HMLT. Results
indicate that properly tuned passive TMDs may be an option to reduce dynamic
response in wind-excited HMLTs.
1
CHAPTER 1. INTRODUCTION
1.1. General
Annoying motions and audible resonant behavior are two common
concerns in structures under service-level dynamic loads. In more serious cases,
excessive vibrations or fatigue which may cause structural failure may exist.
These unwanted vibrations should be eliminated or at least reduced below
corresponding threshold levels to avoid serious structural problems or simply to
meet system performance requirements. It may be easier to modify the design of
a yet to be built structure to mitigate against possible unwanted vibrations
compared to modifying an actual, existing structure. The latter case may be
called for if the existing structure has insufficient design or, is subjected to
change in functionality, or due to changes in environmental conditions.
1.1.1. Sources of Dynamic Excitation
In practical engineering design one of the most important requirements is
to define the sources of dynamic excitation and to analyze their magnitude and
significance by comparing them with the static loads. It is usually much easier to
deal with static loads rather than dynamic loads. Some of the structures,
especially the flexible and lightly damped ones, may exhibit large amplifications
against dynamic loads. The use of a structure, such as a laboratory housing
sensitive equipments or hospitals with sensitive operating rooms such as those
for neurosurgery or microsurgery, is another issue to consider in vibration
analysis. Therefore, the relation between the sources of dynamic excitation, the
structural form and the purpose of the structure should be considered at the
design stage.
2
1.1.2. Dynamic Loadings
Dynamic loads can be categorized as harmonic, periodic, transient and
impulsive as illustrated in Figure 1.1. Rotating machinery can be identified as
harmonic or sinusoidal loads. Rhythmic human activities such as dancing and
aerobics and impactive machinery cause periodic loads. Transient loads consist
of movement of people including walking and running. Single jumps and heel-
drop impacts can be given examples of impulsive loads.
Figure 1.1 Types of Dynamic Loadings (Murray et al.1997)
1.1.3. Consequences of Vibration
Vibration of structures is undesirable for a number of reasons. For
example, overstressing and collapse of structures or simply cracking or other
damages requiring repair can be given as consequences of earthquake-induced
vibrations. To give another example, damage to safety-related equipment is
another problem occurring in nuclear plants during earthquakes. Excessive
structural vibrations in hospitals and other medical facilities can interfere with the
performance of medical procedures, impair the operation of sensitive equipment
and have adverse effects on patient comfort. Adverse human response is also a
3
phenomenon in structures such as health clubs, gymnasiums, stadiums, dance
floors and even office buildings due to the human activities.
Human beings are highly sensitive to vibration. Therefore, adverse human
response should be seriously thought at the design stage although vibrations that
are disturbing for occupants of buildings usually cause small stresses. However,
if the structure is subjected to large number of cycles of loads above certain
thresholds, fatigue fracture problem may occur as another phenomenon. Fatigue
fracture usually occurs in welded steel structures where tiny cracks, which are
initially difficult to see, grow in size under the repetitions of stress until they are
large enough to be seen or cause rupture. For example, high-mast lighting
towers are subjected to wind load which, over time, may cause millions of cycles
of significant stress and could result in structural failure.
1.1.4. Vibration Control
The first step to design the structure which is sensitive to vibrations is to
identify the dynamic loads in terms of frequency and amplitude or measured
variation in time. Analyzing the response of the structure to obtain dynamic
deflections, stresses, frequencies and accelerations come next. Finally, it is
essential to check the calculated or measured performance by using specified
criteria to guarantee that there are no adverse consequences of vibration.
It is important to think ahead during the early stage of conceptual design
and make necessary design adjustments in order to minimize the vibration
susceptibility. Some structures, such as dancing floors, are affected over a
confined frequency range. It may be feasible to increase the structural depth in
dance floors in order to improve stiffness of the structure so as to keep the
frequency of the structure above predominant dancing frequency.
4
Active control over the natural frequency of buildings may be provided by
increasing the stiffness or reducing the mass but this method is usually difficult or
uneconomic to obtain the optimum value. It may be more efficient to design and
use special vibration-absorbing devices, called passive tuned mass dampers
(TMDs) or tuned vibration absorbers (TVAs), as part of the structure to reduce
effects of dynamic loads. Some of the construction techniques, such as welded
steelwork, may be more delicate to vibration because of their lack of inherent
damping capacity. Therefore, it may sometimes be more effective to choose
materials with high damping or to install artificial damping devices.
1.2. Object and Scope
The main motivation of this study is to investigate the effects of tuned
vibration absorbers (TVAs), also called tuned mass dampers (TMDs), in terms of
controlling the dynamic deflections, stresses, frequencies and accelerations of
structural elements or structures which are subjected to different types of
dynamic loadings.
The scope of the study includes (a) analytical and numerical procedures to
solve the free and forced vibration of uniform and non-uniform single and multi-
span beams, which are subjected to different types of dynamic loadings, with
attached spring-mass systems, in particular, passive TMDs (b) some applications
of these analytical procedures in real structural elements.
The free vibration analysis of structural elements, such as beams, is
commonly found in existing literature. A critical aspect of the presented study is
to analyze the forced vibration of these structural elements with attached spring-
mass systems and to check that whether these spring-mass systems, which can
also be called as TVA or TMD, work towards reducing the consequences of
adverse vibration.
5
1.3. Organization
Chapter 2 reviews and describes the existing research on practical
approaches for vibration reduction together with detailed explanation of vibration
excitation sources which are human-induced vibration, machinery induced
vibration and wind-induced vibration.
Chapter 3 discusses the analytical approaches to determine natural
frequencies, mode shapes and responses of beams with attached spring mass
systems under different types of dynamic loads. Applications of passive TVAs to
uniform and non-uniform beams are presented. The effect of single TVA versus
multiple TVAs is discussed.
Chapter 4 includes the numerical results for free and forced vibration of
beams carrying single or multiple spring mass systems. Illustrative numerical
examples are presented for step-function forces (i.e., impact loading), harmonic
forces, moving loads and moving pulsating forces.
Summary and conclusions are presented in Chapter 5.
6
CHAPTER 2. BACKGROUND AND PREVIOUS RESEARCH
2.1. Human-Structure Dynamic Interaction and Human Induced Vibration
Human-induced forces, such as walking, running, jumping, dancing and
other similar activities, cause unwanted vibrations in civil engineering structures,
such as floors, footbridges, grandstands, stairs. These kinds of rhythmical human
activities can generate significant resonant, transient, steady-state or impulsive
responses (Nyawako and Reynolds, 2007). According to Nyawako and Reynolds
(2007), pacing frequency for walking can be considered in the range of 1.5-3.0
Hz and it is above 3 Hz for activities such as running or jogging. However, this
range can decrase to the range of 1.25-1.5 Hz for offices based on Smith’s
(1998) and Murray’s (1998) studies.
Human-structure interaction concept is a considerably important issue for
slender structures which are subjected to human-induced forces (R.Sachse et al.
2003). Today’s new construction techniques provide us to design light, slender,
long-spanned structures, however these opportunities have increased the
susceptibility level of structures to detrimental vibrations (Firth, 2002; Naeim,
1991; Tuan and Saul, 1985; Setareh et al. 2006a). When these lighter and longer
floor systems together with their less damping are considered with the rhythmical
activities of the occupants, there is a significant increase in attention to the
vibration level during the design stage to reduce floor vibration problems because
of the greater possibility of increased vibration annoyance in occupants (Naeim,
1991). Moreover, human-induced vibrations may cause serviceability and safety
problems, in terms of annoying level of vibrations for occupants and fatigue
behavior of structures, respectively (Smith, 1998; Bachmann, 1992). The major
reasons of annoying vibrations in civil engineering structures can be explained
with three main factors which are increased human activities, such as aerobics or
7
audience participation, reduced natural frequency due to longer floor systems
and reduced damping and mass provided by new construction techniques (Allen,
1990). Pedestrian structures, office buildings, footbridges, gymnasiums and sport
halls, shopping malls, airport terminals, dance halls and concert halls can be
given as an example for civil engineering structures which tend to be susceptible
to human-induced vibration (Bachmann, 1992; Kerr and Bishop, 2001; Pavic et
al. 2002a; Hanagan et al. 2003; Ebrahimpour and Sack, 2005).
Another case that designers need to pay attention is to consider the
usability of civil engineering structures for different types of occupation for both
economic and serviceability reasons. For instance, building owners or occupants
might need to convert an office floor into a gymnasium or a dance facility in the
future (Webster and Levy, 1992).
It is also necessary to know how human induced forces can be identified
in order to figure out the response of the structure under human induced
vibration. Human induced force-time histories have been obtained for the last
three decades and these experimental results have been analyzed by Fourier
series. Because of this approximation for many experimental results, most of the
researchers came up with the common idea that human induced forces are
perfectly periodic (Sachse et al., 2003). On the other hand, there are some
opposite assumptions questioning that human induced forces are perfectly
periodic because of the fact that these forces are inherently narrow-band
(Eriksson, 1994). Sometimes, auto-spectral density functions are used to
represent human-induced forces in the frequency domain (Tuan and Saul, 1985;
Mouring and Ellingwood, 1994; Eriksson, 1994). Moreover, according to Murray
et al. (1997), dynamic forces which cause floor vibration problems are generally
repeated forces, such as machinery or human induced forces, and they are
usually sinusoidal or nearly sinusoidal. Thereby, such repeated forces can be
defined as sum of sinusoidal forces. Their forcing frequencies can be considered
as multiples of the fundamental frequency of the force repetition such as step
frequency for human activities. Murray et al. (1997) defined time-dependent
8
harmonic force component matching the fundamental frequency of the floor by
using Fourier series as given in Equation 2.1.
F cos 2 Eq. 2.1
where P (person’s weight) can be taken as 0.7 kN (157 pounds) and
recommended values for are given in Table 2.1.
Table 2.1 Recommended values for α (Murray et al., 1997)
Common Forcing Frequencies (f) and Dynamic Coefficients* (αi)
Harmonic
i
Person Walking Aerobic Class Group Dancing
f, Hz (αi) f, Hz (αi) f, Hz (αi)
1 1.6-2.2 0.5 2.2-2.8 1.5 1.8-2.8 0.5
2 3.2-4.4 0.2 4.4-5.6 0.6 3.6-5.6 0.1
3 4.8-6.6 0.1 6.6-8.4 0.1 - -
4 6.4-8.8 0.05 - - - -
*Dynamic Coefficients = Peak Sinusoidal force/weight of person(s).
Previous researches show that the number of people has an effect on
dynamic loads. Activities of a group of people generate more dynamic loads than
individuals. However, it does not mean that dynamic loads are increasing with
the number of people linearly (Sachse et al., 2003). In addition, synchronization
of people is also an important issue that influences dynamic loads and it can be
classified as deliberate and unintentional. Coordinated human activities such as
in aerobic classes can be given as an example for deliberate synchronization and
these types of human activities has been studied for a long time (Sachse et al.,
2003). However, unintentional synchronization is a recent subject that
researchers have considered to be important. This type of synchronization can
cause serviceability problems due to considerably strong vibration level and
safety problems due to panic (Dallard et al. 2000). There have been many cases
9
observed and reported in civil engineering structures due to unintentional
synchronization (Bachmann, 1992; Fujino et al., 1993; Dallard et al., 2000,
Curtis, 2001, New Civil Engineer, 2001).
Excessive structural vibration is also very important problem for healthcare
facility floors. It might negatively affect the performance of medical procedures,
make highly vibration sensitive equipments unusable and cause negative effects
on patient comfort. These problems pose newer vibration criteria that force
structural engineers to accomplish more strict requirements to provide
serviceability for highly vibration sensitive high-tech equipments than before. Pan
et al. (2008) studied a long-spanned biotechnology laboratory floor which is
supported by reinforced concrete beams and evaluated the performance of the
laboratory instruments while the floor is subjected to human induced vibration,
specifically walking. Footfall forces are applied to finite element model of beams
and floors, and then time history analysis results are compared with the
appropriate vibration criteria. According to results, the floor performed well
enough to satisfy the required vibration criteria. Yazdanniyaz et al. (2004) studied
footfall induced vibrations and discussed different vibration criteria and prediction
methods including AISC method, BBN method and analytical modal analysis.
This study also compares these prediction methods with measured floor vibration
levels for both composite and concrete laboratory floor. According to
Yazdanniyaz et al. (2004), designers should avoid predicting a vibration level
greater than it is or applying more strict vibration criteria than it is needed by
considering possible increase for building cost.
Vibration perceptibility is another subject of human-structure dynamic
interaction and needs to be considered to understand what kind of factors
influence the level of perception and sensitivity of people. This issue provides
some improvement during design stage by considering sensitivity of people and
preventing some possible increase in structural vibration. According to Naeim
(1991), vibration perceptibility mostly depends on position of human body,
characteristics of excitation sources and floor systems, exposure time, level of
10
expectancy and type of activity that people perform. As it is shown in Figure 2.1
human-body coordinate system has three different axes which are x, y and z
showing back to chest direction, right side to left side direction and foot to head
direction, respectively. According to International Standard ISO-2631/1-1985(E)
and International Standard ISO-2631/2-1989(E), humans’ sensitivity to
acceleration is experienced significantly when the frequency range is between 4
to 8 Hz for vibration along the z-axis and 0 to 2 Hz for vibration along the x or y
axes. Human activities being influenced by the vibration along the x or y axes,
such as sleeping, make these two axes as important as the z axis which is more
important than others for the design of office buildings.
Figure 2.1 Directions of coordinate systems for vibrations influencing humans
(Naeim, 1991).
11
Standard heel drop impact test is a practical way to model impulsive
forces caused by walking and to approximate to the real. In this test, a person
raises his heel about 2.5 inches and drops his weight through his heels to the
floor. Before initiating the impulse, the person, who weighs 170 pounds, supports
his weight on his toes. The result of the heel drop impact and a typical floor
response to heel impact are in Figure 2.2 and Figure 2.3 (Naeim, 1991).
Figure 2.2 Average plot of force versus time for heel impact (Naeim, 1991)
Figure 2.3 Typical floor response to heel impact (Naeim, 1991)
12
2.2. Vibration Criteria
2.2.1. ISO International Standard
Naeim (1991) indicates the classification of human response to vibrations
by referring International Standard Organization (ISO-2631/1). ISO-2631/1
categorizes human response with respect to the limitations which increase step
by step by considering the perception levels of humans. These categories can be
defined as the limit for reduced comfort, the limit that affects the working
conditions negatively and the limit that cause health or safety problems. These
categories were conceived depending on the studies related with transportation
industries which provide higher level of tolerance than the acceptable level for
buildings. The magnitudes of these limits are slightly exceeding the minimum
levels of human tolerance and they are determined with respect to minimum
adverse comment level of occupants.
Murray et al. (1997) also states that the perception level of people and
how they react to vibration considerably depend on what they are doing.
According to Murray et al. (1997), the vibration level that disturbs people in
offices and residences is 0.5 percent of the acceleration of gravity, g. On the
other hand, this limit might be approximately 10 times higher than that (5 percent
of g) when people are taking part in an activity. In addition, duration of vibration
and distance from the vibration source are other factors that affect the sensitivity
of the occupants. If people are very close to vibration excitation source, the limit
of annoyance level will be about 1.5 percent of g. The frequency range for these
limits indicated in the above is between 4Hz and 8 Hz. Higher vibration
acceleration levels are accepted for the outside of this frequency range. ISO-
2631/2 provides the recommended peak acceleration limits for different types of
structures and occupancies in terms of “root mean square” (RMS) acceleration
as it is shown in Figure 2.4. The limitations can vary depending on duration and
frequency of vibration by the range between 0.8 and 1.5 times the suggested
values.
13
According to ISO, vibrations can be classified as transient and continuous
(steady-state) in terms of human response depending on the type of excitation
and its duration. If a structural system is subjected to a continuous harmonic
force, the resulting motion has constant frequency and constant maximum
amplitude. This type of vibration is defined as continuous vibration. On the other
hand, transient vibration is defined as an instantaneous increase to a peak
followed by a damped decay in a short time. For example, floors subjected to
operating machines have continuous vibration because machines usually run
continuously for a long period of time. On the contrary, the floor vibration for the
residential and office buildings can be categorized in transient vibration because
of the intermittent movement of a small number of occupants. There are also
some cases that walking-induced excitation is rhythmic with an approximately
constant frequency and this type of vibration can be defined as mostly steady-
state.
Figure 2.4 Recommended peak acceleration for human comfort for vibrations
(Allen and Murray, 1993; ISO 2631/2, 1989).
14
2.2.2. Murray’s Criterion
According to Naeim (1991), the primary aim of Murray’s categorization of
human response is to be more design oriented. Therefore, this factor makes his
categorization more useful than others. There are four categories of human
response defined with respect to perception and discomfort levels of occupants.
For the first category, vibration is not perceived by the occupants and for the
second one, vibration is perceived but does not annoy. These two categories are
acceptable for design. For the third category, vibration just disturbs but for the
fourth one, it also makes occupants ill.
Murray’s method evaluates the potential floor vibration problems by
providing a step by step procedure. Field measurements and human response
studies for approximately 100 floor systems conceived the basis of this method.
Murray’s criteria is a very common and recommended for especially residential
and office buildings (Naeim, 1991).
2.2.3. Other Recommendations and Criteria
Ellingwood et al. (1986) recommended a vibration criterion for commercial
floor systems such as shopping malls. According to this criterion, if the maximum
deflection for anywhere on the floor, which is subjected to 450 lb force, does not
exceed 0.2 inches, the criterion is satisfied. In addition, Canadian Standards
Association and Murray also recommend that the natural frequency of
commercial floor systems should be greater than 8 Hz in order to reduce the
possibility of resonance due to walking (Naeim, 1991).
Another criterion called Wiss-Parmelee (1974) rating factor was based on
a laboratory study which was conceived to investigate human perception to
transient floor vibrations. The perception levels of 40 people were observed by
using platform motions corresponding to walking induced floor vibration. This
criterion was adopted by United States Department of Housing and Urban
Development to verify whether the floor system is in acceptable limits or not.
15
Reiher and Meister (1949) also studied human perceptibility to steady
state vibration in early 1930s. The forcing frequency range and the displacement
amplitude range of this study were 3 to 100 Hz and 0.0004 inches to 0.4 inches,
respectively. In early 1960s, Lenzen (1966) stated that if Reiher and Meister’s
amplitude scale was extended with a factor of 10, the resulting scale would be
applicable to correlate human perceptibility with natural frequency and
displacement amplitude for lightly damped floor systems. The new scale, called
Modified Reiher-Meister scale, shown in Figure 2.5 is usually used with an
additional method, like Murray’s acceptability criterion, in the design environment
to satisfy critical situations. The scale is being criticized negatively due to lack of
explicit consideration of damping which is considered to be quite important
*Density of participants is for commonly encountered conditions. For special events the density of participants can be greater. **Values of α are based on commonly encountered events involving a minimum of about 20 participants. Values of α should be increased for well-coordinated events (e.g. jump dances) or for fewer than 20 participants. ***Suggested revision to the 1985 supplement of CSA code.
Type of floor construction Dance floors*,Gymnasia**
Stadia, arenas**
Composite (steel-concrete) 9 6
Solid Concrete 7 5
Wood 12 8
*Limiting peak acceleration 0.02 g
**Limiting peak acceleration 0.05 g
2.2.4. Recommended Criteria for Sensitive Laboratory and Healthcare
Facility Floors
Because of the highly vibration sensitive high-tech equipments operated in
healthcare facilities or biotechnology laboratories, vibration criteria for these
19
critical areas are much more restrictive than before. This phenomenon was
carried out after the first vibration problem experienced in advanced technology
facilities by Intel in Livermore and Aloha, in late 1970s. Therefore, more
restrictive criteria were adopted by industry with the rapid development of
science and technology and these strict requirements posed new challenges on
structural engineers (Pan et al.2008). Total system approach in a single
analytical model, including both the support system and the equipment, is an
appropriate way to get more accurate response if a particular piece of equipment
was accommodated in a specific space (Medearis, 1995). The vibration criteria
provided by the manufacturer are then compared with the response results of the
equipment. However, sometimes the equipment hasn’t been selected yet or the
supporting structure might be required for more flexible usage. Therefore,
structural engineers want to use more comprehensive criteria which cover the
requirements of all equipments in just one particular category (Pan et al.2008).
Spectrum based generic criterion is one of the criteria that can be
classified into two groups in terms of expression. One of them is discrete
frequencies and the other is frequency bands. Design Guide 11 (AISC 1997) can
be given as an example of discrete frequencies. One-third octave bandwidth
spectra is one of the frequency bands that is commonly adopted by industries.
The one third octave band criteria were first presented in 1983 and then Gordon
and Ungar developed and republished them (Ungar et al. 1990; Gordon 1991).
Institute of Environmental Sciences (IES) also accepted these criteria and
published them in 1993 and 1998 (IES 1998).
As it is shown in Figure 2.7, there is a set of root mean square (RMS)
velocity spectra defined as vibration criteria curves VC-A – VC-E and table 2.5
shows the explanation of each curve of generic vibration criteria (IES 1998).
These criteria were posed by reviewing many particular equipment criteria which
were provided by equipment manufacturers (Pan et al.2008).
20
Figure 2.7 Generic Vibration Criteria of Gordon (Pan et al.2008)
The criteria for healthcare facilities are based on American National
Standard Institute (ANSI) Standard S3.29-1983 titled as ‘Guide to the Evaluation
of Human Exposure to Vibration in Buildings’ which shows base-response curve
values related with human perception. According to ANSI, the base response
curve values correspond to the approximate threshold of perception for the most
sensitive humans and approximately one half of the mean threshold of
perception. This standard has similarities with International Standard ISO 2631-
2:1989(E), ‘Evaluation of Human Exposure to Whole-Body Vibration-Part 2:
Continuous and Shock-Induced Vibration in Buildings’ (Ungar, 2007). The
frequency variations of two base-response-curve RMS accelerations which are in
units of micro-g are shown in Figure 2.8.
21
Table 2.5 Application and Interpretation of Generic Vibration Criteria (Pan et al.2008)
Criterion curve
RMS amplitude* (µm)
Detail size (µm)
Description of use
Office (ISO)
400 N/A** Perceptible vibration. Appropriate for offices and non-sensitive areas.
VC-A 50 8 Adequate in most instances for optical microscopes to 400x, microbalances, optical balances, proximity and projection aligners, etc.
VC-B 25 3 Appropriate standard for optical microscopes to 1000x, inspection and lithography equipment (including steppers) to 3 µm line widths.
VC-C 12.5 1 A good standard for most lithography and inspection equipment (including electron microscopes) to 1 µm detail size.
VC-D 6 0.3
Suitable in most instances for the most demanding equipment, including electron microscopes (TEMs and SEMs) and e-beam systems, operating to the limits of their capability.
VC-E 3 0.1
A difficult criterion to achieve in most instances. Assumed to be adequate for the most demanding of sensitive systems including long-path, laser-based small target systems and other systems requiring extraordinary dynamic stability.
Figure 2.8 Perception criteria (Ungar, 2007)
22
ANSI S3.29-1983 also recommends that the vibrations of operating room
floors should not be more than between 0.7 and 1.0 times the base-response
curve values. However, ISO which has many similarities with ANSI standard
suggests that vibrations of operating rooms should not exceed the base
response values. One-fourth of the limiting values for ordinary operating rooms
have been used for sensitive operating rooms such as neurosurgery and
microsurgery.
For the floor vibration in rooms where the patients, who are very
susceptible to vibration and should not be disturbed, are located, vibrations
should also be within the limits of criterion as indicated in the above. Greater
vibrations may be acceptable in rooms for less sensitive patients. For the footfall-
induced floor vibrations, the RMS footfall-induced vibrations and the peak footfall-
induced vibrations should not exceed 4000 μin/sec and 5600 μin/sec
respectively. The corresponding limiting values for sensitive operating rooms are
1000 μin/sec and 1400 μin/sec respectively (Ungar, 2007).
Floor vibration above the operating rooms can affect sensitive instruments
which are supported from the operating room ceiling. Vibration sensitivity of the
equipment and the transmission characteristics of the instruments which are
used in order to mount the equipment are the main factors to be considered in
order to satisfy adopted criteria. There are not any specific criteria for vibrations
taking place on areas above operating rooms but it is suggested to meet
requirements of criteria for sensitive operating rooms. If these instruments do not
contact with the ceiling directly, there is no need to use such criteria (Ungar,
2007).
For MRI Systems, there are different criteria for different suppliers. Ungar
(2007), summarized some of the MRI supplier’s criteria and corresponding MRI
systems’ criteria shown in his study.
23
2.3. Vibration Mitigation Techniques
Advancements in civil engineering, especially the development of high
strength light-weight materials together with highly developed computer aided
design, have provided to design more light-weight and longer spanned floor
structures. Therefore, decreasing structural mass and damping cause some
serviceability problems in structures. There are some conventional methods that
can be used for mitigation. For example, changing mass and stiffness might be a
solution in order to overcome excessive floor vibration problems. However, there
might not be enough space for new structural elements in order to increase the
stiffness, and also additional mass might create overstress in structural
members. Moreover, it is not usually possible to use these additional
nonstructural elements because of architectural requirements (Setareh et al.
2006).
Vibration mitigation techniques are used in order to improve the
performance of the structures which are susceptible to excessive vibration.
These improvements cover reduction of annoyance level of occupants,
prevention of safety problems or providing convenient spaces for highly vibration
sensitive equipments. Vibration mitigation techniques can be applied in order to
improve vibration serviceability problems of existing structures or considered
during the design stage of new structures tending to be susceptible to vibrations
(Nyawako and Reynolds, 2007). According to Nyawako and Reynolds, vibration
mitigation techniques can generally be classified into three main categories as
passive, active and semi-active.
2.3.1. Passive Vibration Mitigation Techniques
Passive vibration mitigation techniques decrease energy dissipation demands on
primary structures. Instead of dissipating all energy by itself, most of the energy
is absorbed by using these techniques (Nyawako and Reynolds, 2007).
Additional damping devices absorb most of the input energy. These systems
24
improve the performance of primary structures by providing additional flexibility
and energy absorption capability through reduction in energy levels transmitted
to primary structures (Yang, 2001). The most important reason that these
devices differ from others is that there is no change in their dynamic
characteristics and also there is no need for external energy to operate them
(Housner et al., 1997; Tentor, 2001; Setareh, 2002). This factor is one of the
limitations of these devices because of their lack of control the characteristics of
external loading, for example when the excitation frequency changes. Viscous
dampers, friction dampers, viscoelastic treatments, tuned mass
dampers/vibration absorbers, pendulum tuned mass dampers, tuned liquid
The main objective of this study is to investigate the effects of spring-mass
systems (Tuned Vibration Absorbers, or TVAs) attached to Euler-Bernoulli
beams in order to control the response due to excessive vibrations. Effectiveness
of tuned vibration absorbers has been studied and their performance evaluated
through comparisons on an extensive combination of loading dynamic types,
span configurations, and TVA distributions. The proposed method includes the
exact solutions of natural frequencies and mode shapes of uniform and non-
uniform beams carrying any number of passive tuned mass dampers (TMD) and
forced vibration of these beams based on their free vibration data. A
mathematical formulation has been presented for free and forced vibration of
beams. An algorithm has been developed through MATHEMATICA and
numerical results have been obtained for various forcing systems and boundary
conditions.
Several numerical examples have been provided in order to evaluate the
performance of TMDs under free and forced vibration of uniform and non-uniform
beams carrying single or multiple spring-mass systems with various boundary
conditions. Free vibration characteristics of beams carrying elastically attached
point masses are obtained through numerical assembly method (Wu and Chou,
1999). Overall coefficient matrix is generated by combining the coefficient
matrices of each boundaries of the beam and each attaching points for a spring-
mass system through conventional assembly technique for finite element
method. Numerical assembly method is used in order to derive the eigenvalue
equation and then the developed algorithm is used for the solution of the
216
eigenvalues and the corresponding mode shapes. The accuracy of the
developed algorithm in this study is evaluated by comparing its numerical results
with existing literature.
The first part of the present study deals with the determination of natural
frequencies and corresponding mode shapes of single-span uniform beams,
single-span non-uniform beams and multi-span uniform beams carrying any
number of spring-mass systems and the second part calculates the forced
vibration responses of uniform beams under the excitation of step-function forces
(Impact Loading), harmonic forces, moving loads and moving pulsating forces.
First and second part also includes the free and forced vibration of a high mast
lighting tower (HMLT) which is subjected to wind induced dynamic load and
represented as non-uniform cantilever beam. The beams are considered as
continuous structural elements and both free and forced vibration solutions are
analyzed using thin beam (Euler-Bernoulli) theory. For single-span beams, four
boundary conditions are studied including simply supported-simply supported,
clamped-clamped, clamped-simply supported and clamped-free boundaries. On
the other hand, each intermediate support is assumed as simply supported for
multi-span beams. For the force vibration response of the structural elements,
90% of modal mass contribution is considered to be sufficient. Forced vibration of
the entire beam is obtained by using normal mode approach and linear
combination of the normal modes. Displacement, velocity, acceleration and jerk
responses of the entire beam with TMDs are calculated and the resultant
responses are compared with the beam without TMDs.
The illustrative numerical examples presented in this study are based on
human induced loads for single-span and multi-span uniform beams. Moreover,
single-span non-uniform beams are subjected to wind-induced loads to evaluate
the passive vibration control of HMLTs. Harmonic forces are considered as
repeated forces caused by human activities such as walking or dancing and
217
represented as time–dependent sinusoidal forcing. Harmonic moving loads are
also represented as concentrated loads with sinusoidally varying amplitude and
moving with a constant speed v0 which is the average human walking speed,
80m/min. Moreover, footstep impulse vibration has defined as step-function force
and non-harmonic moving load is considered to simulate pedestrian walking load
with a constant speed. The step-function and harmonic forces are represented by
using Dirac delta function and moving load and moving pulsating force are
expressed by using Fourier series. To evaluate the performance of TMDs, the
frequency component of the exciting forces is selected to match with the natural
frequency of the beams in order to simulate and approximate the condition for
resonance which could be the worst scenario causing significant vibration
amplification.
In addition, the performance of TMD under wind induced vibration has been
investigated through an analysis performed for a HMLT structure which is
assumed to be a non-uniform cantilever beam to carry out the proposed method
in this study. Wind induced dynamic loads are estimated using available wind
velocity data obtained at a certain height of the HMLT. The wind velocity profile
has been generated with empirical power-law method based on five selected
points throughout the height of HMLT and the corresponding forcing functions
with respect to time are defined by Fourier series using obtained wind profiles for
each selected points. A TMD is attached to top of the structure and dynamic
response at that point has been compared with the results obtained from bare
HMLT under wind-induced vibration.
5.2. Conclusion
This study presents the free and forced vibration of beams carrying any
number of spring-mass systems and the resultant responses were compared
with the bare uniform and non-uniform beams. Based on the observations from
218
numerical results for single span uniform, single span non-uniform and multi-
span uniform beams, the following conclusions are made;
1. It is observed that TMDs are very effective when they are properly
tuned to the excitation frequency. The effectiveness level of TMD
increases if the excitation frequency converges to the any normal
mode frequency causing the condition for resonance.
2. When TMD is tuned to the exact excitation frequency, it can be
concluded that single TMD application is more effective than multiple
TMDs of the same total mass ratio based on the peak resultant
responses or RMS values obtained from the main structure for
harmonic excitations. On the other hand, it is difficult to estimate the
exact excitation frequency in practice. Therefore it may be more
effective to implement multiple TMDs within a small frequency range in
order to overcome randomly varying excitations such as wind induced
or human induced vibrations.
3. TMD loses its effectiveness when the structure is subjected to non-
harmonic excitations such as step-function forces or constant moving
loads.
4. If the natural frequency of the TMD diverges further from the excitation
frequency or fundamental frequency of the structure, the performance
of TMD significantly decreases.
5. Single or multiple TMD application is more effective and robust when
the attached mass is increased without changing natural frequency of
properly tuned TMD under harmonic excitations.
219
6. Based on the results obtained from wind induced vibration analysis of
HMLT structure, it can be confirmed that a passive TMD can be
effective in reducing the dynamic response of HMLTs when it is
properly tuned to the fundamental frequency of the HMLT it is installed
in.
7. It is also observed that a TMD attached to top of the HMLT structure
having 1% mass of the total mass of the structure decreases the wind
induced dynamic response by about 50%.
8. The results for HMLT structure also show that the dynamic response of
the main structure does not change with the increase of the mass of
TMD under wind induced vibration. The relative motion of TMD is also
in practical limit and able to be accommodated in the actual structure.
9. For multi-span beams, when TMDs are used for each span, one of the
normal mode frequencies being dominated by any of the TMDs may
converge to the fundamental frequency of the structure and this may
cause undesirable responses. Therefore, this case requires a careful
consideration on selecting the final dynamic characteristics of the
TMDs for multi-span beams.
10. Although the first four lowest natural frequencies and corresponding
mode shapes of the structures are found without having any problem
by solving the eigenvalue equation using the developed algorithm in
MATHEMATICA, some numerical difficulties are encountered in finding
the roots of determinant expression of coefficient matrix in the
eigenvalue equation. The precise starting values for finding the roots of
determinant expression increase the computational time in case of
more than two TMDs particularly.
220
5.3. Future Work
No damping characteristics of the structure or the TMDs have been
included in the proposed method of this study. Based on the numerous examples
given in this study, the use of single or multiple TMDs is significantly effective in
reducing the dynamic response of the main structure under harmonic excitations
and wind induced vibration. However, the study may be extended by considering
the effects of damping characteristics of the structure and TMD in order to
generalize the results of this study.
Experimental study is needed to extend this research and to obtain more
information in order to understand the performance of TMD in actuality.
Moreover, experimental results are necessary to study the level of reliability of
the analytical and numerical method proposed in this study.
This study may also be extended for the free and forced vibration of
uniform rectangular plates with attached TMDs located at arbitrary points and the
performance of TMDs can be also evaluated for floor vibration control of
structures subjected to human induced vibration particularly.
In this study, it is also indicated that the use of single TMD has a
disadvantage when the excitation frequency is not known exactly. However,
multiple TMDs within a small frequency range may perform better for randomly
varying excitations. As a result of this, it is recommended to investigate the
optimum parameters of multiple TMDs having different dynamic characteristics in
order to improve the effectiveness of vibration control.
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221
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APPENDICES
228
Appendix A.
A.1. Detailed formulation for the responses of SDOF-TMD system
Figure A.1 SDOF-TMD system
Parameters of primary structure and TMD;
Eq. A.1
2 Eq. A.2
Eq. A.3
2 Eq. A.4
The mass ratio, µ, is defined as;
Eq. A.5
The governing equations of motion for the SDOF-TMD system are as
follows;
0 Eq. A.6
0 Eq. A.7
From equations A.6 and A.7 one obtains;
Eq. A.8
Dividing equations A.6 and A.7 by md the governing equations of motions
are as follows;
229
Primary mass,
1 2 Eq. A.9
Tuned mass,
2 Eq. A.10
The optimal approximation for the damper is assumed as,
Eq. A.11
The stiffness relation between damper and structure is defined as,
Eq. A.12
And the periodic excitation can be shown as follows,
sin Eq. A.13
The responses of the structure and the damper is given by,
sin Eq. A.14
sin Eq. A.15
The critical scenario is the equality of Ω and ω which is resonant condition
and the solutions for this case are as follows,
sin cos Eq. A.16
sin cos Eq. A.17
cos sin Eq. A.18
cos sin Eq. A.19
Eq. A.20
Eq. A.21
sin Eq. A.22
sin Eq. A.23
When equations A.16 and A.17 are introduced into equations A.9 and
A.10, one obtains,
1 sin cos 2 cos sin
sin cos sin sin cos
Eq. A.24
230
sin cos sin cos sin
cos cos sin 0
Eq. A.25
If equations A.24 and A.25 are simplified, one obtains,
1 2
Eq. A.26
1 2 Eq. A.27
0 Eq. A.28
0 Eq. A.29
where,
Eq. A.30
tan Eq. A.31
tan Eq. A.32
2 Eq. A.33
2 Eq. A.34
If equation A.30 is substituted into equations A.26, A.27, A28 and A29,
one obtains,
2 Eq. A.35
2 0 Eq. A.36
2 0 Eq. A.37
2 0 Eq. A.38
The simplified forms of equations A.35, A.36, A.37 and A.38,
2 Eq. A.39
2 0 Eq. A.40
2 0 Eq. A.41
2 0 Eq. A.42
A, B, C and D are as follows after solving equations A.39, A.40, A.41 and
A.42;
231
4
8 4 16 Eq. A.43
2 4
8 4 16 Eq. A.44
4
8 4 16 Eq. A.45
2
8 4 16 Eq. A.46
And the responses are given by;
2
8 16 1 4 Eq. A.47
The simplified form of equation A.47 is given by,
1
1 2 12
Eq. A.48
8 16 1 4 Eq. A.49
12
Eq. A.50
The response for no damper is as follows;
0 Eq. A.51
2sin Eq. A.52
sin cos sin Eq. A.53
cos sin Eq. A.54
232
When equation A.53 is introduced into equation A.52, one obtains,
sin cos 2 cos sin sin
cos sin
Eq. A.55
The simplified form of equation A.55 is given by,
2 0 Eq. A.56
2 Eq. A.57
where , and A and B are as follows after solving equations A.56 and
A.57,
0 Eq. A.58
12
1
2
2
Eq. A.59
And the response for no damper is given by;
2
12
Eq. A.60
Equation A.60 can be expressed in terms of equivalent damping ratio in
order to compare the cases with and without damper.
12
Eq. A.61
where,
21
2 12
Eq. A.62
If equation A.54 substituted in equation A.53, one obtains,
sin cos cos sin Eq. A.63
tan Eq. A.64
tan Eq. A.65
2 Eq. A.66
233
Example
Assume that 0 and 0.1, the relation between and is as
follows
21
2 12
0.1 Eq. A.67
where
12
Eq. A.68
Inserting equation Eq.A.68 into equation Eq.A.67 and assuming that
0 as indicated above gives,
21 0.1 Eq. A.69
Since is greater than 1, equation A.69 can be written as,
20.1 Eq. A.70
If is assumed to be 10, then equation A.70 gives an estimate for ,
2 0.110
0.02 Eq. A.71
and from equation A.68 and A.12,
12
0.12
0.05 Eq. A.72
0.02 Eq. A.73
Therefore, 2% of the primary mass provides an effective damping ratio of
10% as it is shown in the above. On the other hand, the large relative motion of
the damper mass should be considered during design stage in order to control
this motion in a real structure.
234
Appendix B.
B.1. Determination of and for Different Boundary Conditions for
Uniform Beams
a) Clamped-Clamped beam
1 2 3 4
0 1 0 1
1 0 1 0
1
2
Eq.B.1
4 1 4 2 4 3 4 4
sin cos sinh cosh
cos sin cosh sinh
1
Eq.B.2
b) Simply supported-Simply Supported beam
1 2 3 4
0 1 0 1
0 1 0 1
1
2
Eq.B.3
4 1 4 2 4 3 4 4
sin cos sinh cosh
sin cos sinh cosh
1
Eq.B.4
c) Clamped-Simply Supported beam
1 2 3 4
0 1 0 1
1 0 1 0
1
2
Eq.B.5
4 1 4 2 4 3 4 4
sin cos sinh cosh
sin cos sinh cosh
1
Eq.B.6
235
d) Clamped-Free End with Attached Mass (M) beam
0 0 0 0 Eq.B.7
0 Eq.B.8
1 2 3 4
0 1 0 1
1 0 1 0
1
2
Eq.B.9
4 1 4 2 4 3 4 4
sin cos sinh cosh 1
Eq.B.10
where
cos sin Eq.B.11
sin cos Eq.B.12
cosh sinh Eq.B.13
sinh cosh Eq.B.14
B.2. Coefficient Matrix for Uniform Simply Supported Beams Carrying Multiple Spring-Mass Systems
Case 1-Uniform Simply Supported Beam Carrying Two Spring-Mass Systems
1
0 1 0 1 0 0 0 0 0 0 0 0 0 0
0 1 0 1 0 0 0 0 0 0 0 0 0 0
sin 1 cos 1 sinh 1 cosh 1 sin 1 cos 1 sinh 1 cosh 1 0 0 0 0 0 0
cos 1 sin 1 cosh 1 sinh 1 cos 1 sin 1 cosh 1 sinh 1 0 0 0 0 0 0
sin 1 cos 1 sinh 1 cosh 1 sin 1 cos 1 sinh 1 cosh 1 0 0 0 0 0 0
cos 1 sin 1 cosh 1 sinh 1 cos 1 sin 1 cosh 1 sinh 1 0 0 0 0 1 0
sin 1 cos 1 sinh 1 cosh 1 0 0 0 0 0 0 0 0 12 1 0
0 0 0 0 sin 2 cos 2 sinh 2 cosh 2 sin 2 cos 2 sinh 2 cosh 2 0 0
0 0 0 0 cos 2 sin 2 cosh 2 sinh 2 cos 2 sin 2 cosh 2 sinh 2 0 0
0 0 0 0 sin 2 cos 2 sinh 2 cosh 2 sin 2 cos 2 sinh 2 cosh 2 0 0
0 0 0 0 cos 2 sin 2 cosh 2 sinh 2 cos 2 sin 2 cosh 2 sinh 2 0 2
0 0 0 0 sin 2 cos 2 sinh 2 cosh 2 0 0 0 0 0 22 1
0 0 0 0 0 0 0 0 sin 1 cos 1 sinh 1 cosh 1 0 0
0 0 0 0 0 0 0 0 sin 1 cos 1 sinh 1 cosh 1 0 0
236
Case 2-Uniform Simply Supported Beam Carrying Three Spring-Mass Systems
1
0 1 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
0 1 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
sin 1 cos 1 sinh 1 cosh 1 sin 1 cos 1 sinh 1 cosh 1 0 0 0 0 0 0 0 0 0 0 0
cos 1 sin 1 cosh 1 sinh 1 cos 1 sin 1 cosh 1 sinh 1 0 0 0 0 0 0 0 0 0 0 0
sin 1 cos 1 sinh 1 cosh 1 sin 1 cos 1 sinh 1 cosh 1 0 0 0 0 0 0 0 0 0 0 0
cos 1 sin 1 cosh 1 sinh 1 cos 1 sin 1 cosh 1 sinh 1 0 0 0 0 0 0 0 0 1 0 0
sin 1 cos 1 sinh 1 cosh 1 0 0 0 0 0 0 0 0 0 0 0 0 12 1 0 0
0 0 0 0 sin 2 cos 2 sinh 2 cosh 2 sin 2 cos 2 sinh 2 cosh 2 0 0 0 0 0 0 0
0 0 0 0 cos 2 sin 2 cosh 2 sinh 2 cos 2 sin 2 cosh 2 sinh 2 0 0 0 0 0 0 0
0 0 0 0 sin 2 cos 2 sinh 2 cosh 2 sin 2 cos 2 sinh 2 cosh 2 0 0 0 0 0 0 0
0 0 0 0 cos 2 sin 2 cosh 2 sinh 2 cos 2 sin 2 cosh 2 sinh 2 0 0 0 0 0 2 0
0 0 0 0 sin 2 cos 2 sinh 2 cosh 2 0 0 0 0 0 0 0 0 0 22 1 0
0 0 0 0 0 0 0 0 sin 3 cos 3 sinh 3 cosh 3 sin 3 cos 3 sinh 2 cosh 2 0 0 0
0 0 0 0 0 0 0 0 cos 3 sin 3 cosh 3 sinh 3 cos 3 sin 3 cosh 2 sinh 2 0 0 0
0 0 0 0 0 0 0 0 sin 3 cos 3 sinh 3 cosh 3 sin 3 cos 3 sinh 2 cosh 2 0 0 0
0 0 0 0 0 0 0 0 cos 3 sin 3 cosh 3 sinh 3 cos 3 sin 3 cosh 2 sinh 2 0 0 3
0 0 0 0 0 0 0 0 sin 3 cos 3 sinh 3 cosh 3 0 0 0 0 0 0 32 1
0 0 0 0 0 0 0 0 0 0 0 0 sin 1 cos 1 sinh 1 cosh 1 0 0 0
0 0 0 0 0 0 0 0 0 0 0 0 sin 1 cos 1 sinh 1 cosh 1 0 0 0
237
238
Appendix C.
C.1. Determination of and for Different Boundary Conditions for Non-
Uniform Beams
a) Clamped-Clamped beam
1 2 3 4
1
2
Eq.C.1
4 1 4 2 4 3 4 4
√ √ √ √
√ √ √ √
1
Eq.C.2
b) Simply supported-Simply Supported beam
1 2 3 4
1
2
Eq.C.3
4 1 4 2 4 3 4 4
√ √ √ √
√ √ √ √
1
Eq.C.4
c) Simply Supported-Clamped beam
1 2 3 4
1
2
Eq.C.5
4 1 4 2 4 3 4 4
√ √ √ √
√ √ √ √
1
Eq.C.6
239
d) Free End with Attached Mass (M)-Clamped beam
For free end
0 1
therefore
Shear,
1 0 Eq.C.7
Bending,
Eq.C.8
0 Eq.C.9
6
Eq.C.10
1 2 3 4
1
2
Eq.C.11
4 1 4 2 4 3 4 4
√ √ √ √
√ √ √ √
1
Eq.C.12
where
68
1 Eq.C.13
68
1 Eq.C.14
68
1 Eq.C.15
68
1 Eq.C.16
C.2. Coefficient Matrix for Non-Uniform Free-Clamped Beams Carrying Multiple Spring-Mass Systems
Case 1-Non-Uniform Free-Clamped Beam Carrying Two Spring-Mass Systems
1
3 3 3 3 0 0 0 0 0 0 0 0 0 0
1 2 3 4 0 0 0 0 0 0 0 0 0 0
1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 0 0 0 0 0 0
2 1 2 1 2 1 2 1 2 1 2 1 2 1 2 1 0 0 0 0 0 0
3 1 3 1 3 1 3 1 3 1 3 1 3 1 3 1 0 0 0 0 0 0
∆11 ∆21 ∆31 ∆41 ∆51 ∆61 ∆71 ∆81 0 0 0 0 0 0
1
12 1 1 1
12 1 1 1
12 1 1 1
12 1 1 0 0 0 0 0 0 0 0 1
2 1 0
0 0 0 0 1 2 1 2 1 2 1 2 1 2 1 2 1 2 1 2 0 0
0 0 0 0 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 0 0
0 0 0 0 3 2 3 2 3 2 3 2 3 2 3 2 3 2 3 2 0 0
0 0 0 0 ∆12 ∆22 ∆32 ∆42 ∆52 ∆62 ∆72 ∆82 0 0
0 0 0 0 2
12 1 2 2
12 1 2 2
12 1 2 2
12 1 2 0 0 0 0 0 2
2 1
0 0 0 0 0 0 0 0 1 √ 1 √ 1 √ 1 √ 0 0
0 0 0 0 0 0 0 0 2 √ 2 √ 2 √ 2 √ 0 0
240
Appendix D.
D.1. Coefficient Matrix for Multi-Span Uniform Beams Carrying Multiple Spring-Mass Systems
Case 1-Two-Span Uniform Beam Carrying Two Spring-Mass Systems
1
0 1 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0
0 1 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0
sin 1 cos 1 sinh 1 cosh 1 sin 1 cos 1 sinh 1 cosh 1 0 0 0 0 0 0 0 0 0 0
cos 1 sin 1 cosh 1 sinh 1 cos 1 sin 1 cosh 1 sinh 1 0 0 0 0 0 0 0 0 0 0
sin 1 cos 1 sinh 1 cosh 1 sin 1 cos 1 sinh 1 cosh 1 0 0 0 0 0 0 0 0 0 0
cos 1 sin 1 cosh 1 sinh 1 cos 1 sin 1 cosh 1 sinh 1 0 0 0 0 0 0 0 0 1 0
sin 1 cos 1 sinh 1 cosh 1 0 0 0 0 0 0 0 0 0 0 0 0 12 1 0
0 0 0 0 sin 1 cos 1 sinh 1 cosh 1 0 0 0 0 0 0 0 0 0 0
0 0 0 0 0 0 0 0 sin 1 cos 1 sinh 1 cosh 1 0 0 0 0 0 0
0 0 0 0 cos 1 sin 1 cosh 1 sinh 1 cos 1 sin 1 cosh 1 sinh 1 0 0 0 0 0 0
0 0 0 0 sin 1 cos 1 sinh 1 cosh 1 sin 1 cos 1 sinh 1 cosh 1 0 0 0 0 0 0
0 0 0 0 0 0 0 0 sin 2 cos 2 sinh 2 cosh 2 sin 2 cos 2 sinh 2 cosh 2 0 0
0 0 0 0 0 0 0 0 cos 2 sin 2 cosh 2 sinh 2 cos 2 sin 2 cosh 2 sinh 2 0 0
0 0 0 0 0 0 0 0 sin 2 cos 2 sinh 2 cosh 2 sin 2 cos 2 sinh 2 cosh 2 0 0
0 0 0 0 0 0 0 0 cos 2 sin 2 cosh 2 sinh 2 cos 2 sin 2 cosh 2 sinh 2 0 2
0 0 0 0 0 0 0 0 sin 2 cos 2 sinh 2 cosh 2 0 0 0 0 0 22 1
0 0 0 0 0 0 0 0 0 0 0 0 sin 1 cos 1 sinh 1 cosh 1 0 0
0 0 0 0 0 0 0 0 0 0 0 0 sin 1 cos 1 sinh 1 cosh 1 0 0
241
Case 2-Three-Span Uniform Beam Carrying One Spring-Mass System attached to First Span
1
0 1 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0
0 1 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0
sin 1 cos 1 sinh 1 cosh 1 sin 1 cos 1 sinh 1 cosh 1 0 0 0 0 0 0 0 0 0
cos 1 sin 1 cosh 1 sinh 1 cos 1 sin 1 cosh 1 sinh 1 0 0 0 0 0 0 0 0 0
sin 1 cos 1 sinh 1 cosh 1 sin 1 cos 1 sinh 1 cosh 1 0 0 0 0 0 0 0 0 0
cos 1 sin 1 cosh 1 sinh 1 cos 1 sin 1 cosh 1 sinh 1 0 0 0 0 0 0 0 0 1
sin 1 cos 1 sinh 1 cosh 1 0 0 0 0 0 0 0 0 0 0 0 0 12 1
0 0 0 0 sin 1 cos 1 sinh 1 cosh 1 0 0 0 0 0 0 0 0 0
0 0 0 0 0 0 0 0 sin 1 cos 1 sinh 1 cosh 1 0 0 0 0 0
0 0 0 0 cos 1 sin 1 cosh 1 sinh 1 cos 1 sin 1 cosh 1 sinh 1 0 0 0 0 0
0 0 0 0 sin 1 cos 1 sinh 1 cosh 1 sin 1 cos 1 sinh 1 cosh 1 0 0 0 0 0
0 0 0 0 0 0 0 0 sin 2 cos 2 sinh 2 cosh 2 0 0 0 0 0
0 0 0 0 0 0 0 0 0 0 0 0 sin 2 cos 2 sinh 2 cosh 2 0
0 0 0 0 0 0 0 0 cos 2 sin 2 cosh 2 sinh 2 cos 2 sin 2 cosh 2 sinh 2 0
0 0 0 0 0 0 0 0 sin 2 cos 2 sinh 2 cosh 2 sin 2 cos 2 sinh 2 cosh 2 0
0 0 0 0 0 0 0 0 0 0 0 0 sin 1 cos 1 sinh 1 cosh 1 0
0 0 0 0 0 0 0 0 0 0 0 0 sin 1 cos 1 sinh 1 cosh 1 0
242
Case 3-Three-Span Uniform Beam Carrying One Spring-Mass System attached to Mid Span
1
0 1 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0
0 1 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0
sin 1 cos 1 sinh 1 cosh 1 0 0 0 0 0 0 0 0 0 0 0 0 0
0 0 0 0 sin 1 cos 1 sinh 1 cosh 1 0 0 0 0 0 0 0 0 0
cos 1 sin 1 cosh 1 sinh 1 cos 1 sin 1 cosh 1 sinh 1 0 0 0 0 0 0 0 0 0
sin 1 cos 1 sinh 1 cosh 1 sin 1 cos 1 sinh 1 cosh 1 0 0 0 0 0 0 0 0 0
0 0 0 0 sin 1 cos 1 sinh 1 cosh 1 sin 1 cos 1 sinh 1 cosh 1 0 0 0 0 0
0 0 0 0 cos 1 sin 1 cosh 1 sinh 1 cos 1 sin 1 cosh 1 sinh 1 0 0 0 0 0
0 0 0 0 sin 1 cos 1 sinh 1 cosh 1 sin 1 cos 1 sinh 1 cosh 1 0 0 0 0 0
0 0 0 0 cos 1 sin 1 cosh 1 sinh 1 cos 1 sin 1 cosh 1 sinh 1 0 0 0 0 1
0 0 0 0 sin 1 cos 1 sinh 1 cosh 1 0 0 0 0 0 0 0 0 12 1
0 0 0 0 0 0 0 0 sin 2 cos 2 sinh 2 cosh 2 0 0 0 0 0
0 0 0 0 0 0 0 0 0 0 0 0 sin 2 cos 2 sinh 2 cosh 2 0
0 0 0 0 0 0 0 0 cos 2 sin 2 cosh 2 sinh 2 cos 2 sin 2 cosh 2 sinh 2 0
0 0 0 0 0 0 0 0 sin 2 cos 2 sinh 2 cosh 2 sin 2 cos 2 sinh 2 cosh 2 0
0 0 0 0 0 0 0 0 0 0 0 0 sin 1 cos 1 sinh 1 cosh 1 0
0 0 0 0 0 0 0 0 0 0 0 0 sin 1 cos 1 sinh 1 cosh 1 0