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111-450: The similar transit instrument , transit circle , or transit telescope is likewise mounted on a horizontal axis, but the axis need not be fixed in the east–west direction. For instance, a surveyor's theodolite can function as a transit instrument if its telescope is capable of a full revolution about the horizontal axis. Meridian circles are often called by these names, although they are less specific. For many years, transit timings were

222-422: A closet rod sagging under the weight of clothes on clothes hangers is an example of a beam experiencing bending. On the other hand, a shell is a structure of any geometric form where the length and the width are of the same order of magnitude but the thickness of the structure (known as the 'wall') is considerably smaller. A large diameter, but thin-walled, short tube supported at its ends and loaded laterally

333-513: A couple or moment as they are equal in magnitude and opposite in direction. This bending moment resists the sagging deformation characteristic of a beam experiencing bending. The stress distribution in a beam can be predicted quite accurately when some simplifying assumptions are used. In the Euler–Bernoulli theory of slender beams, a major assumption is that 'plane sections remain plane'. In other words, any deformation due to shear across

444-401: A device which allowed matching a vertical crosshair's motion to the star's motion. Set precisely on the moving star, the crosshair would trigger the electrical timing of the meridian crossing, removing the observer's personal equation from the measurement. The field of the wires could be illuminated; the lamps were placed at some distance from the piers in order not to heat the instrument, and

555-418: A fine screw . By this slow motion, the telescope was adjusted until the star moved along the horizontal wire (or if there were two, in the middle between them), from the east side of the field of view to the west. Following this, the circles were read by the microscopes for a measurement of the apparent altitude of the star. The difference between this measurement and the nadir point was the nadir distance of

666-409: A hook or yoke with friction rollers , suspended from a lever supported by the pier, counterbalanced so as to leave only a small fraction of the weight on the precision V-shaped bearings. In some cases, the counterweight pushed up on the roller bearings from below. The bearings were set nearly in a true east–west line, but fine adjustment was possible by horizontal and vertical screws. A spirit level

777-472: A large meridian quadrant. Meridian circles have been used since the 18th century to accurately measure positions of stars in order to catalog them. This is done by measuring the instant when the star passes through the local meridian. Its altitude above the horizon is noted as well. Knowing one's geographic latitude and longitude these measurements can be used to derive the star's right ascension and declination . Once good star catalogs were available

888-427: A meridian circle, fitted with leveling screws. Extremely sensitive levels are attached to the telescope mount to make angle measurements and the telescope has an eyepiece fitted with a micrometer . The idea of having an instrument ( quadrant ) fixed in the plane of the meridian occurred even to the ancient astronomers and is mentioned by Ptolemy , but it was not carried into practice until Tycho Brahe constructed

999-552: A new instrument for the British Ordnance Survey . The Ramsden theodolite was used over the next few years to map the whole of southern Britain by triangulation. In network measurement, the use of forced centering speeds up operations while maintaining the highest precision. The theodolite or the target can be rapidly removed from, or socketed into, the forced centering plate with sub-millimeter precision. Nowadays GPS antennas used for geodetic positioning use

1110-439: A perfectly horizontal mirror, reflecting an image of the crosshairs back up the telescope tube. The crosshairs could then be adjusted until coincident with their reflection, and the line of sight was then perpendicular to the axis. The line of sight of the telescope needed to be exactly within the plane of the meridian. This was done approximately by building the piers and the bearings of the axis on an east–west line. The telescope

1221-455: A shadow to set the Sun's position. It was mounted vertically and aligned with the meridian. The instrument was used to measure the altitude of the Sun at noon in order to determine the path of the ecliptic . A meridian circle enabled the observer to simultaneously determine right ascension and declination , but it does not appear to have been much used for right ascension during the 17th century,

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1332-417: A similar mounting system. The height of the reference point of the theodolite—or the target—above the ground benchmark must be measured precisely. The term transit theodolite , or transit for short, refers to a type of theodolite where the telescope is short enough to rotate in a full circle on its horizontal axis as well as around its vertical axis. It features a vertical circle which is graduated through

1443-510: A single instrument that could measure both angles simultaneously. The first occurrence of the word "theodolite" is found in the surveying textbook A geometric practice named Pantometria (1571) by Leonard Digges . The origin of the word is unknown. The first part of the Neo-Latin theo-delitus might stem from the Greek θεᾶσθαι , "to behold or look attentively upon" The second part

1554-467: A single unit. Triangulation , as invented by Gemma Frisius around 1533, consists of making such direction plots of the surrounding landscape from two separate standpoints. The two graphing papers are superimposed, providing a scale model of the landscape, or rather the targets in it. The true scale can be obtained by measuring one distance both in the real terrain and in the graphical representation. Modern triangulation as, e.g., practiced by Snellius ,

1665-422: A small collimating telescope, as the main telescope was rotated, the shape of the pivots, and any wobble of the axis, could be determined. Near each end of the axis, attached to the axis and turning with it, was a circle or wheel for measuring the angle of the telescope to the zenith or horizon. Generally of 1 to 3  feet or more in diameter, it was divided to 2 or 5 arcminutes , on a slip of silver set into

1776-439: A star of known declination passing from one wire to the other, the pole star being best on account of its slow motion. \ Timings were originally made by an "eye and ear" method, estimating the interval between two beats of a clock. Later, timings were registered by pressing a key, the electrical signal making a mark on a strip recorder . Later still, the eye end of the telescope was usually fitted with an impersonal micrometer ,

1887-471: A theodolite, but it does not measure vertical angles, and is used only for leveling on a horizontal plane (though often combined with medium accuracy horizontal range and direction measurements). Temporary adjustments are a set of operations necessary in order to make a theodolite ready for taking observations at a station. These include its setting up, centering, leveling up and elimination of parallax, and are achieved in four steps: Sightings are taken by

1998-467: A transit telescope could be used anywhere in the world to accurately measure local longitude and time by observing local meridian transit times of catalogue stars. Prior to the invention of the atomic clock this was the most reliable source of accurate time. In the Almagest , Ptolemy describes a meridian circle which consisted of a fixed graduated outer ring and a movable inner ring with tabs that used

2109-443: A transverse load is applied on it. In the quasi-static case, the amount of bending deflection and the stresses that develop are assumed not to change over time. In a horizontal beam supported at the ends and loaded downwards in the middle, the material at the over-side of the beam is compressed while the material at the underside is stretched. There are two forms of internal stresses caused by lateral loads: These last two forces form

2220-531: A very accurate dividing engine of his own design. Ramsden's instruments were used for the Principal Triangulation of Great Britain . At this time the highest precision instruments were made in England by such makers as Edward Troughton . Later the first practical German theodolites were made by Breithaupt together with Utzschneider , Reichenbach and Fraunhofer . As technology progressed

2331-407: Is where J = m I A {\displaystyle J={\tfrac {mI}{A}}} is the polar moment of inertia of the cross-section, m = ρ A {\displaystyle m=\rho A} is the mass per unit length of the beam, ρ {\displaystyle \rho } is the density of the beam, A {\displaystyle A}

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2442-451: Is where E {\displaystyle E} is the Young's modulus, I {\displaystyle I} is the area moment of inertia of the cross-section, w ( x , t ) {\displaystyle w(x,t)} is the deflection of the neutral axis of the beam, and m {\displaystyle m} is mass per unit length of the beam. For

2553-395: Is where I {\displaystyle I} is the area moment of inertia of the cross-section, A {\displaystyle A} is the cross-sectional area, G {\displaystyle G} is the shear modulus , k {\displaystyle k} is a shear correction factor , and q ( x ) {\displaystyle q(x)}

2664-417: Is a long history of theodolite use in measuring winds aloft, by using specially-manufactured theodolites to track the horizontal and vertical angles of special weather balloons called ceiling balloons or pilot balloons ( pibal ). Early attempts at this were made in the opening years of the nineteenth century, but the instruments and procedures weren't fully developed until a hundred years later. This method

2775-495: Is a precision optical instrument for measuring angles between designated visible points in the horizontal and vertical planes. The traditional use has been for land surveying , but it is also used extensively for building and infrastructure construction , and some specialized applications such as meteorology and rocket launching. It consists of a moveable telescope mounted so it can rotate around horizontal and vertical axes and provide angular readouts. These indicate

2886-430: Is an applied transverse load. For materials with Poisson's ratios ( ν {\displaystyle \nu } ) close to 0.3, the shear correction factor for a rectangular cross-section is approximately The rotation ( φ ( x ) {\displaystyle \varphi (x)} ) of the normal is described by the equation The bending moment ( M {\displaystyle M} ) and

2997-430: Is an example of a shell experiencing bending. In the absence of a qualifier, the term bending is ambiguous because bending can occur locally in all objects. Therefore, to make the usage of the term more precise, engineers refer to a specific object such as; the bending of rods , the bending of beams , the bending of plates , the bending of shells and so on. A beam deforms and stresses develop inside it when

3108-458: Is called a resection solution or free station position surveying and is widely used in mapping surveying. Such instruments are "intelligent" theodolites called self-registering tacheometers or colloquially " total stations ", and perform all the necessary angular and distance calculations, and the results or raw data can be downloaded to external processors, such as ruggedized laptops , PDAs or programmable calculators . A gyrotheodolite

3219-436: Is distributed along the length of the beam) The dynamic bending of beams, also known as flexural vibrations of beams, was first investigated by Daniel Bernoulli in the late 18th century. Bernoulli's equation of motion of a vibrating beam tended to overestimate the natural frequencies of beams and was improved marginally by Rayleigh in 1877 by the addition of a mid-plane rotation. In 1921 Stephen Timoshenko improved

3330-422: Is known as a total station where angles and distances are measured electronically, and are read directly to computer memory. In a transit theodolite , the telescope is short enough to rotate about the trunnion axis , turning the telescope through the vertical plane through the zenith ; for non-transit instruments vertical rotation is restricted to a limited arc. The optical level is sometimes mistaken for

3441-527: Is often attributed to an unscholarly variation of the Greek word: δῆλος , meaning "evident" or "clear". Other Neo-Latin or Greek derivations have been suggested as well as an English origin from "the alidade ". The early forerunners of the theodolite were sometimes azimuth instruments for measuring horizontal angles, while others had an altazimuth mount for measuring horizontal and vertical angles. Gregorius Reisch illustrated an altazimuth instrument in

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3552-466: Is only valid if the cross-section is symmetrical. For homogeneous beams with asymmetrical sections, the maximum bending stress in the beam is given by where y , z {\displaystyle y,z} are the coordinates of a point on the cross section at which the stress is to be determined as shown to the right, M y {\displaystyle M_{y}} and M z {\displaystyle M_{z}} are

3663-426: Is possible to calculate the bending stress at any point on the beam cross section regardless of moment orientation or cross-sectional shape. Note that M y , M z , I y , I z , I y z {\displaystyle M_{y},M_{z},I_{y},I_{z},I_{yz}} do not change from one point to another on the cross section. For large deformations of

3774-452: Is taken into account in the choice of measurement procedure in order to eliminate their effect on the measurement results of the theodolite. Prior to the theodolite, instruments such as the groma , geometric square and the dioptra , and various other graduated circles (see circumferentor ) and semicircles (see graphometer ) were used to obtain either vertical or horizontal angle measurements. Over time their functions were combined into

3885-406: Is the area moment of inertia of the cross-section, and M {\displaystyle M} is the internal bending moment in the beam. If, in addition, the beam is homogeneous along its length as well, and not tapered (i.e. constant cross section), and deflects under an applied transverse load q ( x ) {\displaystyle q(x)} , it can be shown that: This

3996-536: Is the Euler–Bernoulli equation for beam bending. After a solution for the displacement of the beam has been obtained, the bending moment ( M {\displaystyle M} ) and shear force ( Q {\displaystyle Q} ) in the beam can be calculated using the relations Simple beam bending is often analyzed with the Euler–Bernoulli beam equation. The conditions for using simple bending theory are: Compressive and tensile forces develop in

4107-466: Is the cross-sectional area, G {\displaystyle G} is the shear modulus, and k {\displaystyle k} is a shear correction factor . For materials with Poisson's ratios ( ν {\displaystyle \nu } ) close to 0.3, the shear correction factor are approximately For free, harmonic vibrations the Timoshenko–Rayleigh equations take

4218-486: Is the neutral axis. Because of this area with no stress and the adjacent areas with low stress, using uniform cross section beams in bending is not a particularly efficient means of supporting a load as it does not use the full capacity of the beam until it is on the brink of collapse. Wide-flange beams ( Ɪ-beams ) and truss girders effectively address this inefficiency as they minimize the amount of material in this under-stressed region. The classic formula for determining

4329-400: Is the same procedure executed by numerical means. Photogrammetric block adjustment of stereo pairs of aerial photographs is a modern, three-dimensional variant. In the late 1780s, Jesse Ramsden , a Yorkshireman from Halifax , England who had developed the dividing engine for dividing angular scales accurately to within a second of arc (≈ 0.0048 mrad or 4.8 μrad), was commissioned to build

4440-405: Is used when the north-south reference bearing of the meridian is required in the absence of astronomical star sights. This occurs mainly in the underground mining industry and in tunnel engineering. For example, where a conduit must pass under a river, a vertical shaft on each side of the river might be connected by a horizontal tunnel. A gyrotheodolite can be operated at the surface and then again at

4551-541: The Everest pattern theodolite with its lower center of gravity. Railway engineers working in the 1830s in Britain commonly referred to a theodolite as a "Transit". The 1840s was the start of a period of rapid railway building in many parts of the world which resulted in a high demand for theodolites wherever railways were being constructed. It was also popular with American railroad engineers pushing west, and it replaced

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4662-551: The Royal Greenwich Observatory (1851) and that at the Royal Observatory, Cape of Good Hope (1855) were made by Ransomes and May of Ipswich. The Greenwich instrument had optical and instrumental work by Troughton and Simms to the design of George Biddell Airy . A modern-day example of this type of telescope is the 8 inch (~0.2m) Flagstaff Astrometric Scanning Transit Telescope (FASTT) at

4773-735: The USNO Flagstaff Station Observatory . Modern meridian circles are usually automated. The observer is replaced with a CCD camera. As the sky drifts across the field of view, the image built up in the CCD is clocked across (and out of) the chip at the same rate. This allows some improvements: The first automated instrument was the Carlsberg Automatic Meridian Circle , which came online in 1984. Attribution: Theodolite A theodolite ( / θ i ˈ ɒ d ə ˌ l aɪ t / )

4884-442: The mural quadrant continued until the end of the century to be employed for determining declinations. The advantages of using a whole circle, it being less liable to change its figure and not requiring reversal in order to observe stars north of the zenith, were then again recognized by Jesse Ramsden , who also improved the method of reading off angles by means of a micrometer microscope as described below. The making of circles

4995-434: The weight of the telescope. Later, it was usually placed in the centre of the axis, which consisted of one piece of brass or gun metal with turned cylindrical steel pivots at each end. Several instruments were made entirely of steel , which was much more rigid than brass. The pivots rested on V-shaped bearings , either set into massive stone or brick piers which supported the instrument, or attached to metal frameworks on

5106-461: The Earth in order to find true north and thus, in conjunction with the direction of gravity, the plane of the meridian. The meridian is the plane that contains both the axis of the Earth's rotation and the observer. The intersection of the meridian plane with the horizontal defines the true north-south direction found in this way. Unlike magnetic compasses , gyrocompasses are able to find true north,

5217-655: The Wild T2 with 3.75 inch circles was not able to provide the accuracy for primary triangulation it was the equal in accuracy to a 12 inch traditional design. The Wild T2, T3, and A1 instruments were made for many years. In 1926 a conference was held at Tavistock in Devon , UK where Wild theodolites were compared with British ones. The Wild product outclassed the British theodolites so manufacturers such as Cooke, Troughton & Simms and Hilger & Watts set about improving

5328-687: The accuracy of their products to match their competition. Cooke, Troughton and Simms developed the Tavistock pattern theodolite and later the Vickers V. 22. Wild went on to develop the DK1, DKM1, DM2, DKM2, and DKM3 for Kern Aarau company. With continuing refinements, instruments steadily evolved into the modern theodolite used by surveyors today. By 1977 Wild, Kern and Hewlett-Packard were all offering "Total stations" which combined angular measurements, electronic distance measurement and microchip functions in

5439-480: The altitude and azimuth scales reading zero degrees. A balloon is released in front of the theodolite, and its position is precisely tracked, usually once a minute. The balloons are carefully constructed and filled, so their rate of ascent can be known fairly accurately in advance. Mathematical calculations on time, rate of ascent, azimuth and angular altitude can produce good estimates of wind speed and direction at various altitudes. In modern electronic theodolites,

5550-486: The angle between the earth's rotation and the direction of gravity is too small for it to work reliably. When available, astronomical star sights are able to give the meridian bearing to better than one hundred times the accuracy of the gyrotheodolite. Where this extra precision is not required, the gyrotheodolite is able to produce a result quickly without the need for night observations. Bending In applied mechanics , bending (also known as flexure ) characterizes

5661-509: The appendix of his 1512 book Margarita Philosophica . Martin Waldseemüller , a topographer and cartographer made the device in that year calling it the polimetrum . In Digges's book of 1571, the term "theodolite" was applied to an instrument for measuring horizontal angles only, but he also described an instrument that measured both altitude and azimuth which he called a topographicall instrument [ sic ]. Possibly

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5772-426: The axis was also as firm as possible, as flexure of the tube would affect declinations deduced from observations. The flexure in the horizontal position of the tube was determined by two collimators —telescopes placed horizontally in the meridian, north and south of the transit circle, with their objective lenses towards it. These were pointed at one another (through holes in the tube of the telescope, or by removing

5883-406: The behavior of a slender structural element subjected to an external load applied perpendicularly to a longitudinal axis of the element. The structural element is assumed to be such that at least one of its dimensions is a small fraction, typically 1/10 or less, of the other two. When the length is considerably longer than the width and the thickness, the element is called a beam . For example,

5994-407: The bending moments about the y and z centroid axes, I y {\displaystyle I_{y}} and I z {\displaystyle I_{z}} are the second moments of area (distinct from moments of inertia) about the y and z axes, and I y z {\displaystyle I_{yz}} is the product of moments of area . Using this equation it

6105-414: The bending stress in a beam under simple bending is: where The equation σ = M y I x {\displaystyle \sigma ={\tfrac {My}{I_{x}}}} is valid only when the stress at the extreme fiber (i.e., the portion of the beam farthest from the neutral axis) is below the yield stress of the material from which it is constructed. At higher loadings

6216-583: The body, the stress in the cross-section is calculated using an extended version of this formula. First the following assumptions must be made: Large bending considerations should be implemented when the bending radius ρ {\displaystyle \rho } is smaller than ten section heights h: With those assumptions the stress in large bending is calculated as: where When bending radius ρ {\displaystyle \rho } approaches infinity and y ≪ ρ {\displaystyle y\ll \rho } ,

6327-433: The circle reading after observing a star and the reading corresponding to the zenith was the zenith distance of the star, and this plus the colatitude was the north polar distance. To determine the zenith point of the circle, the telescope was directed vertically downwards at a basin of mercury , the surface of which formed an absolutely horizontal mirror. The observer saw the horizontal wire and its reflected image, and moving

6438-482: The circle. The error was determined occasionally by measuring standard intervals of 2' or 5' on the circle. The periodic errors of the screw were accounted for. On some instruments, one of the circles was graduated and read more coarsely than the other, and was used only in finding the target stars. The telescope consisted of two tubes screwed to the central cube of the axis. The tubes were usually conical and as stiff as possible to help prevent flexure . The connection to

6549-460: The clocks, recorders, and other equipment for making observations. At the focal plane , the eye end of the telescope had a number of vertical and one or two horizontal wires ( crosshairs ). In observing stars, the telescope was first directed downward at a basin of mercury forming a perfectly horizontal mirror and reflecting an image of the crosshairs back up the telescope tube. The crosshairs were adjusted until coincident with their reflection, and

6660-406: The deviation of the star's path from a great circle, and for the inclination of the horizontal wire to the horizon. The amount of this inclination was found by taking repeated observations of the zenith distance of a star during the one transit, the pole star being the most suitable because of its slow motion. Attempts were made to record the transits of a star photographically. A photographic plate

6771-441: The direction of the beam axis under bending loads. These forces induce stresses on the beam. The maximum compressive stress is found at the uppermost edge of the beam while the maximum tensile stress is located at the lower edge of the beam. Since the stresses between these two opposing maxima vary linearly , there therefore exists a point on the linear path between them where there is no bending stress. The locus of these points

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6882-445: The distance of the circle graduations from the centre of the field of view could be measured. The drum of the screw was divided to measure single seconds of arc (0.1" being estimated), while the number of revolutions were counted by a comb like scale in the field of view. The microscopes were given such magnification and placed at such a distance from the circle that one revolution of the micrometer screw corresponded to 1 arcminute (1') on

6993-459: The dynamic Euler–Bernoulli beam theory by including the effect of rotational inertia of the cross-section of the beam. Timoshenko improved upon that theory in 1922 by adding the effect of shear into the beam equation. Shear deformations of the normal to the mid-surface of the beam are allowed in the Timoshenko–Rayleigh theory. The equation for the bending of a linear elastic, isotropic, homogeneous beam of constant cross-section under these assumptions

7104-588: The early 1920s a step change in theodolite design occurred with the introduction of the Wild T2 made by the Swiss Wild Heerbrugg company. Heinrich Wild designed a theodolite with divided glass circles with readings from both sides presented at a single eyepiece close to the telescope so the observer did not have to move to read them. The Wild instruments were not only smaller, easier to use and more accurate than contemporary rivals but also sealed from rain and dust. Canadian surveyors reported that while

7215-421: The essential features of the modern theodolite was built in 1725 by Jonathan Sisson . This instrument had an altazimuth mount with a sighting telescope. The base plate had spirit levels, compass and adjusting screws. The circles were read with a vernier scale . The theodolite became a modern, accurate instrument in 1787, with the introduction of Jesse Ramsden 's famous great theodolite , which he created using

7326-453: The eye. Gradually these scales were enclosed for physical protection, and finally became an indirect optical readout, with convoluted light paths to bring them to a convenient place on the instrument for viewing. The modern digital theodolites have electronic displays. Index error, horizontal-axis error ( trunnion-axis error ) and collimation error are regularly determined by calibration and are removed by mechanical adjustment. Their existence

7437-457: The face of the circle near the circumference. These graduations were read by microscopes , generally four for each circle, mounted to the piers or a framework surrounding the axis, at 90° intervals around the circles. By averaging the four readings the eccentricity (from inaccurate centering of the circles) and the errors of graduation were greatly reduced. Each microscope was furnished with a micrometer screw, which moved crosshairs , with which

7548-409: The first instrument approximating to a true theodolite was the built by Josua Habemel in 1576, complete with compass and tripod. The 1728 Cyclopaedia compares " graphometer " to "half-theodolite". As late as the 19th century, the instrument for measuring horizontal angles only was called a simple theodolite and the altazimuth instrument, the plain theodolite . The first instrument to combine

7659-461: The following are true: In this case, the equation describing beam deflection ( w {\displaystyle w} ) can be approximated as: where the second derivative of its deflected shape with respect to x {\displaystyle x} is interpreted as its curvature, E {\displaystyle E} is the Young's modulus , I {\displaystyle I}

7770-402: The foot of the shafts to identify the directions needed to tunnel between the base of the two shafts. Unlike an artificial horizon or inertial navigation system, a gyrotheodolite cannot be relocated while it is operating. It must be restarted again at each site. The gyrotheodolite comprises a normal theodolite with an attachment that contains a gyrocompass , a device which senses the rotation of

7881-417: The form This equation can be solved by noting that all the derivatives of w {\displaystyle w} must have the same form to cancel out and hence as solution of the form e k x {\displaystyle e^{kx}} may be expected. This observation leads to the characteristic equation The solutions of this quartic equation are where The general solution of

7992-410: The full 360 degrees and a telescope that could "flip over" ("transit the scope"). By reversing the telescope and at the same time rotating the instrument through 180 degrees about the vertical axis, the instrument can be used in 'plate-left' or 'plate-right' modes ('plate' refers to the vertical protractor circle). By measuring the same horizontal and vertical angles in these two modes and then averaging

8103-461: The glass cases, while their eyepiece ends and micrometers were protected from dust by removable silk covers. Certain instrumental errors could be averaged out by reversing the telescope on its mounting. A carriage was provided, which ran on rails between the piers, and on which the axis, circles and telescope could be raised by a screw-jack, wheeled out from between the piers, turned 180°, wheeled back, and lowered again. The observing building housing

8214-402: The high-precision work for which these instruments are employed: The state of the art of meridian instruments of the late 19th and early 20th century is described here, giving some idea of the precise methods of construction, operation and adjustment employed. The earliest transit telescope was not placed in the middle of the axis, but nearer to one end, to prevent the axis from bending under

8325-427: The light passed through holes in the piers and through the hollow axis to the center, whence it was directed to the eye-end by a system of prisms . To determine absolute declinations or polar distances, it was necessary to determine the observatory's colatitude , or distance of the celestial pole from the zenith , by observing the upper and lower culmination of a number of circumpolar stars . The difference between

8436-448: The line of sight was then perfectly vertical; in this position the circles were read for the nadir point . The telescope was next brought up to the approximate declination of the target star by watching the finder circle. The instrument was provided with a clamping apparatus, by which the observer, after having set the approximate declination, could clamp the axis so the telescope could not be moved in declination, except very slowly by

8547-502: The main telescope needed to be exactly horizontal. A sensitive spirit level , designed to rest on the pivots of the axis, performed this function. By adjusting one of the V-shaped bearings, the bubble was centered. The line of sight of the telescope needed to be exactly perpendicular to the axis of rotation. This could be done by sighting a distant, stationary object, lifting and reversing the telescope on its bearings, and again sighting

8658-426: The meridian circle did not have a rotating dome, as is often seen at observatories. Since the telescope observed only in the meridian, a vertical slot in the north and south walls, and across the roof between these, was all that was necessary. The building was unheated and kept as much as possible at the temperature of the outside air, to avoid air currents which would disturb the telescopic view. The building also housed

8769-439: The method of equal altitudes by portable quadrants or measures of the angular distance between stars with an astronomical sextant being preferred. These methods were very inconvenient, and in 1690, Ole Rømer invented the transit instrument. The transit instrument consists of a horizontal axis in the direction east and west resting on firmly fixed supports, and having a telescope fixed at right angles to it, revolving freely in

8880-432: The middle of the 19th century to be the principal instrument in observatories, the first transit circle constructed there being that at Greenwich (mounted in 1850). However, on the continent, the transit circle superseded them from the years 1818–1819, when two circles by Johann Georg Repsold and Georg Friedrich von Reichenbach were mounted at Göttingen , and one by Reichenbach at Königsberg . The firm of Repsold and Sons

8991-428: The most accurate method of measuring the positions of heavenly bodies, and meridian instruments were relied upon to perform this painstaking work. Before spectroscopy , photography , and the perfection of reflecting telescopes , the measuring of positions (and the deriving of orbits and astronomical constants ) was the major work of observatories . Fixing a telescope to move only in the meridian has advantages in

9102-420: The object. If the crosshairs did not intersect the object, the line of sight was halfway between the new position of the crosshairs and the distant object; the crosshairs were adjusted accordingly and the process repeated as necessary. Also, if the rotation axis was known to be perfectly horizontal, the telescope could be directed downward at a basin of mercury , and the crosshairs illuminated. The mercury acted as

9213-417: The orientation of the telescope, and are used to relate the first point sighted through the telescope to subsequent sightings of other points from the same theodolite position. These angles can be measured with accuracies down to microradians or seconds of arc . From these readings a plan can be drawn, or objects can be positioned in accordance with an existing plan. The modern theodolite has evolved into what

9324-522: The original formula is back: In 1921, Timoshenko improved upon the Euler–Bernoulli theory of beams by adding the effect of shear into the beam equation. The kinematic assumptions of the Timoshenko theory are: However, normals to the axis are not required to remain perpendicular to the axis after deformation. The equation for the quasistatic bending of a linear elastic, isotropic, homogeneous beam of constant cross-section beam under these assumptions

9435-444: The plane of the meridian. At the same time Rømer invented the altitude and azimuth instrument for measuring vertical and horizontal angles, and in 1704, he combined a vertical circle with his transit instrument, so as to determine both co-ordinates at the same time. This latter idea was, however, not adopted elsewhere, although the transit instrument soon came into universal use (the first one at Greenwich being mounted in 1721), and

9546-411: The processor. Many modern theodolites are equipped with integrated electro-optical distance measuring devices, generally infrared based, allowing the measurement in one step of complete three-dimensional vectors —albeit in instrument-defined polar coordinates , which can then be transformed to a preexisting coordinate system in the area by means of a sufficient number of control points. This technique

9657-463: The railroad compass , sextant and octant . Theodolites were later adapted to a wider variety of mountings and uses. In the 1870s, an interesting waterborne version of the theodolite (using a pendulum device to counteract wave movement) was invented by Edward Samuel Ritchie . It was used by the U.S. Navy to take the first precision surveys of American harbors on the Atlantic and Gulf coasts. In

9768-405: The readout of the horizontal and vertical circles is usually done with a rotary encoder . These produce signals indicating the altitude and azimuth of the telescope which are fed to a microprocessor. CCD sensors have been added to the focal plane of the telescope allowing both auto-targeting and the automated measurement of residual target offset. All this is implemented in embedded software of

9879-460: The results, centering and collimating errors in the instrument can be eliminated. Some transit instruments are capable of reading angles directly to thirty arc-seconds (≈ 0.15 mrad ). Modern theodolites are usually of the transit-theodolite design, but engraved plates have been replaced with glass plates designed to be read with light-emitting diodes and computer circuitry, greatly improving accuracy up to arc-second (≈ 0.005 mrad ) levels. There

9990-424: The section is not accounted for (no shear deformation). Also, this linear distribution is only applicable if the maximum stress is less than the yield stress of the material. For stresses that exceed yield, refer to article plastic bending . At yield, the maximum stress experienced in the section (at the furthest points from the neutral axis of the beam) is defined as the flexural strength . Consider beams where

10101-445: The shear force ( Q {\displaystyle Q} ) are given by According to Euler–Bernoulli, Timoshenko or other bending theories, the beams on elastic foundations can be explained. In some applications such as rail tracks, foundation of buildings and machines, ships on water, roots of plants etc., the beam subjected to loads is supported on continuous elastic foundations (i.e. the continuous reactions due to external loading

10212-730: The situation where there is no transverse load on the beam, the bending equation takes the form Free, harmonic vibrations of the beam can then be expressed as and the bending equation can be written as The general solution of the above equation is where A 1 , A 2 , A 3 , A 4 {\displaystyle A_{1},A_{2},A_{3},A_{4}} are constants and β := ( m E I   ω 2 ) 1 / 4 {\displaystyle \beta :=\left({\cfrac {m}{EI}}~\omega ^{2}\right)^{1/4}} In 1877, Rayleigh proposed an improvement to

10323-481: The standard theodolite design. Development of the theodolite was spurred on by specific needs. In the 1820s progress on national surveying projects such as the Ordnance Survey in Britain produced a requirement for theodolites capable of providing sufficient accuracy for large scale triangulation and mapping. The Survey of India at this time produced a requirement for more rugged and stable instruments such as

10434-437: The star. A movable horizontal wire or declination-micrometer was also used. Another method of observing the apparent altitude of a star was to take half of the angular distance between the star observed directly and its reflection observed in a basin of mercury. The average of these two readings was the reading when the line of sight was horizontal, the horizontal point of the circle. The small difference in latitude between

10545-411: The stress distribution becomes non-linear, and ductile materials will eventually enter a plastic hinge state where the magnitude of the stress is equal to the yield stress everywhere in the beam, with a discontinuity at the neutral axis where the stress changes from tensile to compressive. This plastic hinge state is typically used as a limit state in the design of steel structures. The equation above

10656-401: The surface direction toward the north pole. A gyrotheodolite will function at the equator and in both the northern and southern hemispheres. The meridian is undefined at the geographic poles. A gyrotheodolite cannot be used at the poles where the Earth's axis is precisely perpendicular to the horizontal axis of the spinner, indeed it is not normally used within about 15 degrees of the pole where

10767-406: The surveyor, who adjusts the telescope's vertical and horizontal angular orientation so the cross-hairs align with the desired sighting point. Both angles are read either from exposed or internal scales and recorded. The next object is then sighted and recorded without moving the position of the instrument and tripod. The earliest angular readouts were from open vernier scales directly visible to

10878-416: The telescope and the basin of mercury was accounted for. The vertical wires were used for observing transits of stars, each wire furnishing a separate result. The time of transit over the middle wire was estimated, during subsequent analysis of the data, for each wire by adding or subtracting the known interval between the middle wire and the wire in question. These known intervals were predetermined by timing

10989-400: The telescope from its mount) so that the crosshairs in their foci coincided. The collimators were often permanently mounted in these positions, with their objectives and eyepieces fixed to separate piers. The meridian telescope was pointed to one collimator and then the other, moving through exactly 180°, and by reading the circle the amount of flexure (the amount the readings differed from 180°)

11100-401: The telescope to make these coincide, its optical axis was made perpendicular to the plane of the horizon, and the circle reading was 180° + zenith point. In observations of stars refraction was taken into account as well as the errors of graduation and flexure. If the bisection of the star on the horizontal wire was not made in the centre of the field, allowance was made for curvature, or

11211-582: The theory further by incorporating the effect of shear on the dynamic response of bending beams. This allowed the theory to be used for problems involving high frequencies of vibration where the dynamic Euler–Bernoulli theory is inadequate. The Euler-Bernoulli and Timoshenko theories for the dynamic bending of beams continue to be used widely by engineers. The Euler–Bernoulli equation for the dynamic bending of slender, isotropic, homogeneous beams of constant cross-section under an applied transverse load q ( x , t ) {\displaystyle q(x,t)}

11322-408: The tops of the piers. The temperature of the instrument and local atmosphere were monitored by thermometers. The piers were usually separate from the foundation of the building, to prevent transmission of vibration from the building to the telescope. To relieve the pivots from the weight of the instrument, which would have distorted their shape and caused rapid wear, each end of the axis was supported by

11433-443: The vertical partial circle was replaced with a full circle, and both vertical and horizontal circles were finely graduated. This was the transit theodolite . This type of theodolite was developed from 18th century astronomical Transit instruments used to measure accurate star positions. The technology was transferred to theodolites in the early 19th century by instrument makers such as Edward Troughton and William Simms and became

11544-487: Was extensively used in World War II and thereafter, and was gradually replaced by radio and GPS measuring systems from the 1980s onward. The pibal theodolite uses a prism to bend the optical path by 90 degrees so the operator's eye position does not change as the elevation is changed through a complete 180 degrees. The theodolite is typically mounted on a rugged steel stand, set up so it is level and pointed north, with

11655-510: Was for a number of years eclipsed by that of Pistor and Martins in Berlin, who furnished various observatories with first-class instruments. Following the death of Martins, the Repsolds again took the lead and made many transit circles. The observatories of Harvard College , Cambridge University and Edinburgh University had large circles by Troughton and Simms . The Airy Transit Circles at

11766-446: Was found. Absolute flexure, that is, a fixed bend in the tube, was detected by arranging that eyepiece and objective lens could be interchanged, and the average of the two observations of the same star was free from this error. Parts of the apparatus, including the circles, pivots and bearings, were sometimes enclosed in glass cases to protect them from dust. These cases had openings for access. The reading microscopes then extended into

11877-516: Was placed in the focus of a transit instrument and a number of short exposures made, their length and the time being registered automatically by a clock. The exposing shutter was a thin strip of steel, fixed to the armature of an electromagnet. The plate thus recorded a series of dots or short lines, and the vertical wires were photographed on the plate by throwing light through the objective lens for one or two seconds. Meridian circles required precise adjustment to do accurate work. The rotation axis of

11988-558: Was shortly afterwards taken up by Edward Troughton , who constructed the first modern transit circle in 1806 for Groombridge 's observatory at Blackheath , the Groombridge Transit Circle (a meridian transit circle). Troughton afterwards abandoned the idea and designed the mural circle to take the place of the mural quadrant. In the United Kingdom, the transit instrument and mural circle continued until

12099-420: Was spent in perfecting it. In practice, none of these adjustments were perfect. The small errors introduced by the imperfections were mathematically corrected during the analysis of the data. Some telescopes designed to measure star transits are zenith telescopes designed to point straight up at or near the zenith for extreme precision measurement of star positions. They use an altazimuth mount , instead of

12210-402: Was then brought into the meridian by repeatedly timing the (apparent, incorrect) upper and lower meridian transits of a circumpolar star and adjusting one of the bearings horizontally until the interval between the transits was equal. Another method used calculated meridian crossing times for particular stars as established by other observatories. This was an important adjustment, and much effort

12321-405: Was used to monitor for any inclination of the axis to the horizon. Eccentricity (an off-center condition) or other irregularities of the pivots of the telescope's axis was accounted for, in some cases, by providing another telescope through the axis itself. By observing the motion of an artificial star, located east or west of the center of the main instrument, and seen through this axis telescope and

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