University of Wisconsin Milwaukee UWM Digital Commons Theses and Dissertations December 2020 The Instability of Verge-Foliot Clocks in Contrast with the Isochronicity of Pendulum Clocks: A Study of Verge-Foliot Kinematics Aaron Seth Blumenthal University of Wisconsin-Milwaukee Follow this and additional works at: https://dc.uwm.edu/etd Part of the Mechanical Engineering Commons Recommended Citation Blumenthal, Aaron Seth, "The Instability of Verge-Foliot Clocks in Contrast with the Isochronicity of Pendulum Clocks: A Study of Verge-Foliot Kinematics" (2020). Theses and Dissertations. 2461. https://dc.uwm.edu/etd/2461 This Thesis is brought to you for free and open access by UWM Digital Commons. It has been accepted for inclusion in Theses and Dissertations by an authorized administrator of UWM Digital Commons. For more information, please contact [email protected]. THE INSTABILITY OF VERGE-FOLIOT CLOCKS IN CONTRAST WITH THE ISOCHRONICITY OF PENDULUM CLOCKS: A STUDY OF VERGE-FOLIOT KINEMATICS by Aaron Seth Blumenthal A Thesis Submitted in Partial Fulfillment of the Requirements for the Degree of Master of Science in Engineering at The University of Wisconsin-Milwaukee December 2020 ABSTRACT THE INSTABILITY OF VERGE-FOLIOT CLOCKS IN CONTRAST WITH THE ISOCHRONICITY OF PENDULUM CLOCKS: A STUDY OF VERGE-FOLIOT KINEMATICS by Aaron Seth Blumenthal The University of Wisconsin-Milwaukee, 2020 Under the Supervision of Professor Michael Nosonovsky The tower clocks of Europe starting in the 13th century CE employed a verge-foliot mechanism, which was low in both precision and accuracy due to the sensitivity of frictional losses in the system. The pendulum and Huygens’ introduction of the Theory of Involutes is shown to be a mechanism with a linear relationship to frictional losses in the system, instead of the non-linear relationship between frictional losses in verge-foliot mechanisms. The isochrone nature of vibrations of the pendulum are discussed in contrast to the variance in oscillation seen in verge-foliots. The introduction of the pendulum resulted in an improvement of accuracy of around π/µ≈31 times, where µ=0.1 is the coefficient of friction. In this thesis a CAD model is developed for both the verge-foliot and pendulum with a verge, and it is found that FEA analysis produces a sufficiently approximate result to the theoretical models. A discussion of CAD and DFM principles in relation to clocks is also discussed. ii © Copyright by Aaron Seth Blumenthal, 2020 All Rights Reserved iii To my wife, my family, my friends, and my best friend Zorro iv TABLE OF CONTENTS LIST OF FIGURES .......................................................................................................................................... vii LIST OF ABBREVIATIONS .............................................................................................................................. ix ACKNOWLEDGMENTS ................................................................................................................................... x 1. Objective ................................................................................................................................................... 1 2. Introduction .............................................................................................................................................. 1 3. The Medieval and Early Modern Tower Clocks ........................................................................................ 4 3.1 Pre-Pendulum Tower Clocks ............................................................................................................... 5 3.2 Impact of the Invention of the Pendulum on Time Measurement ..................................................... 9 4. Dynamic Modeling of The Verge-And-Foliot Mechanism ....................................................................... 12 4.1 Mechanics of the Verge-and-Foliot Mechanism ............................................................................... 12 4.2 Mathematical Model of the Verge Escapement Mechanism with Foliot ......................................... 15 4.3 Mathematical Model of the Verge Escapement Mechanism with Foliot ......................................... 17 4.4 Stability Analysis ................................................................................................................................ 19 4.5 Comparison with the Stability of the Pendulum Mechanism ........................................................... 21 5. Experimental Modeling of The Verge-And-Foliot Mechanisms .............................................................. 23 5.1 Methods Used ................................................................................................................................... 23 5.2 Set-Up of Experiment ........................................................................................................................ 26 5.3 Results ............................................................................................................................................... 27 5.4 Discussion .......................................................................................................................................... 30 v 6. Conclusions ............................................................................................................................................. 32 References .................................................................................................................................................. 34 APPENDIX .................................................................................................................................................... 37 Appendix A .............................................................................................................................................. 38 Appendix B .............................................................................................................................................. 40 vi LIST OF FIGURES Figure 1 The Elephant Clock made by Ismail al-Jazari was a clepsydra that utilized a primitive escapement, Metropolitan Museum of Art (al-Razzaz al-Jazari & `Abd al-Latif, 1315) (Creative Commons CC0 1.0) ...... 2 Figure 2 Schematic of Verge-Foliot ............................................................................................................... 4 Figure 3 Villard de Honnecourt Foliot 44 illustrating a supposed escapement (Honnecourt, 1230) (Public Domain) ......................................................................................................................................................... 5 Figure 4 Astrarium astronomical clock was finished 2 years later in 1364 by Giovanni Dondi dell’Orologio, note the use of a balance wheel instead of a foliot (de Dondi, 1461) (Public Domain) ............................... 6 Figure 5 Salisbury Cathedral Clock (Llewelyn, 2014) (Creative Commons 3.0) ............................................ 7 Figure 6 Cotehele Clock with its inverted foliot (Denning, 2007) (Creative Commons 3.0) ......................... 9 Figure 7 Tautochrone curve (Bisgaard, 2006) (Public Domain) .................................................................. 11 Figure 8 Comparison of the verge mechanism with a foliot and a pendulum. Note the difference in escapement wheel orientation ................................................................................................................... 14 Figure 9 System block diagram of a verge-foliot system ............................................................................ 19 Figure 10 System block diagram of a verge escapement with a pendulum ............................................... 21 Figure 11 CAD of clock using a proven going train ..................................................................................... 26 Figure 12 Angular Displacement of the Verge-Foliot .................................................................................. 27 Figure 13 Angular Velocity of the Verge-Foliot ........................................................................................... 28 Figure 14 Angular Acceleration of the Verge-Foliot ................................................................................... 28 Figure 15 Angular Displacement of the Pendulum-Verge .......................................................................... 29 Figure 16 Angular Velocity of the Pendulum-Verge ................................................................................... 29 Figure 17 Angular Acceleration of the Pendulum-Verge ............................................................................ 30 Figure 18 Dependency of Period of Oscillation on the Coefficient of Friction ........................................... 30 vii Figure 19 Fourier Transform of Angular Displacement .............................................................................. 40 Figure 20 High Pass Filter at 63 mHz Application of Verge-Foliot Acceleration ......................................... 41 Figure 21 High Pass Filter at 63 mHz Application of Verge-Foliot Displacement in the Frequency Domain .................................................................................................................................................................... 42 viii LIST OF ABBREVIATIONS CAD Computer-Aided Design CAM Computer-Aided Manufacturing FEA Finite Element Analysis FEM Finite Element Method CAE Computer-Aided