Unbalance measuring method and unbalance measuring apparatus

Abstract

To provide an unbalance measuring method and an unbalance measuring device capable of accurately determining correction unbalance of an elastic rotor having a non-uniform cross-sectional shape in the axial direction without rotating the elastic rotor at high speed.SOLUTION: In unbalance measurement, an elastic rotor 3 having a non-uniform cross-sectional shape at the X1 in the axial direction is supported at two points by bearings 7. The vibration of the elastic rotor 3 when the elastic rotor 3 is rotated is detected by a vibration detection part. The correction unbalance of the elastic rotor 3 in the correction surface C1, the correction surface C2, and the correction surface C3 is determined based on the positions z1, z2, z3 of the correction surface C1, the correction surface C2, and the correction surface C3 of the elastic rotor 3, the fourth or higher order approximate expression of the first bending eigenfunction (Φ 1 (z)) of the elastic rotor 3, and the detected vibration of the elastic rotor 3.SELECTED DRAWING: Figure 6

Description

[Technical Field]

[0001] The present invention relates to an unbalance measuring method and an unbalance measuring device. [Background technology]

[0002] In addition to rigid rotors, flexible rotors are also subject to balancing (Patent Document 1). Flexible rotors include not only those with a uniform cross-sectional shape in the axial direction, but also those with a non-uniform cross-sectional shape in the axial direction.

[0003] Conventionally, high-speed balancing has been employed to balance elastic rotors having a non-uniform cross-sectional shape in the axial direction (hereinafter sometimes referred to as "non-uniform cross-section elastic rotors"), in which balancing is performed while rotating the non-uniform cross-section elastic rotor at the same high rotation speed as during actual operation (for example, approximately 5000 rpm or more and approximately 8000 rpm or less). [Prior art documents] [Patent documents]

[0004] [Patent Document 1] Japanese Patent Application Laid-Open No. 04-351348 (Fig. 4) Summary of the Invention [Problem to be solved by the invention]

[0005] However, a high-speed balancing machine capable of performing high-speed balancing is a large device equipped with a vacuum chamber and the like, and is very expensive. Furthermore, since only a limited number of testing institutions are equipped with high-speed balancing machines, there is also the cost of transporting the non-uniform cross-section elastic rotor to the testing institution. In other words, performing high-speed balancing requires a great deal of cost.

[0006] The present inventors are studying how to balance a non-uniform cross-section flexible rotor with high precision by low-speed balancing, in which the measurement object is rotated at a low speed in the range of approximately 200 rpm to approximately 1000 rpm. If a non-uniform cross-section flexible rotor can be balanced with high precision by low-speed balancing, there is no need to use the high-speed balancing machine described above. In this case, the cost required for balancing a non-uniform cross-section flexible rotor can be reduced.

[0007] Therefore, one object of the present invention is to provide an unbalance measuring method and an unbalance measuring device that can accurately determine the corrected unbalance of a flexible rotor having an axially non-uniform cross-sectional shape without rotating the flexible rotor at high speed. [Means for solving the problem]

[0008] One embodiment of the present invention provides an unbalance measurement method including the steps of supporting a flexible rotor (3) having a non-uniform cross-sectional shape in an axial direction (X1) at two points, detecting vibration of the elastic rotor when the elastic rotor is rotated, and an unbalance determination step of determining corrected unbalance (U1, U2, U3) of the elastic rotor on the first correcting plane (C1), the second correcting plane (C2), and the third correcting plane (C3) based on positions (z1, z2, z3) of the first correcting plane (C1), the second correcting plane (C2), and the third correcting plane (C3) of the elastic rotor, a fourth-order or higher approximation of an eigenfunction (Φ1(z)) of the first bending of the elastic rotor, and the detected vibration of the elastic rotor.

[0009] According to this method, the eigenfunction of the first-order bending of the flexible rotor can be expressed by an approximate expression of fourth order or higher. The approximate expression of the eigenfunction of the first-order bending is obtained, and the corrected unbalance of the flexible rotor on the first, second, and third correcting planes is determined based on the approximate expression, the positions of the first, second, and third correcting planes, and the detected vibration of the flexible rotor. This allows the corrected unbalance of the flexible rotor to be determined with high accuracy. This allows the corrected unbalance of the flexible rotor to be determined with high accuracy without rotating the flexible rotor, which has an axially non-uniform cross-sectional shape, at high speed.

[0010] In one embodiment of the present invention, the unbalance determining step determines the corrected unbalance of the flexible rotor on the first correction plane, the second correction plane, and the third correction plane using the following determinant (1).

[0011]

number

[0012] Here, U1, U2, and U3 are the corrected imbalances of the flexible rotor on the first correcting plane, the second correcting plane, and the third correcting plane, respectively. Φ1(z) is an approximate expression of the eigenfunction of the first bending of the flexible rotor, and is a function of the axial position z of the flexible rotor (distance from the axial reference position). z1, z2, and z3 are the axial positions z of the first correcting plane, the second correcting plane, and the third correcting plane, respectively, and all represent the distance from the axial reference position. U S is the static unbalance occurring in the flexible rotor. l is the axial distance between the two supports (axial distance). Note that in equation (1), the influence of the overhanging part of the flexible rotor is ignored.

[0013] In one embodiment of the present invention, the unbalance determining step determines the corrected unbalance of the flexible rotor on the first correcting plane, the second correcting plane, and the third correcting plane using the following determinant (2).

[0014]

number

[0015] where U1, U2, U3, Φ1(z), z1, z2, z3, U S and l are as already explained. Furthermore, L is the axial length of the flexible rotor. Note that equation (2) takes into account the influence of the overhanging portion of the flexible rotor.

[0016] In one embodiment of the present invention, the unbalance determining step determines the corrected unbalance of the flexible rotor on the first correcting plane, the second correcting plane, and the third correcting plane using the following determinant (3).

[0017]

number

[0018] where U1, U2, U3, Φ1(z), z1, z2, z3, U S , l, and L are as already explained. Furthermore, a and b are constants. Note that equation (3) takes into consideration the influence of the overhanging portion of the flexible rotor.

[0019] In one embodiment of the present invention, the approximate expression of the eigenfunction of the first-order bending of the flexible rotor is preferably a fifth-, sixth-, or seventh-order polynomial.

[0020] "Method of measuring imbalance" may be read as "method of determining corrected imbalance."

[0021] One embodiment of the present invention provides an unbalance measuring device (1) including: a support unit that supports at two points a flexible rotor (3) having a non-uniform cross-sectional shape in the axial direction (X1); a vibration detection unit (9) that detects vibration of the elastic rotor when the elastic rotor is rotated; and an unbalance determination unit (13) that determines corrected unbalance (U1, U2, U3) of the elastic rotor in the first correcting plane (C1), the second correcting plane (C2), and the third correcting plane (C3) based on positions (z1, z2, z3) of the first correcting plane (C1), the second correcting plane (C2), and the third correcting plane (C3) of the elastic rotor, a fourth-order or higher approximation of an eigenfunction (Φ1(z)) of the primary bending of the elastic rotor, and the detected vibration of the elastic rotor.

[0022] Furthermore, one embodiment of the present invention provides an unbalance measurement method including the steps of supporting an elastic rotor (3) having a non-uniform cross-sectional shape in the axial direction (X1) at two points, detecting vibration of the elastic rotor when the elastic rotor is rotated, and determining positions (z1, z3) of the first and second correcting planes and corrected unbalance (U1, U3) of the elastic rotor on the first and second correcting planes based on a fourth-order or higher approximation of an eigenfunction (Φ1(z)) of the primary bending of the elastic rotor and the detected vibration of the elastic rotor, so that the unbalance of the primary bending of the elastic rotor can be corrected using only the first and second correcting planes (C1) and C3.

[0023] According to this method, the eigenfunction of the first-order bending of the flexible rotor can be expressed by an approximate expression of fourth order or higher. By obtaining the approximate expression of the eigenfunction of the first-order bending and based on the obtained expression and the detected vibration of the flexible rotor, the positions of the first and second correcting planes and the respective corrected unbalances are determined so that the unbalance of the first-order bending of the flexible rotor can be corrected using only the first and second correcting planes. This makes it possible to accurately determine the positions of the first and second correcting planes and the respective corrected unbalances without rotating the flexible rotor, which has a cross-sectional shape that is non-uniform in the axial direction, at high speed.

[0024] In one embodiment of the present invention, the unbalance determination step uses the following determinant (4), finds positions of the first correction plane and the second correction plane that minimize U2 (so that U2 becomes zero) in equation (4), and determines the corrected unbalance of the flexible rotor on the first correction plane and the second correction plane.

[0025]

number

[0026] Here, U1 and U3 are the corrected unbalance of the flexible rotor at the first correcting plane and the second correcting plane, respectively. Φ1(z) is an approximate expression of the eigenfunction of the first bending of the flexible rotor, and is a function of the axial position z of the flexible rotor (distance from a reference position in the axial direction). z1 and z3 are the axial positions z of the first correcting plane and the second correcting plane, respectively, and both represent the distance from a reference position in the axial direction. z2 is the axial position z of an intermediate correcting plane between the axial positions of the first correcting plane and the second correcting plane. U2 is the corrected unbalance of the flexible rotor at the intermediate correcting plane. U S is the static unbalance occurring in the flexible rotor. l is the axial distance between the two supports (axial distance). Note that in equation (4), the influence of the overhanging part of the flexible rotor is ignored.

[0027] More specifically, the imbalance determination step determines the positions of the first and second correction planes that minimize the corrected imbalance U2 by solving the optimization problem shown in the following equation (5) based on equation (4).

[0028]

number

[0029] In addition, in one embodiment of the present invention, the unbalance determination step uses the following determinant (6), finds positions of the first correcting plane and the second correcting plane that minimize U2 (so that U2 becomes zero) in equation (6), and determines the corrected unbalance of the flexible rotor on the first correcting plane and the second correcting plane.

[0030]

number

[0031] where U1, U2, U3, Φ1(z), z1, z2, z3, U Sand l are as already explained. Furthermore, L is the axial length of the flexible rotor. Note that equation (6) takes into account the influence of the overhanging portion of the flexible rotor.

[0032] In addition, in one embodiment of the present invention, the unbalance determination step uses the following determinant (7), finds positions of the first correcting plane and the second correcting plane that minimize U2 (so that U2 becomes zero) in equation (7), and determines the corrected unbalance of the flexible rotor on the first correcting plane and the second correcting plane.

[0033]

number

[0034] where U1, U2, U3, Φ1(z), z1, z2, z3, U S , l, and L are as explained above. Furthermore, a and b are constants. Note that equation (7) takes into consideration the influence of the overhanging portion of the flexible rotor.

[0035] More specifically, the imbalance determination step determines the positions of the first and second correction planes that minimize the corrected imbalance U2 by solving the optimization problem shown in the following equation (8) based on equations (6) and (7).

[0036]

number

[0037] In one embodiment of the present invention, the approximate expression of the eigenfunction of the first-order bending of the flexible rotor is preferably a fifth-, sixth-, or seventh-order polynomial.

[0038] Furthermore, one embodiment of the present invention provides an unbalance measuring device (1) including: a support unit (7) that supports at two points an elastic rotor (3) having a non-uniform cross-sectional shape in the axial direction (X1); a vibration detection unit (9) that detects vibration of the elastic rotor when the elastic rotor is rotated; and a corrected imbalance determination unit (13) that determines positions (z1, z3) of the first correcting plane (C1) and the second correcting plane (C3) and the corrected unbalance (U1, U3) of the elastic rotor on the first correcting plane (C1) and the second correcting plane (C3) based on a fourth-order or higher approximation of an eigenfunction (Φ1(z)) of the first bending of the elastic rotor and the detected vibration of the elastic rotor, so that the unbalance of the first bending of the elastic rotor can be corrected using only the first correcting plane (C1) and the second correcting plane (C3).

[0039] Furthermore, in the above, numbers in parentheses represent reference symbols of corresponding components in the embodiments described below, but these reference symbols are not intended to limit the scope of the claims. [Brief explanation of the drawings]

[0040] [Figure 1A] FIG. 1A is a schematic front view of an unbalance measuring device used in an unbalance measuring method according to one embodiment of the present invention. [Figure 1B] FIG. 1B is a schematic right side view of the imbalance measuring device. [Figure 2] FIG. 2 is an electric circuit diagram for explaining the configuration of the control device. [Figure 3] FIG. 3 is a schematic front view showing the configuration of the elastic rotor. [Figure 4] FIG. 4 is a schematic perspective view for explaining three-surface correction of the elastic rotor. [Figure 5] FIG. 5 is a graph showing an example of an eigenfunction of the first bending of the flexible rotor. [Figure 6] FIG. 6 is a flow chart showing the flow of the first imbalance measurement of the elastic rotor by the imbalance measuring device. [Figure 7]FIG. 7 is a diagram showing the results of a first simulation of the resonance amplitude before and after the modification of the flexible rotor. [Figure 8] FIG. 8 is a flow chart showing the flow of the second imbalance measurement of the elastic rotor by the imbalance measuring device. [Figure 9] FIG. 9 is a schematic diagram showing a first example of the positional relationship between the pair of bearings and the correction surfaces C1 and C3. [Figure 10] FIG. 10 is a schematic diagram showing a second example of the positional relationship between the pair of bearings and the correction surfaces C1 and C3. [Figure 11] FIG. 11 is a schematic diagram showing a third example of the positional relationship between the pair of bearings and the correction surfaces C1 and C3. [Figure 12] FIG. 12 is a schematic diagram showing a fourth example of the positional relationship between the pair of bearings and the correction surfaces C1 and C3. [Figure 13] FIG. 13 is a diagram showing the results of a second simulation of the unbalance correction effect of the flexible rotor. [Figure 14] FIG. 14 is a diagram showing the results of a third simulation of the unbalance correction effect of the flexible rotor. DETAILED DESCRIPTION OF THE INVENTION

[0041] An embodiment of the present invention will be described in detail below. Fig. 1A is a schematic front view of an imbalance measuring device 1 according to one embodiment of the present invention. Fig. 1B is a schematic right side view of the imbalance measuring device 1. Fig. 2 is an electric circuit diagram for explaining the configuration of a control device 10.

[0042] 1A and 1B, the unbalance measurement device 1 is a device that measures the unbalance (dynamic unbalance) of a test object 2 (elastic rotor 3). The unbalance measurement device 1 may also be called a balancing machine.

[0043] The unbalance measurement device 1 is a horizontal balancing machine that measures the unbalance of a test object 2 in a horizontal position. The direction in which the rotation axis 2a that forms the center of rotation of the test object 2 extends (axial direction) is along the horizontal direction. The test object 2 is shaft-shaped with the rotation axis 2a that forms the center of rotation. The elastic rotor 3 that serves as the test object 2 has a non-uniform cross-sectional shape in the axial direction.

[0044] The imbalance measuring device 1 includes a base 4, a spring 5, a vibration frame 6, a pair of bearings 7 as support units, a drive motor 8, a vibration detection unit 9, and a control unit 10. The base 4 is fixed to the floor. The vibration frame 6 is disposed above the base 4. The vibration frame 6 is, for example, in the shape of a rectangular ring.

[0045] The springs 5 ​​are plate- or rod-shaped and extend vertically. The springs 5 ​​are installed between the vibration frame 6 and the base 4. As a result, the vibration frame 6 is supported by the base 4 via the springs 5 ​​so that it can vibrate in the horizontal direction. For example, a total of four springs 5 ​​are arranged, one at each of the four corners of the vibration frame 6 in a plan view.

[0046] The pair of bearings 7 are arranged on the upper surface of the vibration frame 6 with an axial distance 1 between them in the left-right direction X along the horizontal direction. The pair of bearings 7 support both axial end portions 2b of the test piece 2 so that the test piece 2 can rotate freely around a rotation axis 2a along the left-right direction X. The bearings 7 may also be called bearing stands.

[0047] The vibration frame 6 is a flat plate extending horizontally. The vibration frame 6 is disposed directly above the base 4. The vibration frame 6 is connected to the upper ends of a plurality of springs 5. In other words, a plurality of springs 5 ​​are disposed between the vibration frame 6 and the base 4. As a result, the vibration frame 6 is supported by the plurality of springs 5 ​​so as to be able to vibrate in the horizontal direction.

[0048] Referring to FIG. 1B, each bearing 7 has a receiving portion 7a. Each bearing 7 has a plate-shaped bearing body that is elongated in the vertical direction. Each receiving portion 7a is formed so as to curve downward in a concave shape on the upper end surface of the bearing body of each bearing 7. Both end portions 2b of the test subject 2 are fitted into the two receiving portions 7a from above. In this way, the test subject 2 is rotatably supported by each bearing 7, and is indirectly set on the vibration frame 6 from above via the bearings 7.

[0049] The drive motor 8 includes, for example, an electric motor. The drive force of the drive motor 8 is transmitted to the test subject 2 via a power transmission member (not shown) such as a drive belt, causing the test subject 2 to rotate. If the test subject 2 is unbalanced, the vibration frame 6 vibrates along with the test subject 2. The drive motor 8 is attached to the vibration frame 6 as shown in FIG. 1B, but may be attached to another member (base 4) as long as it does not affect the accuracy of the unbalance measurement.

[0050] 1A and 1B, the vibration detection unit 9 detects vibrations of the vibration frame 6. The vibration detection unit 9 may also be referred to as a vibration pickup. In this embodiment, a pair of vibration detection units 9 is provided. Each of the pair of vibration detection units 9 is attached to the upper end of a pair of left and right fixing members 11 that are fixed to the base 4 and extend upward. Of the two vibration detection units 9, the one on the left side shown in FIG. 1A is referred to as vibration detection unit 9A, and the one on the right side shown in FIG. 1A is referred to as vibration detection unit 9B. In this embodiment, the vibration detection unit 9A is located to the right of the left bearing unit 7, and the vibration detection unit 9B is located to the left of the right bearing unit 7.

[0051] 1B, each vibration detection unit 9 is connected to the front end (left end in FIG. 2) of a vibration transmission rod 12 that extends horizontally toward the detection target 6a, which is the front end face of the vibration frame 6. The rear end (right end in FIG. 2) of the vibration transmission rod 12 abuts against the detection target 6a of the vibration frame 6. Vibrations generated in the vibration frame 6 are transmitted to each vibration detection unit 9 via the vibration transmission rod 12 and detected by each vibration detection unit 9.

[0052] Referring to Fig. 2, the control device 10 includes a calculation unit 13, a memory unit 14, and an input unit 15. The calculation unit 13 includes a processor such as a CPU. The calculation unit 13 may be configured by a microcomputer or the like including a CPU, a ROM, a timer, etc. The memory unit 14 is electrically connected to the calculation unit 13. The memory unit 14 stores various information used for measurement in the imbalance measuring device 1. The input unit 15 is electrically connected to the calculation unit 13 and the memory unit 14. The input unit 15 includes an operation unit such as a liquid crystal touch panel or a keyboard.

[0053] 2, the control device 10 further includes an A / D converter, an amplifier, a memory, a display, etc. The control device 10 controls the driving of the drive motor 8 described above.

[0054] 3 is a schematic front view showing the configuration of the elastic rotor 3. The axial direction of the elastic rotor 3 is defined as the axial direction X1. The elastic rotor 3 has a non-uniform cross-sectional shape in the axial direction X1. In other words, the cross-sectional shape of the elastic rotor 3 changes (is not uniform) in the axial direction X1.

[0055] Referring to Figure 3, an elastic rotor 3 refers to a rotor in which significant elastic deflection occurs in the rotating shaft within the range of rotational speeds used. The distribution of couple unbalance in an elastic rotor 3 is uneven. Examples of elastic rotors 3 include compressor rotors, pump rotors, steam turbine rotors, and turbine generator rotors. The unbalance of an elastic rotor 3 must be corrected by balancing three or more correction planes (multi-plane correction of three or more planes).

[0056] The elastic rotor 3 includes end portions 3b on both sides and a central portion 3a sandwiched between the end portions 3b in the axial direction X1. The central portion 3a has a uniform cross-sectional shape in the axial direction X1. In other words, the cross-sectional shape of the central portion 3a varies (is not uniform) in the axial direction X1.

[0057] The central portion 3a includes a plurality of cylindrical portions (first diameter portion 31 to ninth diameter portion 39) arranged in the axial direction X1 and having different diameters (radii). The diameters of the cylindrical portions (first diameter portion 31 to ninth diameter portion 39) adjacent to each other in the axial direction X1 are different from each other. In this embodiment, the plurality of cylindrical portions includes, from left to right in FIG. 3, the first diameter portion 31, the second diameter portion 32, the third diameter portion 33, the fourth diameter portion 34, the fifth diameter portion 35, the sixth diameter portion 36, the seventh diameter portion 37, the eighth diameter portion 38, and the ninth diameter portion 39. The diameters of the first diameter portion 31, the second diameter portion 32, the third diameter portion 33, the fourth diameter portion 34, the fifth diameter portion 35, the sixth diameter portion 36, and the seventh diameter portion 37 increase in this order. The diameters of the seventh diameter portion 37, the eighth diameter portion 38 and the ninth diameter portion 39 decrease in this order. The first diameter portion 31 and the ninth diameter portion 39 have the same diameter.

[0058] In the example shown in Figure 3, the length L (i.e., axial length) of the elastic rotor 3 in the axial direction X1 is approximately 4725 mm, and the axial distance l (lowercase "L"; the distance between the pair of bearing portions 7) is approximately 4275 mm. The longitudinal elastic modulus of the elastic rotor 3 is 211 GPa, and the transverse elastic modulus of the elastic rotor 3 is 81.2 GPa.

[0059] As described above, the unbalance of the flexible rotor 3 needs to be corrected by balancing three or more correction planes (multi-plane correction of three or more planes). In the first imbalance measurement (first unbalance measurement method, first corrected unbalance determination method) performed by the imbalance measurement device 1, three correction planes (correction plane C1, correction plane C2, and correction plane C3) are set on the flexible rotor 3, and the unbalance of the flexible rotor 3 is corrected by corrections on these three correction planes C1, C2, and C3. The correction planes C1, C2, and C3 are arranged in this order from the left side of FIG. 3 and are spaced apart from each other in the axial direction X1. In the first unbalance measurement, the axial positions of the correction planes C1, C2, and C3 are determined in advance prior to the unbalance measurement.

[0060] With the elastic rotor 3 supported by two bearing portions 7, the axial direction X1 of the elastic rotor 3 coincides with the left - right direction X (FIGS. 1A and 1B) of the unbalance measuring device 1. The position z in the axial direction X1 of one bearing portion 7 (the left - hand bearing portion 7 shown in FIG. 3) is 0 (z = 0), and the position z in the axial direction X1 of the other bearing portion 7 (the right - hand bearing portion 7 shown in FIG. 3) is L (z = l). That is, the positions of the bearing portions 7 are the reference positions. At this time, the position z of the correction plane C1 is z1 (z = z1), the position z of the correction plane C2 is z2 (z = z2), and the position z of the correction plane C3 is z3 (z = z3) (0 < z1 < z2 < z3 < l). "l" is the axial distance between the pair of bearing portions 7.

[0061] FIG. 4 is a schematic perspective view for explaining the three - plane correction of the elastic rotor 3. FIG. 5 is a graph of an example of the first - order bending eigenfunction Φ1(z) of the elastic rotor 3. Although the elastic rotor 3 has a non - uniform cross - sectional shape in the axial direction X1, for the sake of clarity, in FIG. 4, the elastic rotor 3 is shown as having a uniform (constant) cross - sectional shape in the axial direction X1 (cylindrical).

[0062] Referring to FIG. 4, the correction unbalance U1 at the correction plane C1 (z = z1) is a vector directed radially from the rotation axis 3c of the elastic rotor 3. The correction unbalance U2 at the correction plane C2 (z = z2) is a vector directed radially outward from the rotation axis 3c of the elastic rotor 3. The correction unbalance U3 at the correction plane C3 (z = z3) is a vector directed radially outward from the rotation axis 3c of the elastic rotor 3. The general unit of correction unbalance is g·mm. The same applies to other unbalances.

[0063] The correction unbalances U1, U2, and U3 can be expressed by the following determinants respectively.

[0064]

Equation

[0065] Here, Φ1(z) is an approximation of the eigenfunction Φ1(z) of the primary bending of the elastic rotor 3, and is a function of the axial position z (distance from the reference position in the axial direction X1). z1, z2, and z3 are the positions z in the axial direction X1 of the correction plane C1, correction plane C2, and correction plane C3, and all represent the distance from the reference position in the axial direction X1. U S is the static unbalance occurring in the elastic rotor 3. 1 is the axial distance 1 between the pair of bearing portions 7.

[0066] When the flexible rotor 3 has a uniform cross-sectional shape in the axial direction X1, the eigenfunction of the first bending can be expressed as a sine function. However, in the case of an elastic rotor 3 having a non-uniform cross-sectional shape in the axial direction X1, the bending and shear deformation is not uniform in the axial direction and is not symmetrical, so the eigenfunction Φ1(z) of the first bending of the elastic rotor 3 cannot be expressed as a sine function. In this case, the eigenfunction Φ1(z) of the first bending of the elastic rotor 3 can be expressed using an approximate equation of a polynomial of fourth order or higher.

[0067] When the flexible rotor 3 has a primary bending eigenfunction Φ1(z) as shown in FIG. 5, this primary bending eigenfunction Φ1(z) can be expressed by the following approximate formula (10). The approximate formula (10) is, for example, a quintic polynomial approximate formula. z is the position in the axial direction X1. l is the axial distance.

[0068]

number

[0069] 6 is a flow chart showing the flow of the first imbalance measurement of the elastic rotor 3 by the imbalance measurement device 1. The first imbalance measurement will be described with reference to FIGS. 1A to 4 and 6.

[0070] An operator carries the elastic rotor 3 into the unbalance measuring device 1. As shown in FIG. 3, the end portions 3b of the carried-in elastic rotor 3 are supported by the pair of bearings 7 (step S1 in FIG. 6).

[0071] Furthermore, the positions of three correction planes (position z1 of correction plane C1, position z2 of correction plane C2, and position z3 of correction plane C3) are set on the elastic rotor 3. In the first unbalance measurement, the correction planes C1, C2, and C3 may be referred to as the first correction plane, the second correction plane, and the third correction plane, respectively.

[0072] Specifically, the worker inputs three positions z1, z2, and z3 by operating the input unit 15. The input three positions z1, z2, and z3 are stored in the storage unit 14 of the control device 10 (step S2 in FIG. 6).

[0073] The positions z1, z2, and z3 of the three correction planes may be provided to the control device 10 prior to the unbalance measurement. That is, they may be stored in advance in the memory unit 14 of the control device 10.

[0074] Furthermore, an approximation formula for the eigenfunction Φ1(z) of the first bending of the elastic rotor 3 to be measured (an approximation formula of a polynomial of fourth degree or higher, for example, the approximation formula of the above formula (10)) is provided to the control device 10. Specifically, the operator inputs the approximation formula for the eigenfunction Φ1(z) of the first bending of the elastic rotor 3 via the input unit 15. The input approximation formula is stored in the memory unit 14 of the control device 10 (step S3 in FIG. 6).

[0075] An approximation of the eigenfunction Φ1(z) of the primary bending of the elastic rotor 3 may be provided to the control device 10 prior to unbalance measurement. In other words, it may be stored in advance in the storage unit 14 of the control device 10.

[0076] Next, the unbalance measurement device 1 starts unbalance measurement (step S4 in FIG. 6). Specifically, the control device 10 drives the drive motor 8 to rotate the test object 2 (i.e., the elastic rotor 3) (step S4 in FIG. 6). At this time, the rotation speed of the test object 2 (elastic rotor 3) is approximately 200 rpm or more and approximately 1000 rpm or less.

[0077] Furthermore, the calculation unit 13 calculates the static unbalance U generated in the elastic rotor 3 based on the vibration detected by the vibration detection unit 9 during rotation of the test object 2 (elastic rotor 3). S and the couple imbalance U C Calculate.

[0078] Then, the calculation unit 13 calculates the corrected unbalance U1 on the correction plane C1, the corrected unbalance U2 on the correction plane C2, and the corrected unbalance U3 on the correction plane C3 (step S5 in FIG. 6). Specifically, the calculation unit 13 calculates the corrected unbalance U1 on the correction plane C1, the corrected unbalance U2 on the correction plane C2, and the corrected unbalance U3 on the correction plane C3 by using the approximate expressions of the positions z1, z2, z3 and the eigenfunction Φ1(z) of the primary bending stored in the storage unit 14, and the calculated static unbalance U S is substituted into equation (9), which is the determinant. As a result, the corrected unbalance U1, the corrected unbalance U2, and the corrected unbalance U3 are calculated, and the corrected unbalances U1, U2, and U3 are determined (step S5 in FIG. 6). This completes the first unbalance measurement shown in FIG. 6.

[0079] For example, let us assume that the approximate expression of the eigenfunction Φ1(z) of the first bending of the flexible rotor 3 given to the control device 10 is the above-mentioned expression (10). In this case, if the positions z1, z2, and z3 of the above-mentioned three correction planes are "0.157894×l", "0.4210053×l", and "0.842105×l", respectively, then

[0080]

number

[0081]

number

[0082] Thereafter, based on the results of the unbalance measurement, the elastic rotor 3 is corrected on the correction plane C1, the correction plane C2, and the correction plane C3 by an unbalance correction device (not shown) (three-plane correction). Referring to FIG. 4, by performing unbalance corrections on the correction plane C1 (z=z1), the correction plane C2 (z=z2), and the correction plane C3 (z=z3) with the corrected unbalance U1, the corrected unbalance U2, and the corrected unbalance U3, respectively, the primary bending of the elastic rotor 3 can be corrected. Furthermore, on the correction plane C1, the correction plane C2, and the correction plane C3, U1' (=U1+U C , see Figure 4), U2 and U3´ (=U3-U C 4), the dynamic unbalance of the rigid rotor and the primary bending unbalance of the flexible rotor 3 can be corrected simultaneously.

[0083] As described above, according to the first unbalance measurement, an approximate expression of the eigenfunction Φ1(z) of the primary bending is obtained, and based on this, the positions z1, z2, z3 of the correcting planes C1, C2, and C3, and the detected vibration of the elastic rotor 3, the corrected unbalances U1, U2, and U3 on the correcting planes C1, C2, and C3, respectively, are determined.

[0084] In this case, the unbalance of the elastic rotor 3 can be corrected with high precision by low-speed balancing, which is performed while rotating the elastic rotor 3 at a low speed (approximately 200 rpm or more and approximately 1000 rpm or less). This makes it possible to accurately determine the corrected unbalance of the elastic rotor 3 without rotating the elastic rotor 3, which has a non-uniform cross-sectional shape in the axial direction X1, at high speed.

[0085] FIG. 7 shows the results of a first simulation of the resonance amplitude of the flexible rotor 3 (FIG. 3) before and after correction. FIG. 7 shows the values ​​of the resonance amplitude of the elastic primary bending at node P1 (FIG. 3) and node P2 (FIG. 3). "Initial" in FIG. 7 indicates the case where no correction is performed, "2-plane" in FIG. 7 indicates two-plane balancing (dynamic balancing), and "3-plane correction" in FIG. 7 indicates the case where unbalance correction is performed using the corrected unbalance determined using the first unbalance measurement of the present disclosure (hereinafter referred to as "unbalance correction based on the first unbalance measurement"). Note that the influence of the overhanging portion of the flexible rotor 3 is ignored.

[0086] The results of the first simulation shown in Figure 7 show that unbalance correction based on the first unbalance measurement reduces the resonance amplitude of the elastic primary bending to less than 1 / 20. The results also show that the resonance amplitude of the elastic primary bending after unbalance correction based on the first unbalance measurement is reduced to less than 1 / 3 of that in the case of two-plane correction. In other words, it was found that unbalance correction based on the first unbalance measurement is extremely effective.

[0087] The present inventors are considering performing highly accurate two-plane balancing of a non-uniform cross-section elastic rotor (elastic rotor 3) by two-plane adjustment instead of three-plane adjustment. Of course, they are considering performing highly accurate two-plane balancing of a non-uniform cross-section elastic rotor (elastic rotor 3) by low-speed balancing, in which the measurement object is rotated at a low speed in the range of approximately 200 rpm to approximately 1000 rpm, rather than high-speed balancing.

[0088] In the above equation (9), if z1, z2, and z3 can be found that minimize the corrected unbalance U2 (so that the corrected unbalance U2 becomes zero), then the unbalance of the flexible rotor 3 can be corrected by making corrections on the correction plane C1 and the correction plane C3 (two-plane correction). In other words, by solving the optimization problem shown in the following equation (13), it is possible to find positions z1, z2, and z3 that minimize the corrected unbalance U2.

[0089]

number

[0090] The objective function is U2 2 The independent variables z1, z2, and z3 to be optimized are included in the constraint matrix [A]. Any initial values ​​that satisfy the constraints (for example, z1 = 0.1 × l, z2 = 0.5 × l, z2 = 0.9 × l) can be given to the independent variables z1, z2, and z3 to be optimized. l is the axial distance.

[0091] The imbalance measuring device 1 can perform the second imbalance measurement (second imbalance measurement method, second correction amount determination method) in addition to the first imbalance measurement (first imbalance measurement method, first correction amount determination method). It is sufficient for the imbalance measuring device 1 to be able to perform at least one of the first imbalance measurement and the second imbalance measurement.

[0092] FIG. 8 is a flow chart showing the flow of the second imbalance measurement of the flexible rotor 3 by the imbalance measurement device 1. The second imbalance measurement will be described with reference to FIGS. 1A to 4 and 8. The second imbalance measurement differs from the first imbalance measurement (FIG. 6) in that the positions of the correcting planes (two correcting planes C1 and C3) are determined by the imbalance measurement device 1. The second imbalance measurement also differs from the first imbalance measurement in that it determines corrected imbalances U1 and U3 for two-plane correction rather than three-plane correction. In the second imbalance measurement, the correcting planes C1 and C3 may be referred to as the first correcting plane and the second correcting plane, respectively. The correcting plane C2 may also be referred to as the intermediate correcting plane.

[0093] An operator carries the elastic rotor 3 into the unbalance measuring device 1. As shown in FIG. 3, the end portions 3b of the carried-in elastic rotor 3 are supported by the pair of bearings 7 (step S11 in FIG. 8).

[0094] Furthermore, an approximation formula for the eigenfunction Φ1(z) of the first bending of the elastic rotor 3 to be measured (an approximation formula of a polynomial of fourth degree or higher, for example, the approximation formula of formula (10)) is provided to the control device 10. Specifically, the operator inputs the approximation formula for the eigenfunction Φ1(z) of the first bending of the elastic rotor 3 via the input unit 15. The input approximation formula is stored in the memory unit 14 of the control device 10 (step S12 in FIG. 8).

[0095] An approximation of the eigenfunction Φ1(z) of the primary bending of the elastic rotor 3 may be provided to the control device 10 prior to unbalance measurement. In other words, it may be stored in advance in the storage unit 14 of the control device 10.

[0096] Next, the unbalance measurement device 1 starts unbalance measurement (step S13 in FIG. 8). Specifically, the control device 10 drives the drive motor 8 to rotate the test object 2 (i.e., the elastic rotor 3) (step S13 in FIG. 8). At this time, the rotation speed of the test object 2 (elastic rotor 3) is approximately 200 rpm or more and approximately 1000 rpm or less.

[0097] Furthermore, the calculation unit 13 calculates the static unbalance U generated in the elastic rotor 3 based on the vibration detected by the vibration detection unit 9 during rotation of the test object 2 (elastic rotor 3). S and the couple imbalance U C Calculate.

[0098] Then, the calculation unit 13 calculates the position z1 of the correction plane C1 and the position z3 of the correction plane C3 by correcting the two correction planes C1 and C3 (two-plane correction) so that the unbalance of the primary bending of the elastic rotor 3 can be corrected by correcting only the correction plane C1 and the correction plane C3. Then, the correction unbalance U1 on the correction plane C1 and the correction unbalance U3 on the correction plane C3 at this time are calculated.

[0099] Specifically, the calculation unit 13 finds positions z1, z2, and z3 that minimize the corrected unbalance U2 by solving the optimization problem shown in the above equation (13). Positions z1 and z3 that minimize the corrected unbalance U2 are calculated, and the corrected unbalance U1 and corrected unbalance U3 at this time are calculated. This determines the position z1 of the correcting plane C1 and the position z3 of the correcting plane C3, as well as the respective corrected unbalances U1 and U3 (step S14 in FIG. 8). This completes the measurement of the unbalance shown in FIG. 8.

[0100] For example, suppose the approximate expression of the eigenfunction Φ1(z) of the primary bending of the flexible rotor 3 given to the control device 10 is the above expression (10). In this case, the positions z1, z2, z3 that minimize the corrected unbalance U2 (so that the corrected unbalance U2 becomes zero), and the corrected unbalance U1 and corrected unbalance U3 at that time are obtained as the following expressions (14) and (15), respectively.

[0101]

number

[0102]

number

[0103] Thereafter, based on the results of the unbalance measurement, the elastic rotor 3 is corrected on the correction plane C1 and the correction plane C3 by an unbalance correction device (not shown) (two-plane correction). Referring to FIG. 4, by performing unbalance corrections on the correction plane C1 (z=z1) and the correction plane C3 (z=z3), respectively, with the corrected unbalance U1 and the corrected unbalance U3, the primary bending of the elastic rotor 3 can be corrected. Also, on the correction plane C1 and the correction plane C3, respectively, U1' (=U1+U C ) and U3´=U3-U C By correcting the unbalance in the above, the dynamic unbalance of the rigid rotor and the primary bending of the flexible rotor 3 can be corrected simultaneously.

[0104] As described above, according to the second unbalance measurement, an approximate expression of the eigenfunction Φ1(z) of the primary bending is obtained, and based on the obtained approximation and the detected vibration of the elastic rotor 3, the positions z of the correction planes C1 and C3 are determined so that the elastic rotor 3 can be corrected only by the correction planes C1 and C3. 1, z3, as well as the respective corrective imbalances U1 and U3 are determined.

[0105] Low-speed balancing, which is performed while rotating the flexible rotor 3 at a low speed (approximately 200 rpm or more and approximately 1000 rpm or less), allows for highly accurate unbalance correction of the flexible rotor 3. Moreover, the unbalance of the flexible rotor 3 can be corrected by two-plane correction, not three-plane correction. In other words, low-speed balancing makes it possible to achieve highly accurate two-plane correction of the elastic rotor 3 having a non-uniform cross-sectional shape in the axial direction X1. This allows for accurate determination of the corrected unbalance (two-plane correction) of the elastic rotor 3 without rotating the elastic rotor 3 having a non-uniform cross-sectional shape in the axial direction X1 at high speed.

[0106] Although one embodiment of the present invention has been described above, the present invention is not limited to the above-described embodiment. Note that the present invention is not limited to the above-described embodiment or the first and second modified examples described below, and various modifications are possible within the scope of the claims.

[0107] For example, the approximate expression of the eigenfunction of the first-order bending of the flexible rotor 3 may be a sixth-order or seventh-order polynomial instead of a fifth-order polynomial.

[0108] The calculation basis for the above determinant (9) is as follows: The elastic bending modes up to the kth order of the flexible rotor are calculated as the corrected unbalance U i The balance conditions when balancing (i=1, 2, . . . , k+2) are expressed by the following equations (16) to (18).

[0109]

number

[0110]

number

[0111]

number

[0112] The above equations (16) to (18) are excerpts from the prior art document (Miwa Shuzo and Shimomura Gen, "Balancing of Rotating Machines," Corona Publishing, July 30, 1976, first edition, pp. 621-625 (section "Balancing of Flexible Rotors"). Φ in equation (18) n is the eigenfunction representing the nth eigenmode. Note that the effect of the overhang on the imbalance is ignored.

[0113] Hereinafter, the first bending mode of the flexible rotor 3 having a non-uniform cross-sectional shape in the axial direction X1 will be considered. Referring to Fig. 4, in this case, by substituting k = 1 into each of the above equations (16) to (18), the following equations (19) to (21) are obtained.

[0114]

number

[0115]

number

[0116]

number

[0117] When equations (19) to (21) are expressed in matrix form, the following determinant (22) is obtained.

[0118]

number

[0119] Then, by rearranging the determinant (22), the above equation (9) is obtained.

[0120] The first and second modified examples of the present application are described below. The first and second modified examples of the present application differ from the above-described embodiment in that the first and second imbalance measurements are performed on the elastic rotor 3 in consideration of the overhang portions OH1 and OH2 of the elastic rotor 3.

[0121] 9 to 12 are schematic diagrams showing the positional relationship between the pair of bearing portions 7 and the correction surfaces C1, C3. FIGS. 9 to 12 show first to fourth modes, respectively. The correction surface C2 is disposed between the correction surfaces C1 and C3 in the axial direction X1. The correction surface C2 is not shown in FIGS. 9 to 12.

[0122] In the first example mode (FIG. 9), the correction surfaces C1 and C3 are disposed between the pair of bearing portions 7 in the axial direction X1. Since the correction surface C2 is disposed between the correction surfaces C1 and C3 in the axial direction X1, the correction surface C2 is also disposed between the pair of bearing portions 7 in the axial direction X1. In other words, in the first example mode (FIG. 9), no correction surfaces are disposed on the overhang portions OH1 and OH2.

[0123] In a second example mode (FIG. 10), one of the correction surfaces C1 and C3 is disposed between the pair of bearing portions 7 in the axial direction X1. The other of the correction surfaces C1 and C3 is not disposed between the pair of bearing portions 7 in the axial direction X1. The other of the correction surfaces C1 and C3 is disposed on an overhang portion. In the example of FIG. 10, the correction surface C1 is disposed between the pair of bearing portions 7 in the axial direction X1, and the correction surface C3 is disposed on the overhang portion OH2. The correction surface C3 may be disposed between the pair of bearing portions 7 in the axial direction X1, and the correction surface C1 may be disposed on the overhang portion OH1.

[0124] In a third example mode (FIG. 11), the correction surfaces C1 and C3 are not disposed between the pair of bearing portions 7 in relation to the axial direction X1. The correction surfaces C1 and C3 are disposed in the overhang portion of the elastic rotor 3. In the third example mode (FIG. 11), the correction surfaces C1 and C3 are disposed in the overhang portion on one side in the axial direction X1. In the example of FIG. 11, the correction surfaces C1 and C3 are disposed in the overhang portion OH2. The correction surfaces C1 and C3 may also be disposed in the overhang portion OH1.

[0125] In the fourth embodiment (FIG. 12), the correction surfaces C1 and C3 are not disposed between the pair of bearing portions 7 in the axial direction X1. The correction surfaces C1 and C3 are disposed on the overhang portions of the elastic rotor 3. In the fourth embodiment (FIG. 12), the correction surfaces C1 and C3 are disposed on the overhang portions OH1 and OH2, respectively.

[0126] <First Modification> In the first modified example, in the first imbalance measurement (FIG. 6) and the second imbalance measurement (FIG. 8), the following equation (23) is used instead of equation (9). Also, in the second imbalance measurement (FIG. 8), the following equation (24) is used instead of equation (13).

[0127]

number

[0128]

number

[0129] The elastic bending modes up to the third order of the elastic rotor 3 are corrected by the unbalance U i The balance condition when balancing (i=1,2,3) is expressed by the following equation (25).

[0130]

number

[0131] Hereinafter, the first bending mode of the flexible rotor 3 having a non-uniform cross-sectional shape in the axial direction X1 will be considered. In this case, the following determinant (26) is obtained.

[0132]

number

[0133] Then, by rearranging the determinant (26), the above equation (23) is obtained.

[0134] <Second Modification> In the second modified example, in the first imbalance measurement (FIG. 6) and the second imbalance measurement (FIG. 8), the following equation (27) is used instead of equation (9): In the second imbalance measurement (FIG. 8), the following equation (28) is used instead of equation (13):

[0135]

number

[0136]

number

[0137] The elastic bending modes up to the third order of the flexible rotor are corrected by the unbalance U i The balance condition when balancing (i=1,2,3) is expressed by the following equation (29).

[0138]

number

[0139] Hereinafter, the first bending mode of the flexible rotor 3 having a non-uniform cross-sectional shape in the axial direction X1 will be considered. In this case, the following determinant (30) is obtained.

[0140]

Number

[0141] And, when the determinant (30) is arranged, the above equation (27) is obtained.

[0142] In Equation (27), "a" and "b" are constants. "a" may be greater than 0 and less than 1 (0 < a < 1). "b" may be greater than 0 and less than 1 (0 < b < 1).

[0143] "a" and "b" are calculated based on, for example, an optimal calculation. Let the "a" of the first embodiment example (Figure 9), the second embodiment example (Figure 10), the third embodiment example (Figure 11), and the fourth embodiment example (Figure 12) be "a1", "a2", "a3", and "a4", respectively. In this case, "a2" and "a3" may be greater than "a1" and "a4" (a2, a3 > a1, a4). More specifically, "a3" may be greater than "a2", "a2" may be greater than "a4", and and "a4" may be greater than "a1" (a3 > a2 > a4 > a1).

[0144] Also, let the "b" of the first embodiment example, the second embodiment example, the third embodiment example, and the fourth embodiment example be "b1", "b2", "b3", and "b4", respectively. In this case, "a2" and "a3" may be less than "a1" and "a4" (a2, a3 < a1, a4). More specifically, "a3" may be less than "a2", "a2" may be less than "a4", and and "a4" may be less than "a1" (a3 < a2 < a4 < a1).

[0145] Figure 13 is a diagram showing the results of the second simulation of the unbalance correction effect of the elastic rotor 3. In the second simulation, the correction amounts at three correction surfaces C1, C2, C3 were calculated based on the first unbalance measurement according to the first modification example (3-surface correction), and the unbalance at a predetermined node when corrected with that correction amount was obtained.

[0146] 14 is a diagram showing the results of a third simulation of the unbalance correction effect of the flexible rotor 3. In the third simulation, the correction amounts on three correction planes C1, C2, and C3 were calculated based on the first unbalance measurement according to the second modified example (three-plane correction), and the unbalance at a predetermined node when corrected by the correction amounts was determined.

[0147] Second and third simulations were performed for each of the first to fourth embodiments (FIGS. 9 to 12). The results of the second and third simulations are shown in FIGS. 13 and 14, along with the results of simulations based on one-plane and two-plane corrections. The second and third simulations calculated the unbalance for the cases where random variations were introduced into the mass distribution of the elastic rotor 3 in the axial direction X1, and for the cases where no such variations were introduced. In FIGS. 13 and 14, the unbalance values ​​are shown as a ratio with the initial unbalance (one-plane correction) set to 1.

[0148] From the results shown in FIGS. 13 and 14, it was found that the imbalance value was reduced in the three-surface corrections of the second simulation (first modified example) and the third simulation (second modified example).

[0149] In particular, it was found that the unbalance value was significantly reduced in the three-plane correction of the third simulation (second modified example) shown in Fig. 14. In all of the first to fourth embodiments, the unbalance was significantly reduced compared to the initial unbalance, and it was found that a high correction effect could be obtained by three-plane correction regardless of the positional relationship between the pair of bearing parts 7 and the correction surfaces C1, C3. [Explanation of symbols]

[0150] 1:Unbalance measuring device 3: Flexible rotor 7: Bearing part (support part) 9: Vibration detection unit C1: Correction surface (1st correction surface) C2: Correction surface (second correction surface, intermediate correction surface) C3: Correction plane (3rd correction plane, 2nd correction plane) U1: Fixed the problem of not fishing together U2: Correction of non-fishing combination U3: Fixed the problem of not catching fish z1: Position z2: Position z3: Position

Claims

1. supporting a flexible rotor having an axially non-uniform cross-sectional shape at two points; detecting vibration of the elastic rotor when the elastic rotor is rotated; and an unbalance determination step of determining corrected imbalance of the flexible rotor in the first correction plane, the second correction plane, and the third correction plane based on positions of a first correction plane, a second correction plane, and a third correction plane in the flexible rotor, a fourth-order or higher approximation of an eigenfunction of a first-order bending of the flexible rotor, and detected vibrations of the flexible rotor.

2. a support portion that supports an elastic rotor having a non-uniform cross-sectional shape in an axial direction at two points; a vibration detection unit that detects vibration of the elastic rotor when the elastic rotor is rotated; and a corrected imbalance determiner that determines corrected imbalance of the flexible rotor in the first corrected plane, the second corrected plane, and the third corrected plane based on positions of a first corrected plane, a second corrected plane, and a third corrected plane in the flexible rotor, a fourth-order or higher approximation of an eigenfunction of a first-order bending of the flexible rotor, and detected vibrations of the flexible rotor.

3. supporting a flexible rotor having an axially non-uniform cross-sectional shape at two points; detecting vibration of the elastic rotor when the elastic rotor is rotated; and a corrected imbalance determination step of determining positions of the first and second correction planes and corrected imbalance of the flexible rotor on the first and second correction planes, based on a fourth-order or higher approximation of an eigenfunction of the primary bending of the flexible rotor and the detected vibration of the flexible rotor, so that the unbalance of the primary bending of the flexible rotor can be corrected using only the first and second correction planes.

4. a support portion that supports an elastic rotor having a non-uniform cross-sectional shape in an axial direction at two points; a vibration detection unit that detects vibration of the elastic rotor when the elastic rotor is rotated; and a correction imbalance determination unit that determines positions of the first and second correction planes and a correction imbalance of the flexible rotor on the first and second correction planes, based on a fourth-order or higher approximation of an eigenfunction of the primary bending of the flexible rotor and the detected vibration of the flexible rotor, so that the unbalance of the primary bending of the flexible rotor can be corrected using only the first and second correction planes.

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