SOME ISSUES ON INERTIA PROPULSION MECHANISMS USING TWO CONTRA-ROTATING MASSES

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1 Преподавание ТММ УДК 61.1 С. G. PROVATIDIS SOE ISSUES ON INERTIA PROPULSION ECHANISS USING TWO CONTRA-ROTATING ASSES 1. INTRODUCTION Among several physial priniples ha are poenially appliable o produe spaial moion, inerial fores possess an imporan posiion. This paper performs he mahemaial analysis of he earlies paens ha use onra-roaing masses moving along wo irumferenes on a verial plane. In addiion, he analysis is followed by many ommens and lemmas ha may be useful in he fuure researh on his ehnial field. A repor published by NASA in Deember 6, whih refers o researh wihin he years 1996-, has onluded ha alhough i is possible o ahieve he emporal lif of an obje hrough several mehanial means, he oal impulse beomes zero and, finally, he graviy fores all he obje bak o he earh [1]. In he wesern lieraure, he firs relevan paen is probably due o Norman L. Dean [], whih has been alled Dean s drive (hp://en.wikipedia.org/wiki/dean_drive). However, a areful survey reveals ha Russian sieniss had earlier sared and sysemaially ondued relevan researh [3-7], whih remains sill alive by ohers [8]. The onep is o uilize onra-roaing eenri masses for ahieving propulsion. However, when he masses roae along he irumferene of a irle on a verial plane, when hey move along he upper semi-irumferene a a onsan angular angle, he impulse is posiive, while when hey move along he lower par i has he same absolue value bu he opposie sign. Therefore, he ne impulse equals o zero hus no lifing fore is aniipaed. This paper onribues on hree opis as follows. Firs, i performs a sudy where i is assumed ha he eleri moors produe a variable angular veloiy desribed by wo losed-form mahemaial expressions. Seond, he mos general ase of a variable radius is also invesigaed. Third, he possibiliy of ahieving a monoonially upward ne impulse is disussed. Despie hese novel feaures and he useful onlusions derived, whih are appliable o several praial fields suh as shor-ime anigraviy oys, he problem of inerial propulsion is sill open.. THEORETICAL ASPECTS.1. Problem definiion 1 Le us onsider an obje (body B) of mass on whih wo onra-roaing rigid rods (No.1 and No.), of he same radius r, are ariulaed (a he poins C 1 and C, respeively) as shown in Figure 1; he rods bring a heir ends onenraed masses, of equal size m. The wo aforemenioned eenri masses are driven by eleri moors and roae a onsan or variable angular veloiies of equal and opposie magniude, i.e. ω ω ω. Obviously, he mass of he eleri moors is inluded ino he body B. The iniial posiion of he rods are denoed by respeively, for whih i is assumed ha ; herefore, for a laer ime insane, i holds 1 ha. Wihou loss of generaliy, we assume ha he ariulaion of he rigid 1 rods is hosen a he level of he enroid G of he mass, where also he axis origin is onsidered. For he sake of briefness, he problem is simplified as follows: 1. The shape of he obje B appears no variaion along he x- and y-axis.. The wo masses m have he same z-oordinae. 3. The moors are fixed o he obje and heir shafs are parallel o he y-axis. 4. The mass of he rods and he relevan momens of ineria are negleed. 1 and, 34 hp://mm.spbsu.ru

2 Some issues on ineria propulsion mehanisms 5. A he iniial ime insane, =, he obje is suddenly lef o fall. 6. The effe of he air is negleed. Due o he abovemenioned assumpions, he omponens of he enrifugal fores along he x-axis are perfely anelled, and any possible moion of he obje will be in he z-direion only. In oher words, no roaion of he obje B will our, hus is verial posiion an be wrien in he simple form: z, z. (1) The deerminaion of he abovemenioned funion z, and pariularly he ondiions for whih he aliude ould be inreased or remain onsan, is he aim of his work. I was found ha he opimum ondiion o ahieve a lif is o leave he obje fall when is rods are a he horizonal posiion wih he endeny of he rods o roae in he upward direion. oreover, we also invesigae he possibiliy of using a ime-varying angular veloiy or/and a ime-varying radius. Figure 1: The seup of he obje on whih wo onra-roaing masses (No.1 and No.) are aahed: (a) he iniial posiion and (b) he arbirary posiion of he obje.. Equaions of moion Due o he assumpion of massless rods, in his paper he Cenre of ass Theorem will be applied. In he general ase, in order o desribe roaing moion he Lagrange equaion had o be used, as was done in [4,5] (hese papers also used advaned mahemaial analysis). Le f f ˆ e sand for he reaion fore, whih he ground imposes o he obje, wih e ˆ z z denoing he uni veor of he z-axis. Aording o he Newon s Seond Law, i holds m m r f g. () where m is he oal mass, g ge ˆ z is he aeleraion of graviy veor, and r is he aeleraion of he ener of mass (C) of he sysem of whih he ordinae is given as Теория Механизмов и Машин Том 8. 35

3 Преподавание ТММ z mz z m m. (3) In Eq(3), z, zm and z are he ordinaes of he enroid of mass, he masses m and he overall enroid, respeively. The relaionship beween z and z m (mass No.1) is: z z rsin. (4) m Eliminaing he ordinae z m beween Eq(3) and Eq(4), one reeives: m z z rsin. (5) m In he general ase of a variable radius verial aeleraion is given by r r, and aking he seond emporal derivaive, he mz m z mrsin. (6) Sine he erm m z represens he sum of he exernal fores, whih onsis of only he graviaional ones, m g, Eq(6) beomes sin m z m r m g. (7) Considering ha he insananeous angular veloiy is defined as d. (9) d he equaion of moion of he free obje B, an ordinary differenial equaion (ODE), is wrien in one of he following equivalen forms: m z r g f g. sin m f (1a) Provided he involved derivaives are oninuous (smooh variaion), he firs ime inegraion of Eq(1a) leads o: m z v g r r r r m sin os sin os while he seond inegraion leads o:. (1b) 36 hp://mm.spbsu.ru

4 Some issues on ineria propulsion mehanisms z 1 m z v g r sin r sin r sin r os m. (1) I is remarkable ha he aliude 1 z inludes he graviaional erm z v g responsible for he free fall, a small ompound erm sin sin m m r r, as well as anoher linear erm r sin r os. When he obje sars from he res ondiion ( z, v ), and he laer oeffiien of is posiive, he aliude iniially inreases unil he square erm 1g dominaes..3. Pariular ase of onsan radius.3.1. General In he ase of a onsan radius, r r, he equaions of moion obain one of he following forms: z z z sin g, os g, sin os g. (11a) (11b) (11) where and and erms are relaed o he enrifugal and he angenial omponens, respeively, Then, he general soluion beomes: mr m. (1) 1 os sinsin z z v g os os. z v g, (13).3.. Analyial soluion fulfilling he moion equaion, o fulfill he equaion of moion In he pariular ase ha we demand he angular veloiy, (Eq(11a)), he general soluion of his ordinary differenial equaion is given in he form 1 g sin, 1 (14) Теория Механизмов и Машин Том 8. 37

5 Преподавание ТММ Assuming iniial ondiions and ( ), respeively, hus Eq(14) beomes, i implies ha sin and 1 os g sin 1 os sin, (15) oreover, in erms of he ime, he angular veloiy is found as g os, 1 os sin g (16).3.3. Exponenial angular veloiy Alernaively, we hoose ha e 1, (17) whene e, (18) where is a onsan, while and are he iniial angular veloiy and he iniial angular (polar) posiion of he rods, respeively. Obviously, he relaionship beween ime and polar angle is. (19) 1 ln 1 Also, he emporal derivaive of he angular veloiy beomes:. () e 3. NET IPULSE 3.1. Definiion of Impulse Using he foring funion f of Eq(1a), i is realled ha he quaniy (1) I f d 38 hp://mm.spbsu.ru

6 Some issues on ineria propulsion mehanisms,. Based on his definiion, apar from he graviaional erm is alled impulse and represens he hange in momenum of he obje during he ime inerval v g, he addiionally produed (propulsive) veloiy will be given as z sin sin propulsive I d r d r m m m m. () 3.. Theorems and Lemmas Based on he above definiions and findings, i is now possible o inrodue some elemenary heorems and lemmas o assis he researher saving his/her effors from useless furher sudies. In he lak of available spae, he ex has been ompressed as muh as possible. Lemma-1: In ase of onsan angular veloiy and radius, he impulse of he enrifugal fores is zero. Lemma-: Under he ondiions of Lemma-1, despie he fa ha he impulse is zero, i is possible o obain a signifianly high aliude. In fa, he laer is due o he iniial ondiions and leads o a erm of he firs degree in ime, i.e. r sin r os in Eq(1). Unforunaely, afer a erain ime insane he seond degree graviaional erm 1g dominaes and he obje reurns o he ground. Theorem-1: When he roaing mass follows a permanen orbi and obains he same veloiy a he same posiion, afer a period of ime T, he ne impulse beomes zero ( I ). ne In fa, for an arbirary ime insane, he orresponding impulse is I m m r r m m sin d sin. (3) while for a omplee irumferene ( ), i holds ha d I ne T m m m m T r sin d rsin. (4) Aording o he above assumpions i holds: r T r and sin T sin sin, r sin r sin, and herefore I. ne Lemma-3: The variaion of he radius r does no lead o a ne impulse. Theorem-: In ase of a onsan radius r, he hange of veloiy of he obje depends on he hange of he angular veloiy. In fa, he laer beomes: os os z v g d v g free fall erm free fall erm. (5) Therefore, he variaion of he obje s veloiy beween a erain posiion ref along he irumferene drawn by he rigid rods and he same posiion afer a ime inerval will be: Теория Механизмов и Машин Том 8. 39

7 Преподавание ТММ g os z, (6a) ref ref while for an enire period of ime will be given as z k gt os, k 1,,, ref ref ref k wih k kt ref ref (6b) Lemma-4: In ase of a onsan radius r, he ne impulse for a omplee round of he rigid rods (measured from he iniial posiion ) is I os. ne Lemma-5: In ase of a onsan radius r and a repeaed angular veloiy (i.e. ), i is onluded ha a he end of every subsequen round, he obje will obain he veloiy i would have when i would be freely lef o fall. Lemma-6: In ase of a onsan radius r, he obje veloiy may inrease when he angular veloiy hanges. 4. NUERICAL RESULTS We onsider an obje of mass = 5kg, on whih wo roaing masses (m = 1kg eah) are aahed a a disane r=.1m. In all ases, he iniial angular veloiy is aken equal o s 3 rpm. Typial graphs of he obje s veloiy z and he orresponding are illusraed in Figure and Figure 3, respeively. Finally, he verial displaemen of he obje for he firs four rounds of he rigid rods is shown in Figure 4. Veloiy (m/s) LADA = Exponenial Veloiy (m/s) LADA = Exponenial Veloiy (m/s) Exponenial LADA = Figure : Obje veloiy, z, for hree differen exponens λ= -1, -1, and +1 [Eq(17)-Eq()] Angular Veloiy (1/s) LADA = Angular Veloiy (1/s) LADA = Angular Veloiy (1/s) LADA = Figure 3: Angular veloiy versus ime,, for he hree differen exponens λ of Figure. 4 hp://mm.spbsu.ru

8 Some issues on ineria propulsion mehanisms Aliude (m) Lamda = Lamda = -1 Lamda = Aliude (m) Lamda 9 = Lamda = -1 Lamda = Angle (deg) Figure 4: Verial displaemen z versus ime for he firs four rounds of he rigid rods (lef) and he firs one (righ) 5. CONCLUSIONS I was shown ha using wo onra-roaing masses (driven by eleri moors) whih have reahed a suffiienly high angular veloiy, a emporal lif of he obje o whih hey are aahed may be ahieved. The laer appears even if he ne impulse equals o zero. Every andidae orbi of he roaing masses, whih is haraerized by a repeaed ime hisory of he angular veloiy (he same value a he same posiion), leads o a zero ne impulse; herefore, he obje an no remain ino he air for a long ime. Furhermore, despie he fa ha a losed-form variable angular veloiy fulfilling he equaion of moion was found, unforunaely i beame singular in he neighborhood of he verial posiions of he rigid arms; herefore, using his formula he obje an no remain sill ino he air. In onras, a variable angular veloiy of exponenial ype an ahieve a ne impulse for only he half of he swep irle, hus i an no solve he problem; however i is apable of onrolling he veloiy of he obje so as i beomes adequaely smooh. Finally, i was shown ha a hypoheial ne impulse leads o a sandard quadrai erm in ime, whih furher leads o a permanen upward anigraviy moion. However, he mehanism for produing suh a ne impulse remains sill an open problem. СITED LITERATURE 1. illis.g., Thomas N.E. Responding o ehanial Anigraviy, NASA/T , AIAA , Deember 6. Available a: hp://glrs.gr.nasa.gov/repors/6/t pdf.. Dean N.L. Sysem for onvering roary moion ino unidireional moion, US Paen,886,976 (Filed Jul. 13, 1954, graned ay 19, 1959). 3. Kapisa P.L. Dynami sabiliy of he pendulum when he poin of suspension is osillaing. J. Exp. Theor. Phys. 1, (1951) (in Russian). 4. Blekhman I.I. Synhronizaion of Dynamial Sysems. osow: Nauka, 1971 (in Russian). 5. Kononenko V.O. Osillaory Sysems wih Limied Exiaion. osow: Nauka, Belesky V.V. Oherki o dvizhenii kosmiheskikh el (Sudies on he oion of Spae Bodies). osow: Nauka, 1977 (in Russian). 7. Belesky V.V. Essays on he moion of elesial bodies, Birkhäuser, Basel (1) (original Russian ediion published by Nauka, osow). 8. Zhao C., Zhu H., Zhang Y., Wen B., Synhronizaion of wo oupled exiers in a vibraing sysem of spaial moion, Aa ehania Sinia (DOI 1.17/s ) Поступила в редакцию После доработки Теория Механизмов и Машин Том 8. 41