Methods and apparatus for determining the location of a shaft within a vessel

• US 20020189378A1
• Filed: 05/31/2002
• Published: 12/19/2002
• Est. Priority Date: 11/02/1999
• Status: Active Grant

• Current Assignee: Agilent Technologies, Inc.
• Original Assignee: Varian Incorporated (Agilent Technologies, Inc.)
• Inventors: Haw, Michael R., Fernando, C.J. Anthony, Duckett, Gregory S.
• Granted Patent: US 6,546,821 B2
• Time in Patent Office: Days
• US Class Current: 73/865.9
• CPC Class Codes: B01F 27/805 wherein the stirrers or the receptacles are moved in order to bring them into operative position; Means for fixing the receptacle
• CPC Class Codes: B01F 35/20 Measuring; Control or regulation
• CPC Class Codes: G01B 21/18 for measuring depth
• CPC Class Codes: G01B 21/24 for testing alignment of axes
• CPC Class Codes: G01D 5/26 characterised by optical transfer means, i.e. using infrared, visible, or ultraviolet light
• CPC Class Codes: G01N 13/00 Investigating surface or boundary effects, e.g. wetting power; Investigating diffusion effects; Analysing materials by determining surface, boundary, or diffusion effects scanning-probe techniques or apparatus G01Q
• CPC Class Codes: G01N 2013/006 Dissolution of tablets or the like
• CPC Class Codes: G01N 33/15 Medicinal preparations ; Physical properties thereof, e.g. dissolubility

Abstract

Apparatus and methods for measuring the amount by which the centerline of a shaft disposed in a vessel is offset from the central vertical axis of the vessel, and for measuring the height of such shaft above the inside bottom of the vessel. Apparatus includes a shaft centerline offset measurement device, a shaft height measurement device, and a control/display console. Each measurement device includes a transducer or optical encoder for sensing a displaced position of a biased plunger to which a code strip is mounted. The devices may be combined into a single shaft offset and height measurement device. Improved methods include calculating shaft offset based on a plurality of readings from the transducer, and applying trigonometric relationships. The apparatus and methods are particularly useful in the verification of paddle or basket shafts utilized in dissolution testing stations, so that the dissolution testing protocol complies with government agency guidelines.

Claims

1. An apparatus mountable to a shaft disposed within a vessel and adapted for measuring the magnitude by which the centerline of the shaft is offset from the central axis of the vessel, comprising:

(a) a housing;

(b) a plunger slidably mounted to the housing, the plunger having an outer section extending radially outwardly beyond a wall of the housing, and having means for biasing the plunger radially outwardly;

(c) a transducer operatively mounted to the housing and adapted to encode positions of the plunger and to produce an electrical signal proportional to a change in position resulting from displacement of the plunger; and

(d) means for transferring the signal to means for interpreting the signal.

2. The apparatus according to claim 1 wherein the housing includes a rear face and a longitudinal recess defined by a recess wall extending inwardly toward an interior of the housing from the rear face.

3. The apparatus according to claim 2 wherein the recess wall has a cylindrical profile.

4. The apparatus according to claim 2 wherein the longitudinal recess includes an upper section extending downwardly from a top face of the housing, the upper section having a width greater than a width of an adjacent section of the longitudinal recess, the apparatus further comprising means disposed in the upper section for removably securing the housing to the shaft.

5. The apparatus according to claim 4 wherein the removably securing means includes a clip extending outwardly from the recess wall within the upper section, the clip including a pair of resilient prongs.

6. The apparatus according to claim 5 wherein each prong has an inside surface, and the clip and inside surfaces of the prongs cooperatively define a cylindrical profile.

7. The apparatus according to claim 1 wherein the housing has a bottom face and a groove extending inwardly toward an interior of the housing from the bottom face.

8. The apparatus according to claim 1 wherein the plunger is slidably disposed within the housing and the outer section of the plunger extends radially outwardly through a hole in the wall of the housing.

9. The apparatus according to claim 1 further comprising a plate mounted to the plunger and including a plurality of equally spaced lines readable by the transducer, wherein the transducer is an optical encoder disposed within the housing and the number of lines read by the transducer corresponds to a magnitude of the change in position of the plunger.

10. The apparatus according to claim 1 wherein the signal transferring means includes an electrical conduit and the signal interpreting means includes a console disposed in remote relation to the housing, wherein the electrical conduit communicates with the console.

15. The apparatus according to claim 4 wherein the removably securing means includes a clip extending outwardly from the recess wall within the upper section, the clip including a pair of resilient prongs.

11. An apparatus mountable to a shaft having a paddle or basket disposed within a vessel, the vessel having a central axis and a hemispherical end region, the apparatus adapted for measuring the distance from a distal surface of the paddle or basket to a lowermost point on the inside surface of the hemispherical end region and comprising:

(a) a housing;

(b) a plunger slidably mounted to the housing, the plunger having an outer section extending outwardly beyond a wall of the housing, means for biasing the plunger outwardly, and an end portion extending transversely from the plunger beneath the housing and substantially centered about a central portion of the housing;

(c) a transducer operatively mounted to the housing and adapted to encode positions of the plunger and to produce an electrical signal proportional to a change in position resulting from displacement of the plunger; and

(d) means for transferring the signal to means for interpreting the signal.

12. The apparatus according to claim 11 wherein the housing includes a rear face and a longitudinal recess defined by a recess wall extending inwardly toward an interior of the housing from the rear face.

13. The apparatus according to claim 12 wherein the recess wall has a cylindrical profile.

14. The apparatus according to claim 12 wherein the longitudinal recess includes an upper section extending downwardly from a top face of the housing, the upper section having a width greater than a width of an adjacent section of the longitudinal recess, the apparatus further comprising means disposed in the upper section for removably securing the housing to the shaft.

16. The apparatus according to claim 15 wherein each prong has an inside surface, and the clip and inside surfaces of the prongs cooperatively define a cylindrical profile.

17. The apparatus according to claim 11 wherein the housing has a bottom face and a groove extending inwardly toward an interior of the housing from the bottom face.

18. The apparatus according to claim 11 wherein the plunger is slidably disposed within the housing and the outer section of the plunger extends outwardly through a hole in the wall of the housing.

19. The apparatus according to claim 11 further comprising a plate mounted to the plunger and including a plurality of equally spaced lines readable by the transducer, wherein the transducer is an optical encoder disposed within the housing and the number of lines read by the transducer corresponds to a magnitude of the change in position of the plunger. .

20. The apparatus according to claim 11 wherein the signal transferring means includes a electrical conduit and the signal interpreting means includes a console disposed in remote relation to the housing, wherein the electrical conduit communicates with the console.

21. A system for determining the location of a rotatable shaft in relation to a vessel, the vessel mounted to a rack of a dissolution testing station, the shaft having a first end mounted to the testing station above the vessel, a second end disposed within the vessel and an operative component secured to the second end, the system comprising:

(a) a housing including means for removably mounting the housing to the shaft;

(b) a plunger slidably mounted to the housing, the plunger having an outer section extending radially outwardly beyond a wall of the housing and extendable to an inside lateral surface of the vessel, and having means for biasing the plunger radially outwardly;

(c) a transducer operatively mounted to the housing, the transducer adapted to encode positions of the plunger and to produce an electrical signal proportional to a distance from a reference position to an extended position at which the plunger is in contact with the inside lateral surface of the vessel; and

(d) means for transferring the signal to means for interpreting the signal.

22. A system for determining the location of a rotatable shaft in relation to a vessel, the vessel having a central axis and a hemispherical end region and mounted to a rack of a dissolution testing station, the shaft having a first end mounted to the testing station above the vessel, a second end disposed within the vessel and an operative component secured to the second end, the system comprising:

(a) a spherical object removably disposed in a lowermost point on an inside surface of the hemispherical end region of the vessel;

(b) a housing including means for removably mounting the housing to the shaft;

(c) a plunger slidably mounted to the housing, the plunger having an outer section extending beyond a wall of the housing and extendable to the spherical object, means for biasing the plunger outwardly, and an end portion having an upper surface and a lower surface, the end portion extending transversely from the plunger and between the operative component and the spherical object;

(d) a transducer operatively mounted to the housing, the transducer adapted to encode positions of the plunger and to produce an electrical signal proportional to a distance from a reference position at which the top surface of the end portion of the plunger is biased against the operative component to an extended position at which the lower surface is in contact with the spherical object; and

(e) means for transferring the signal to means for interpreting the signal.

25. The apparatus according to claim 24 wherein the mounting assembly includes a centering ring.

26. The apparatus according to claim 24 further comprising a lateral mounting bracket attached to the mounting assembly, wherein the lateral plunger is slidably mounted to the lateral mounting bracket.

27. The apparatus according to claim 26 wherein the lateral mounting bracket includes a lower bearing member and the lateral plunger includes a lower bearing track slidably mounted to the lower bearing member, and the apparatus further comprises means for biasing the lateral plunger inwardly with respect to the lateral mounting bracket.

28. The apparatus according to claim 26 wherein the lateral transducer is mounted to the lateral mounting bracket and the apparatus further comprises a code strip, the code strip mounted to the lateral plunger and slidable with the lateral plunger in operative relation to the lateral transducer.

29. The apparatus according to claim 26 wherein the mounting assembly includes an upper bearing track, the lateral mounting bracket includes an upper bearing member disposed in movable engagement with the upper bearing track, and the means for biasing the lateral plunger radially outwardly is a means for biasing the lateral mounting bracket radially outwardly along the upper bearing track.

30. The apparatus according to claim 24 further comprising a code strip mounted to the lateral plunger and slidable with the lateral plunger in operative relation to the lateral transducer.

31. The apparatus according to claim 24 further comprising a plurality of lateral plungers and a plurality of corresponding lateral transducers.

32. The apparatus according to claim 31 wherein the plurality of lateral plungers are three in number, the plurality of corresponding lateral transducers are three in number, and each lateral plunger is disposed 120°

from the other lateral plungers.

33. The apparatus according to claim 24 further comprising a code strip mounted to the vertical plunger and slidable with the vertical plunger in operative relation to the vertical transducer.

34. The apparatus according to claim 24 further comprising a linear bearing mounted to the mounting assembly, wherein the vertical plunger is mounted to the linear bearing.

35. The apparatus according to claim 24 wherein the vertical plunger includes a first arm and a second arm, the second arm slidably mounted to the mounting assembly, wherein the upper end portion of the vertical plunger extends transversely from the first arm and the second arm includes a lower end portion extending transversely from the second arm below the upper end portion, the apparatus further comprising means for slidably mounting the first arm in relation to the second arm and means for biasing the upper end portion in spaced-apart relation to the lower end portion.

36. The apparatus according to claim 35 further comprising a vertical transducer mounting bracket secured to the second arm, wherein the vertical transducer is attached to the vertical transducer mounting bracket.

37. The apparatus according to claim 36 further comprising a code strip mounted to the first arm and slidable with the first arm in operative relation to the vertical transducer.

38. The apparatus according to claim 35 wherein the first arm includes a first arcuate section, the upper end portion extends transversely from the first arcuate section, the second arm includes a second arcuate section disposed substantially adjacent to the first arcuate section, and the lower end portion extends transversely from the second arcuate section.

39. The apparatus according to claim 35 further comprising a first linear bearing attached to the mounting assembly and a vertical rail slidably mounted to the first linear bearing, wherein the second arm is mounted to the vertical rail and depends downwardly therefrom, the means for slidably mounting the first arm in relation to the second arm includes a second linear bearing movably connected to the vertical rail, the first arm is attached to the second linear bearing, and the means for biasing the plunger downwardly is disposed in operative communication with the vertical rail.

40. The apparatus according to claim 39 further comprising a vertical transducer mounting bracket connected to the vertical rail, wherein the vertical transducer is attached to the vertical transducer mounting bracket.

41. The apparatus according to claim 40 wherein the second arm is mounted to the vertical transducer mounting bracket and depends downwardly below the vertical transducer mounting bracket and the vertical rail.

42. The apparatus according to claim 24 wherein the signal interpreting means includes a console having logic means for effecting interpretations of the signals produced respectively by the lateral and vertical transducers and having means for displaying the interpretations in human-readable form, and wherein the signal transferring means communicates with the console.

43. The apparatus according to claim 24 wherein the mounting assembly has a bore and is mounted over an open end of the vessel, and the shaft is insertable into the vessel through the bore.

23. A system for determining the location of a shaft in relation to a vessel in which the shaft is disposed, the vessel having a central axis and a hemispherical end region, the system comprising:

(a) a shaft offset measurement device including;

(i) a first housing;

(ii) a first plunger slidably mounted to the first housing, the first plunger having an outer section extending radially outwardly beyond a wall of the first housing, and having means for biasing the first plunger radially outwardly; and

(iii) a first transducer operatively mounted to the first housing and adapted to encode positions of the first plunger and to produce a first electrical signal proportional to a change in position resulting from displacement of the first plunger;

(b) a shaft height measurement device including;

(i) a second housing;

(ii) a second plunger slidably mounted to the second housing, the second plunger having an outer section extending outwardly beyond a wall of the second housing, means for biasing the second plunger outwardly, and an end portion extending transversely from the second plunger beneath the second housing and substantially centered about a central portion of the second housing; and

(iii) a second transducer operatively mounted to the second housing and adapted to encode positions of the second plunger and to produce a second electrical signal proportional to a change in position resulting from displacement of the second plunger;

(d) a console including logic means for effecting interpretations of the first and second electrical signals and means for displaying the interpretations in human-readable form; and

(e) means for transferring the first and second signals to the logic means.

24. An apparatus adapted for measuring the magnitude by which the centerline of a shaft is offset from the central axis of a vessel in which the shaft is disposed, and for measuring the distance from a distal end of the shaft to the lowermost point on an inside surface of a hemispherical end region of the vessel, the apparatus comprising:

(a) a mounting assembly;

(b) a lateral plunger slidably mounted to the mounting assembly and having means for biasing the lateral plunger radially outwardly;

(c) a lateral transducer operatively disposed with respect to the mounting assembly and to the lateral plunger, the lateral transducer adapted to encode positions of the lateral plunger and to produce an electrical signal proportional to a change in position resulting from displacement of the lateral plunger;

(d) a vertical plunger slidably mounted to the mounting assembly, the vertical plunger having means for biasing the vertical plunger downwardly with respect to the mounting assembly and including an upper end portion extending transversely from the vertical plunger;

(e) a vertical transducer operatively disposed with respect to the mounting assembly and to the vertical plunger, the vertical transducer adapted to encode positions of the vertical plunger and to produce an electrical signal proportional to a change in position resulting from displacement of the vertical plunger; and

(f) means for transferring the signals produced respectively by the lateral and vertical transducers to means for interpreting the signals.

44. A method for measuring the amount by which the centerline of a shaft is offset from the central axis of a vessel in which the shaft is to be disposed, comprising the steps of:

(a) mounting a measurement device including a radially outwardly biased plunger to the shaft, the plunger having a settable zero reference position;

(b) inserting the shaft into the vessel at a normal operating position of the shaft, wherein a distal end of the plunger is in contact with a lateral inside surface of the vessel at a first distal plunger position;

(c) defining a first displaced plunger position as a position on the plunger located a distance by which the plunger has moved in relation to the zero reference position, the distance being equal a first displacement magnitude;

(d) measuring the first displacement magnitude by encoding the first displaced plunger position and interpreting the first displaced plunger position in relation to the zero reference position, wherein the first displacement magnitude is proportional to the shaft centerline offset amount;

(e) calculating a value for the shaft centerline offset amount based on the measured first displacement magnitude; and

(f) producing a signal indicative of the shaft centerline offset amount.

45. The method according to claim 44 further comprising the steps of:

(a) rotating the measurement device 120°

after measuring the first displacement magnitude, wherein the distal end of the plunger is in contact with the lateral inside surface of the vessel at a second distal plunger position;

(b) defining a second displaced plunger position as a position on the plunger located a distance by which the plunger has moved in relation to the zero reference position, the distance being equal to a second displacement magnitude;

(c) measuring the second displacement magnitude by encoding the second displaced plunger position and interpreting the second displaced plunger position in relation to the zero reference position;

(d) rotating the measurement device 120°

after measuring the second displacement magnitude, wherein the distal end is in contact with the lateral inside surface at a third distal plunger position

(e) defining a third displaced plunger position as a position on the plunger located a distance by which the plunger has moved in relation to the zero reference position, the distance being equal to a third displacement magnitude;

(f) measuring the third displacement magnitude by encoding the third displaced plunger position and interpreting the third displaced plunger position in relation to the zero reference position; and

(g) wherein the step of calculating the shaft centerline offset amount is based on the measured first, second and third displacement magnitudes.

46. The method according to claim 45 wherein the step of calculating the shaft centerline offset amount includes the steps of:

(a) calculating a first value AB according to a first equation;

$\mathrm{AB}=\sqrt{{\left(d_1\right)}^2+{\left(d_2\right)}^2-2·<br><br>d_1·<br><br>d_2·<br><br>\cos <br><br>\left(\frac{2·<br><br>\mathrm{\pi <br><br>}}{360}·<br><br>120\right)},$

 

wherein d₁ is taken as equal to zero and d₂ is equal to a value for the measured second displacement magnitude relative to the measured first displacement magnitude;

(c) calculating a second value AC according to a second equation;

$\mathrm{AC}=\sqrt{{\left(d_1\right)}^2+{\left(d_3\right)}^2-2·<br><br>d_1·<br><br>d_3·<br><br>\cos <br><br>\left(\frac{2·<br><br>\mathrm{\pi <br><br>}}{360}·<br><br>120\right)},$

 

wherein d₃ is equal to a value for the measured third displacement magnitude relative to the measured first displacement magnitude;

(d) calculating a third value BC according to a third equation;

$\mathrm{BC}=\sqrt{{\left(d_3\right)}^2+{\left(d_2\right)}^2-2·<br><br>d_3·<br><br>d_2·<br><br>\cos <br><br>\left(\frac{2·<br><br>\mathrm{\pi <br><br>}}{360}·<br><br>120\right)};<br><br>$

(e) calculating a fourth value R according to a fourth equation;

$R=\frac{\mathrm{AB}·<br><br>\mathrm{AC}·<br><br>\mathrm{BC}}{4·<br><br>\sqrt{S·<br><br>\left(S-\mathrm{AB}\right)·<br><br>\left(S-\mathrm{AC}\right)·<br><br>\left(S-\mathrm{BC}\right)}},<br><br>\mathrm{wherein}<br><br>\text{\hspace{1em}<br><br>}<br><br>S=\frac{\mathrm{AB}+\mathrm{AC}+\mathrm{BC}}{2};<br><br>$

(f) calculating a fifth value AOB according to a fifth equation;

$\mathrm{AOB}={\cos }^{-1}<br><br>\left(\frac{{\left(\mathrm{AO}\right)}^2+{\left(\mathrm{BO}\right)}^2-{\left(\mathrm{AB}\right)}^2}{2·<br><br>\mathrm{AO}·<br><br>\mathrm{BO}}\right)·<br><br>\frac{360}{2·<br><br>\mathrm{\pi <br><br>}};<br><br>$

(g) calculating a sixth value ABT according to a sixth equation;

$\mathrm{ABT}={\sin }^{-1}<br><br>\left(\frac{d_1·<br><br>\sin <br><br>\left(\frac{120·<br><br>2·<br><br>\mathrm{\pi <br><br>}}{360}\right)}{\mathrm{AB}}\right)·<br><br>\frac{360}{2·<br><br>\mathrm{\pi <br><br>}};<br><br>$

(h) calculating a seventh value ABO according to a seventh equation;

$\mathrm{ABO}=\frac{180-\mathrm{AOB}}{2};<br><br>$

(i) calculating an eighth value OBT according to an eighth equation;

OBT=ABT−

ABO;

and

(j) defining the shaft centerline offset amount as a distance OT according to a ninth equation;

$\mathrm{OT}=\sqrt{{\left(\mathrm{BO}\right)}^2+{\left(d_2\right)}^2-\left(2·<br><br>\mathrm{BO}·<br><br>d_2·<br><br>\cos <br><br>\left(\frac{\mathrm{OBT}·<br><br>2·<br><br>\mathrm{\pi <br><br>}}{360}\right)\right)},$

 

wherein the signal produced is indicative of the calculated distance OT.

47. The method according to claim 45 wherein the step of calculating the shaft centerline offset amount includes the steps of:

(a) defining the lateral inside surface of the vessel as being substantially coincident with a circle ABC, the circle ABC having points A, B and C, wherein the point A corresponds to the first distal plunger position, the point B corresponds to the second distal plunger position, and the point C corresponds to a third distal plunger position;

(b) calculating a first chordal distance AB between the first and second distal plunger positions according to a first equation;

$\mathrm{AB}=\sqrt{{\left(d_1\right)}^2+{\left(d_2\right)}^2-2·<br><br>d_1·<br><br>d_2·<br><br>\cos <br><br>\left(\frac{2·<br><br>\mathrm{\pi <br><br>}}{360}·<br><br>120\right)},$

 

wherein d₁ is equal to a first radial distance AT defined from the centerline of the shaft to the first distal plunger position and is linearly proportional to the measured first displacement magnitude, and d₂ is equal to a second radial distance BT defined from the centerline of the shaft to the second distal plunger position and is linearly proportional to the measured second displacement magnitude;

(c) calculating a second chordal distance AC between the first and third distal plunger positions according to a second equation;

$\mathrm{AC}=\sqrt{{\left(d_1\right)}^2+{\left(d_3\right)}^2-2·<br><br>d_1·<br><br>d_3·<br><br>\cos <br><br>\left(\frac{2·<br><br>\mathrm{\pi <br><br>}}{360}·<br><br>120\right)},$

 

wherein d₃ is equal to a third radial distance CT defined from the centerline of the shaft to the third distal plunger position and is linearly proportional to the measured third displacement magnitude;

(d) calculating a third chordal distance BC between the second and third distal plunger positions according to a third equation;

$\mathrm{BC}=\sqrt{{\left(d_3\right)}^2+{\left(d_2\right)}^2-2·<br><br>d_3·<br><br>d_2·<br><br>\cos <br><br>\left(\frac{2·<br><br>\mathrm{\pi <br><br>}}{360}·<br><br>120\right)};<br><br>$

(e) calculating a theoretical radius R of the circle ABC according to a fourth equation;

$R=\frac{\mathrm{AB}·<br><br>\mathrm{AC}·<br><br>\mathrm{BC}}{4·<br><br>\sqrt{S·<br><br>\left(S-\mathrm{AB}\right)·<br><br>\left(S-\mathrm{AC}\right)·<br><br>\left(S-\mathrm{BC}\right)}},<br><br>\mathrm{wherein}<br><br>\text{\hspace{1em}<br><br>}<br><br>S=\frac{\mathrm{AB}+\mathrm{AC}+\mathrm{BC}}{2};<br><br>$

(f) calculating an angle AOB defined between a radius AO and a radius BO, the radius AO being defined as a radial distance from a theoretical center O of the circle ABC to the point A and the radius BO being defined as a radial distance from the theoretical center O to the point B, each of the radii AO and BO being taken as equal to the theoretical radius R, according to a fifth equation;

$A<br><br>\text{\hspace{1em}<br><br>}<br><br>O<br><br>\text{\hspace{1em}<br><br>}<br><br>B={\cos }^{-1}<br><br>\left(\frac{{\left(A<br><br>\text{\hspace{1em}<br><br>}<br><br>O\right)}^2+{\left(B<br><br>\text{\hspace{1em}<br><br>}<br><br>O\right)}^2-{\left(A<br><br>\text{\hspace{1em}<br><br>}<br><br>B\right)}^2}{2·<br><br>A<br><br>\text{\hspace{1em}<br><br>}<br><br>O·<br><br>B<br><br>\text{\hspace{1em}<br><br>}<br><br>O}\right)·<br><br>\frac{360}{2·<br><br>\mathrm{\pi <br><br>}};<br><br>$

(g) calculating an angle ABT defined between the first chordal distance AB and the radial distance BT according to a sixth equation;

$A<br><br>\text{\hspace{1em}<br><br>}<br><br>B<br><br>\text{\hspace{1em}<br><br>}<br><br>T={\sin }^{-1}\left(\frac{d_1·<br><br>\sin <br><br>\left(\frac{120·<br><br>2·<br><br>\mathrm{\pi <br><br>}}{360}\right)}{A<br><br>\text{\hspace{1em}<br><br>}<br><br>B}\right)·<br><br>\frac{360}{2·<br><br>\mathrm{\pi <br><br>}};<br><br>$

(h) calculating an angle ABO defined between the first chordal distance AB and the radius BO according to a seventh equation;

$A<br><br>\text{\hspace{1em}<br><br>}<br><br>B<br><br>\text{\hspace{1em}<br><br>}<br><br>O=\frac{180-A<br><br>\text{\hspace{1em}<br><br>}<br><br>O<br><br>\text{\hspace{1em}<br><br>}<br><br>B}{2};<br><br>$

(i) calculating an angle OBT defined between the radius BO and the radial distance BT according to an eighth equation;

OBT=ABT−

ABO; and

(j) defining the shaft centerline offset amount as a distance OT between the theoretical center O and the centerline of the shaft, and calculating the distance OT according to a ninth equation;

$O<br><br>\text{\hspace{1em}<br><br>}<br><br>T=\sqrt{{\left(B<br><br>\text{\hspace{1em}<br><br>}<br><br>O\right)}^2+{\left(d_2\right)}^2-\left(2·<br><br>B<br><br>\text{\hspace{1em}<br><br>}<br><br>O·<br><br>d_2·<br><br>\cos <br><br>\left(\frac{O<br><br>\text{\hspace{1em}<br><br>}<br><br>B<br><br>\text{\hspace{1em}<br><br>}<br><br>T·<br><br>2·<br><br>\mathrm{\pi <br><br>}}{360}\right)\right)},$

 

wherein the signal produced is indicative of the calculated distance OT.

48. The method according to claim 44 wherein:

(a) the plunger is permitted to contact the lateral inside surface of the vessel at a plurality of distal plunger positions, wherein each distal plunger position is circumferentially spaced from the other distal plunger positions;

(b) the step of defining the first displaced plunger position includes defining a plurality of displaced plunger positions in addition to the first displaced plunger position, each displaced plunger position being defined when the plunger is disposed at a corresponding one of the plurality of distal plunger positions, and the plurality of displaced plunger positions being positions on the plunger located a plurality of respective distances by which the plunger has moved in relation to the zero reference position, wherein each of the respective distances is equal to a corresponding one of a plurality of displacement magnitudes;

(c) the step of measuring the first displacement magnitude includes measuring a plurality of displacement magnitudes in addition to the first displacement magnitude by encoding each of the displaced plunger positions in relation to the zero reference position; and

(d) the step of calculating the value of the shaft centerline offset amount is based on the plurality of measured displacement magnitudes.

49. The method according to claim 48 further comprising the steps of:

(a) rotating the measurement device 360°

;

(b) measuring the first displacement magnitude and the plurality of displacement magnitudes by sampling, at predetermined time intervals while the measurement device is rotating, a reading from a transducer operatively disposed in relation to the plunger;

(c) selecting the maximum measured displacement magnitude d_MAX and the minimum measured displacement magnitude d_MIN; and

(d) calculating a value OT equal to the shaft centerline offset amount according to the following equation;

$O<br><br>\text{\hspace{1em}<br><br>}<br><br>T=\frac{d_{\mathrm{MAX}}-d_{\mathrm{MIN}}}{2},$

 

wherein the signal produced is indicative of the calculated value OT.

50. The method according to claim 44 further comprising the step of displaying an indication of the shaft centerline offset amount in human-readable form.

51. A method for measuring a shaft height defined as the distance between the distal end of a shaft and the inside lowermost surface of a hemispherical end region of a vessel in which the shaft is to be disposed, comprising the steps of:

(a) mounting a measurement device including a downwardly biased plunger to the shaft, the plunger including an end portion extending below the shaft, the end portion having a predetermined end portion height;

(b) defining a zero reference position of the plunger by urging the end portion against the distal end of the shaft and encoding the zero reference position;

(c) locating the inside lowermost surface of the hemispherical.end region of the vessel by inserting a spherical object having a predetermined diameter into the vessel;

(d) inserting the shaft into the vessel at a normal operating position of the shaft;

(e) permitting the end portion of the plunger to contact the spherical object;

(f) defining a displaced plunger position as a position on the plunger located a distance by which the plunger has moved in relation to the zero reference position in order to contact the spherical object, the distance being equal to a displacement magnitude;

(g) measuring the displacement magnitude by encoding the displaced plunger position and interpreting the displaced plunger position in relation to the zero reference position, wherein the displacement magnitude is proportional to the shaft height;

(h) calculating a value for the shaft height based on the measured displacement magnitude; and

(i) producing a signal indicative of the shaft height.

52. The method according to claim 51 wherein the step of calculating a value for the shaft height includes adding the measured displacement magnitude to a first value representing the height of the end portion of the plunger and a second value representing the diameter of the spherical object.

53. The method according to claim 51 further comprising the step of displaying an indication of the shaft height in human-readable form.

54. A method for measuring the amount by which the centerline of a shaft is offset from the central axis of a vessel in which the shaft is to be disposed, and for measuring a shaft height defined as the distance between the distal end of the shaft and the inside lowermost surface of a hemispherical end region of the vessel, comprising the steps of:

(a) locating the inside lowermost surface of the hemispherical end region of the vessel by inserting a spherical object into the vessel;

(b) mounting a measurement device over the vessel, the measurement device including a lateral plunger and a vertical plunger, the vertical plunger including an end portion;

(c) inserting the shaft into the vessel at a normal operating position of the shaft;

(d) permitting a distal end of the lateral plunger to contact a lateral inside surface of the vessel;

(e) defining a displaced lateral plunger position as a position on the lateral plunger located a lateral distance by which the lateral plunger has moved in relation to a settable zero reference position of the lateral plunger, the lateral distance being equal to a lateral displacement magnitude;

(f) measuring the lateral displacement magnitude by encoding the displaced lateral plunger position and interpreting the displaced lateral plunger position in relation to the zero reference position of the lateral plunger, wherein the lateral displacement magnitude is proportional to the shaft centerline offset amount;

(g) calculating a value for the shaft centerline offset amount based on the measured lateral displacement magnitude;

(h) producing a signal indicative of the shaft centerline offset amount;

(i) permitting the end portion of the vertical plunger to contact the spherical object;

(j) defining a displaced vertical plunger position as a position on the vertical plunger located a vertical distance by which the vertical plunger has moved in relation to a predetermined zero reference position of the plunger, the vertical distance being equal to a vertical displacement magnitude;

(k) measuring the vertical displacement magnitude by encoding the displaced vertical plunger position and interpreting the displaced vertical plunger position in relation to the zero reference position of the vertical plunger, wherein the vertical displacement magnitude is proportional to the shaft height;

(l) calculating a value for the shaft height based on the measured vertical displacement magnitude; and

(m) producing a signal indicative of the shaft height.