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Frequency Tuning of Vibration Absorber Using Topology FREQUENCY TUNING OF VIBRATION ABSORBER USING TOPOLOGY OPTIMIZATION by SWAPNIL SUBHASH HAREL Presented to the Faculty of the Graduate School of The University of Texas at Arlington in Partial Fulfillment of the Requirements for the Degree of MASTER OF SCIENCE IN MECHANICAL ENGINEERING THE UNIVERSITY OF TEXAS AT ARLINGTON Copyright © by Swapnil Subhash Harel 2017 All Rights Reserved 2 Acknowledgements I would like to express my gratitude towards Dr. Robert Taylor for guiding and helping me throughout my thesis research. He always had time to discuss my work and helped solve any problem that I faced during my research work. I would like to thank Dr. Bo Wang for constantly supporting my research work with his ideas which were second to none. I would also like to thank Dr, Kent Lawrence for finding time out of their busy schedule to serve on my thesis defense committee. Lastly, I would like to thank my family. I would especially like to support my father, Subhash Harel, my mother Surekha Harel and my younger sister Apurva Harel for their support all these years. It was their continued support and belief that I have been able to do all that I have done so far. I would also like to offer regards to all those who have supported or helped me in any regards this period. May 04, 2017 3 Abstract FREQUENCY TUNING OF VIBRATION ABSORBER USING TOPOLOGY OPTIMIZATION by Swapnil Subhash Harel, MS The University of Texas at Arlington, 2017 Supervising Professors: Dr. Robert Taylor and Dr. Bo Wang A tuned mass absorber is a system for reducing the amplitude in one oscillator by coupling it to a second oscillator. If tuned correctly, the maximum amplitude of the first oscillator in response to a periodic driver will be lowered, and much of the vibration will be ’transferred’ to the second oscillator. The tuned vibration absorber (TVA) has been utilized for vibration control purposes in many sectors of Civil/Automotive/Aerospace Engineering for many decades since its inception. Time and again we come across a situation in which a vibratory system is required to run near resonance. In the past, approaches have been made to design such auxiliary spring mass tuned absorbers for the safety of the structures. This research focuses on the development and optimization of continuously tuned mass absorbers as a substitute to the discretely tuned mass absorbers (spring- mass system). After conducting the study of structural behavior, the boundary condition and frequency to which the absorber is to be tuned are determined. The Modal analysis approach is used to determine mode shapes and frequencies. The absorber is designed and optimized using the topology optimization tool, which simultaneously designs, optimizes and tunes the absorber to the desired frequency. The tuned, optimized absorber, after post processing, is attached to the target structure. The number of the absorbers are increased to amplify 4 bandwidth and thereby upgrade the safety of structure for a wide range of frequency. The frequency response analysis is carried out using various combinations of structure and number of absorber cell. 5 Table of Contents Acknowledgements……………………………………………………………………………… 3 Abstract…………………………………………………………………………………………… 4 List of Illustration………………………………………………………………………………… 8 List of Tables…………………………………………………………………………………… 11 Chapter-1 – Introduction……………………………………………………………………..…12 1.1 Motivation……………………………………………………………………..……12 1.2 Research Objective………………………………………………….……………13 1.3 Theory of Increasing Bandwidth…………………………………………………14 Chapter-2- Background…………………………………………..…………………………… 20 2.1 Resonance…………………………………………………………………………20 2.2 Tuned Mass Absorber……………………………………………………….……23 2.3 Design Optimization…………………………...…………………….……………28 2.4 Topology Optimization ………………………………………….………………..29 Chapter -3 Methodology……………………………………………………….…….…………32 3.1 Unit Absorber Cell Pre-Optimized Model……………………….………………33 3.2 Boundary Conditions and Meshing…………………………...…………………35 3.3 Setting up Topology Optimization Problem…………………………….………37 Chapter-4 – Results……………………………………………………….….…………………42 4.1 Topology Optimization Results……………………………..……………………42 4.2 Modal analysis of optimized cell…………………………………………………43 4.3 Topology Optimization Results for Various Geometry…………………...……45 4.4 Topology Optimization Results for Various Concentrated Masses……….…46 Chapter-5 Testing……………………………………………………………………………….48 5.1 Case-I………………………………………………………………………………52 6 5.2 Case-II………………………………………………………………………………58 5.3 Case-III…………………………………………………………………………..…66 Chapter-6 – Conclusion……………………………………………………………………...…74 Chapter-7 Limitations and Future Work………………………………………………………76 References ……………………………………………………………………………..............77 Biological data……………………………………………………………………………..........79 7 List of Illustrations Figure 1‑1 Conversion of classical spring –mass structure to continues absorber…...…13 Figure 1‑2 Spring –Mass system…………………..………………….……………………...14 Figure 1‑3 Frequency response analysis of spring-mass system…………………………15 Figure 1‑4 Spring –Mass system with absorber ..…………....….............…………………16 Figure 1‑5 Frequency response analysis with one absorber ...….....................………….16 Figure 1‑6 Frequency response analysis of structure with two absorbers ……………... 18 Figure 1‑7 Frequency response analysis of structure with 5, 10, and 20 absorbers…....18 Figure 2‑1 Resonance in spring-mass system ……...….....................….................….21 Figure 2‑2 Structures affected due to resonance ………...….....................…...........……22 Figure 2‑3 (a) Spring –Mass system………...….....................…..........................……….24 Figure 2‑3 (b) Frequency response analysis of spring mass structure ……………..……25 Figure 2‑3 (c) Frequency response analysis of spring mass structure with an absorber 26 Figure 3‑1 Pre-Optimized unit absorber cell …...….....................…........…..................…32 Figure 3‑3 Cell Dimensions ………………………………………………………………….. 34 Figure 3‑4 Boundary Conditions on top and bottom edge ……………….………….……35 Figure 3‑5 Boundary Conditions on Right and Left Edges …………………………….…..35 Figure 3-7 Optimization Problem ………………………………………………………….….38 Figure 3-8 Setting up Design Variable ……...….....................…........…...................... 38 Figure 3-9 Setting up Design Response…...….....................…........….....................……39 Figure 3-10 Setting up Manufacturing Constraints…………………………………..………41 Figure 3-11 Setting up Optimization Constraints on Response ……...….............……….41 8 Figure 3-12 Setting up Optimization Objective……...….....................…..................……..41 Figure 4-1 Topology Optimization Results…...….....................…........….....................….42 Figure 4-2 Topology Optimization Results for Various Geometry …………………….….45 Figure 4-3 Topology Optimization Results for Various Concentrated Masses ……….….46 Figure 5-1 Testing Procedure …...….....................…........….....................….......…….48 Figure 5-2 (a) Mode Shape of Plate …...….....................…........….....................…….….52 Figure 5-2 (b) Mode Shape Frequencies …...….....................…........…................………53 Figure 5-3 Transformation of Absorber Cell ……...….....................…........…...................54 Figure 5-4 Frequency Response Analysis of Plate Without Absorber …...…..............….56 Figure 5-5 Frequency Response Analysis of Plate with Absorber ……...….................…56 Figure 5-6 (a) Frequency Response Analysis of Plate with 2 Absorber…...…................ 57 Figure 5-6 (b) Frequency Response Analysis of Plate with 5 Absorber…...….................57 Figure 5-6 (c) Frequency Response Analysis of Plate with 10 Absorber…...…...............57 Figure 5-7 Case II- Beam …...….....................…........….....................….................…… 59 Figure 5-8 (a) Mode Shapes of Beam ...….....................….........….................…………. 60 Figure 5-8 (b) Mode Shapes Frequencies of Beam …...….....................…........…..........60 Figure 5-9 Transformation of Absorber Cell ………...….....................…........…….………61 Figure 5-10 Frequency Response Analysis of Beam Without Absorber ……….….….….63 Figure 5-11 Frequency Response Analysis of Beam with Absorber ……………...………63 Figure 5-12 (a) Frequency Response Analysis of Beam with 2 Absorber ……..........… 64 Figure 5-12 (b) Frequency Response Analysis of Beam with 5 Absorber ……….….….65 Figure 5-12 (c) Frequency Response Analysis of Beam with 10 Absorber ………….…. 65 Figure 5-12 (d) Frequency Response Analysis of Beam with 20 Absorber …….………. 65 Figure 5-13 Case III Beam ...….....................…....….....................…...…….….…........... 67 9 Figure 5-14 (a) Mode Shapes of Beam …....................….....................…..................…. 68 Figure 5-14 (b) Mode Shapes Frequencies of Beam ....................…….....................…. 68 Figure 5-16 Transformation of Absorber Cell....................….……....................……........ 69 Figure 5-17 Frequency Response Analysis of Beam Without Absorber ....................…. 71 Figure 5-18 Frequency Response Analysis of Beam with Absorber ……...………….…. 71 Figure 5-19 Frequency Response Analysis of Beam with 10 Absorber ………………… 72 10 List of Tables Table 3-2 Material Property …………….. …………………………………………………… 34 Table 3-6 Summary of Constraints ………………………………………………………….. 36 11 Chapter 1 Introduction A tuned mass absorber is a system for reducing the amplitude in one oscillator by coupling it to a second oscillator. If tuned correctly, the maximum amplitude of the primary system
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