Flexure Pivot Oscillator with Intrinsically Tuned Isochronism
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Floating and Platform Balances an Introduction
Floating and Platform Balances An introduction ©Darrah Artzner 3/2018 Floating and Platform Balances • Introduce main types • Discuss each in some detail including part identification and function • Testing and Inspecting • Cleaning tips • Lubrication • Performing repairs Balance Assembly Type Floating Platform Floating Balance Frame Spring stud Helicoid spring Hollow Tube Mounting Post Regulator Balance wheel Floating Balance cont. Jewel Roller Pin Paired weight Hollow Tube Safety Roller Pivot Wire Floating Balance cont. Example Retaining Hermle screws Safety Roller Note: moving fork Jewel cover Floating Balance cont. Inspecting and Testing (Balance assembly is removed from movement) • Inspect pivot (suspension) wire for distortion, corrosion, breakage. • Balance should appear to float between frame. Top and bottom distance. • Balance spring should be proportional and not distorted in any way. • Inspect jewels for cracks and or breakage. • Roller pin should be centered when viewed from front. (beat) • Rotate balance wheel three quarters of a turn (270°) and release. It should rotate smoothly with no distortion and should oscillate for several (3) minutes. Otherwise it needs attention. Floating Balance cont. Cleaning • Make sure the main spring has been let down before working on movement. • Use non-aqueous watch cleaner and/or rinse. • Agitate in cleaner/rinse by hand or briefly in ultrasonic. • Rinse twice and final in naphtha, Coleman fuel (or similar) or alcohol. • Allow to dry. (heat can be used with caution – ask me how I would do it.) Lubrication • There are two opinions. To lube or not to lube. • Place a vary small amount of watch oil on to the upper and lower jewel where the pivot wire passed through the jewel holes. -
Pioneers in Optics: Christiaan Huygens
Downloaded from Microscopy Pioneers https://www.cambridge.org/core Pioneers in Optics: Christiaan Huygens Eric Clark From the website Molecular Expressions created by the late Michael Davidson and now maintained by Eric Clark, National Magnetic Field Laboratory, Florida State University, Tallahassee, FL 32306 . IP address: [email protected] 170.106.33.22 Christiaan Huygens reliability and accuracy. The first watch using this principle (1629–1695) was finished in 1675, whereupon it was promptly presented , on Christiaan Huygens was a to his sponsor, King Louis XIV. 29 Sep 2021 at 16:11:10 brilliant Dutch mathematician, In 1681, Huygens returned to Holland where he began physicist, and astronomer who lived to construct optical lenses with extremely large focal lengths, during the seventeenth century, a which were eventually presented to the Royal Society of period sometimes referred to as the London, where they remain today. Continuing along this line Scientific Revolution. Huygens, a of work, Huygens perfected his skills in lens grinding and highly gifted theoretical and experi- subsequently invented the achromatic eyepiece that bears his , subject to the Cambridge Core terms of use, available at mental scientist, is best known name and is still in widespread use today. for his work on the theories of Huygens left Holland in 1689, and ventured to London centrifugal force, the wave theory of where he became acquainted with Sir Isaac Newton and began light, and the pendulum clock. to study Newton’s theories on classical physics. Although it At an early age, Huygens began seems Huygens was duly impressed with Newton’s work, he work in advanced mathematics was still very skeptical about any theory that did not explain by attempting to disprove several theories established by gravitation by mechanical means. -
Computer-Aided Design and Kinematic Simulation of Huygens's
applied sciences Article Computer-Aided Design and Kinematic Simulation of Huygens’s Pendulum Clock Gloria Del Río-Cidoncha 1, José Ignacio Rojas-Sola 2,* and Francisco Javier González-Cabanes 3 1 Department of Engineering Graphics, University of Seville, 41092 Seville, Spain; [email protected] 2 Department of Engineering Graphics, Design, and Projects, University of Jaen, 23071 Jaen, Spain 3 University of Seville, 41092 Seville, Spain; [email protected] * Correspondence: [email protected]; Tel.: +34-953-212452 Received: 25 November 2019; Accepted: 9 January 2020; Published: 10 January 2020 Abstract: This article presents both the three-dimensional modelling of the isochronous pendulum clock and the simulation of its movement, as designed by the Dutch physicist, mathematician, and astronomer Christiaan Huygens, and published in 1673. This invention was chosen for this research not only due to the major technological advance that it represented as the first reliable meter of time, but also for its historical interest, since this timepiece embodied the theory of pendular movement enunciated by Huygens, which remains in force today. This 3D modelling is based on the information provided in the only plan of assembly found as an illustration in the book Horologium Oscillatorium, whereby each of its pieces has been sized and modelled, its final assembly has been carried out, and its operation has been correctly verified by means of CATIA V5 software. Likewise, the kinematic simulation of the pendulum has been carried out, following the approximation of the string by a simple chain of seven links as a composite pendulum. The results have demonstrated the exactitude of the clock. -
1 Functions, Derivatives, and Notation 2 Remarks on Leibniz Notation
ORDINARY DIFFERENTIAL EQUATIONS MATH 241 SPRING 2019 LECTURE NOTES M. P. COHEN CARLETON COLLEGE 1 Functions, Derivatives, and Notation A function, informally, is an input-output correspondence between values in a domain or input space, and values in a codomain or output space. When studying a particular function in this course, our domain will always be either a subset of the set of all real numbers, always denoted R, or a subset of n-dimensional Euclidean space, always denoted Rn. Formally, Rn is the set of all ordered n-tuples (x1; x2; :::; xn) where each of x1; x2; :::; xn is a real number. Our codomain will always be R, i.e. our functions under consideration will be real-valued. We will typically functions in two ways. If we wish to highlight the variable or input parameter, we will write a function using first a letter which represents the function, followed by one or more letters in parentheses which represent input variables. For example, we write f(x) or y(t) or F (x; y). In the first two examples above, we implicitly assume that the variables x and t vary over all possible inputs in the domain R, while in the third example we assume (x; y)p varies over all possible inputs in the domain R2. For some specific examples of functions, e.g. f(x) = x − 5, we assume that the domain includesp only real numbers x for which it makes sense to plug x into the formula, i.e., the domain of f(x) = x − 5 is just the interval [5; 1). -
A Royal 'Haagseklok'
THE SPLIT (Going and Strike) BARREL (top) p.16 Overview Pendulum Applications. The Going-wheel The Strike-wheel Ratchet-work Stop-work German Origins Hidden Stop Work English Variants WIND-ME MECHANISM? p.19 An "Up-Down" Indicator (Hypothetical Project) OOSTERWIJCK'S UNIQUE BOX CASE p.20 Overview Show Wood Carcass Construction Damage Control Mortised Hinges A Royal 'Haagse Klok' Reviewed by Keith Piggott FIRST ASSESSMENTS p.22 APPENDIX ONE - Technical, Dimensions, Tables with Comparable Coster Trains - (Dr Plomp's 'Chronology' D3 and D8) 25/7/1 0 Contents (Horological Foundation) APPENDIX TWO - Conservation of Unique Case General Conservation Issues - Oosterwijck‟s Kingwood and Ebony Box INTRODUCTION First Impressions. APPENDIX SIX - RH Provenance, Sir John Shaw Bt. Genealogy HUYGENS AUTHORITIES p.1 Collections and Exhibitions Didactic Scholarship PART II. „OSCILLATORIUM‟ p.24 Who was Severijn Oosterwijck? Perspectives, Hypotheses, Open Research GENERAL OBSERVATIONS p.2 The Inspection Author's Foreword - Catalyst and Conundrums Originality 1. Coster‟s Other Contracts? p.24 Unique Features 2. Coster‟s Clockmakers? p.24 Plomp‟s Characteristic Properties 3. Fromanteel Connections? p.25 Comparables 4. „Secreet‟ Constructions? p.25 Unknown Originator : German Antecedents : Application to Galilei's Pendulum : Foreknowledge of Burgi : The Secret Outed? : Derivatives : Whose Secret? PART I. „HOROLOGIUM‟ p.3 5. Personal Associations? p.29 6. The Seconds‟ Hiatus? p.29 The Clock Treffler's Copy : Later Seconds‟ Clocks : Oosterwijck‟s Options. 7. Claims to Priority? p.33 THE VELVET DIALPLATE p.3 8. Valuation? p.34 Overview Signature Plate APPENDIX THREE - Open Research Projects, Significant Makers Chapter Ring User-Access Data Comparable Pendulum Trains, Dimensions. -
The Pope's Rhinoceros and Quantum Mechanics
Bowling Green State University ScholarWorks@BGSU Honors Projects Honors College Spring 4-30-2018 The Pope's Rhinoceros and Quantum Mechanics Michael Gulas [email protected] Follow this and additional works at: https://scholarworks.bgsu.edu/honorsprojects Part of the Mathematics Commons, Ordinary Differential Equations and Applied Dynamics Commons, Other Applied Mathematics Commons, Partial Differential Equations Commons, and the Physics Commons Repository Citation Gulas, Michael, "The Pope's Rhinoceros and Quantum Mechanics" (2018). Honors Projects. 343. https://scholarworks.bgsu.edu/honorsprojects/343 This work is brought to you for free and open access by the Honors College at ScholarWorks@BGSU. It has been accepted for inclusion in Honors Projects by an authorized administrator of ScholarWorks@BGSU. The Pope’s Rhinoceros and Quantum Mechanics Michael Gulas Honors Project Submitted to the Honors College at Bowling Green State University in partial fulfillment of the requirements for graduation with University Honors May 2018 Dr. Steven Seubert Dept. of Mathematics and Statistics, Advisor Dr. Marco Nardone Dept. of Physics and Astronomy, Advisor Contents 1 Abstract 5 2 Classical Mechanics7 2.1 Brachistochrone....................................... 7 2.2 Laplace Transforms..................................... 10 2.2.1 Convolution..................................... 14 2.2.2 Laplace Transform Table.............................. 15 2.2.3 Inverse Laplace Transforms ............................ 15 2.3 Solving Differential Equations Using Laplace -
The Tautochrone/Brachistochrone Problems: How to Make the Period of a Pendulum Independent of Its Amplitude
The Tautochrone/Brachistochrone Problems: How to make the Period of a Pendulum independent of its Amplitude Tatsu Takeuchi∗ Department of Physics, Virginia Tech, Blacksburg VA 24061, USA (Dated: October 12, 2019) Demo presentation at the 2019 Fall Meeting of the Chesapeake Section of the American Associa- tion of Physics Teachers (CSAAPT). I. THE TAUTOCHRONE A. The Period of a Simple Pendulum In introductory physics, we teach our students that a simple pendulum is a harmonic oscillator, and that its angular frequency ! and period T are given by s rg 2π ` ! = ;T = = 2π ; (1) ` ! g where ` is the length of the pendulum. This, of course, is not quite true. The period actually depends on the amplitude of the pendulum's swing. 1. The Small-Angle Approximation Recall that the equation of motion for a simple pendulum is d2θ g = − sin θ : (2) dt2 ` (Note that the equation of motion of a mass sliding frictionlessly along a semi-circular track of radius ` is the same. See FIG. 1.) FIG. 1. The motion of the bob of a simple pendulum (left) is the same as that of a mass sliding frictionlessly along a semi-circular track (right). The tension in the string (left) is simply replaced by the normal force from the track (right). ∗ [email protected] CSAAPT 2019 Fall Meeting Demo { Tatsu Takeuchi, Virginia Tech Department of Physics 2 We need to make the small-angle approximation sin θ ≈ θ ; (3) to render the equation into harmonic oscillator form: d2θ rg ≈ −!2θ ; ! = ; (4) dt2 ` so that it can be solved to yield θ(t) ≈ A sin(!t) ; (5) where we have assumed that pendulum bob is at θ = 0 at time t = 0. -
The Cycloid Scott Morrison
The cycloid Scott Morrison “The time has come”, the old man said, “to talk of many things: Of tangents, cusps and evolutes, of curves and rolling rings, and why the cycloid’s tautochrone, and pendulums on strings.” October 1997 1 Everyone is well aware of the fact that pendulums are used to keep time in old clocks, and most would be aware that this is because even as the pendu- lum loses energy, and winds down, it still keeps time fairly well. It should be clear from the outset that a pendulum is basically an object moving back and forth tracing out a circle; hence, we can ignore the string or shaft, or whatever, that supports the bob, and only consider the circular motion of the bob, driven by gravity. It’s important to notice now that the angle the tangent to the circle makes with the horizontal is the same as the angle the line from the bob to the centre makes with the vertical. The force on the bob at any moment is propor- tional to the sine of the angle at which the bob is currently moving. The net force is also directed perpendicular to the string, that is, in the instantaneous direction of motion. Because this force only changes the angle of the bob, and not the radius of the movement (a pendulum bob is always the same distance from its fixed point), we can write: θθ&& ∝sin Now, if θ is always small, which means the pendulum isn’t moving much, then sinθθ≈. This is very useful, as it lets us claim: θθ&& ∝ which tells us we have simple harmonic motion going on. -
A Tale of the Cycloid in Four Acts
A Tale of the Cycloid In Four Acts Carlo Margio Figure 1: A point on a wheel tracing a cycloid, from a work by Pascal in 16589. Introduction In the words of Mersenne, a cycloid is “the curve traced in space by a point on a carriage wheel as it revolves, moving forward on the street surface.” 1 This deceptively simple curve has a large number of remarkable and unique properties from an integral ratio of its length to the radius of the generating circle, and an integral ratio of its enclosed area to the area of the generating circle, as can be proven using geometry or basic calculus, to the advanced and unique tautochrone and brachistochrone properties, that are best shown using the calculus of variations. Thrown in to this assortment, a cycloid is the only curve that is its own involute. Study of the cycloid can reinforce the curriculum concepts of curve parameterisation, length of a curve, and the area under a parametric curve. Being mechanically generated, the cycloid also lends itself to practical demonstrations that help visualise these abstract concepts. The history of the curve is as enthralling as the mathematics, and involves many of the great European mathematicians of the seventeenth century (See Appendix I “Mathematicians and Timeline”). Introducing the cycloid through the persons involved in its discovery, and the struggles they underwent to get credit for their insights, not only gives sequence and order to the cycloid’s properties and shows which properties required advances in mathematics, but it also gives a human face to the mathematicians involved and makes them seem less remote, despite their, at times, seemingly superhuman discoveries. -
Mathematics, the Language of Watchmaking
View metadata, citation and similar papers at core.ac.uk brought to you by CORE 90LEARNINGprovidedL by Infoscience E- École polytechniqueA fédérale de LausanneRNINGLEARNINGLEARNING Mathematics, the language of watchmaking Morley’s theorem (1898) states that if you trisect the angles of any triangle and extend the trisecting lines until they meet, the small triangle formed in the centre will always be equilateral. Ilan Vardi 1 I am often asked to explain mathematics; is it just about numbers and equations? The best answer that I’ve found is that mathematics uses numbers and equations like a language. However what distinguishes it from other subjects of thought – philosophy, for example – is that in maths com- plete understanding is sought, mostly by discover- ing the order in things. That is why we cannot have real maths without formal proofs and why mathe- maticians study very simple forms to make pro- found discoveries. One good example is the triangle, the simplest geometric shape that has been studied since antiquity. Nevertheless the foliot world had to wait 2,000 years for Morley’s theorem, one of the few mathematical results that can be expressed in a diagram. Horology is of interest to a mathematician because pallet verge it enables a complete understanding of how a watch or clock works. His job is to impose a sequence, just as a conductor controls an orches- tra or a computer’s real-time clock controls data regulating processing. Comprehension of a watch can be weight compared to a violin where science can only con- firm the preferences of its maker. -
Technical Data Sheet RM
TECHNICAL SPECIFICATIONS OF THE TOURBILLON RICHARD MILLE RM 053 PABLO MAC DONOUGH Limited edition of 15 pieces. CALIBER RM053: manual winding tourbillion movement tilted 30° with hours, minutes and seconds. Dimensions: 50 mm x 42.70 mm x 20.00 mm. MAIN FEATURES POWER RESERVE Circa 48 hours. BASEPLATE AND BRIDGES IN GRADE 5 TITANIUM The manufacturing of these components in grade 5 titanium gives the whole assembly a high degree of rigidity and achieves excellent surface flatness which is essential for ensuring that the gear train functions perfectly. The skeletonized baseplate and the bridges have been subjected to intensive and complete validation tests to optimize their resistance capacities. It is tilted 30° providing optimal conditions for users to check the time on their wrist. Both compact and lightweight, the RM053 movement is ultra-resistant to the centrifugal and centripetal forces generated during a match. FREE SPRUNG BALANCE WHEEL WITH VARIABLE INERTIA The free-sprung balance provides better reliability in the event of shocks, movement assembly and disassembly. It also guarantees better chronometric results over an extended period of time. There is no need for an index plate, allowing more precise and frequent adjustment using 4 setting screws. FAST ROTATING BARREL (6 hours per revolution instead of 7.5 hours) This type of barrel provides the following advantages: - The phenomenon of periodic internal mainspring adhesion is significantly diminished, thereby increasing performance, - Provision of an excellent mainspring delta curve with an ideal power reserve/performance/regularity ratio. BARREL PAWL WITH PROGRESSIVE RECOIL This device permits a profitable winding gain (circa 20 %), especially during the winding start. -
LECTURE NOTES (SPRING 2012) 119B: ORDINARY DIFFERENTIAL EQUATIONS June 12, 2012 Contents Part 1. Hamiltonian and Lagrangian Mech
LECTURE NOTES (SPRING 2012) 119B: ORDINARY DIFFERENTIAL EQUATIONS DAN ROMIK DEPARTMENT OF MATHEMATICS, UC DAVIS June 12, 2012 Contents Part 1. Hamiltonian and Lagrangian mechanics 2 Part 2. Discrete-time dynamics, chaos and ergodic theory 44 Part 3. Control theory 66 Bibliographic notes 87 1 2 Part 1. Hamiltonian and Lagrangian mechanics 1.1. Introduction. Newton made the famous discovery that the mo- tion of physical bodies can be described by a second-order differential equation F = ma; where a is the acceleration (the second derivative of the position of the body), m is the mass, and F is the force, which has to be specified in order for the motion to be determined (for example, Newton's law of gravity gives a formula for the force F arising out of the gravitational influence of celestial bodies). The quantities F and a are vectors, but in simple problems where the motion is along only one axis can be taken as scalars. In the 18th and 19th centuries it was realized that Newton's equa- tions can be reformulated in two surprising ways. The new formula- tions, known as Lagrangian and Hamiltonian mechanics, make it eas- ier to analyze the behavior of certain mechanical systems, and also highlight important theoretical aspects of the behavior of such sys- tems which are not immediately apparent from the original Newtonian formulation. They also gave rise to an entirely new and highly use- ful branch of mathematics called the calculus of variations|a kind of \calculus on steroids" (see Section 1.10). Our goal in this chapter is to give an introduction to this deep and beautiful part of the theory of differential equations.