Table of Contents Chapter 1: Research Objectives and Literature Review..1 1.1 Introduction...1 1.2 Literature Review......3 1.2.1 Describing Vibration......3 1.2.2 Vibration Isolation.....6 1.2.2.1 Overview. 6 1.2.2.2 Types of Isolators...7 1.2.2.3 Single-Degree-of-Freedom System...7 1.2.2.4 Stiffness in Vertical Vibration Isolators...8 1.2.3 Elastically Buckled Struts as Isolators...9 1.3 Research Scope.. 10 1.3.1 Overview..10 1.3.2 Research Scope...10 1.3.2.1 Two Buckled Struts Supporting a Rigid Bar.... 11 1.3.2.2 Two Buckled Struts Supporting an Asymmetric Rigid Bar......11 1.3.2.3 Two Pairs of Pre-bent Struts with an Intermediate Bonded Filler, Supporting a Rigid Bar... 12 1.3.3 Objectives...13 Chapter 2: Rigid Bar Supported by Two Buckled Struts under Axial, Harmonic, Displacement Excitation..14 2.1 Introduction. 14 2.2 General Model Description.14 2.3 Symmetric Bar 15 2.3.1 Equilibrium Analysis Procedure.. 15 2.3.2 Dynamic Analysis Procedure... 19 2.3.3 Results for Symmetric Bar...26 2.4 Asymmetric Bar..28 2.4.1 Equilibrium Analysis Procedure..28 iv
2.4.2 Equilibrium Results.31 2.4.3 Dynamic Analysis Procedure... 37 2.4.3.1 Strut Equations...... 37 2.4.3.2 Rigid Body Equations... 38 2.4.3.3 Boundary Conditions......41 2.4.4 Results for Asymmetric Bar 44 2.4.5 Asymmetric Bar with No Static Rotation 55 2.4.5.1 Equilibrium Analysis Modifications.56 2.4.5.2 Equilibrium Results... 57 2.4.5.3 Dynamic Analysis Modifications.. 58 2.4.5.4 Dynamic Results...58 2.5 Summary and Conclusions. 62 Chapter 3: Symmetric Rigid Bar Supported by Two Pairs of Pre-bent Struts with Intermediate Bonded Filler 64 3.1 Introduction. 64 3.2 Equilibrium Analysis..65 3.3 Equilibrium Analysis Results. 69 3.4 Dynamic Analysis..72 3.5 Dynamic Analysis Results.. 75 3.6 Dimensional Example.79 3.7 Summary and Conclusions.84 Chapter 4: Summary and Conclusions...86 4.1 Summary. 86 4.2 Conclusions. 87 4.3 Future Research.. 90 References...92 v
Appendix A: Equilibrium Program for Single Strut - Symmetric Bar...94 Appendix B: Dynamic Program for Single Strut Symmetric Bar.96 Appendix C: Equilibrium Program for Two Struts - Asymmetric Bar...98 Appendix D: Dynamic Program for Two Struts Asymmetric Bar.101 Appendix E: Additional Transmissibility Plots Asymmetric Bar.105 Appendix F: Rotation vs. Frequency Plots Asymmetric Bar....106 Appendix G: Additional Adjusted Strut Stiffness Plots... 109 Appendix H: Equilibrium Program for Pre-bent Struts with Filler...115 Appendix I: Dynamic Program for Pre-bent Struts with Filler.. 117 Vita..119 vi
List of Figures Figure 1.1 Two Struts Supporting Rigid Bar.. 2 Figure 1.2 Two Struts Supporting Asymmetric Bar... 2 Figure 1.3 Two Pairs of Struts with Intermediate Bonded Filler, Supporting Rigid Bar 3 Figure 1.4 Waveform of Simple Harmonic Motion 5 Figure 1.5 Single-Degree-of-Freedom System... 8 Figure 2.1 Rigid Bar Supported by Fixed-End Struts Undeformed Shape 15 Figure 2.2 Post-Buckled Strut Equilibrium Symmetric Bar..16 Figure 2.3 Single Buckled Strut Under Static Load P o 16 Figure 2.4 Free Body Diagram of Element of Strut in Equilibrium.17 Figure 2.5 Strut Under Forced Harmonic Vibration.20 Figure 2.6 Free Body Diagram of Element of Strut Under Forced Harmonic Vibrations 22 Figure 2.7 Transmissibility vs. Frequency with Vibration Shapes...27 Figure 2.8 Post-Buckled Equilibrium State Asymmetric Case..28 Figure 2.9 Free Body Diagram of Static Forces on Rigid Bar..29 Figure 2.10 Left Strut Axial Load p 1 vs. b 1..33 Figure 2.11 Shear Force of Left Strut q 1 vs. b 1. 34 Figure 2.12 Left Strut Moment m 1 vs. b 1...35 Figure 2.13 Right Strut Moment m 2 vs. b 1...35 Figure 2.14 Rotation of Rigid Bar vs. b 1..36 Figure 2.15 Deformed Shape of Model for Different Values of b 1.. 37 Figure 2.16 Free Body Diagram of Rigid Bar Under Forced Harmonic Vibrations.. 39 Figure 2.17 Transmissibility vs. Frequency for Different Values of b 1 44 Figure 2.18 Transmissibility vs. Frequency and Vibration Shapes for b 1 =0.52... 46 Figure 2.19 Transmissibility vs. Frequency and Vibration Shapes for b 1 =0.60... 47 Figure 2.20 Transmissibility vs. Frequency and Vibration Shapes for b 1 =0.70... 48 Figure 2.21 Vibration and Equilibrium Shape for Struts of b 1 =0.52 Model at ω = 3.32 (Second Peak Frequency)..50 vii
Figure 2.22 Vibration and Equilibrium Shape for Struts of b 1 =0.60 Model at ω = 45.1 (Third Peak Frequency).....51 Figure 2.23 Vibration and Equilibrium Shape for Struts of b 1 =0.70 Model at ω = 99.5 (Fourth Peak Frequency)...52 Figure 2.24 Peak Frequency ω 1 vs. b 1..53 Figure 2.25 Peak Frequency ω 2 vs. b 1..53 Figure 2.26 Peak Frequency ω 3 vs. b 1..54 Figure 2.27 Peak Frequency ω 4 vs. b 1..54 Figure 2.28 Peak Frequency ω 5 vs. b 1..55 Figure 2.29 Transmissibility vs Frequency Adjusted Strut Stiffness for No Rotation at Equilibrium.59 Figure 2.30 Transmissibility vs. Frequency for Two Cases of b 1 = 0.55. 60 Figure 2.31 Transmissibility vs. Frequency for Two Cases of b 1 = 0.60. 61 Figure 2.32 Transmissibility vs. Frequency for Two Cases of b 1 = 0.65. 61 Figure 2.33 Transmissibility vs. Frequency for Two Cases of b 1 = 0.70. 62 Figure 3.1 Symmetric Rigid Bar Supported by Two Pairs of Pre-bent Struts With Intermediate Bonded Filler (in Nondimensional Terms) 64 Figure 3.2 Undeformed Strut with Initial Curvature and Intermediate Bonded Filler...65 Figure 3.3 Deformed Shape of Pair of Struts....66 Figure 3.4 Free Body Diagram of Element of Strut in Equilibrium. 67 Figure 3.5 Mid-height Deflection vs. Filler Stiffness for Different Values of a o.....70 Figure 3.6 Mid-height Deflection vs. Filler Stiffness for Different Values of p o.....71 Figure 3.7 Strut Under Harmonic Axial Base Excitation.72 Figure 3.8 Free Body Diagram of Element of Strut Including Inertia and Damping Forces.. 73 Figure 3.9 Transmissibility vs. Frequency for Pre-bent Struts with Filler Material k = 0.01, 0.1, 1, and 10...76 Figure 3.10 Transmissibility vs. Frequency for Pre-bent Struts with Filler Material a o = 0.01, 0.05, and 0.1...77 viii
Figure 3.11 Transmissibility vs. Frequency for Pre-bent Struts with Filler Material p o = 10, 20, 30, and 40...78 Figure 3.12 Transmissibility vs. Frequency Dimensional Example..83 Figure E.1 Transmissibility vs. Frequency for b 1 = 0.55...105 Figure E.2 Transmissibility vs. Frequency for b 1 = 0.65...105 Figure F.1 Rotation vs. Frequency for b 1 = 0.52 106 Figure F.2 Rotation vs. Frequency for b 1 = 0.55 106 Figure F.3 Rotation vs. Frequency for b 1 = 0.60 107 Figure F.4 Rotation vs. Frequency for b 1 = 0.65 107 Figure F.5 Rotation vs. Frequency for b 1 = 0.70 108 Figure G.1 Rotation vs. Frequency for b 1 = 0.55, Adjusted Strut Stiffness... 109 Figure G.2 Rotation vs. Frequency for b 1 = 0.60, Adjusted Strut Stiffness... 109 Figure G.3 Rotation vs. Frequency for b 1 = 0.65, Adjusted Strut Stiffness... 110 Figure G.4 Rotation vs. Frequency for b 1 = 0.70, Adjusted Strut Stiffness... 110 Figure G.5 Peak Frequency ω 1 vs. b 1, Adjusted Strut Stiffness. 111 Figure G.6 Peak Frequency ω 2 vs. b 1, Adjusted Strut Stiffness. 111 Figure G.7 Peak Frequency ω 3 vs. b 1, Adjusted Strut Stiffness. 111 Figure G.8 Peak Frequency ω 4 vs. b 1, Adjusted Strut Stiffness. 112 Figure G.9 Peak Frequency ω 5 vs. b 1, Adjusted Strut Stiffness. 112 Figure G.10 Peak Frequency ω 6 vs. b 1, Adjusted Strut Stiffness... 112 Figure G.11 Transmissibility vs. Frequency and Vibration Shapes for b 1 = 0.60, Adjusted Strut Stiffness..113 Figure G.12 Transmissibility vs. Frequency and Vibration Shapes for b 1 = 0.70, Adjusted Strut Stiffness..114 ix
List of Tables Table 2.1 Values of a and the Corresponding b 1 Value.... 32 Table 2.2 Peak Frequencies for Varying Values of b 1. 52 Table 2.3 Stiffness Adjustment Factors and Strut Load Ratios...57 Table 2.4 Peak Frequencies for Struts with Adjusted Stiffness... 59 Table 3.1 First Two Frequency Peaks for Different p o, k=0.1, a o =0.1, c=1, r=1..79 Table 3.2 Values of Transmissibility and c for Given ζ Single-Degree-of-Freedom System.. 82 x