Jeodezide jiroskop hareketinin uygulanması hakkında öneriler

Abstract

ÖZET Bu çalışmanın I. bölümünde rijit cisim dinamiği kısaca incelenmiştir. Bundan sonra bu bilgiler jiros koplu teodolitin hareketlerini incelemek için kulla nılmıştır,, Jiroskoplu teodolitin hareketi incelenirken şu iki durum incelenmiştir, a® Jiroskop rotorunun dönmemesi hali b. Jiroskop rotorunun dönmesi hali Jiroskop rotorunun dönmemesi halinde rotorun si metri ekseninin hareketinin diferansiyel denklemi bu lunmuş ve bu diferansiyel denklem çözülerek hareketin az sönümlü bir harmonik hareket olduğu gösterilmiştir. Bu az sönümlü harmonik hareketin salınım eğrisi en küçük kareler yöntemine göre dengelenmiş ve hareketin parametreleri elde edilmiştir. Jiroskop rotorunun dönmesi halinde ise hareketin diferansiyel denklemleri elde edilmiş ve bunlar çözül düğünde hareketin meridyen etrafında az sönümlü bir harmonik hareket olduğu görülmüştür. II. bölümde, bu âz sönümlü harmonik hareketin ö- zellikleri kullatiilâtak elde edilen azimut saptama yöntemleri incelenmiştir. Jiroskoplu teödolitle azimut saptanırken birkaç işlem ard arda yapılmaktadır. Bunlar kaba, ön ve kesin yönelmelerdir. Kaba ve ön yönelme işlemleri açıklan dıktan sonra kesin yönelme yöntemleri incelenmiştir. Bundan sonra kesin yönelme yöntemlerinden olan dö nüm noktasi yöntemi incelenmiştir. Dönüm noktalarını değerlendirmede kullanılan Schuler ortalaması sönüm katsayısı Taylor serisine açılarak ve ilk iki terimi â- İmarak elde edilmektedir. Aynı Taylor serisinin ilk üç terimi alınırsa Thomas ortalaması elde edilmektedir.Alınan terini sayısı arttıkça kuzey doğrultusunu veren diğer ortalamalar elde edilmektedir. Bu ortalamaların katsayıları Paskal üçgeninin elemanlarıdır,, Latif eşitlikleri olarak adlandırılan ortalamalar dönüm noktalarına en küçük kareler yöntemini uygulaya rak elde edilmiştir. Bu ortalamalarda dönüm noktası sa yısı çifttir. Bu çalışmada bu değerlendirme bir adım öteye götü rülmüş ve dönüm noktası sayısının tek olması Halinde de kuzey doğrultusunu veren ortalamalar elde edilmiştir. Diğer taraftan ortalamalardan biri dönüm noktala rını değerlendirmek için uygun olsa bile ikinci adımda bir genel aritmetik ortalama almak uygun değildir,, Bu nedenle ortalamalar matematiksel korrelasyon gözönüne alınarak en küçük kareler yöntemine göre dengelenmiş ve ortalamaların ortalamaları olan eşitlikler verilmiştir. Bunun ardından geçiş yöntemi ve teorisi incelen- mıştiro Bundan sonra elektronik kayıt yapan bir alet gru bunun jiroskoplu teodolit ile beraber nasıl kullanıla cağı açıklanmıştır. Âyirica. bu bölümde yapılan teorik a- raştırmalar sonucu jiroskoplu teodolitle yapılan ölçme lere etki eden faktörlerin sıcaklık, akım, histerisis etkisi ve manyetik etkiler olduğu gösterilmiştir. Ayrıca jiroskoplu. t eoüöl itlerin iç ve dış presiz-^- yonları incelenmiş ve aletin iç presizyonunun brahtl katsayısına bağlı olduğu gösterilmiştir. III* Bölümde jiroskoplu teodolitin sal mimi arından coğrafik enlemin ne biçimde saptanacağı açıklanmıştır. Bu amaçla dönüm noktalar! en küçük kareler yöntemine göre dengelenmekte bu dengeleme sonucu coğrafik enlem, yerçekimi ivmesi ve kuzey doğrultusu elde edilmektedir»IVe Bölümde Foucault jiroskopları ve doğu-batı doğrultusunu veren sistemler eleştirici bir gözle ince lenmektedir.

SUM MA RY lö the first part of this work dynamics of rigid bodies which is necessary for the investigation of the gyroscopic phenomenon has been investigated. Regarding a rigid body as' constituting of mass particles firstly the motion of a mass particle has been investigated. Following this the kinematics of a rigid body have been examined » In investigating the motion of a mass particle or of a rigid body it is appropriate to examine the motion with respect to rotating axes. Because of this the theory of rotating axes has been explained. Using the results thus obtained the motion of a mass particle moving near the earth's surface has been explained. In order to obtain the equations of motion of â rigid body the equations of motion of a system of par ticles have been derived. Using these results the equ ations of motion öf â rigid body have been obtained. Then the Euler equations of motion of a rigid body with one point fixed have been obtained. Following the above derivations the motion of Fou- cault's pendulum has been investigated. Foucault's pen dulum is used to demonstrate that the earth is rotating around its axis. Besides this the motion of a gyrosc0pe with 3 deg rees of freedom has been investigated. Applying Euler equations the equations of motion of gyroscope have been obtained. When these differential equations have been integrated it has been seen that the rotation axis of the gyroscope will seek a North-South direction. When the stability of this motion has been investigated it is apparent that a gyroscope with 3 degrees of free dom may be used for the two following purposes %ae to find North direction hB To find South direction Following this the motion of gyro-*theod olite has been investigated 6 In investigating the motion of gyro- theodolite the following two situations have been handled? SLS Gyro-theodolite with non-spinning rotor be Gyro-theodolite with spinning rotor The equation of motion of gyro-theodolite with non-spinning rotor has been derived by using the prin ciple stating that the rate of change of angular momen tum with respect to time ise equal to the moment acting on the system. The differential equation of motion is a second order linear differential equation with constant coefficients a This differential equation has been in tegrated by reducing it to a system of linear homoge nous differential equations of the first order. As a result it has been seen that the motion of gyro-theo dolite with non-rotating rotor is a weakly -jumped har monic oscillation*, The oscillation curve' has been adjusted according to the least squares principle and as a result the pa rameters of the motion have been determined» Secondly the differential equations of motion of gyrd^theodolite with rotating rotor have been derived and when they have been integrated it has been found that the motion is a weakly damped harmonic oscilla tion about the meridian* Gyro``theodoiites are divided into two groups ac cording to their construction. The mechanical construc tion of gyro-theodolites has been explained. In the second part of the work azimuth determinâ-t ion methods have been thorougly examined by using the properties of these weakly damped harmonic oscilla tions. In determining azimuth by using gyro-theodolites several operations are done in succession» These are rough, primary and precise orientations. After explaining rough and primary orientation methods precise orientation methods have been discussed. Firstly the turning point method which is one of the precise orientation methods has been investigated» If the damping coefficient is expressed with a Taylor series and the first two terms are used,Schuler Mean is obtained. On the other hand if the first three terms are used Thomas. Mean is obtained. As the number of used terms increase corresponding Means giving North direction are obtained. The coefficients of these various means are the elements of the elements of the Pascal triangle. Means known as tauf Means have been obtained.by using the least squares principle* In these Means. the number of turning points are evens Following this transit method which is one of the precise orientation methods and its theory have been discussed* Laiif Means have been obtained by using the least squares principle, but the number of turning points is even. Turning points have been evaluated for nod d and even by using the least squares principle and corres ponding means have been derived» On the other hand even if öne of the means is app ropriate for the evaluation of the turning points, to take a grand mean in the second step is not appropri ate. Because of this the elementary means obtained havebeen adjusted by the least squares principle by taking into account mathematical correlation. As a result in stead of taking simple arithmetic mean the means ob tained as a result of this adjustment should be used. Following this, for the registration of time an electronic registrating device is proposed. The most important part of this proposal is the usage of photo- transistors for registrating time. As a result of the theoretical investigation it has been shown that the factors affecting the measure ments made with the gyro-theodolites are temperature» electric current, hysterisis, magnetic effects. The inner arid outer precision of gyro-theodolites have been discussed indetail and it hâs been shown that the inner precision of the instrument in the transit method is dependent on the proportionality factor. The azimuth determination is not the oniy appli cation of gyro-^theodolites in geodesy. The same instru ment may be used to determine geographical latitude arid gravity acceleratiott. In the third part of this work a method has been developed for the determination of the two unknows namely geographical latitude and gravity acceleration. Iri this method weakly damped harmonic os cillations of the rotatirig gyro-theodolite have been used» These oscillations have been adjusted according to the least squares principle and as a result geograp hical latitude and gravity acceleration have been ob tained. In the last part of this work it has been shown that Foueault gyroscopes with two degrees of freedom might be used to determine geographical latitude.On the other hand the dynamical analysis of the sphere within a sphere system used to give East-West direction has been made.