US 3605625 A Abstract available in Claims available in Description (OCR text may contain errors) Sept. Filed 1971 o. s. BENCSICS 3,605,625 CURVE LINERS Sept. 15, 1969 4 Sheets-Sheet l RADIOID CLOTHOID 8 Y m Q a Y X0 Xb INVENTOR ODON S. BENCSICS ATTORNEYS. 20, 1971 o. s. BENCSICS 3,605,625 . CURVE LINERS Filed Sept. 15. 1969 A Sheets-Sheet 2 INVENTQR QDON S. BENCSICS ATORNEYS. P 1971 o. s. BENCSICS 3,605,625 CURVE LINERS iled Sept. 15. 1969 4 Sheets-Sheet 5 INVENTOR ODON SBENCSICS ATTORNEYS. United States Patent 01 Patented Sept. 20, 1971 fee US. Cl. 104-8 17 Claims ABSTRACT OF THE DISCLOSURE A method and apparatus for the alignment of railroad track which provides a smooth transition from tangent to circular curve track through a spiral curve and which enable the apparatus to be used in lining on a single pass through a section of track. The method comprises steps of measuring parameters of a length of track, establishing from the measured parameters a curve form to which the length of track approximates and which provides a constant rate of change of curvature and adjusting for some point in the length of track the position of the point to lie on the established curve form. According to a preferred embodiment disclosed the required curve form is determined by the measurement of parameters determinative of the curvature at two points in the length of track. BACKGROUND OF THE INVENTION This invention relates to the lateral alignment of railroad tracks and particularly to the correction of errors occurring in previously laid track. Railroad track comprises three basic curve forms. These are known as straight or tangent track, transition or spiral track, and circular curved track. Transition track is found between stretches of tangent and circular curved track and its function is to reduce the impact loading on the wheels of railroad vehicles as they proceed from tangent track to circular track or vice versa. n tangent track there is little or no lateral force on the wheels of a railroad vehicle. On circular curved track there is a lateral force which is a function of the radius of the curve, the mass of the vehicle, and the velocity of the vehicle. The use of spiral track results in a gradual increase of the lateral force from zero on the tangent track to the maximum on entering the circular track. With the increasing speeds of railroad vehicles, it has become important that the application of the lateral force should be as smooth as possible. This requires not only that the rails of the track be free from lateral bumps but that the curve form of the track be as close as possible to that required theoretically to provide the smooth transition of forces. Railroad track is initially laid to a given curve form. However, during use this curve form is lost. There are two approaches in the art to the problem of rectifying this loss of initial curve form. One approach is to realign the track to its original curve form, working from records. The second approach is to re-align the track to provide the best curve from the existing state of the track. The present invention is concerned with this secand approach. Apparatus has been developed which can satisfactorily align both tangent track and circular curved track. However, the alignment of spiral track to a curve form which provides a smooth transition of forces still poses certain problems. Prior art apparatus has only been able to approximate spiral curves to the theoretical best curve. For example, apparatus is known in which a spiral curve is aligned as a series of arcs of circles. In another prior art apparatus, a lining machine is first passed over a length of spiral curve to measure variations of the curve. A graph of these variations is prepared and an average curve is prepared from this graph. The lining machine then retraverses the length of track and aligns the track to the average curve. As will be shown hereinafter, a preferred form of curve for spiral track is a curve which provides a constant rate of change of curvature between the tangent track and the circular curved track. This form of curve provides a smooth transition of the lateral forces on the wheels of a railroad vehicle. SUMMARY OF THE INVENTION It is an object of the present invention to provide a method of aligning spiral track to a curve form providing a constant rate of change of curvature, which method is also applicable to tangent and circular curved track. It is a further object of this invention to provide such a method of aligning railroad track which does not require the provision of external information, such as the form of the track as initially laid down or the desired form of the track. It is a further object of the inevntion to provide such a method of aligning track which does not require more than one pass over a length of the track for alignment purposes. It is a still further object of the invention to provide apparatus for carrying out such methods. The present invention lies in the discovery that, from an analysis of a preferred form of spiral track, it is possible to derive at least one control function which can be readily utilized in a method and apparatus for the alignment of tangent and circular curved track as well as spiral track. The control function is valid for the three basic curve forms and will not introduce errors into those curve forms. The control function is not valid for sections of track which comprise either part tangent and part spiral or part spiral and part circular curved track. However, the errors introduced into such sections of track by use of the control function are negligible in operating conditions of railroad track. Essentially, the invention lies in a method of aligning railroad track comprising the steps of selecting a length of track; measuring parameters of that length of track; establishing from the measured parameters a curve form to which the length of track approximates and which provides a constant rate of change of curvature; establishing, for some point in the length of track, the required position of the point to lie on the curve form; and moving the track at the point towards said position. More particularly, the method comprises the steps of taking measurements, determinative of a first angle located between the tangent to a first point and a chord extending between the first point and a second point and a second angle located between the tangent at a third point and a chord extending between the third point and a fourth point, said first, second, third, and fourth points being in said length of track and said first and second and said third and fourth points having the same straight line spacing. The invention also provides apparatus for carrying out the method of the invention. In general terms, said apparatus comprises means for mounting the apparatus for movement along a track; means for measuring parameters of a selected length of the track; control means for determining from the measured parameters the curve form to which the length of the track approximates and which provides a constant rate of change of curvature; means for establishing for some point in said length of track, the required position of said point on said curve form; and track moving means for moving said track at said point. In a more particular form of the apparatus of the invention, the apparatus comprises reference line establishing means and control means for said reference line establishing means, said control means operable to establish a survey reference line representing a chord of a length of track and including means for making measurements of the parameters of the length of track and means for determining from the measured parameters the curve form to which the length of track approximates and which provides a constant rate of change of curvature and operable to establish a lining reference line representing a chord of said curve form and passing through a point on said curve form representing the required position of a corresponding point of the track and means for moving said track at the corresponding point to move said point towards said required position. Hereinafter there will be given an explanation of the derivation of a number of control functions which may be used in carrying out the invention. This derivation will include an analysis of the spiral curve form. Hereinafter there will also be described particular embodiments of the invention which are given by way of example. BRIEF DESCRIPTION OF THE DRAWINGS The derivation and the embodiments of the invention described are given in conjunction with the accompanying drawings, in which: FIG. 1 shows the geometrical relationship of two points on a transition spiral; FIG. 2 is a drawing illustrating the derivation of a general control function; FIG. 3 is a schematic illustration of the arrangement of elements utilizing a first specific control function; FIG. 4 is a schematic illustration of the arrangement of elements utilizing a second specific control function; FIG. 5 is a general outline drawing in plan view of apparatus embodying the invention; FIG. 6 is a side elevation of the apparatus shown in FIG. 5; and FIG. 7 is a block diagram of control means of the apparatus shown in FIGS. 4 to 6. DESCRIPTION OF THE PREFERRED EMBODIMENTS Hereinafter there will be reference to curves and curve forms. Such terms include a straight line, a circular curve, and various other curve forms. Also, there will be use of the term constant rate of change of curvature. For the purposes of this specification, a straight line which has zero curvature and zero rate of change of curvature and a circle which has a constant curvature and zero rate of change of curvature, are considered to have a constant rate of change of curvature. It is well known, from a determination of the forces applied on a curve by a railroad vehicle that for a circular curve, the required super-elevation of the track, that is the height of the outside rail above the inside rail, can be obtained from the formula: EP=N or where O is the radius of the curvature at a point P on the transition curve. 4 For a smooth transition in passing from tangent track to circular curve track along a transition curve it is necessary that the rate of increase of superelevation from zero to E is linear with the distance along the transition curve, i.e. an E L where L is the length of the transition curve from the end of the tangent track to the beginning of the circular curve track and S is the distance from the end of the tangent track to the point P in the transition curve. From Equations 1 and 2 it is readily shown that: From Equation 3 the general definition of the required transition curve therefore becomes: Q LR (4) L and R are constants of the particular spiral so the where a is a constant, and the curvature of the curve at a point is equal to the reciprocal of the radius at that point. Therefore, since where K is the curvature at the point under consideration, Equation 4 can be rewritten as: K=aS (5) Equation K=aS is the equation of a radioid clothoid curve. In words, this equation states that the curvature of a point on the curve is proportional to the distance of the point from the origin of the curve. This curve fulfills the requirement for a smooth transition of forces, ie a constant increase or decrease of forces and is a curve which has a constant rate of change of curvature along its length. It will be seen that a straight line is a special form of radioid clothoid curve in which the constant a is zero and a circle is a special case of the radioid clothoid curve in which the curvature is constant. Turning now to FIG. 1, there is shown a section of a radioid clothoid curve commencing with zero curvature at origin 0 at the intersection of rectangular co-ordinates X and Y and terminating in a circular curve having a radius of curvature R. Arbitrary points A and B are noted on the curve. In the following analysis: S: the arclength from the origin; L: the full length of the section of the radioid clothoid curve from the origin to the circular curve; Q: the radius of curvature of the radioid clothoid; 6a is the angle between the tangent to the curve at point A and the X axis; 7 is the angle between the chord joining A and B and the X axis; a is the angle between the chord joining A and B and the tangent at A; and A0 is the angle between tangents to the curve at A and B. As is established above, a preferred form for a spiral curve is a radioid clothoid curve having the equation K=aS. Such a curve is completely defined once the c0nstant a is known. From K=aS and s) Thus, if it is possible to measure the curvature at two points a known distance apart, the curve will be completely defined. However, it is extremely difiicult to make an accurate determiation of the curvature of a curve at a point. The purpose of the following analysis is to show that if a tangent is drawn to any radioid clothoid curve at an arbitrary point A and this point is connected by a chord to another arbitrary point B on the clothoid, the angle between the chord and the tangent defines the curvature of the spiral at a third related point C. Therefore, if a similar angle is formed with points A and B at dilferent positions on the clothoid, the angles so obtained determine the constant a of the curve and so completely define the curve. If, therefore, the measurement of such angles is made on a length of track, the radioid clothoid curve to which that length of track approximates can be established Whether one or more further points in that length of track lie on the established radioid clothoid curve for that length of track. For the purposes of calculation, it is necessary to establish the relationship of a point on a radioid clothoid curve in terms of rectangular X and Y co-ordinates. Now Therefore, de K 3;: as and integrating this relationship: aS 6= afSds C where C=zero for the equation to be valid at the origin. Accordingly, Diiferentiating results in the relationship: s a) v5 (7) and dX=ds cos dY=ds sin 0 (8) Substituting the relationships of Equation 8 in Equation 7 and integrating gives the relationships: 00s 6 X= 2 I d0 0 F0 and sin 6 -r/2 Y (2a) 0 0 d0 (10) and (s 42 12 There is inaccuracy due to neglecting the powers of the expressions of sin 0 and cos 0 above 5. From FIG. 1 tan 'y= and, therefore: The errors involved in this derivation are due solely to approximations of the series expansions of tan 7, sin 0, and cos 0. In the railroad art the magnitude of 'y and 0 are such that the errors in ignoring higher powers of these angles in the expansions are negligible. Accordingly, it can be stated that: Yb Ya 1 Sb0b-Sa0a =--m Xb-Xa) 3 Sb-Sa 17 From Equations 6 and 17 it can be shown that =2 2 as use 2' (6 6 3) (18) From FIG. 1: 7=0a+a (19) and from Equation 6 2 2 8a 0a Thus S12 SaSb S112 T T3 Sb-Sa SbSa 2 a since K=aS, then Sb-Sa Sb-Sa a(Sa+ 3 )K |:Sa+ and f'\ A S(AB) I: (AB)] f'\ Where =half the arclength between A and B 5121 and K ]=the curvature at a point one-third the arelength between A and B, measured from A. Equation 22 states that the angle measured between thetangent line at a point A and the chord If! on the radioid clothoid is equal to half the arclength between A and B multiplied by the curvature of the clothoid at a point C where C is always the distance A an 3 from A between A and B. It can be shown by a similar development that the above relationship is also valid when the tangent line is drawn at the point B and the angle is measured at B in which case the curvature in Equation 22 refers to a point from B between A and B. In this latter case, the measurements are being taken towards the origin. Equation 22 is also valid for circular curves. The rate of change of curvature of a clothoid having the relationship K =aS is as follows: d s)-a, a constant In a circular curve the curvature is a constant, i.e. the reciprocal of the radius, therefore, dK a? which, for the present purposes makes the circle a special case of a radioid clothoid. Therefore: E AS Sy-Sz for the clothoid (for the circle this expression is undefined). Again for tangent track the rate of change of curvature is which is constant which makes the tangent track a special case also of the relationship stated in Equation 22. A Cubical Parabola may be represented by the equation: y=aX (26) where, for transition curves, a is a constant and H G=Z5 where L=length of parabola, from origin to the circular curve, measured along the X abscissa, i.e. the projected length of the parabolic curve, and H =the maximum offset, that is Y at the point where the parabola joins the circle. The curvature of a parabola is: Where (l+9a x is always very nearly equal to 1 owing to the fact that for most railroad parabolic curves. Thus K is approximately equal to 6ax and The expression 6a is a constant, which makes Equation 22 approximately valid when applied to cubic parabolae, for 8 railroad work, where the constant a in a radioid clothoid curve has the value 6a in a parabolic curve. It is to be noted that in the analyses of these clothoid and parabolic curves a is a ditferent constant to each analysis. Thus, it has been shown that by determining the curvature at two points in a length of track, by obtaining the values of the angle a at two adjacent points on the curve, it is possible to determine not only whether the track is tangential, circular curved, or transition curved, but also the exact curve form to which the length of track approximates between the points by which the angles a are determined and which provides a constant rate of change of curvature for the length of track. Tangent track is identified because K and K are zero and circular track is identified because K =K It will thus be seen that a lining machine able to determine these angles need have no external information provided to it for it to be able to determine the characteristics of any length of track on which it is placed. Since, in deriving this relationship, no significant approximations have been used, a lining machine adapted to make use of the relationship can be used on tangent, circular curved, and parabolic curve track as well as spiral track without introducing errors except on those lengths of track including part tangent and part transition, or part transition and part circular, or part tangent and part circular track. Even in these latter cases, the errors introduced may be made insignificant for railroad work by suitable design of apparatus. The above analysis has been based upon the angle between the tangent at a point and the chord between that point and another point on the curve. It is to be expected that other relationships can be established for such curves by utilizing a difi'erent base, for example the angle between two chords from a given point. It has been shown so far that the curve form. to which a length of track approximates and which provides a constant rate of change of curvature can be established by determining the curvature, by measuring a certain angle, at two points on the length of track. The next step is to find a way of determining whether another point on the track in fact lies on the required curve form and of moving that point to lie on the curve form if it is found that, in fact, the point does not lie on the curve form. Turning now to FIG. 2, there is shown a method of making this further determination. The points A, B and D on the track in FIG. 2 represent components of a liner system moving on the curve in the direction indicated by the arrow G. The points are assumed to be moving in a manner such that arclengths (Sb-Sa) and (Sd'Sa), between A and B, and between A and D, respectively, remain unchanged. This in practice cannot be completely achieved, but with railroad ergipment it is possible to maintain constant chord lengths AB and X17 and this is suflicient for the desired accuracy of measurements. As indicated, the curve at D is in the wrong position, the lateral error at that point being DD. This error, however, is considered small enough to allow the assumption that ZDEZF. In FIG. 2, a indicates the angle between the tangent to A and the chord AB and ,8 represents the angle between the chord AB and the chord BD where D is the point at which the track location D should be to lie on a transition curve of radioid clothoid form including the length AB. If it can be shown that B=f(vt,a) and if means of measuring a at A and of locating a chord DB that satisfies the function B=f(u,a) are available, the error DD can be corrected by moving the track at D until D is at D on DB in which case, since A ZT=A I7, W: 0. The function p=f(a,a) is shown to exist from an analysis of Equation 22. Considering B, BY, 5, of FIG. 2 being the angle be tween chord BD and the tangent at B), and that Equation 22 is also valid in conditions where the angle between the tangent and chord is oriented toward the origin of the spiral, then But from Equation 22 Sa= and Sb= (Kb-KG) a a a (Sb-8a) and Ka Sit- (Sd-Sa) (32) Furthermore, m Sb-Sa=E and ,S'd-Sa=AD Therefore, a 3Ka A A B-AD +AB+AD) (33) From Equation 22: Sb-Sa 2a 2a. KAT 3 Sb Sa 2E Also KAT [Sa+ ]-Ka =a (Sa-l- -Sot)= AB From Equations 34 and 35: 2a. a m Ka= AB Therefore, substituting in Equation 33: A A A A 2 two, 3 9% 11 Equation 37 then becomes 6 f( uU- (38) To obtain the curve constant a, two values of a, a and 11 corresponding to two positions of the liner system on the curve, are required. Assuming a change, h, of position along the are between the two points at which a is measured, then: Utilizing this equation in a practical machine requires that an initial measurement of :1 be made with the machine in a first position, and that the information be stored. Then the machine must be moved a known distance, h, along the track to a second position at which a measurement of a, a is made. At this time the stored information can be used to evaluate the equation. Similarly, for operation in subsequent positions, each measured value of 05 must be retained for computation purposes at the next station. For simplicity, obviously h should be made a constant value. It can be seen that (a oc may 'be a positive or negative quantity or zero, depending on whether the machine enters the spiral from straight or circular track or operates on a circle or tangent track. If (X2(Z1-=0 then the equation is reduced to AD (1 l A B 2 which represents conditions on a circular curve. If both 0: and a, are 0, then [i=0 representing conditions on a tangent track. The practical machine, therefore, requires means for measuring the angle on between the tangent at a point A and a chord between A and B; means for storing the previous measurement of the angle a; means for evaluating the Equation 43 to determine the additional angle 3 which is required to establish a chord BD passing through a point D on the preferred form of curve extending between A and B; and means for moving the track in the region of the corresponding point D towards the point D until the point D of the track coincides with the point D of the required curve. In practice, the measurement of angles between various reference lines is difiicult. It is preferred that measurements of distances which are determinative of the various angles should be made. The measurement of distances from a datum to a reference line is common practice in the art. FIG. 3 illustrates a possible arrangement of components in a system adapted to use the method described above, but making measurements of distance, rather than of angle. The discussion of this arrangement hereinbelow also shows how the angle measurements may be replaced bydistance measurements. A length of track 10 has mounted thereon a number of dollies located at predetermined distances apart. The first dolly 11 comprises a platform which is held in a predetermined relationship to the track 10 such that a line, R A, on the platform, is at all times tangential to the track at the point A which lies on the track. Mounted on the platform and movable along a line PR perpendicular to the line RA and passing through the point R are two radiation receivers Ru and R5. The system includes means for moving ROE and RB along the line PR on either side of the line R A'. The line PR is at a predetermined distance, W, from the point A. At the point A there is an optical pivot constituting reference line detection means which may be in the form of a shadow board. Forward of the platform in a direction of travel of the dollies is another dolly 12 which incorporates track moving devices. Fixed to this dolly 12 is reference line detection means located on the track at a point D. Ahead of the dolly 12 is another dolly 13 on which is mounted a radiation transmitter at a point B and able to transmit one or more beams of radiation in the direction of the dollies 11 and 12. The point B is on the track. The transmitter and receiver constitute reference line establishing means. Means are provided for moving receivers Rat and R 3 along the line PR and for measuring the distance R Ru and for computing a predetermined distance R RB using the relationship B=f(a,a) In operation of the apparatus, with the apparatus located in one position of the dollies 11, 12 and 13, on the track, the distance Um between R and Rat is measured when Ru and the points A and B are in a straight line. This first measurement of Uot, Uzx is then stored in the computation device. The system is moved a distance, it, along the track and a second measurement of Ua, U01 is made when receiver Roz is in a straight line with the points A and B. The computation device then determines the distance Us which receiver Rf) is to be from receiver Ra so that the chord between receiver RB and the transmitter at B shall intersect a radioid clothoid curve form to which the length of track between A and B approximates at the point D" which is spaced a distance AD" from A. When the computing device has made this calculation, the information is transmitted to the means for moving receiver RB and receiver R is located in the required position relative to receiver Roz. When the position of receiver RB has been determined and located, the receiver R13 determines whether in fact the shadowboard at point D" on the dolly 12 is in a straight line between RI? and B. If the shadowboard is not in the correct position then the lining mechanism is brought into operation by a signal from receiver R6 to move the track towards the chord extending from receiver Re to B. In most instances the lining mechanism will have a capacity to move the track to the required position where the preferred curve form between A and B and the chord between receiver RB and B intersects. However, the lining mechanism has a limited force and in some instances the correction required may be so large as to be beyond the capability of the lining machine. The function U5 is a function of U41 and is obtained from Equation 43, i.e. where W is the distance between A and the line PR substituting Equations 44 in Equation 43 results in the device ever starts functioning. Accordingly, it will be seen that the computing device, once it has been set up with the constants, need only be able to store the value Ua and to subtract this value from Uu multiply this value by the machine constant, and add the resulting value to the value of U11 It will be seen that comparatively simple apparatus is required. There are practical difficulties in maintaining the line A'R in the above described embodiment as a tangent to the track at the point A. Accordingly, FIG. 4 shows an alternative arrangement which does not require that a tangent at point A be set up. In FIG. 4 there is shown a transmitter at the point B", the optical pivot mounted on a dolly at A, the shadowboard mounted on a dolly at D together with the track moving devices, and mounted on a dolly at E receivers Ro' and Rfi. Receivers R0 and Rfi can move in a line perpendicular to the track at E. The principle of operation is very similar to the operation described with reference to the components of FIG. 3. The basic diiference is that now the datum point for measurement of the distances of the receivers is a point on the track noted as E instead of the point R on the tangent of the curve at the point A. FIG. 4, Se denotes the length of the are from the origin of the spiral to E, and the chord EA" is assumed to have a length of W. In this case the quantity which is measured and stored is the quantity U0- which is proportional to the angle 0' which equals a-t-a, where 5 is the angle between the chord EA" and the tangent at point A. Accordingly the distance U5 which receiver Rfi must be located from R0 to set up the chord RBB" is a function of U0 in the manner UB=G (U r). This function can be obtained by considering 6 in relation to 0:. From Equation 22 and using A, B, D, and W for simplicity: sagwS'e Sa;Se] (46) Furthermore, from Equation 36: KF MEF B and from Equation 5: K=as (48) Combining Equations 46, 47 and 48: bi g +W) (49) Accordingly, W W It'F-l-W r=oz+6=(fi+1)aatherefore, IF WE? W+E" 'T 51 Therefore, But from Equation 42: and therefore, W+IE h From Equations 51 and 54: E WZF 47 -0; W+ZF (W+ZF) h 13 but from Equation 43: Ji l. M- Zn 323 h From Equations 43, 52 and 55, therefore: 5: It? WE L (EV -6 W+ZB" (W+It) (W+2t?) h now considering: U H1 ZF+W and W (58) the: A D Ii (Z77) U0'g-U0' U5Gl(Uv) W Ur 3 W Equation 59 is thus the required control function for a system in which the datum is taken to be a point on the track and not a point on the tangent to the track at the optical pivot. It will be seen that control Function (59) is in the same terms as control Function (45), that is it is in the form of a constant times the second measurement of [L1, U0 plus a constant times the dilference between the second measurement of U0, U62, and the first measurement of Ucr, U6 divided by the distance between the points at which the two measurements were taken. Thus, the same equipment is required in the computing device to determine the distance U 3. In other words, the control Functions (43) and (59) can be written as: Where C and C are constants; h, representing the distance moved on the track between the first and second positions can also be reduced to a constant by assigning an arbitrary value to it. FIGS. 5 and 6 illustrate a practical form of apparatus embodying the present invention. The apparatus illustrated combines means for levelling railroad track and means for lining railroad track. The apparatus comprises a main car 21 having supporting wheels 22 running on a track 23. A levelling projector 24 is located ahead of the car 21 and levelling receivers 25 are located in the body of the car behind a shadowboard 25a. Such apparatus is known. The radiation beam transmitting means in the present embodiment of the invention is mounted on a dolly 26 ahead of the car 21 and takes the form of an infrared transmitter 27. The dolly 26 is held a fixed distance ahead of the car 21 by a strut 28. Trailing the car 21 is a receiver dolly 29 which carries a survey receiver 30 and a liner receiver 31. The dolly 29 is constrained to follow the car 21 at a fixed distance by a strut 32. Mounted to the frame of the car 21 is a dolly 33 carrying reference line detection means in the form of a shadowboard 34. The dolly 33 is constrained to travel with the car 21 but is free of lateral restraint with respect to the car 21 so that it may be held against the rails of the track. Between the dolly 33 and the dolly 27 are track moving means 35. These track moving means are mounted on the main frame of the car 21 and include reference line detection means in the form of a shadowboard 36. If desired, the track moving means for laterally moving the track can be associated with means for levelling the track. The shadowboard 36 is so mounted that it moves laterally with lateral movement of the track. The shadowboard 36 is maintained at a fixed distance from the projector 27 and the shadowboard 34 so that the elements 30, 31, 34, 36 and 27 are maintained in fixed spatial relationship. The receivers 30 and 31 are mounted on the end of individual shafts 37, 38 and the shafts are movable laterally by motor means (not shown) in response to control 14 signals. A distance measuring device 39 is secured to the dolly 29 for measuring the distance moved along the track by the apparatus. FIG. 7 illustrates the main components in a control system for the receivers. The diagram is shown in block form as the components are standard equipment. For determining the function Uo', a closed null seeking servo circuit is provided. This is shown in the upper part of FIG. 5. The loop consists of the receiver, Ra; an amplifier Aa; and a motor unit Me, which controls a linear actuator for moving the receiver Ra along the line ER an optical pivot is provided between the receiver and the source of radiation. This optical pivot may be, for example, a semi-transparent sheet with transparency gradually decreasing towards a pivot line. The null seeking servo unit functions to locate the receiver R0- in a position in which no radiation is received. As will have been apparent, the feed-back loop of this circuit is optical. The output from the linear actuator is abstracted from the closed loop and is converted into an analog signal by a potentiometer type rotary transducer, To. The output, f(Ua' from transducer T is supplied to a storage circuit which holds the output for a set interval of h as described above. The value of h is determined by means of a counter connected to the reference wheel 39 running along the track. When the counter determines that the predetermined distance h has been reached, the storage unit is actuated to feed an output function f(h, U0 to a summing circuit which also receives f(U 2), which is derived from the null servo seeking unit at the end of the interval h. In the summing circuit, the function C1U0'2JFC2 W is established. The output from the summing circuit is supplied to a closed loop servo for controlling the position of the receiver R5. The closed loop control for receiver R5 comprises: an amplifier A5; a motor MB for controlling the linear actuator of receiver RB; and a feedback of function U'B through a rotary transducer TB. There is provided a subtracting unit, 4:, which relates distance Ufl to the function Having now established the required position of the line RBB, the final step is to determine whether the shadowboard D and thus the track at D lie on the line RaB" and the required curve, or whether to move the track and shadowboard by use of the moving mechanism mounted on the dolly fixedly carrying the shadowboard D". A system such as that described in United Kingdom Pat. No. 1,067,826 may be employed for carrying out the determination and the track movement, as well as other known systems for moving a point on a track to lie on a predetermined line. In discussing the apparatus shown in FIGS. 5 to 7, it will be noted that it has been suggested to use two receivers. It will be apparent that it would be possible to use a single receiver provided that suitable switching apparatus were incorporated in the electrical circuitry so that the single receiver could be used initially in the surveying mode, for the determination of the angles a and a and subsequently in the lining mode. In the apparatus described it will also be necessary to provide radiation transmitter means, reference line detection means, and receiver means for right hand curves located on the other side of the apparatus from the location shown in the drawings. In the alternative, it will be possible to mount all the elements above the head of the car and to arrange for the elements to be movable to either the left hand or right hand side of the track. What I claim as my invention is: 1. A method of laterally aligning railroad track comprising: where K and K denote respectively the curvature at the two points and '(S S is the distance between the two points; (d) establishing, for some point in the length of track, the required position of the point to lie on the curve form; and (e) moving the track at the point towards said required position. 2. A method according to claim 1 wherein, to determine the curvature (Kc) at a point (C) in the length of track, the method includes the steps of measuring a parameter determinative of the angle (a) between the tangent to the track at a first point (A) and a chord passing through the first point and a second point (B), where point C is one third the distance between points A and B in the direction of B, and solving the relationship where S is the distance between A and B. 3. A method according to claim 1 comprising measuring a distance parameter determinative of the angle a. 4. A method according to claim 1 wherein, to establish said required position of said some point (D), the method includes setting up a first chord of the track through a point (B) at an angle 8) to a second chord extending between (B) and another point (A), where where f\ A AB and AD are, respectively, the arc lengths between A and B and A and D and is the angle between the chord AB and the tangent at A, whereby, if said first chord intersects said track at said point D, said point is at said required position. 5. A method according to claim 1 including determining the curvature at a first of said points, storing information concerning the curvature at said first point, determining the curvature at the second of said points, and finding the constant a after determining the curvature at the second point after recalling the information on the curvature at the first point. 6. A method according to claim 1 including the steps of: (a) at a first station of said points measuring first parameters determinative of a first angle between the chord joining the points and the tangent at one of those points; (b) storing said first parameters; (c) moving said points to a second station a given distance along said length of track; (d) at said second station measuring parameters of a second angle between the chord joining the points and the tangent at said one point. 7. A method according to claim 6 including the further steps of: (e) establishing at said second station, a chord passing through the second of said two points and at an angle (,8) to the chord passing through the two points, where the angle is given by where (1 is said first angle a is said second angle h is said given distance and C and C are constants relating to the spacing of the said two points and the said some point; and (f) moving the track at said some point towards said required position which is the location of said some point on said chord at angle {3. 8. A method according to claim 1 wherein the parameters measured are determinative of a first angle located between the tangent to a first point and a chord extending between the first point and a second point and of a second angle located between the tangent at a third point and a chord extending between the third point and a fourth point, said first second, third and fourth points being in said length of track and said first and second and said third and fourth points having the same straight line spacmg. 9. A method according to claim 8 comprising measuring the distances between said tangents and chords. 10. A method according to claim 8 comprising establishing reference line means through said first and second points and measuring a first distance between the reference line means and a fifth point on one of said tangent to the first point and the track, and establishing reference line means through said third and fourth points and measuring a second distance between the reference line means and a sixth point on one of said tangent to the third point and the track, the distance between said first and fifth points and between said third and sixth points being equal. 11. A method according to claim 8 comprising the further step of establishing a third angle located between the chord passing through the third and fourth points and the chord passing through said fourth point and said required position from the relationship where ,9 is said third angle a is said first angle a is said second angle IE is the distance between said first and second and between said third and fourth points 15 is the distance between said third point and said some point and L h is the distance between the first and third points. 12. A method according to claim 11 including establishing reference line means passing through said fourth point and at said third angle to said chord through said third and fourth points and moving the track at said some point towards a position in which it lies on said reference line means. 13. A method according to claim 10 comprising establishing reference line means passing through said fourth point and at a third distance from said sixth point determined by the relationship tm=o uta+o2 where U6 is said third distance Uot is said first distance Uu is said second distance h is the distance between said first and third points and C and C are constants related to the distances between various points. 1 7 14. A method according to claim 13 wherein said first, second, and third distances are measured from said tangent to said third point and wherein and where E is the distance from the third point to said some point and W is the distance from said sixth point to said third point. 16. A method according to claim 13 including the step of moving said some point to lie on said reference line means. 17. Method of aligning railroad track comprising: (a) locating apparatus components comprising a transmitter, receiver means, and first and second shadowboards in a fixed spaced-apart manner along a length of railroad track, said first shadowboard being operatively connected to track moving means; (b) aligning said transmitter, receiver means and second shadowboard along a first line; (c) making measurements at a first position of said components determinative of the angle (04 between said first line and the tangent to the track at the second shadowboard; (d) moving the aforementioned components along the track a distance (h) to a second position; (e) repeating step (b); (f) locating the receiver means in a position such that a second line joining the transmitter and receiver means is at an angle (6) to the line joining the second shadowboard and transmitter where C and C being constants related to the spacing of the components, and (g) moving the track and the first shadowboard to said second line. References Cited UNITED STATES PATENTS 3,165,073 1/1965 Blix et a1 104-8 3,314,373 4/1967 Plasser et al. 104-7 3,343,496 9/1967 Warwick 104-7 ARTHUR L. LA POINT, Primary Examiner R. A. BERTSCH, Assistant Examiner Referenced by
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