Shaft alignment data acquisition

Methods and systems for acquiring offset data for aligning co-rotatable in-line machine shafts. Two sets of offset data, each set for example representing relative displacement in a radial direction between a reference point referenced to one of the shafts and a particular point on the other of the shafts at given angular positions, are collected at a plurality of measurement angular positions. The measurement angular positions are not necessarily the 0.degree., 90.degree., 180.degree. and 270.degree. angular positions at which measurements are traditionally taken. Preferably, measured offset data is collected during continuous rotation of the shafts in their normal direction of rotation or at convenient positions where the rotation is halted. Measured offset data may be collected at as few as three angular positions, but preferably is collected at more than three angular positions, and typically at a multiplicity of angular positions such as thirty or forty. A respective sine function is fit to each set of measured offset data. Data for the traditional 0.degree., 90.degree., 180.degree. and 270.degree. angular positions are then determined from the sine function. This data is thus determined in an indirect predictive manner, and is used in subsequent calculations to determine machine moves as though the calculation data had been determined by direct measurement at the 0.degree., 90.degree., 180.degree. and 270.degree. angular positions.

BACKGROUND OF THE INVENTION 
The present invention relates generally to the art of aligning co-rotatable 
in-line machine shafts which are coupled together for operation by means 
of a shaft coupling. More particularly, the invention relates to methods 
and systems for acquisition of data from which the amount of misalignment 
can be determined, and from which machine moves to bring the shafts into 
alignment can be determined. 
As is well known, whenever two rotating machine shafts are coupled 
together, such as the shaft of an electric motor and the shaft of a pump, 
it is important that the shafts be aligned within predetermined 
tolerances. Such shafts, when in perfect alignment, have their extended 
center lines (axes of rotation) coinciding along a straight line. 
Misalignment can lead to vibration, excessive wear, and ultimate 
destruction of couplings, bearings, seals, gears and other components. 
There are two relevant misalignment components, and either or both may be 
present in a given situation. One misalignment component is offset 
misalignment, also termed parallel misalignment or simply offset. In the 
case of offset misalignment, shaft center lines may be parallel, but they 
do not intersect. The other misalignment component is angular 
misalignment, and occurs when shafts intersect at an angle. Angular 
misalignment is also termed angularity, and is manifested as a difference 
in distance between coupling hub faces across a diameter of the coupling 
hub faces. 
A number of shaft alignment methods are known, which generally have in 
common the use of suitable alignment fixtures, also termed alignment 
brackets. The alignment brackets are employed to measure particular 
relative displacements (also termed offsets) as the shafts are rotated 
together through one revolution, while stopping at 0.degree., 90.degree., 
180.degree. and 270.degree. angular rotation positions to take readings. 
Each relative displacement is measured between a point referenced to one 
of the shafts by means of the alignment bracket and a point on the other 
shaft. Dial indicators are often employed, these dial indicators having a 
plunger which moves a hand on the face of the dial indicator. 
The readings are then used to calculate machine moves which will bring the 
shafts into alignment. The 0.degree., 90.degree., 180.degree. and 
270.degree. angular positions at which readings are conventionally taken 
lie in geometric planes in which either of the machines, for example the 
motor, may be moved for purposes of alignment. In particular, the mounting 
bolts of the machine may be loosened and the machine may be either moved 
in a horizontal plane; or the machine may be moved in a vertical plane by 
placing or removing shims under one or more of the feet of the machine, or 
both. There are well developed calculation methods and procedures known in 
the art for determining what machine moves to make to achieve an aligned 
condition based on measurement of relative displacement (offset) data at 
the 0.degree., 90.degree., 180.degree. and 270.degree. positions 
mentioned, which may be termed calculation angular positions. 
Although mechanical dial indicators are referred to above, it will be 
appreciated that other forms of measurement devices may be employed, 
including various optical and mechanical transducers. Also, although it is 
relative displacement which is actually determined, it will further be 
appreciated that absolute readings may be taken, referenced to a 
particular angular position, and a simple subtraction operation performed 
to determine relative displacement. 
There are various points where relative displacements may be measured, 
depending upon the particular alignment geometry employed. However, a 
commonly employed method is the reverse indicator method wherein a pair of 
relative displacements in a radial direction are measured at each of the 
calculation angular positions. 
As usually practiced, the reverse indicator method employs either one or 
two alignment brackets. An alignment bracket has a base firmly clamped or 
otherwise affixed to one shaft, and an extension bar or arm extends 
laterally from the base in a direction generally parallel to the shafts 
across the coupling over to a reference point adjacent a point on the 
periphery of the other shaft. A device for measuring displacement, such as 
a dial indicator, is positioned so as to measure relative displacement in 
a radial direction (offset) from the reference point to the point on the 
periphery of the other shaft as the shafts are rotated together while 
stopping at the 0.degree., 90.degree., 180.degree. and 270.degree. angular 
positions to take and record readings. The position of the alignment 
bracket is then reversed so as to be fixedly referenced to the other 
shaft, establishing a reference point adjacent a point on the periphery of 
the one shaft, and the procedure is repeated. Alternatively, a pair of 
alignment brackets may be employed for simultaneous readings. 
From the geometry just described, it will be appreciated that the reference 
point on the alignment bracket attached to the one shaft rotates about the 
projected centerline (axis of rotation) of the one shaft to define a 
circle centered on that projected centerline, and vice versa for the other 
shaft, and that the distance and direction of the distance between the two 
shaft centerlines as projected can be determined at any transverse plane 
along the shaft axes. From the thus measured distances and directions of 
the distances between the two shaft centerlines as projected in two 
transverse planes, both the offset misalignment component and the angular 
misalignment component may be calculated. 
Another method which is sometimes employed is known as the face-and-rim 
method. The "rim" part of this method is measurement of a relative 
displacement in a radial direction as just described, and the "face" part 
of this method is measurement of a relative displacement in an axial 
direction, again at each of the predetermined angular positions which lie 
in geometric planes in which either of the machines connected to the 
shafts may be moved in order to achieve an aligned condition. Typically, 
but not necessarily, the "face" and "rim" measurements are taken generally 
in the same transverse plane along the shaft axes. While the face-and-rim 
method thus directly measures angular misalignment, it nevertheless is 
generally considered to be less accurate than the reverse indicator 
method. 
The techniques of the present invention are applicable to either the 
reverse indicator method or the face-and-rim method, as well as the other 
related methods where relative displacement measurements are made at a 
plurality of angular positions, and are then used in subsequent 
calculations, particularly to determine machine moves for alignment 
purposes. 
SUMMARY OF THE INVENTION 
It is an object of the invention to provide methods and systems for more 
accurately acquiring offset data for aligning co-rotatable in-line machine 
shafts. 
Briefly, in accordance with the invention, it is recognized that offset 
data from each measurement device collected as described briefly above 
during one revolution of the shafts has the form of a sine function, where 
offset magnitude is a function of shaft angular position. As is known, 
when a sine function is plotted in a Cartesian coordinate system, the 
result is what is known as a sine wave. When plotted in a polar coordinate 
system, the result is a circle. 
In accordance with one overall aspect of the invention, data for each of 
the offsets is collected at a plurality of arbitrary measurement angular 
positions, which are not necessarily, and likely are not, the 0.degree., 
90.degree., 180.degree. and 270.degree. angular positions at which 
measurements are traditionally taken. A respective sine function for each 
of the offsets is then fitted to the measured data, and data for the 
traditional 0.degree., 90.degree., 180.degree. and 270.degree. positions 
are determined from the sine function in an indirect predictive manner, 
rather than by direct measurement at these positions. Advantageously, 
actual measurement data may be taken at as few as three angular positions 
of the shafts. Preferably, however, measurement data is taken at more than 
three angular positions, and typically is taken at a multiplicity of 
angular positions, perhaps thirty or forty. This has the effect of 
reducing the overall error which may be caused by an error in an 
individual reading. 
There are thus two different sets of data. An initial set of data results 
from actual measurement at arbitrary angular positions, at least three in 
number. This initial data is also referred to herein as measured offset 
data. A subsequent set of data is determined from the sine functions, for 
example based on the sine function values at the predetermined 0.degree., 
90.degree., 180.degree. and 270.degree. angular positions. This subsequent 
data is also referred to herein as calculation offset data, because it is 
employed in later calculations to determine actual machine moves to bring 
the shafts into alignment. 
Another overall aspect of the present invention is the collection of 
measured offset data during continuous rotation of the shafts in their 
normal direction of rotation. An acceptable alternative is to smoothly 
rotate the shafts, momentarily pausing the shaft rotation at any arbitrary 
angular position to take a reading, but avoiding any counter-rotation 
forces. In the traditional method, where the shafts are rotated and 
stopped at each of the four angular positions, the shaft position is often 
adjusted back and forth to achieve the desired angular position as 
accurately as possible. A number of factors can induce error, such as 
torsional play in the coupling. There is no guarantee that a coupling will 
be engaged in the same manner each time shaft position is adjusted to the 
desired angle. This error can be on the order of several mils, adding time 
and effort to the alignment process. In addition, collecting data in the 
normal direction of rotation, avoiding any counter-rotation, more closely 
approximates the conditions present during equipment operation. 
In a more particular aspect, the invention provides a method for developing 
a pair of data sets respectively indicative of a pair of offsets at each 
of a plurality of predetermined calculation angular positions of first and 
second co-rotatable in-line shafts. The predetermined calculation angular 
positions advantageously lie in geometric planes in which either of first 
and second machines connected to the first and second shafts may be moved 
in order to achieve an aligned condition. Typically, the predetermined 
calculation angular positions are 0.degree., 90.degree., 180.degree. and 
270.degree.. In one embodiment, the offsets are radial offsets, and the 
offset data comprises relative displacement in a radial direction between 
a reference point referenced to one of the shafts and a particular point 
on the other of the shafts at given angular positions. Alternatively, in 
an implementation of the face-and-rim method, one of the offsets may be a 
radial offset and the other an axial offset. 
An initial step in the method comprises employing at least one measurement 
device to acquire measured offset data for each of the pair of offsets at 
each of at least three measurement angular positions of the shaft. 
Preferably, two measurement devices are employed to simultaneously acquire 
measured offset data. The measurement angular positions may be the same 
for each of the pair of offsets, which is facilitated by employing the two 
measurement devices simultaneously, but this is not necessary, since the 
data sets are fit independently. Although measured offset data is acquired 
at each of at least three measurement angular positions of the shafts, 
preferably measured offset data for each of the pair of offsets is 
acquired at each of a plurality greater than three measurement angular 
positions of the shafts. 
Preferably, measured offset data is collected as the shafts are turned in 
their normal direction of rotation in a manner such that no 
counter-rotation occurs, even if shaft rotation is paused to collect data. 
It will be appreciated that, since measured data is collected at arbitrary 
angular positions, there is no need to adjust the shafts to achieve a 
particular shaft angular position. Where a data acquisition system 
permits, data may be collected during continuous rotation of the shafts, 
again in their normal direction of rotation. 
Where the offsets are radial offsets, the measured offset data comprises 
relative displacement in a radial direction between a reference point 
referenced to one of the shafts and a particular point on the other of the 
shafts at a series of angular positions corresponding to the measurement 
angular positions. 
The method continues with the step of fitting a respective pair of sine 
functions for each of the pair of offsets to the measured offset data, the 
sine functions each being a function of shaft angular position. 
A pair of calculation offset data sets are then determined from the sine 
functions based on the respective values of the pair of sine functions at 
each of the predetermined calculation angular positions. The calculation 
offset data sets may be viewed as predicted data sets, since they 
represent values which presumably would be found by traditional direct 
measurement at the 0.degree., 90.degree., 180.degree. and 270.degree. 
angular positions. 
In accordance with another more particular aspect of the invention, there 
is provided a system for aligning first and second co-rotatable in-line 
shafts connected respectively to first and second machines. The system 
comprises an alignment fixture having first and second fixture elements 
respectively mountable to the first and second shafts. The alignment 
fixture includes a pair of displacement transducers for measuring a 
respective pair of offsets, and producing respective offset data signals. 
The alignment fixture additionally includes an angular transducer for 
measuring angular position of the shafts and producing an angular position 
data signal. The system also includes an alignment calculator, and a data 
link for transmitting the data signals from the transducers to the 
alignment calculator. The data link may comprise either a conventional 
cable, or a wireless link such as a radio link or an optical data link. 
The alignment calculator includes elements which serve to record measured 
offset data based on offset data signals from each of the pair of 
displacement transducers at each of at least three measurement angular 
positions as determined from the angular position data signal, to fit a 
respective pair of sine functions as a function of shaft angular position 
to the measured offset data for the pair of offsets, and to determine a 
pair of calculation offset data sets based on respective values at each of 
a plurality of predetermined calculation positions. Elements within the 
alignment calculator additionally employ the calculation offset data sets 
to generate instructions for moving at least one of the machines to 
improve alignment of the shafts. 
In a more particular embodiment, one of the displacement transducers 
measures a system wherein one of the displacement transducers measures 
relative displacement in a radial direction between a reference point 
referenced to one of the shafts and a particular point on the other of the 
shafts, and the other of the displacement transducers measures relative 
displacement in a radial direction between a reference point referenced to 
the other of the shafts and a particular point on the one shaft. 
As discussed above, the predetermined calculation angular positions lie in 
geometric planes in which either of the machines may be moved in order to 
achieve an aligned condition, and these predetermined calculation angular 
positions are typically 0.degree., 90.degree., 180.degree. and 
270.degree.. 
It will be appreciated that the term "shaft" employed herein includes, in 
addition to the shafts per se, various attached elements such as coupling 
hubs and flanges. Although the methods of the invention are preferably 
implemented in an automatic system, it will be appreciated that readings 
may be taken manually, for example employing conventional dial indicators 
and an angular position gauge, and manually entered into a suitably 
programmed calculator.

DETAILED DESCRIPTION 
Referring initially to FIG. 1, first and second rotating machines 10 and 
12, in the representative form of a motor 10 driving a pump 12, have 
respective first and second in-line shafts 14 and 16 connected to each 
other by means of a coupling 18. The machines 10 and 12 are secured to a 
floor or other underlying support by means of bolts 20. As is well known, 
for alignment purposes, the bolts 20 can be loosened, and either or both 
of the machines, typically the motor 10, can be moved in horizontal and 
vertical planes in order to achieve alignment between the two shafts 14 
and 16 within predetermined tolerances. Although not shown in FIG. 1, 
shims are usually employed to selectively raise and lower mounting points 
for the machines 10 and 12 during a alignment procedure. 
Also shown in FIG. 1 is a typical alignment fixture 22, comprising a base 
24 fixed to the first shaft 14, an extension bar 26 extending generally 
parallel to the shafts 14 and 16 over the coupling 18, and a dial 
indicator 27 having a plunger 28 contacting the periphery of a portion of 
the second shaft 16 at a point 30, which portion happens to be a hub of 
the coupling 18. It will be appreciated that the base 24 and extension bar 
26 together serve to define a reference point over the point 30 on the 
second shaft 16, which reference point is referenced to the first shaft 
14. As the shafts 14 and 16 are rotated together, relative displacement in 
a radial direction (radial offset) between the reference point and the 
point 30 on the shaft 16 is measured at various angular positions, to thus 
collect a set offset data. This offset data directly reflects distance and 
direction of the distance between the extended centerlines of the two 
shafts 14 and 16 in the transverse plane containing the measurement point. 
In accordance with the reverse indicator alignment method, in order to have 
sufficient data for determining alignment moves, a second set of offset 
data must be collected in another transverse plane, in addition to the 
transverse plane containing the point 30. Although such data could be 
taken by extending the extension bar 26, conveniently the second set of 
data is taken by an alignment fixture extending in the opposite direction, 
that is, fixed to the second shaft 16 and extending laterally to a 
reference point over a point on the first shaft 14. Either a single 
alignment fixture 22 may be employed, and moved from one side to the 
other, or a pair of alignment fixtures may be employed for simultaneous 
readings. When a pair of alignment fixtures 22 are employed, they often 
are positioned on opposite sides of the shafts, that is, 180.degree. apart 
for "out of phase" readings. However, they may also be positioned 
immediately adjacent each other for "in phase" readings, and this "in 
phase" orientation is preferred in the practice of the present invention. 
Although the alignment fixturing in FIG. 1 is for the reverse indicator 
method, the invention is not limited to this particular method, and 
various forms of the face-and-rim method may be employed wherein one 
offset is a radial offset, and the other is an axial offset. Analysis 
procedures for a number of specific methods are well developed. 
It should be noted that another relevant factor is known as "sag", which is 
a result of beam deflection of the extension bar 26 under force of 
gravity. To achieve an accurate alignment, sag is a factor which must be 
independently measured and taken into account in making calculations, as 
is well known in the art. One method of determining sag is disclosed in 
commonly-assigned related application Ser. No. 07/893,102 filed Jun. 3, 
1992 concurrently herewith by Kenneth R. Piety and Daniel L. Nower 
entitled "Alignment Bracket Assembly Integrity Check and Sag 
Determination." 
FIG. 2 illustrates the misalignment component alternatively known as offset 
misalignment, offset, or parallel misalignment. FIG. 3 illustrates the 
misalignment component alternatively known as angular misalignment or 
angularity. Either or both of these misalignments may be present in a 
given situation, and they may exist in any plane. In most situations, the 
misalignment can be corrected by proper moves of one of the machines in 
vertical and horizontal planes, including the use of shims to raise and 
lower individual mounting feet of a particular machine. 
Referring now to FIGS. 4 and 5, shown in greater detail is a system 40 in 
accordance with the invention for aligning the first and second 
co-rotatable in-line shafts 14 and 16 connected respectively to the first 
and second machines 10 and 12. In FIG. 4, an alignment fixture 42 has 
first and second fixture elements 44 and 44' respectively mountable to the 
first and second shafts 14 and 16. In particular, the first fixture 
element 44 comprises a suitably configured mounting block 48 secured to 
the shaft 14 by means of a chain 50 and swing link 52 adjustably connected 
to one end of the chain, and a tightening device 54 connected to the other 
end of the chain 50. The second fixture element 44' correspondingly 
comprises a suitably configured mounting block 48', a chain 50', a swing 
link 52' and a chain tightening device 54'. 
Attached to the mounting blocks 48 and 48' are respective sensor heads 60 
and 60'. Although not depicted in FIG. 4, spacer blocks may be employed 
between the sensor heads 60 and 60' and the mounting blocks 48 and 48' in 
order to provide clearance around a large coupling 18. Such spacer blocks 
(not shown) may be provided as a set of different height spacer blocks for 
selective use to adopt the fixture 42 to various coupling situations. 
The particular fixture 42 depicted in FIG. 4 measures a pair of radial 
offsets at each of a plurality of measurement angular positions as the 
shafts 14 and 16 are rotated together. Thus, a first extension bar 62 is 
firmly affixed to the first sensor head 60 by means of an extension bar 
clamp 64, and the first extension bar 62 extends over the coupling 18, 
generally parallel to the shafts 14 and 16, to a point over the second 
shaft 16. Secured to the distal end 66 of the extension bar 62 by an 
adjustable attachment element 68 is an adjustable tip element 70, the end 
of which is fixedly referenced, neglecting sag, to the first shaft 14. The 
tip element 70 thus defines a reference point referenced to the first 
shaft 14, and is positioned over a particular point on the second shaft 
16. The extension bar clamp 64 and the attachment 68 for the tip element 
70 are adjustable to adopt the fixture 42 to various coupling situations. 
Within the second sensor head 60', and as shown in greater detail in FIG. 
5, is a displacement transducer 72 having a connecting rod 74 terminating 
in a transducer tip 76 which engages the extension bar tip element 70. The 
connecting rod 74 of the displacement transducer 72 is lightly spring 
loaded such that the extension bar tip element 70 and the displacement 
transducer tip 76 are in contact at all times during a measurement 
operation, and the transducer tip 76 and connecting rod 74 translate as 
the distance between the extension bar tip element 70 and the second shaft 
16 varies in a redial direction during rotation of the shafts 14 and 16. 
Firmly affixed to the second sensor head 60' is a second extension bar 62'. 
The second sensor head 60' and second extension bar 62' are essentially 
identical to the first sensor head 60 and first extension bar 62, but in 
the opposite orientation. Elements associated with the second extension 
bar 62' and corresponding with like elements of the first extension bar 62 
as described hereinabove include a second extension bar clamp 64', a 
distal end 66', an adjustable attachment element 68', and an adjustable 
tip element 70' which defines a reference point referenced to the second 
shaft 16 and positioned over a particular point on the first shaft 14. 
Although not visible in FIGS. 4 and 5, it will be appreciated that, within 
the first sensor head 44, is a displacement transducer like the 
displacement transducer 72 having a connecting rod 74' (partially visible 
in FIG. 4) terminating in a second transducer tip 76'. The second 
transducer tip 76' engages the adjustable tip element 70', and the 
transducer tip 76' and connecting rod 74' translate as the distance 
between the second extension bar tip element 70' and the first shaft 14 
varies in a radial direction during rotation of the shafts 14 and 16. 
The system 40 additionally includes an angular position transducer 80 (FIG. 
5) for measuring angular position of the shafts 14 and 16 for alignment 
purposes. The angular position transducer 80 produces an angular position 
data signal. 
While mechanical transducers 72 and 72' are employed for producing the 
measured offset data, it will be appreciated that optical or other forms 
of transducer may be employed, for example employing a beam of laser 
light. 
Another element of the system 40 is an alignment calculator 82 and a 
representative data link 84 for transmitting data signals from the 
transducers 72, 72' and 80 to the alignment calculator 82. In FIG. 4, the 
data link 84 comprises a conventional wire cable. However, various forms 
of wireless data links may be employed, using radio or optical signals to 
transmit the data. Wireless data links have the advantage that the shafts 
14 and 16 may be freely rotated for any number of revolutions without 
entangling the cable 84. (However, this is not necessarily a practical 
disadvantage even with a cable 84.) 
The alignment calculator 82 is microprocessor based and includes suitable 
conventional elements which serve to record measured offset data signals 
based on offset data signals from the displacement transducers 72 and 72' 
at each of at least three measurement angular positions as determined from 
the angular position data signal from the transducer 80. Elements within 
the alignment calculator 82 additionally serve to fit a respective pair of 
sine functions as a function of shaft angular position to the measured 
offset data for the pair of offsets, as is described in greater detail 
hereinbelow with reference to FIGS. 6A and 6B. Elements within the 
alignment calculator 82 also serve to determine, from the sine functions, 
a pair of calculation offset data sets based on respective values of the 
sine functions at each of a plurality of predetermined calculation angular 
positions. Finally, elements within the alignment calculator 82 employ the 
calculation offset data sets to generate instructions for moving at least 
one of the machines 10 and 12 to improve alignment of the shafts 14 and 
16. 
FIG. 6A is a plot, in a Cartesian coordinate system, of measured offset 
data points 90 based on data from the displacement transducer 72, and FIG. 
6B is a corresponding plot of measured offset data points 90' based on 
data from the other displacement transducer, both as a function of angular 
position as determined by the angular position transducer 80. The data 
points 90 and 90' are taken at a plurality of arbitrary angular positions, 
approximately every 10.degree.. The particular data represented in FIGS. 
6A and 6B were taken by two in-phase sensors at a misaligned condition 
characterized as a combination of thirty mils offset misalignment with 
three mils/inch of angular misalignment. 
Although the data points 90 and 90' of FIGS. 6A and 6B were taken at the 
same angular positions, such is not necessary in the practice of the 
invention. As mentioned hereinabove, in the practice of the method of the 
invention, it is preferable to acquire the measured offset data during 
continuous rotation of the shafts 14 and 16 in their normal direction of 
rotation. Nevertheless, the invention may be practiced by manually 
acquiring and recording the data points 90 and 90', using conventional 
dial indicators. 
Respective sine functions represented as sine wave curves 92 and 92' are 
fitted to the sets 90 and 90' of data points. In the Cartesian coordinate 
system plots of FIGS. 6A and 6B, the curves 92 and 92' appear as sine 
waves, as noted. If a polar coordinate system were employed, the curves 92 
and 92' would be circles, with the points 90 and 90' lying generally on 
such circles. 
In a Cartesian coordinate system, the equation for a sine wave is y=B+A sin 
(.theta.+.phi.), where .theta. is the angle from 0.degree. to 360.degree., 
A is the peak amplitude, .phi. is the constant phase offset of the sine 
wave, and B is the constant offset for the amplitude of the curve. The 
values of B, A and .phi. thus define a particular sine wave. 
Various known techniques can be used for fitting the sine functions to the 
data, and the present invention does not require any particular method. By 
way of example, a least square technique may be employed. As another 
example, phase offset could be estimated by locating the zero crossing and 
peak values, and estimating the peak amplitude by calculating the power 
integral of the area under the curve. If a partial revolution of the 
shafts 14 and 16 is all that can be achieved, then a non-linear, iterative 
least square fit to the data may be employed. 
In FIGS. 6A and 6B, it is evident that the data points 90 and 90' do not 
all lie exactly on the plotted sine curves 92 and 92'. These deviations 
from a perfect sine wave are caused by the various error sources mentioned 
hereinabove, such as torsional play in the coupling. By employing multiple 
sample points, overall errors are greatly reduced because a single 
erroneous reading, for example, has far less influence on the overall 
result. Field engineers applying traditional techniques often report that 
the four measured values do not follow the expected theoretical 
relationship. 
At the same time, it will be appreciated that the present invention, which 
involves fitting a sine function to data points, can be employed where 
only three sampled data points for each of the two offsets are available, 
because mathematically a sine function can be fit to three data points. 
Since the measurement angular positions can be the same for each of the 
pair of measured offsets, three measurement positions are all that are 
required to implement the method. 
After the sine functions represented by the curves 92 and 92' of FIGS. 6A 
and 6B are developed, the values of the functions at the predetermined 
calculation angular positions are determined. As explained hereinabove, 
the predetermined calculation angular positions preferably lie in 
geometric planes in which either of the machines 10 and 12 connected to 
the shafts 14 and 16 may be moved. These planes of movement are typically 
horizontal and vertical planes, and the predetermined calculation angular 
positions are thus traditionally 0.degree., 90.degree., 180.degree. and 
270.degree.. 
In FIGS. 6A and 6B, vertical lines 94, 96, 98 and 100 and 94', 96', 98' and 
100' are drawn at the 0.degree., 90.degree., 180.degree. and 270.degree. 
angular positions, intersecting the curves 92 and 92' at points 102, 104, 
106 and 108, and 102', 104', 106' and 108'. The four values of each of the 
calculation offset data sets are thus determined at the intersection 
points 102, 104, 106 and 108, and 102', 104', 106' and 108' as the 
function values at these points. The calculation offset data sets may also 
be viewed as predicted values, since they represent values which would 
presumably be determined by traditional direct measurement at the 
0.degree., 90.degree., 180.degree. and 270.degree. angular positions, but 
more accurately. 
By way of example, the following tables present the results of analysis of 
the data of FIGS. 6A and 6B, taking into account also the measurement of 
sag as is well-documented in the art and described in greater detail in 
the above-identified concurrently-filed application Ser. No. 07/893,102. 
______________________________________ 
DATA ANALYSIS FOR FIG. 6A 
SAG: 0.0010 
FITTED AMETERS: 
OFFS: -0.0604 
AMPL: 0.0571 
PHAS: 75.46 
% ERR: 3 
PREDICTED VALUES: 
0: -0.0051 
90: -0.0461 
180: -0.1157 
270: -0.0748 
SAG ADDED BACK: 
0: -0.0046 
90: -0.0461 
180: -0.1162 
270: 0.0748 
DATA ANALYSIS FOR FIG. 6B 
SAG: 0.0010 
FITTED AMETERS: 
OFFS: 0.0357 
AMPL: 0.0336 
PHAS: 230.76 
% ERR: 8 
PREDICTED VALUES: 
0: 0.0096 
90: 0.0144 
180: 0.0617 
270: 0.0570 
SAG ADDED BACK: 
0: 0.0101 
90: 0.0144 
180: 0.0612 
270: 0.0570 
______________________________________ 
In the above "Data Analysis" charts, and with reference to the sine wave 
equation y=B+A sin (.theta.+.phi.), "OFFS" corresponds to B, "AMPL" 
corresponds to A, and "PHAS" corresponds to .phi.. "%ERR" indicates how 
closely the actual data points fit a sine wave. 
Given the data thus determined or predicted for the 0.degree., 90.degree., 
180.degree. and 270.degree. angular positions, calculation of the required 
machine moves for alignment proceeds in a conventional manner as is well 
known in the art. 
While specific embodiments of the invention have been illustrated and 
described herein, it is realized that numerous modifications and changes 
will occur to those skilled in the art. It is therefore to be understood 
that the appended claims are intended to cover all such modifications and 
changes as fall within the true spirit and scope of the invention.