Chapter 2: Rigid Bar Supported by Two Buckled Struts under Axial, Harmonic, Displacement Excitation..14

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1 Table of Contents Chapter 1: Research Objectives and Literature Review Introduction Literature Review Describing Vibration Vibration Isolation Overview Types of Isolators Single-Degree-of-Freedom System Stiffness in Vertical Vibration Isolators Elastically Buckled Struts as Isolators Research Scope Overview Research Scope Two Buckled Struts Supporting a Rigid Bar Two Buckled Struts Supporting an Asymmetric Rigid Bar Two Pairs of Pre-bent Struts with an Intermediate Bonded Filler, Supporting a Rigid Bar Objectives...13 Chapter 2: Rigid Bar Supported by Two Buckled Struts under Axial, Harmonic, Displacement Excitation Introduction General Model Description Symmetric Bar Equilibrium Analysis Procedure Dynamic Analysis Procedure Results for Symmetric Bar Asymmetric Bar Equilibrium Analysis Procedure..28 iv

2 2.4.2 Equilibrium Results Dynamic Analysis Procedure Strut Equations Rigid Body Equations Boundary Conditions Results for Asymmetric Bar Asymmetric Bar with No Static Rotation Equilibrium Analysis Modifications Equilibrium Results Dynamic Analysis Modifications Dynamic Results Summary and Conclusions. 62 Chapter 3: Symmetric Rigid Bar Supported by Two Pairs of Pre-bent Struts with Intermediate Bonded Filler Introduction Equilibrium Analysis Equilibrium Analysis Results Dynamic Analysis Dynamic Analysis Results Dimensional Example Summary and Conclusions.84 Chapter 4: Summary and Conclusions Summary Conclusions Future Research.. 90 References...92 v

3 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 Appendix G: Additional Adjusted Strut Stiffness Plots Appendix H: Equilibrium Program for Pre-bent Struts with Filler Appendix I: Dynamic Program for Pre-bent Struts with Filler Vita..119 vi

4 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 Figure 2.11 Shear Force of Left Strut q 1 vs. b Figure 2.12 Left Strut Moment m 1 vs. b Figure 2.13 Right Strut Moment m 2 vs. b Figure 2.14 Rotation of Rigid Bar vs. b Figure 2.15 Deformed Shape of Model for Different Values of b 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 = Figure 2.19 Transmissibility vs. Frequency and Vibration Shapes for b 1 = Figure 2.20 Transmissibility vs. Frequency and Vibration Shapes for b 1 = Figure 2.21 Vibration and Equilibrium Shape for Struts of b 1 =0.52 Model at ω = 3.32 (Second Peak Frequency)..50 vii

5 Figure 2.22 Vibration and Equilibrium Shape for Struts of b 1 =0.60 Model at ω = 45.1 (Third Peak Frequency) 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 Figure 2.25 Peak Frequency ω 2 vs. b Figure 2.26 Peak Frequency ω 3 vs. b Figure 2.27 Peak Frequency ω 4 vs. b Figure 2.28 Peak Frequency ω 5 vs. b 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 = Figure 2.31 Transmissibility vs. Frequency for Two Cases of b 1 = Figure 2.32 Transmissibility vs. Frequency for Two Cases of b 1 = Figure 2.33 Transmissibility vs. Frequency for Two Cases of b 1 = 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 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 Figure 3.6 Mid-height Deflection vs. Filler Stiffness for Different Values of p o 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 Figure 3.10 Transmissibility vs. Frequency for Pre-bent Struts with Filler Material a o = 0.01, 0.05, and viii

6 Figure 3.11 Transmissibility vs. Frequency for Pre-bent Struts with Filler Material p o = 10, 20, 30, and Figure 3.12 Transmissibility vs. Frequency Dimensional Example..83 Figure E.1 Transmissibility vs. Frequency for b 1 = Figure E.2 Transmissibility vs. Frequency for b 1 = Figure F.1 Rotation vs. Frequency for b 1 = Figure F.2 Rotation vs. Frequency for b 1 = Figure F.3 Rotation vs. Frequency for b 1 = Figure F.4 Rotation vs. Frequency for b 1 = Figure F.5 Rotation vs. Frequency for b 1 = Figure G.1 Rotation vs. Frequency for b 1 = 0.55, Adjusted Strut Stiffness Figure G.2 Rotation vs. Frequency for b 1 = 0.60, Adjusted Strut Stiffness Figure G.3 Rotation vs. Frequency for b 1 = 0.65, Adjusted Strut Stiffness Figure G.4 Rotation vs. Frequency for b 1 = 0.70, Adjusted Strut Stiffness 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 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

7 List of Tables Table 2.1 Values of a and the Corresponding b 1 Value Table 2.2 Peak Frequencies for Varying Values of b Table 2.3 Stiffness Adjustment Factors and Strut Load Ratios...57 Table 2.4 Peak Frequencies for Struts with Adjusted Stiffness 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

Analysis of Buckled and Pre-bent Columns Used as Vibration Isolators

Analysis of Buckled and Pre-bent Columns Used as Vibration Isolators by Jenny E. Sidbury Thesis submitted to the Faculty of the Virginia Polytechnic Institute and State University In partial fulfillment