@ © The International Maintenance Seminar 2002 MIDTH Institute” Kiln Axis by Erik Reinhardt Lecture 04-04-01The International Maintenance Seminar m IDTH Institute” Table of Contents 2. Rotary Kiln 2.1 What is ment by the term Kiln AXIS... 2 2.2. Why take Interest in the Kiln Axis. Hts senctcoetec 2.3. Distribution of the Weight Load between the Supports ......00c0ne 14 2.4 Kiln Stiffness... 2.5. Distribution of Load between the Rollers... 2.6 Longitudinal Bending Stress Distribution.. 2.7 What are the Results of a faulty Kiln Axis... 2.8 What may cause the Kiln Axis to change . 2.9. What are the Symptoms of a faulty Kiln Axis... ee 26 2.10 How to measure the Kiln Axis..... 27 2.10.1 Mechanical measurement method... Hissasuamininiosawarscaneses oO <2IT0 DEPOTS? MEOROG s.cssnccresp mcusiseaeeconmmesiinincsion ne 31 2.103 Phillips's Method. 33 2.104 FL. Smidth’s Method . 234 Kiln Axis ERe —IMS 2002The International Maintenance Seminar 5MIDTH Institute" @ 2. ROTARY KILN AXIS ® @ Figure 11 @ Kiln Axis u ERe -IMS 2002miIDTH Institute” The International Maintenance Seminar 2.1 | What is ment by the term Kiln Axis Theoretically, the kiln axii defined as the axis, which for any cross-section along the lengih of the kiln connects the intersecting points of the cross-section’s neutral axis for bending in vertical and horizontal planes. For the practical application of the term the definition is simplified to cover the collection of straight lines which connect the intersecting points of the neutral axis for bending in ver al and horizontal planes of the kiln shell cross-section over the support centrelines. > Itis common practice to speak about the horizontal kiln axis and the vertical kiln axis meaning the images of the kiln axis reflected on the horizontal and vertical planes respectively. Horicantal axis a - Vertical asis Figure 12 ‘The two figures, 1] and 12, show the vertical and horizontal axis of a four-support satellite cooler kiln and a three-support grate cooler kiln respectively. By default the straight reference Kiln Axis 12 ERe ~IMS 2002The International Maintenance Seminar mMIDTH Institute” line is chosen through the first and the last of the supports where, consequently, the corresponding axis deflections are zero. Horizontal axis Vertical axis Figure 13 Any pair of supports can, however, be chosen as references. If the meshing conditions between girth gear and pinion(s) are acceptable, the support nearest to the girth gear normally, be among the chosen in order not to complicate any subsequent roller adjustment to an unnecessary degree aba Berrdinne, Shure ¢ 2.2 (Why take Interest in the Kiln Axis | ‘To maintain the kiln axis geometry within acceptable limitations constitutes a very important part of the maintenance of the availability of the rotary kiln itself, The extent of what may be the effect of faulty kiln axis geometry is, however, often overestimated. The widespread attention to e.g. popular terms like “hot kiln alignment” bears witness of this. Many companies order @ kiln axis alignment measurement or -analysis made in the belief that it will expose all essential threats to their rotary kiln(s). And still more companies order it on the background of already occurred problems or even faults Kiln Axis 1B ERe ~IMS 2002mIDTH Institute” The International Maintenance Seminar Itis important to realise and establish that an alignment measurement or alignment analysis i a preventive maintenance tool which is able, at an early stage, to disclose inappropriate mechanical operating conditions and hereby, if followed up by adequate countermeasures. avoid that these conditions develop into a certain group of faults or break-downs (see figure 10), It follows trom this that the alignment measurement tool ought to be brought into use at a time well before any inappropriate becomes a recognised reality, To be more specific; the above-mentioned certain group of faults or break-downs are those related to an inappropriate distribution of the weight load of the kiln shell with internals, charge and crust among the individual supports and only those! The immediate consequences of an inappropriate weight load distribution are exceeding of the imitations of the longitudinal bending fatigue stress in kiln shell and support roller shafis or of the contact pressure between tyre and support roller. Cireumferential fatigue crack growth in kiln shell and support roller shaft and piuing formation on roller surfaces of tyre and support rollers will be the visual proofs if not avoided in time, 2.3 | Distribution of the Weight Lond b tween the Supports / For the present purpose the kiln shell is considered as being a long, clastic beam which is suspended in a number of simple supports. The number of supports is, however, important, Consider a two-suppot kiln (figure 14), The position of the common centre of inertia of the kiln shell with refractory lining, internals (if any), charge and crust is calculated to be in the distance, (from the left support. This position is not affected by the rotation of the kiln. It may, however, vary as result of accidental variations in eg. the chemistry of the charge and ctust size ete, Such variations are considered of negligible importance and will not be drawn into the following considerations. ‘The inclination of the kiln implies the existence of an axial component of the weight force. Ey, equal to Ey tanfa) where ais the inclination angle. This axial force is transferred to the foundation through the combined effect of guide ‘Kiln Axts 14 ERe -IMS 2002 @The International Maintenance Seminar miIDTH Institute” Ri+Rr= Ew R= EW(L-) RIS Ew (LAL & Rr= Ew- Rl Rr= Ew (UL) Z Figure 14 Roller arrangement and support roller thrust bearings. Being of no importance to the present discussion this effect is implicitly understood here only. It will be treated in details elsewhere, By combining the static equilibrium condition and st ‘moment equilibrium condition of the vertical forces (see figure 14) the two reactions, Rand R,, is derived. It is observed that they are determined by the two distances Zand J only and, of course, the weight of the kiln. To put it in another way: the support reactions of a two-base kiln are determined by static ‘equilibrium conditions only. Two-base kilns are statically determined structures. | Applying the definition of kiln axis mentioned in section 2./ it can be concluded that the axis ‘of a two-base Kiln is always staight. It is, therefore, meaningless to adjust the axis of a two: base kiln and it has no sense to artange for a hot kiln alignment of a two-base kiln. The situation is quite different for rotary kilns with more than two Supports, As demonstrated in the following it is impossible to derive explicit expressions or formulas for the size of the support reactions, corresponding to the expressions of figure 14, by applying static Kiln Axis 15 ERe-IMS 2002MIDTH Institute” The International Maintenance Seminar equilibrium conditions onl) statically indeterminate structures. | 6 | WA+R2+ RS Ew Figure 15 ‘The equation shown in figure 15 is, of course, of general vali of the vertical forces. MtR2e RS Ew Figure 16 Consequentially, rotary kilns with more than two supports are Ic ensures static equilibrium Kila Axis 16 ERe -IMS 2002The International Maintenance Seminar MIDTH Institute” Gytin-e RAtR2+RO=Ew bred vballa [I Figure 17 DI tip nd A PURELY INTELLECTUAL EXPERIMENT (SEE THE FIGURES 16 AND 17) WILL BE USED TO DEMONSTRATE THE MECHANISM OF LOAD DISTRIBUTION BETWEEN THE SUPPORTS OF ROTARY KILNS WITH MORE THAN TWO C SUPPORTS. The experiment assumes a three-support kila as the one shown on the figures. The validity of its conclusions is, however, not limited 10 a three-base kiln. Imagine that the intermediate support is itself carried by a hydraulic jack, the force of which equals the load on that support. By slowly increasing the hydraulic pressure the load on this support will increase and the vertical position of the support will increase, i.e. the kiln axis will change. Not shown on the figures is the corresponding bending of the elastic kiln in the vertical plane. Simultaneously, and because of the general validity of the equation of static, equilibrium between the vertical forces, the load on the two other supports must decrease correspondingly (depicted as the transfer from figure 15 to figure 16) Kiln Axis 17 ERe ~IMS 2002mIDTH Institute” The International Maintenance Seminar By continued increase of the hydraulic pressure (see fig. 17) at a certain point the intermediate support carries the entire weight load, E.. At this point the two other supports carry no weight load and the kiln axis has reached a geometry with the ultimate curvature of the kiln. Continued supply of hydraulic liquid to the jack will only increase the vertical position of this support without to change the curvature of the kiln Some kilns reach this point at é.g. 15 mm increase of the vertical position of the intermediate support in relation to a straight line between the two other supports, another at e.g. 70 mm vertical inerease depending on what could be termed the flexibility or stiffness of that individual kiln: By means of the static equilibrium conditions (ic. one vertical- and one horizontal equilibrium equation and one static moment equation) two and only two relations can be established between the unknown vertical support reactions and the likewise vertical weight loads. Because wo-support kilns contain just (vo unknown support reactions the static equilibrium conditions suffices for those kilns. which, consequentially, are called statically determinate. To calculate kilns with more than two supports i.e. with more than two unknowns a number of supplementary conditions equal to the number of supports minus two are needed. These supplementary conditions must be found outside the already exhausted static equilibrium conditions and those kilns ate called statically indeterminate. Among the often-used methods of establishing supplementary condition are e.g. Castigliano 's method and the method of angular deformation. It will be outside the scope of this presentation t go into details with any of these methods, Instead reference is made to engineering school basic theory of elasticity. Kiln Axis 18 ERe -IMS 2002 e@The International Maintenance Seminar MIDTH 24 Kiln Stiffness. at Ruins The intellectual experiment mentioned above has produced some qualitative knowledge about a connection between the geometry of the kiln axis and the so-called stiftness of the kiln ie. a conncction between the distribution of the weight forces between the supports and the kiln stifiness. Qualitatively, the stiffness of the kiln is apprehended as the kiln’s resistance to longitudinal bending, Expressed in a different way; the stiffer the kiln the higher the force necessary to bend it (see fig. 18). The stiffness characteristic of a rotary kiln is an important design feature. In the drawing office it is handled in the form of the so-called stiffness mairtx Figure 18, Figure 19 depicts a stiffness matrix belonging to a specific three-support kiln. It is read in rows and each row contains information about the consequences on the load distribution between the supports of changing the vertical position of the support mentioned in the first, column, Kiln Axis 19, ERe -IMS 2002mipTH Institute” The International Maintenance Seminar CHANGE QF REACTIONS IN % BY 10 381 LOWERING OF ‘SUBTORT 40. =r 4.3 a1 oy UO ~The load on sagpoio, 10 navensoiy 43% olla lenon sion, 20M dasreasely 35%, and Figure 19 general, the correction of a kiln axis involves adjustment of the position of one or more support roller bearings. This apparently simple operation is, however, often accompanied by the increased risk of entering into a hot bearing problem. This is especially the case by older Kilns and the reason why many plants choose not to carry this operation out by themselves. A decision of correcting a specific kiln axis should, therefore, not be based only upon its deviation from being straight but upon wether the results are worth the effort. Will we get value for money? Does the achieved decrease in bending stress or Hertz pressure compensate the risks involved? Will the decision increase the availability to a degree worth the expenses? In telling, relatively, what will be achieved in terms of changes in the longitudinal bending stress in the kiln shell by a certain correction of the rotary kiln axis the stiffness matrix represents an important tool in relation to this kind of evaluations. Without sufficient knowledge of the stiffness characteristic of the kiln such a decision is drawn in blindness Some kilas are very flexible. Consider é.g. the three-support 6,0 x 96 metre kilns for 10,000 1/24h production with stiffness matrix as shown in the lefi side of figure 20. It is seen that a 10 mum offset of support no. Lor IIT will cause some 3 % change of the support loads only, while the same offset of support no. II will cause some 6 % change only. Clearly, the attention to Kiln Axis 20, ERe -IMS 2002The International Maintenance Seminar MIDTH Institute” changes in the kiln axis need not be especially high in this case meaning that the alignment check frequency may resemble something like one every four year Figure 20 Other kilns are very stiff, Typically, elder wet process kilns with many supports belong to this group. The stiffness matrix to the right in figure 20 belongs to a foursupport 4,14 x 64 metre kiln, By comparison with the values from the afore-mentioned 6.0 metre kiln the vulnerability towards changes in the kiln axis geometry appears clearly. A 10 mm change of the vertical position of support nos. [lor IIL will result in loss of contact between tyre and rollers at one or some of the supports. Very stiff kilns demand that high aitention be given towards changes in the kiln axis geometry, Aligament check frequency may very well amount to one per year. Between the two above-mentioned extremes are, of course, a great number of kilns with normal stiffness where, depending upon experience, the alignment check frequeney may vary between one every two or three years. The middle stiffness matrix of figure 20 belongs to such a kiln, The main dimensions are equal to those of the stiff kiln, however, the number of supports is now only three ic. one less. It is ebserved that the magnitude of the stiffness matrix members is one less than those of the very stiff kiln and one higher than those of the Kiln Axis 2 ERe-IMS 2002miDTH Institute” The International Maintenance Seminar Yery flexible kiln. Information concerning the stiffness of a specific kiln must be gathered through the kiln manufacturer. * {) 10 Mey ye St. 2.5 Distribution of Load between the Rollers A rotary kiln support consists of two support rollers, four support bearings and a base plate, Its function is to transfer the combined weight load from kiln and tyre, Eto the foundation. Dons eqn Risin(a) «Rr sinfa) = 0-> Ri cos(a) + Rr cos(a) - Ew=0-> 2R cos{30) = Ew-> R=RI=Rr= Ewisqr(3) igure 21 The first step is to split the vertical force, E,., between the two support rollers. Fig, 21 depicts situation where the horizontal component of the kiln axis is straight ie. where no elastic forces originating from deformation of the kiln appear. Neglecting the influence from friction in the two support bearings é is balanced by the two unknown radial reactions, Rj and &,. ‘Through the static conditions of equilibrium in vertical- and horizontal directions respective two necessary relation are achieved from which the results are derived Itappears that the two reactions in this situation are of equal numerical value. in other words: the support load is divided equally between the two support rollers when the horizontal component of the kiln axis is straight. Kiln Axis 22 ERe -IMS 2002 oThe International Maintenance Seminar MIDTH Institute” Ri sin(a)- Rrsin(a)-F = 0-> R- Rr= Fisin(a) Usivall equiva Ri cos(a) + Rr cosa) Ew = 0> mane Figure22. Gyn) bn 7 bes fleryovitod ‘ ‘ nee } Ay, [2 DaXoonee In order to bend the horizontal kiln axis component, obviously. (a horizontal force, F)] must be present (see Fig. 22), The unknowns are still the two support roller reactions, Rand R, . for which reason expressions of their individual numerical values are derived the same way as above. ILis seen that when horizontal forces appear, because 2-sin(@) = 1.00, they are directly added to the reaction of one of the support rollers and subtracted from the reaction of the other while, if the same force had been vertical, it would have been divided equally with 58 % to ia of the two support rollers. 2.6 Longitudinal Bending Stress Distribution Having calculated the distribution of the weight load on the supports Le. the reaction from each of the supports belonging to a specific kiln axis geometry, the corresponding bending moment distribution and subsequently the bending stress distribution along the length of the kiln can finally be calculated. Kiln Axis 2BMIDTH Institute” The International Maintenance Seminar Figure 23 Fig, 23 shows a 4,75 x 77 metre ILC kiln with a kiln axis where support No, Il is elevated by 10 mm over the straight connecting the two other supports. Figure 24 Fig, 24 shows the same kiln with straight kiln axis. By comparing the corresponding bending stress distributions it is scen this kiln axis correction has caused a reduction from 42 N/mm’ to Kiln Axis 24 ERe -IMS 2002 OoThe International Maintenance Seminar MIDTH i Institute” 25 Nimm’ of the bending stress amplitude in the circumferential welding between the intermediate and the light kiln shell section on the outlet side of support No. I. Simultaneously, the amplitude in the middle of both of the light sections, The example suggesis how different choices of kiln axis geometry makes it possible to chose that distribution of the weight load between the supports which is most convenient in order to fulfil a specific purpose. As in the above-mentioned example this purpose may be special stress considerations, however, optimising the utilisation of the support’s load carrying. capacities or choosing the optimal Hertz pressure distribution between tyres and rollers are commen purposes too. It must be mentioned that the recommended kiln axis geometry need not, necessarily, be straight. Even that the original axis was straight changes in lining quality and crust formation ete, may cause new recommendation. 2.7 What are the Results of a faulty Kiln Axis 1 Sapte pie lrad 2, hea Sub — Laty Sachs Indy Tem hd peli Creda. 3 Btn Wh bone + Pelt _4 Raton ipo, Gack Kk Shap 3B Berd WS a & = —, @ Sel) Kolm Avo 8 AX Pa Crm) a = ter H 4 Kiln Axisfismivtn Institute” The International Maintenance Seminar 2.8 What may cause the Kiln Axis to change te Wena ite ia daw bong 4 Pll pe @, Supysd Urs Pal Adyrrobirct 3. Flmeedacbion Silfiey of ffigrven dann 4. Wea blo tyne t fad 2.9 What are the Symptoms of a faulty Kiln Axis (a A Belew Brawn — Tewered Wj ene rs Hasanstves a i Gack Tot ike Sapp” Folin Seo Cninthe ‘i Oeurnoh asin val Kaden “Foal Cuseiuin Lanmahii dln SET 1ECon ‘Kiln Axis 26 ERe -1MS 2002The International Maintenance Seminar MIDTH 2.10 How to measure the Kiln Axis The execution of an alignment analysis of a specific kiln including the measurement of the kiln axis is a maintenance tool which is applied in order to ensure the availability of the kiln in question, The ambition has always been to perform all necessary measurements without interruption of the kiln operation, and, since itis in hot operating condition that the kiln eventually may fail, itis this situation which really is of interest. ‘The difficulty has long been that the fulfilment of this ambition demanded the possibility to measure to and to fix in space a point on a moving surface. Modern technology has made this possible and a number of different approaches are met on the market. These approaches can be grouped into the three below described and principally different methodologies. Common to these methodologies are, however, that they apply relatively advanced equipment of which the utilisation demands special knowledge and routine, In practice, therefore, they are reserved for special groups or organisations, This doesn’t, however, exclude the individual cement plant from making its own alignment check. The traditional kiln axis measurement method, sometimes called the mechanical kiln axis method, is still available even its accuracy doesn't match that of the advanced methods. Figure 25 Institute” Kiln Axis 27 ERe IMS 2002MIDTH Institute™ The International Maintenance Seminar 2.10.1 Mechanical measurement method ‘The mechanical measurement method falls into two phases: the vertical kiln axis phase and the horizontal kiln axis phase, ‘pasggpmerr2. Oreos eae evans © Figure 26 ‘The sequence of calculations by determination of the vertical kiln axis appears from figure 26 where the tyre is assumed to be of the traditional migrating type and of cireular geometry. Ttappears that, first, the height of the centre of the tyre, F, is calculated. Then, because of the so-called tap play, 5. i.e. the flatiening out of the kiln shell on top, a centre correction. s. is introduced in order to get down to the position of the kiln shell cross-section’s horizontal neutral axis. ‘The correction, s, may be determined in two ways. Either by measuring through the top play. which, of course, can be measured only by cold kiln, or through the migration Kiln Axis 28 ERe IMS 2002The International Maintenance Seminar MIDTH Institute” _e —— —_____ = “1,25-10°* - (shelfiemp.—20) Migration between tre and shell (mm/ Rev] ‘The method of measuri the horizontal kiln axis is depicted on fig. 27. A theodolite is placed at one end of the kiln with its binocular directed towards the other end and with the line of sight outside and in convenient distance from the support rollers of the same side. Measurements of distances from each support roller to the line of sight are performed with a — measuring scale which, by means of a spirit level and a convenient suspension, is kept horizontally and exactly in level with the horizontal diameter of the support roller, & SITTIN Mee ASIP ea NT nZONLANCIStAN (iets Figure 27 Having measured these distances at all supports a diagram as the one shown on fig. 28 is drawn, By connecting the midpoints of two of the supports —in fig. 28 support nos. I and TV ~ with a straight line the deviations between the midpoints of the resting supports and this line | @ raja Kiln Axis, 29 ERe—IMS 2002MIDTH Institute” The International Maintenance Seminar neasureinentine Figure 28 ‘The mechanical alignment method described is relatively simple to execute and demands only very basic knowledge of theodolite operation. It is, however, based upon a great number of measurements performed with the kiln in cold condition and takes no considerations of any foundation tilting. Consequently, its accuracy is should be considered as somewhat limited and its conclusions handled with caution, ‘Kiln Axis 30 ERe IMS 2002The International Maintenance Seminar MIDTH Institute” 2.10.2 Polysius' Method Figure 29 The problem of measuring to a point on a moving surface is solved through choosing the the support roller shaft centres as target points. The choise of these points who perform rotations only allows the inspector to focus his theodolite on the so-called sightening targets, which are fixed to the shafts end faces through magnets (see fig. 29), while the support roller rotates. Polysius makes use of different measuring methods in order to adapt best possible to actual site conditions. Al ihe polar setting out method (sce fig. 30) the vertival and horizontal angles to all sightening targets are measured by an electronic theodotite as well as the corresponding, distances from the theodolite to the sightening targets the latter either by the build-in infrared tclemeter or by a tape measure, The collected data which is used for determination of the spatial coordinates of the sightening targets are stored in a computer for later calculation of adjustments Kiln Axis 31 ERe -IMS 2002MIDTH Institute” The International Maintenance Seminar fine pte poker sein ou ® © Figure 30 Polar setting out instrument positioning ‘Two theodolites are used in the stepwise intersection method (see fig 31). While one of the theodolites is positioned in the same way as for the polar setting out method, the position of. the second one is determined relative to the first by means of a very precisely dimensioned base line bar positioned between the Ovo instruments. The measurements involved in this positioning procedure is controlled and monitored by a computer. Figure 31 Stepwise intersection Kiln Axis ERe -IMS 2002The International Maintenance Seminar MIDTH Institute” Based on the determination of the positions of the two theodolites, the coordinates of the sightening targets can now be established using the stepwise intersection measuring method. This method determines the position of a point through the interseretion of two lines of sigth based on known positions of the two comesponding theodolites. 2.10.3 Phillips's Method The basic principle behind Phillips’ method is that knowledge of the relative position of three points on the surface of a circle allows determination of the relative position of the centre of the circle (see fig. 32) eit Rotten ton, 68) ea Seceae ‘ee cele ot ne niueciel dene - brnean pone si aise comune, PUUMips Figure 32 In practice the measurements are performed by means of a measuring bridge which integrates threc laser distance transducers and two prism-reflectors in known relative positions (see fig. 33). With the measuring bridge positioned under the rotating kiln and the three laser distance transducers constantly measuring the distances to the moving kiln shell at their specific Kiln Axis 33 ERe IMS 2002mIDTH Institute” The International Maintenance Seminar position the transducer signals are logged to a computer 180 times per revolution of the shell. Figure 33 Having finished one rotation of the kiln the mean values of the 180 measurements from each transducer is calculated and the correspond luree mean positions shown on fig. 32 is calculated and on this basis the centre of rotation of the kiln shell at that particular position. Measurements are performed at either side of each tyre in order to allow for interpolation of the centre of rotation for the centre of each tyre support. The corresponding. positions of the measuring bridge are determined by the polar setting out method by means of a conveniently placed total station i.e, an electronic theodolite equipped vith a distance telometer focussed on each of the two reflectors of the measuring bridge. 2.10.4 FL. Smidth’s Method ‘The F.L.Smidth method uses two electronic theodolites. The moving surface problem is x solved by having one of the two theodolites equipped with a laser having its beam adjuster to Kiln Axis 34 ERe IMS 2002@ The International Maintenance Seminar mMmIDTH Institute” coincide exactly with the line of sight. The two theodolites are connected to a computer which monitors and controls the whole measurement procedure through specially developed software (fig. 34). Figure 34 The method determines the spatial coordinates of two sets of three points each on the rolling surface of every tyre. During each measurement the theadolites are placed in measuring points opposite to the tyre in question. All measuring points are situated on a base line set out along the kiln ex in the intersection between the concrete floor and an imagined, straight, vertical plane (see fig.35). ‘The spatial coordinates refer to a coordinate system, (X.Y. orthogonal and with the X- and H-axis in the above-mentioned vertical plane respectively horizontal and vertical and the ¥ aksis perpendicular to the plane and directed towards the kiln. The origo height is determined by a reference point (fig 37) Kiln Axis 35 ERe ~IMS 2002MIDTH Institute” The International Maintenance Seminar Figure 35 ‘The distances between the individual measuring points on the hase line are measured with a tape. By focussing the laser-equipped theodolite on a point, P, of the tyre surface (see fig. 36) the position of this point becomes fixed and visible by the small laser dot. When, subsequently, the other theodbolite is focussed on the same point simultaneous reaclings of the corresponding, vertical and horizontal angles of the two theodolites makes calculation of P's co-ordinates possible, Figure 36 Kiln Axis 36 ERe ~IMS 2002The International Maintenance Seminar MIDTH Institute” ‘The accuracy of the measurements is controlled frequently throughout the measuring procedure. The height, ie. the //-co-ordinate, of cach of the theodolites is e.g. determined by focussing on the reference point (fig. 37) and reading of the two vertical angles whereupon the heights are calculated by simple trigonometric. Figure 37 By focusing one of the theodolites agains! the centre mark of the other theodolite and reading the corresponding vertical angle the difference between the two heights is subsequently checked, Again simple trigonometric calculations leads to the desired result. The procedure will be continued only if the two results are in agreement, Similarly (see fig, 38), the difference between the heights of the two theodolites also appears. in the calculations of the H-co-ordinate of point P. Only if the calculated difference is in ‘agreement with the corresponding difference referred to fig, 38, the P-measurements are accepted. Kiln Axis, 37 ERe-IMS 2002mipTH Institute” The International Maintenance Seminar Figure 38 As mentioned above, tvo sets of three measurements each, are made to point P. In the first set the position of P is chosen below the horizontal diameter of the tyre and in the second set above this same diameter, Having measured its perimeter, ie. its nominal radius, and knowing from calculations the eccentricity of the elliptic tyre the two times three surface point co-ordinates allows nine | calculations of tyre centre co-ordinates. After checking that all nine centre expressions are @ situated within a very small circle, the resulting tyre centre co-ordinates are, finally, i, calculated from these nine sets by the least square method ‘The measurements and calculations described leads to the tyré centre axis of which the horizontal component, i.e. its mirror on a horizontal plane, coincides with the kiln axis. To arrive at the vertical kiln axis, i.c. the acutral plane for yectical bending of the kiln cross- section, the tyre centre heights at each support have to be reduced an amount that depends on the corresponding tyre migration, | All needed values are input to the computer program, which, finally, presents a graphics showing the measured kiln axis. | @ | es a Kiln Axis 38 ERe IMS 2002i ———E|®HRM RSEHt)cosa) rca) -EW=0> Real) E> ReMl=Bes Ene)il)Tleniunee st LoanNich Attar t L 2 IL I Was O g ole | 2 2 ' 4 = hee alsu Sees Wh ile Oe WhKe o a % e eo ® clowwilet SeviriCentur PS = Vg fe Shp = vey (42 -12)KH2 4 74 ees (eShop Slum « + we tof oo, oe axes W (PRD is (= . we Vio I mm, Wy cw) Une Axi nye aot e @ _f