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Multibody Dynamics Based Simulation Studies of Escapement Mechanisms in Mechanical Watch Movement

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Multibody Dynamics Based Simulation Studies of Escapement Mechanisms in Mechanical Watch Movement Multibody Dynamics Based Simulation Studies of Escapement Mechanisms in Mechanical Watch Movement FU，Kin Chung Denny A Thesis Submitted in Partial Fulfilment of the Requirements for the Degree of Master of Philosophy in Automation and Computer-Aided Engineering © The Chinese University of Hong Kong September 2008 The Chinese University of Hong Kong holds the copyright of this thesis. Any person(s) intending to use a part or whole of the materials in the thesis is a proposed publication must seek copyright release from the Dean of the Graduate School. UNIVERSITY ~yM/J NgSXLIBRARY SYSTEM Thesis/Assessment Committee Professor Hui, Kin-chuen (Chair) Professor Yam, Yeung (Thesis Supervisor) Professor Kong, Ching Tom (Committee Member) Professor Chen, Yonghua (External Examiner) Abstract Simulation is an important tool to enhance efficiency in optimizing mechanism design. A quality simulation capable of accurately revealing complex physical phenomena serves to enable design engineers in understanding the different aspects of kinematics and dynamics, parametric dependence, tolerance analysis, etc., without the need to construct the actual full scale model beforehand. Generally, a mechanism is composed of more than two bodies. Complex dynamical phenomena may arise in their multiple interactions depending on the frictional conditions during impact. Classical Newton's Laws are inadequate to tackle these multiple contacts problems. In this regard, some theories of multibody dynamics have been formulated to model these complex phenomena. Simulations based on such theories were shown to be successfully applicable in many cases of complex mechanisms. They provided adequate performance in the length of simulation time and the quality of the attained solutions. The main objective of this thesis is to apply multibody dynamics to the simulation of a specific type of complex mechanism, namely the Swiss lever escapement in mechanical watches. We adopted the approach by P. Pfeiffer and Ch. Glocker in our work. A collision detection algorithm was specifically developed to facilitate the computations. Basic graphical output and movie visualization of the results were included in the simulation package for motion analyses, dynamics and kinematics studies. The revised numerical treatments helped to improve the quality of solutions. A simple example was conducted for verification before actual application to the escapement problem. Our technique developed, however, can be applied to a broad scope of complex mechanism simulation. This thesis is part of a research project supported by the Innovation and Technology Fund of the Government of Hong i Kong SAR to develop technologies for making core movement of mechanical watches. • r . ‘ :.才. ：.: • I- .....•< '‘’：；. ....� - , . •V • , ： % •‘...，：：：. “if?;-- '-‘,•.. 1J"；‘； ....‘.{ ,- ； ..•‘’ ’•+ . .•. ‘ > \ I . ii 摘要 模擬軟件在機械機件優化的過程中是很重要的工具。—個優良的模擬軟件能夠 準確地模擬出機械在運作中的複雜情況’爲設計工程師在該機械機件被製造出 來前，進行運動學，動力學，各參量的從屬關係，容限的分析。一般的機械機 件通常是由超過兩件以上的零件’同一時間有超過兩點以上的碰撞或接觸點是 常見的事情，而磨擦力的存在更使機件當中的運作變得更爲複雜。傳統的牛頓 力學理論是不足以應付多點接觸的問題，多體力學便是爲此而被硏究開發出 來。多體力學能夠爲多點接觸問題中的複雜變化以數學的方式表示出來。很多 多體力學的理論已經被成功地應用在模擬複雜機件的運作，有著令人滿意的模 擬的計算時間和答案的質量和準確度。 本文目的在應用多體力學於複雜機械機件，特別應用在機械手錶中的靈魂—— 著名的瑞士槓桿擒縱機構°本文基於學者P. Pfeiffer and Ch. Glocker的多體力學 理論，硏發出複雜機械的模擬工具。過程中特別專門硏發了模擬中的碰撞檢測 方法，以及爲更準確和穩定的計算在運算數據處理作出修訂。硏發出的模擬工 具的輸出除了基本的運動學分析圖表外，還爲機件的運作分析配備了模擬影像 輸出。本文特地運用了一個簡單例子，證明了工具的準確度。除了瑞士槓桿擒 縱機構，本文中的模擬工具能爲其他複雜的機械進行運動學和動力學等分析。 本硏究是“機械錶芯硏究及發展”硏究計劃的其中之一部分，由香港政府的創 新及科技基金資助。 iii Acknowledgements I would like to express my sincere thanks to my supervisor, Professor Yeung Yam, for his guidance, support and invaluable advices. Thank for giving me the chance to engage in career of research. Thanks are given to my colleagues in the Intelligent Control Systems Laboratory for their support, encouragement and the working atmosphere created. A special acknowledgment needs to go to my family, for their love, care and patience. Dedicated to my Grandaunts iv Table of Contents Abstract i 薩 iii Acknowledgements iv Table of Contents v List of Figures viii List of Tables xi Chapter 1 Introduction 1 1.1 Objective ；. 1 1.2 Fundamental knowledge of multibody dynamics 2 1.3 Escapement mechanisms 5 1.3.1 Time keeping accuracy and stability factors 7 1.3.2 Estimations of moment of inertia 9 1.3.3 Other simulations and analyses 15 1.4 Thesis outlines 15 1.5 Chapter summary 17 Chapter 2 Multibody Dynamics 18 2.1 The unilateral comer law of impact 18 2.2 The Coulomb's friction 19 2.3 Slip, stick, and slip reversal phenomena 20 2.4 The coefficients of restitution 20 2.5 Ways of formulating multiple contacts 22 2.6 Integration procedure 22 2.7 The P. Pfeiffer and Ch. docker's approach 23 2.7.1 Kinematics calculation 23 V 2.7.2 Configuration index 26 2.7.3 Motion without contact 27 2.7.4 Motion for detachment and slip-stick transition and LCP formulation 27 2.7.5 Motion for impact and LCP formulation 37 2.8 Solving LCP 50 2.9 Chapter summary 52 Chapter 3 Development of the Simulation Tool 54 3.1 Kinematics calculation 54 3.1.1 Geometric definitions 55 3.1.2 Line-to-line contact 59 3.1.3 Arc-to-line contact 62 3.1.4 Kinematics calculation procedures 67 3.2 Obtaining the solutions 72 3.3 Revised numerical treatment for LCP solving 73 3.4 Integration procedure of simulation 74 3.5 Verification example 76 3.5.1 Classical mechanics approach 76 3.5.2 Pre-calculation before application 79 3.5.3 Simulation results 81 3.6 Chapter summary 83 Chapter 4 Application to Swiss Lever Escapement 84 4.1 Working principle of Swiss lever escapement 84 4.2 Simulation of Swiss lever escapement 87 4.2.1 Pre-calculation of kinematics 88 4.2.2 Simulation results 89 vi 4.3 More simulations 102 4.3.1 Theoretical optimal peak amplitudes 102 4.3.2 Simulation of coaxial escapement 103 4.3.3 Simulations with different simulation parameters 109 4.3.4 Relation of input complexity and computational time 111 4.4 Chapter summary 113 Chapter 5 Conclusions and Future works 114 5.1 Conclusions 114 5.2 Future works 117 Bibliography 119 vii List of Figures Figure 1.1 The verge-and-foliot escapement (figure from [18]) 5 Figure 1.2 Swiss lever escapement (Isometric view) 6 Figure 1.3 Swiss lever escapement (exploded view) 6 Figure 1.4 Impulse mechanism of Daniel's coaxial escapement (left) and Swiss lever escapement (right) 8 Figure 1.5 Working principle of tourbillon (from [24]) 8 Figure 1.6 Balance wheel 10 Figure 1.7 Impulse roller 11 Figure 1.8 Safety roller 11 Figure 1.9 Balance staff. 12 Figure 1.10 Hairspring 14 Figure 2.1 Impact corner law 19 Figure 2.2 General configuration of two rigid bodies 24 Figure 2.3 General configuration of two bodies and its contact forces. 28 Figure 2.4 Contact law of a sliding contact /• e /�(g；^ =, 0) in the normal direction 31 Figure 2.5 Contact law of a sliding contact / e ( g灿== 0 ) in the tangential direction 32 Figure 2.6 Left and right characteristics after decomposition 33 Figure 2.7 The four characteristics after decomposition 33 Figure 2.8 Impact law at the end of compression in normal direction 38 Figure 2.9 Impact law at the end of compression in tangential direction 39 Figure 2.10 Impact law at the end of restitution in normal direction 43 Figure 2.11 Impact laws at the end of restitution in tangential direction 44 viii Figure 2.12 Impact characteristics for different values of v and s飞 46 Figure 2.13 The asymmetric characteristic of impact law at the end of restitution in tangential direction 48 Figure 3.1 A line vector and its normal and tangent vector 55 Figure 3.2 Convex and concave arc representations and their normal and tangent vectors 57 Figure 3.3 Origin of an three points arc 57 Figure 3.4 Distances from points Q and 60 Figure 3.5 Line to line contact case 1 60 Figure 3.6 Line to line contact case 2 : 61 Figure 3.7 Line to line contact case 3 61 Figure 3.8 Normal vector in endpoint-endpoint contact 62 Figure 3.9 Distances from points Q and 63 Figure 3.10 Possible convex arc-line contact configurations 65 Figure 3.11 Possible concave arc-line contact configurations 66 Figure 3.12 Line-to-line distance and contact points pair calculation 67 Figure 3.13 Arc-to-line distance and contact points pair calculation 68 Figure 3.14 Normal and tangent vector calculation 69 Figure 3.15 Input groups of geometries 70 Figure 3.16 Collision detection procedures 71 Figure 3.17 Integration procedure of the simulation 75 Figure 3.18 A bouncing and rolling ball on an inclined plane 76 Figure 3.19 Free body diagram of the ball 77 Figure 3.20 Displacement-time graph of the ball in y-direction simulated by classical mechanic approach 82 ix Figure 3.21 Displacement-time graph of the ball in y-direction by our simulation tool 82 Figure 4.1 Working mechanism of impulse phase 85 Figure 4.2 Working mechanism of draw phase 85 Figure 4.3 Working mechanism of lock phase 86 Figure 4.4 Working mechanism of unlock phase 86 Figure 4.5 Initial input of the design profile 87 Figure 4.6 Angular displacement-time graph and angular velocity-time graph of the escape wheel 91 Figure 4.7 Angular displacement-time graph and angular velocity-time graph of the escape lever 92 Figure 4.8 Angular displacement-time graph and angular velocity-time graph of the balance wheel 93 Figure 4.9 Daniel's coaxial escapement 103 Figure 4.10 Plot of peak amplitude versus input torque 110 Figure 4.11 Plot of computational time versus number of possible contact pairl 12 V List of Tables Table 1.1 Material properties 13 Table 4.1 Motion timeline of the Swiss lever escapement during t =179.4s to t = 179.75s 94 Table 4.2 Motion timeline of Daniel's coaxial escapement during t =179.6779s to t = 179.981s 104 Table 4.3 Comparison of peak amplitudes between the Swiss lever escapement and the Daniel's coaxial escapement 108 Table 4.4 Relationship of input torque and peak amplitudes 109 Table 4.5 Period versus spring constant 110 Table 4.6 Period versus moment of inertia Ill Table 4.7 Relation of input complexity and computational time Ill xi G_®�11 Chapter 1 Introduction 1.1 Objective In design optimization of complex mechanisms, a reliable simulation tool is very important in kinematics and dynamics studies.
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