CH358958A

CH358958A - Apparatus for illustrating the elliptical motion of a point in a plane, especially in the case of motion governed by Kepler's first two laws

Info

Publication number: CH358958A
Authority: CH
Inventors: Prior Eugene
Original assignee: Prior Eugene
Priority date: 1958-03-25
Filing date: 1958-03-25
Publication date: 1961-12-15

Description

Appareil pour illustrer 1e mouvement elliptique d'un point dans un plan, notamment dans 1e eas d'un mouvement regi par les deux premieres lois de Kepler 0n connait divers appareils ou installations sous forme de modeles mecaniques servant ä muntrer au public ou aux jeunes gens, sans avoir ä proceder ä des observations directes ainsi qu'ä des calculs labo- rieux, les principales particularites des mouvements dans 1e systeme solaire, voire des etoiles et de leurs constellations. Apparatus for illustrating the elliptical movement of a point in a plane, in particular in the case of a movement governed by the first two laws of Kepler We know various apparatus or installations in the form of mechanical models serving to show the public or young people , without having to proceed to direct observations as well as to laborious calculations, the main particularities of the movements in the solar system, even of the stars and their constellations.

Lobjet de 1'invention, inspiK des conceptions de Ptolem6e, est un appareil pour illustrer 1e mouve- ment elliptique d'un point dans un plan, notamment dans le cas d'un mouvement regi par les deux pre- mieres lois de Kepler. Cet appareil est caracterisd en ce qu'il comporte une manivelle tournant autour d'un axe perpendiculaire audit plan et passant par un point fixe S de ce dernier, cette manivelle portant ä son extr6mite un coulisseau agence pour glisser dans une coulisse qui se prolonge par une Barre mont6e rotativement sur un axe egalement perpendi- culaire audit plan, cette Barre portant un pignon satellite qui roule sur une roue dentee de Base, de meine axe que celui portant ladite Barre, et un pignon-epicycle entraine par 1e satellite, les centres de ces trois e16ments dentes etant alignes, 1e rayon primitif du pignon-epicycle etant egal ä la moitie de celui de la roue de Base, 1e tout 6tant agence de fa@on que lorsque laxe portant ladite Barre est immo- bilis6 de maniere ä passer par un point fixe O dudit plan, un point d'un bras solidaire du pignon-epicycle decrive une ellipse dont 1e point O est 1e centre et 1e point S est 1e foyer. The object of the invention, inspired by the designs of Ptolemy, is an apparatus for illustrating the elliptical motion of a point in a plane, in particular in the case of a motion governed by Kepler's first two laws. This device is characterized in that it comprises a crank rotating around an axis perpendicular to said plane and passing through a fixed point S of the latter, this crank carrying at its end a slide arranged to slide in a slide which is extended by a Bar rotatably mounted on an axis also perpendicular to said plane, this Bar carrying a satellite pinion which rolls on a Base toothed wheel, of the same axis as that carrying said Bar, and an epicycle pinion driven by the first satellite, the centers of these three toothed elements being aligned, the primitive radius of the pinion-epicycle being equal to half that of the Base wheel, the whole being arranged so that when the axis carrying the said Bar is immobilized so as to pass by a fixed point O of said plane, a point of an arm integral with the pinion-epicycle describes an ellipse of which the first point O is the first center and the first point S is the first focus.

11 est opportun de rappeler les enances des deux premieres lois de K6pler, qui interviennent dans 1e present brevet: Premiere loi : chaque planete decrit une ellipse dont 1e centre du soleil occupe 1'un des foyers. Seconde loi : 1e rayon vecteur allant du soleil ä la planete balaye des aires egales en des temps egaux. It is appropriate to recall the beginnings of K6pler's first two laws, which are used in the present patent: First law : each planet describes an ellipse whose center of the sun occupies 1 one of the homes. Second law : the ray vector from the sun to the planet sweeps out equal areas in equal times.

Le dessin annexe represente, ä tiire d'exemple, une forme d'ex6cution de l'appareil selon 1'invention. La fig. 1 represente un ensemble schematique de 1'appareil. The appended drawing shows, by way of example, one embodiment of the apparatus according to the invention. fig. 1 represents a schematic assembly of the apparatus.

La fig. 2 est un trace de la trajectoire d6crite par une planete, avec RTI ID="0001.0275" WI="11" HE="4"LX="1445" LY="1544"> avance du perih6lie, et la fig. 3, un trace de ladite trajectoire perme2- tant de passer d'un tour au suivant. fig. 2 is a trace of the trajectory described by a planet, with RTI ID="0001.0275" WI="11" HE="4"LX="1445" LY="1544"> advance of the perihelion, and fig. 3, a trace of said trajectory permitting to pass from one turn to the next.

A la fig. 1, O designe un pivot central mon te sur 1e panneau vertical d'un bäti nun represente. Une Barre porte-pignons 1 peut tourner autour du pivot O ä une vitesse angulaire 62. Elle se prolonge axialement par une coudisse 2 et poxite un satellite 3 qui roule sur une roue dent6e de bare 4 solidaire du bäti par 1'intermediaire d'un montant vertical Don represent6. La Barre 1 porte egalement, ä son extre- mite, un pignon-epicycle 5 dont 1e rayon primitif r est egal ä la moitie du rayon primitif R' de la roue dent6e de Base 4. Le satellite 3 a pour but de renverser 1e sens de rotation de la Barre de fagon que 1e pignon-6picycle 5 tourne ä vitesse double de la Barre et en sens inverse, donc ä la vitesse -2 - S2 relativement au corps de la Barre 1. Le rayon du satellite est quelconque, 1e module des roues dentees 4, 3 et 5 etant 1e meine. La Barre 1 est actionnee par une manivelle 6, de -longueur egale au 1/2 grand axe a de 1'ellipse, et qui tourne ä la vitesse cons- tante w. Ladite manivelle porte ä son extremit6 un galet ou maneton N qui sert ä entrainer la Barre 1 en s'engageant dans la coulisse rotative 2 de lon- gueur int6rieure au moins egale ä 2e, soit au double de 1'excentricit6 lineaire e de 1'elhpse. Un bras porte-planete 7, solidaire du pignon- epicycle 5 et toumant par consequent ä une vitesse angulaire egale ä -2-Q, poste un feu rouge P qui represente la planere situee ä une distance Q (r6gla- ble ä volonte) sur 1e bras 7. Si M est 1e centre du pignon-6picycle et du cerele 6picycle, la distance 0M est egale au rayon R du cercle dir d6ferent , tandis que la distance MP est egale au rayon o du cercle epicycle . Le demi grand axe de Fellipse decrite par P est Egal ä a = R + o, tandis que 1e demi petit axe est Egal ä b = R-o ; 1'excentricite lineaire a pour valeur In fig. 1, O denotes a central pivot mounted on the first vertical panel of a frame represented. A pinion bar 1 can rotate around the pivot O at an angular speed 62. It is extended axially by a elbow joint 2 and near a satellite 3 which rolls on a toothed wheel of bar 4 fixed to the frame by means of a vertical upright Don represented. The Bar 1 also carries, at its end, an epicycle pinion 5 whose pitch radius r is equal to half the pitch radius R' of the Base toothed wheel 4. The purpose of the satellite 3 is to reverse the direction of rotation of the Bar so that the pinion-6picycle 5 rotates at twice the speed of the Bar and in the opposite direction, therefore at the speed -2 - S2 relative to the body of the Bar 1. The radius of the satellite is arbitrary, the 1st module toothed wheels 4, 3 and 5 being the first mine. Bar 1 is actuated by a crank 6, of length equal to the 1/2 major axis a of the ellipse, and which rotates at the constant speed w. Said crank carries at its end a roller or crank pin N which serves to drive the bar 1 by engaging in the rotary slide 2 of internal length at least equal to 2e, i.e. twice the linear eccentricity e of 1' elhpse. A planet-carrying arm 7, integral with the pinion-epicycle 5 and consequently rotating at an angular speed equal to -2-Q, posts a red light P which represents the planet located at a distance Q (adjustable at will) on the arm 7. If M is the center of the epicycle pinion and the epicycle circle, the distance 0M is equal to the radius R of the circle dir deferent , while the distance MP is equal to the radius o of the epicycle circle . The semi-major axis of the ellipse described by P is equal to a=R+o, while the semi-minor axis is equal to b=R-o; the linear eccentricity has the value

Le cerele de rayon b et de centre O touche 1'epi- cycle en E, tandis que 1'epicycle et 1e cercle de rayon a avec O comme centre se touchent en Q, les points O, E, M et Q etant alignes sur Faxe geome- trique de la Barre 1. Le point P decrit dans Fespace une ellipse th6oriquement exacte, de sorte que la premiere loi de K6pIer se trouve satisfaite. Quant ä la loi des aires ou 2e loi de K6pler, elle est rea- lis6e de maniere pratiquement exacte, F6cart angu- laire moyen entre la position du point P de 1'appareil et la position d'un point conjugue qui suivrait rigou- reusement cette loi 6tant inf6rieur en valeur absolue ä un degre (pour une excentricite specifiqiie tres 61evee de -, = 0,6), au cours de chaque demi-revo- lution allant du periMlie [-J ä 1'aphelie A et inver- sement. The circle of radius b and center O touches the epicycle at E, while the epicycle and the circle of radius a with center O touch at Q, the points O, E, M and Q being aligned on Geometric fax of Bar 1. The point P describes in space a theoretically exact ellipse, so that K6pIer's first law is satisfied. As for the law of areas or Kepler's 2nd law, it is carried out in a practically exact manner, the average angular difference between the position of the point P of the apparatus and the position of a conjugate point which would follow rigorously this law being lower in absolute value than one degree (for a very high specific eccentricity of -, = 0.6), during each half-revolution going from the periMlia [-J to the aphelion A and vice versa .

0n pourrait aussi (fig. 2), avec des moyens appropries, faire tourner 1e centre O de la roue den- t & de Base sur une circonf6rence ayant 1e foyer S comme centre, ä une vitesse angulaire d'entrainement w = Q/ q proportionnelle par consequent ä Gelle, rela tive, S2, de la Barre porte-pignons autour de O', position variable de O. Dans Ges conditions, 1e point mobile P representant la planete et tournant autour de M ä la vitesse -2 - S. par rapport ä la Barre porte- pignons decrirait, comme trajectoire ou orbite, une ellipse alt6ree T qui possederait encore un perih6lie, au sens d'un point de distance minimum par rapport au foyer, autrement dit, pour chaque tour relatif de la Barre porte-pignons, un p6rih61ie decale en avant de 360o/q, avec q> 1. A chaque tour de la Barre porte-pignons, la trajectoire tournerait par points de ip = 360,Ilq autour de S, de sorte que 1e perihelie avancerait chaque fois de 3600/q. It would also be possible (fig. 2), with appropriate means, to make the center O of the Base tooth wheel rotate on a circumference having the focus S as its center, at an angular driving speed w = Q/q proportional therefore to Gelle, relative, S2, of the pinion bar around O', variable position of O. Under these conditions, the 1st mobile point P representing the planet and rotating around M at the speed -2 - S relative to the Pinion Bar would describe, as a trajectory or orbit, an altered ellipse T which would still possess a perihelion, in the sense of a point of minimum distance from the focus, in other words, for each relative revolution of the Bar carrier, a perihelion shifted forward by 360o/q, with q > 1. At each turn of the carrier bar, the trajectory would rotate by points of ip = 360,Ilq around S, so that the perihelion would advance each time 3600/q.

Ainsi compl6te, 1'appareil d6erit permettrait d'il- lustrer 1e ph6nomene dit de Favance du periMlie. Completed in this way, the apparatus described would make it possible to illustrate the phenomenon known as the advance of the periMlia.

A la fig. 2, 1e pivot O est venu en O', 1e rayon SO', de longueur egale ä e, ayant decrit pendant un temps t et par des moyens non repr6sentis, un angle de deflexion ip proportionnel ä w ; au bout du meme temps t, la Barre 1 fait avec SO' un angle (p dans 1e sens de sa rotation S2 et proportionnel ä celle-ci ; ip <I≥ cplq, oü q est une constante sup6rieure ä 1. Le point P, dopt la construction ressort directement de 1'examen de la figure, decrit une ellipse alt6ree T. In fig. 2, the pivot O has come to O', the radius SO', of length equal to e, having described during a time t and by means not shown, a deflection angle ip proportional to w; at the end of the same time t, Bar 1 makes with SO' an angle (p in the direction of its rotation S2 and proportional to this; ip <I≥ cplq, where q is a constant greater than 1. The point P, adopts the construction, comes directly from the examination of the figure, describes an altered ellipse T.

La fig. 3 montre comment an peut passer direc- tement du ler au 2e tour en faisant tourner autour de S la trajectoire du 1,r tour d'un angle constant fig. 3 shows how an can pass directly from the 1st to the 2nd turn by rotating around S the trajectory of the 1st turn by a constant angle

les lettres non accentu6es se rapportant au ler tour, alors que les memes lettres accentuees se rapportent au 2F tour. the unaccented letters refer to the 1st round, while the same accented letters refer to the 2F round.

En outre, pour illustrer une ellipse ä la Lissa- jous, c'est-ä-dire une ellipse oü 1'acceleration est diri- gee vers 1e centre de 1'ellipse et a une grandeur pro portionnelle ä la distance du point mobile qui decrit 1'ellipse au centre de celle-ci, 1'appareil pourrait com- porter des moyens pour entrainer la Barre porte- pignons ä vitesse de rotation constante autour du centre de 1'ellipse et pour, simultanement, laisser tourner librement la manivelle autour dudit foyer. Furthermore, to illustrate a Lissajous ellipse, i.e., an ellipse where the acceleration is directed towards the center of the ellipse and has a magnitude proportional to the distance from the moving point which describes the ellipse at the center thereof, the apparatus could include means for driving the pinion bar at constant rotational speed around the center of the ellipse and for, simultaneously, allowing the crank to rotate freely around it. of said hearth.

Claims

CLAIM Apparatus for illustrating the elliptical motion of a point in a plane, in particular in the Gas of a motion governed by Köpler's first two laws, characterized in that it comprises a crank rotating around a perpendicular axis to said plane and passing through a fixed point S of the latter, this crank carrying at its end a slide arranged to slide in a slide which is extended by a bar mounted rotatably on an axis also perpendicular to said plane, this bar carrying a satellite pinion which rolls on a Base toothed wheel, on the same axis as that carrying the said Bar, and an epicycle pinion driven by the satellite, the centers of the three toothed elements being aligned, the pitch radius of the epicycle pinion being equal half of that of the Base wheel, the first turn being arranged so that when the axis carrying the said Bar is immobilized so as to pass through a fixed point O of the said plane, a point of an arm integral with the pinion-6picycle d & rivets an ellipse whose 1st point O is the 1st center and 1st point S is the 1st focus. SUB-CLAIMS 1. Apparatus according to claim, characterized in that it comprises RTI ID="0002.0549" WI="8"HE="4" LX="1676" LY="2065"> means for running Faxe of the base wheel around Faxe passing through said focus S, in order to achieve the movement of a planetary orbit with mobile perihelion.

2. Apparatus according to the claimed claim, characterized in that, in order to illustrate a Lissajous ellipse, it comprises means for driving the pinion bar at a constant speed of rotation around the axis passing through the first center O and for, simultaneously, let the crank turn freely around Faxe passing through said focus S.