CN109506555A - The measuring device and measurement method of the U-shaped frame bearing hole concentricity of two axis - Google Patents
The measuring device and measurement method of the U-shaped frame bearing hole concentricity of two axis Download PDFInfo
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- CN109506555A CN109506555A CN201811520185.3A CN201811520185A CN109506555A CN 109506555 A CN109506555 A CN 109506555A CN 201811520185 A CN201811520185 A CN 201811520185A CN 109506555 A CN109506555 A CN 109506555A
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- G—PHYSICS
- G01—MEASURING; TESTING
- G01B—MEASURING LENGTH, THICKNESS OR SIMILAR LINEAR DIMENSIONS; MEASURING ANGLES; MEASURING AREAS; MEASURING IRREGULARITIES OF SURFACES OR CONTOURS
- G01B5/00—Measuring arrangements characterised by the use of mechanical techniques
- G01B5/24—Measuring arrangements characterised by the use of mechanical techniques for measuring angles or tapers; for testing the alignment of axes
- G01B5/25—Measuring arrangements characterised by the use of mechanical techniques for measuring angles or tapers; for testing the alignment of axes for testing the alignment of axes
- G01B5/252—Measuring arrangements characterised by the use of mechanical techniques for measuring angles or tapers; for testing the alignment of axes for testing the alignment of axes for measuring eccentricity, i.e. lateral shift between two parallel axes
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Abstract
The present invention provides the measuring devices and measurement method of a kind of U-shaped frame bearing hole concentricity of two axis, wherein measuring device is used to measure the bearing hole concentricity of the tested U-shaped frame of two axis (1), including left side mandrel (2), rear end cap (3), expansion sleeve (4), sleeve (5), housing screw (6), right side mandrel (7), drive end bearing bracket (8) and turntable (9);Measurement method includes measuring device installation steps, measuring process and calculating analytical procedure.Measuring device of the invention has structurally reasonable, easy to use, the high advantage of measurement accuracy;The advantages of it is low that there is measurement method device travel to require, and measurement result is not influenced by aperture size deviation.
Description
Technical field
The present invention relates to field of measuring technique, and in particular, to the measuring device and survey of the U-shaped frame bearing hole concentricity of two axis
Amount method, more particularly to a kind of by using expansion sleeve in pairs, the method for realizing the coaxality measurement of long span bearing hole.
Background technique
U-shaped frame structure is widely used in the high-precisions Inertial Navigation Testing Device such as two-axle rotating table, three-axle table, and the U-shaped frame of two axis is general
Comprising two rotation axis in orientation and pitching, pitch axis both ends are bearing mounting hole.As main body load-carrying construction, U-shaped frame is to carry
Optical device stable workbench is provided, so that equipment is more effectively worked.
To meet equipment high-precision attitude requirement, U-shaped frame pitch axis two end axles bearing bore must assure that corresponding concentricity is wanted
It asks, is the main performance index of U-shaped frame, directly affects the operating accuracy of equipment, it need not can when being U-shaped frame development and checking and accepting
Few measure the item.
For large-scale high-precision Inertial Navigation Testing Device, U-shaped frame two end axles bearing bore span is more than 1.5m even more big, and coaxial
Degree requires then to reach 15 μm or lower of Φ, therefore, is accurate evaluation machining accuracy, chooses reasonable measuring device and measurement method
It is particularly important.
The measurement of U-shaped frame long span bearing hole concentricity is difficult the part as and fills since volume of part is big, weight weight
It is clipped on chuck and does System of Rotating about Fixed Axis to measure.Currently, generalling use three-dimensional coordinates measurement method, optical telescope self-collimation measurement
Method and laser tracker mensuration etc..Three-dimensional coordinates measurement is using the method that workpiece is fixed, gauge head is got ready, respectively in two bearings hole
Axially bored line is got and fitted in inside ready, carries out resolving concentricity, but need to use star probe, and three coordinates, three axis is mobile, reduces
Measurement accuracy;Optical telescope auto-collimation method can measure the folder between line and optical axis between the center of two bearings hole
Angle, by being manually calculated concentricity size, but measurement result includes bore dia dimensional discrepancy, and when while measuring needs human eye long
Between observe, easily cause error in reading;Laser tracker measurement, as converse microscope group, passes through laser using three orthogonal plane mirrors
Ranging and pitch angle, deflection angle establish spherical coordinate system, resolve target ball centre coordinate, and then fit axially bored line and acquire concentricity
Deviation, but measurement accuracy is influenced by angular errors, target ball error, laser ranging error, and integrated measurement accuracy is lower.The present invention mentions
For a kind of measurement method of novel U-shaped frame long span bearing hole concentricity of measurement, there is measurement accuracy height, device travel is wanted
Ask the features such as low, measurement result is not influenced by aperture size deviation.
Summary of the invention
For the defects in the prior art, the object of the present invention is to provide a kind of surveys of the U-shaped frame bearing hole concentricity of two axis
Measure device and measurement method.
The measuring device of the U-shaped frame bearing hole concentricity of two axis of one kind provided according to the present invention, for measuring tested two axis U
The bearing hole concentricity of type frame, including left side mandrel, rear end cap, expansion sleeve, sleeve, housing screw, right side mandrel, drive end bearing bracket with
And turntable;
The tested U-shaped frame of two axis is mounted on turntable;The left side mandrel, right side mandrel are separately mounted to tested two axis
In the left side of U-shaped frame, right side bearing hole;The sleeve is mounted at first position, and two sides are additionally provided with expansion sleeve;It is described swollen
Fixed and sleeve is fixed on first position by rear end cap and drive end bearing bracket;The rear end cap and drive end bearing bracket pass through housing screw
It is fixed;
The first position is the position and the right side between left side outside diameter of mandrel and the left side bearing hole internal diameter of the tested U-shaped frame of two axis
Position between side outside diameter of mandrel and the right side bearing hole internal diameter of the tested U-shaped frame of two axis.
Preferably, the left side mandrel, right side mandrel pass through left side, the right side of expansion sleeve and the tested U-shaped frame of two axis respectively
Bearing hole blind-mate, and the left side of the respectively greater than tested U-shaped frame of two axis of the axial length of left side mandrel, right side mandrel,
The axial length of right side bearing hole.
Preferably, the left side mandrel, right side mandrel are supported compression using two pairs of expansion sleeves.
Preferably, the bearing hole of the left side mandrel, right side mandrel, expansion sleeve and the U-shaped frame of two axis is fitted close, cooperation
Precision meets the requirement of setting.
Preferably, the sleeve, rear end cap and drive end bearing bracket and other parts clearance fit, do not influence measurement accuracy.
The measurement method of the U-shaped frame bearing hole concentricity of two axis of one kind provided according to the present invention, it is U-shaped using two above-mentioned axis
The measuring device of frame bearing hole concentricity, comprising:
Measuring device installation steps: the tested U-shaped frame of two axis is installed on turntable, and it is flat with turntable working surface to adjust datum level
Row, azimuth axis and turntable shaft are coaxial, by left side mandrel, rear end cap, expansion sleeve, sleeve, housing screw, right side mandrel and
Drive end bearing bracket is installed to the two end axles bearing bore of the tested U-shaped frame of two axis;
Measuring process: along left side, mandrel axis direction beats periphery and side periphery on the mandrel of table measurement left side and reads and remember
Record, turntable rotate 180 degree, right side mandrel are transferred to measured zone;Along right side, mandrel axis direction beats table measurement right side mandrel
Upper periphery and side periphery read and record;
Calculate analytical procedure: two sides measurement data compares, and resolves concentricity.
Preferably, in the measuring device installation steps, the left end bearing hole for measuring and recording the tested U-shaped frame of two axis is also needed
Diameter of phi D1, right end bearing hole diameter of phi D2, left side mandrel diameter of phi d1And the diameter of phi d of right side mandrel2。
Preferably, in the measuring process:
Along left side mandrel axis direction, periphery highest point reading n times on the mandrel of table measurement left side are beaten, and are denoted as δzsi, edge
Left side mandrel axis direction repeatedly beats table measurement left side mandrel side periphery highest point reading n times, and is denoted as δzci, wherein i be
Positive integer no more than n;
After turntable rotation, along right side mandrel axis direction, beats periphery highest point on the mandrel of table measurement right side and reads m times,
And it is denoted as δysj, along right side mandrel axis direction, beat table measurement right side mandrel side periphery highest point and read m times, and be denoted as δycj,
Wherein j is the positive integer no more than m.
Preferably, δ is taken respectivelyzsiMinimum value δzs-min、δzciMinimum value δzc-min、δysjMaximum value δys-maxAnd
δycjMaximum value δyc-maxAs measurement data, resolved for calculating analytical procedure.
Preferably, in the calculating analytical procedure, concentricity t is calculated by the first formula:
In formula, δsFor two axial lines short transverse maximum deviation, can be calculated by the second formula, δcFor two axial lines side
To maximum deviation, can be calculated by third formula;
Second formula are as follows:
δs=| δys-max-δzs-min|-Δ
Third formula are as follows:
δc=| δyc-max-δzc-min|-Δ
Wherein, Δ is that axle diameter is poor, can be calculated by the 4th formula:
Δ=| Φ d1-Φd2|。
Compared with prior art, the present invention have it is following the utility model has the advantages that
1, the measuring device of the U-shaped frame bearing hole concentricity of two axis provided by the invention, have it is structurally reasonable, it is easy to use,
The high advantage of measurement accuracy;
2, the measurement method of the U-shaped frame bearing hole concentricity of two axis provided by the invention there is device travel to require low, measurement
As a result the advantages of not influenced by aperture size deviation.
Detailed description of the invention
Upon reading the detailed description of non-limiting embodiments with reference to the following drawings, other feature of the invention,
Objects and advantages will become more apparent upon:
Fig. 1 is that two axis U are tested in the measuring device and measurement method of the U-shaped frame bearing hole concentricity of two axis provided by the invention
The structural schematic diagram of type frame;
Fig. 2 is the structural schematic diagram of the measuring device of the U-shaped frame bearing hole concentricity of two axis provided by the invention;
Fig. 3 is the signal that mandrel plays table measurement in the measurement method of the U-shaped frame bearing hole concentricity of two axis provided by the invention
Figure.
It is shown in figure:
The tested U-shaped frame 1 of two axis
Left side mandrel 2
Rear end cap 3
Expansion sleeve 4
Sleeve 5
Housing screw 6
Right side mandrel 7
Drive end bearing bracket 8
Turntable 9
Specific embodiment
The present invention is described in detail combined with specific embodiments below.Following embodiment will be helpful to the technology of this field
Personnel further understand the present invention, but the invention is not limited in any way.It should be pointed out that the ordinary skill of this field
For personnel, without departing from the inventive concept of the premise, several changes and improvements can also be made.These belong to the present invention
Protection scope.
The measuring device of the U-shaped frame bearing hole concentricity of two axis of one kind provided according to the present invention, for measuring tested two axis U
The bearing hole concentricity of type frame 1, including left side mandrel 2, rear end cap 3, expansion sleeve 4, sleeve 5, housing screw 6, right side mandrel 7,
Drive end bearing bracket 8 and turntable 9;The tested U-shaped frame 1 of two axis is mounted on turntable 9;The left side mandrel 2, right side mandrel 7 are distinguished
It is mounted on the left side of the U-shaped frame 1 of tested two axis, in the bearing hole of right side;The sleeve 5 is mounted at first position, and two sides are also set
It is equipped with expansion sleeve 4;The expansion sleeve 4 and sleeve 5 pass through rear end cap 3 and drive end bearing bracket 8 is fixed on first position;The rear end cap
3 and drive end bearing bracket 8 be fixed by housing screw 6;The first position is 2 outer diameter of left side mandrel and the tested U-shaped frame 1 of two axis
Left side bearing hole internal diameter between position and the position between 7 outer diameter of right side mandrel and the right side bearing hole internal diameter of the tested U-shaped frame 1 of two axis
It sets.
Preferably, the left side mandrel 2, right side mandrel 7 respectively by the left side of expansion sleeve 4 and the tested U-shaped frame 1 of two axis,
Right side bearing hole blind-mate, and the axial length of left side mandrel 2, right side mandrel 7 is respectively greater than the tested U-shaped frame 1 of two axis
Left side, right side bearing hole axial length.The left side mandrel 2, right side mandrel 7 are supported using two pairs of expansion sleeves 4
It compresses.The bearing hole tight fit of the left side mandrel 2, right side mandrel 7, expansion sleeve 4 and the U-shaped frame 1 of two axis, cooperation precision are full
The requirement set enough.The sleeve 5, rear end cap 3 and drive end bearing bracket 8 and other parts clearance fit, do not influence measurement accuracy.
The measurement method of the U-shaped frame bearing hole concentricity of two axis of one kind provided according to the present invention, it is U-shaped using two above-mentioned axis
The measuring device of frame bearing hole concentricity, comprising:
Measuring device installation steps: the tested U-shaped frame 1 of two axis is installed on turntable 9, and adjusts datum level and 9 worksheet of turntable
Face is parallel, and azimuth axis and 9 shaft of turntable are coaxial, by left side mandrel 2, rear end cap 3, expansion sleeve 4, sleeve 5, housing screw 6, the right side
Side core axis 7 and drive end bearing bracket 8 are installed to the two end axles bearing bore of the tested U-shaped frame 1 of two axis;
Measuring process: periphery and side periphery on the mandrel 2 of table measurement left side are beaten along 2 axis direction of left side mandrel and is read simultaneously
Record, turntable 9 rotate 180 degree, right side mandrel 7 are transferred to measured zone;Table measurement right side is beaten along 7 axis direction of right side mandrel
Periphery and side periphery read and record on mandrel 7;
Calculate analytical procedure: two sides measurement data compares, and resolves concentricity.
Specifically, in the measuring device installation steps, the left end bearing for measuring and recording the U-shaped frame 1 of tested two axis is also needed
Bore dia Φ D1, right end bearing hole diameter of phi D2, left side mandrel 2 diameter of phi d1And the diameter of phi d of right side mandrel 72。
In the measuring process:
Along 2 axis direction of left side mandrel, periphery highest point reading n times on the mandrel 2 of table measurement left side are beaten, and are denoted as δzsi,
Along 2 axis direction of left side mandrel, table measurement left side 2 side periphery highest point of mandrel reading n times are repeatedly beaten, and are denoted as δzci, wherein
I is the positive integer no more than n;
After turntable 9 rotates, along 7 axis direction of right side mandrel, periphery highest point reading m on the mandrel 7 of table measurement right side is beaten
It is secondary, and it is denoted as δysj, along 7 axis direction of right side mandrel, beat table measurement right side 7 side periphery highest point of mandrel and read m times, and remember
For δycj, wherein j is the positive integer no more than m.
δ is taken respectivelyzsiMinimum value δzs-min、δzciMinimum value δzc-min、δysjMaximum value δys-maxAnd δycjMaximum
Value δyc-maxAs measurement data, resolved for calculating analytical procedure.
In the calculating analytical procedure, concentricity t is calculated by the first formula:
In formula, δsFor two axial lines short transverse maximum deviation, can be calculated by the second formula, δcFor two axial lines side
To maximum deviation, can be calculated by third formula;
Second formula are as follows:
δs=| δys-max-δzs-min|-Δ
Third formula are as follows:
δc=| δyc-max-δzc-min|-Δ
Wherein, Δ is that axle diameter is poor, can be calculated by the 4th formula:
Δ=| Φ d1-Φd2|。
More specifically, the technical problem to be solved is that realize bearing hole air plug gauge and bearing hole for preference of the present invention
Blind-mate converts hole central axis to mandrel periphery.Used technical solution are as follows: mandrel passes through swollen with bearing hole
Fixed 4 connection, each mandrel use two pairs of expansion sleeves 4, are separately mounted to two side position of bearing hole, pass through drive end bearing bracket 8 and rear end
Lid 3 and housing screw 6 are compressed, and realize blind-mate.Expansion sleeve 4, mandrel need to guarantee size by Ultra-precision Turning
Precision, sleeve 5 are gap-matched with bearing hole and mandrel, are realized that mandrel two o'clock stablizes support, are reduced unnecessary measurement
Measurement accuracy is improved in region.The reading of periphery and side periphery on table measurement left end mandrel along generatrix direction is first beaten when measurement,
Turntable rotates 180 degree and guarantees running accuracy by mechanical compensation, then beats periphery and side periphery on table measurement right end mandrel
Along the reading of generatrix direction, concentricity can be resolved to obtain by comparing and calculating.
Further, below in conjunction with the implementation steps of Detailed description of the invention preference of the present invention:
Step 1: as shown in Figure 1, the U-shaped frame 1 of tested two axis is mounted on turntable, and adjusting datum level and 9 work of turntable
It is parallel to make surface, azimuth axis and 9 shaft of turntable are coaxial.Concentricity need to adjust repeatedly, until thinking it not influences left end bearing
Hole Φ D1With right end bearing hole Φ D2Concentricity measurement accuracy.
Step 2: measuring and recording left side 2 diameter of phi d of mandrel1With 7 diameter of phi d of right side mandrel2, the two diameter difference Δ=| Φ
dl-Φd2|。
Step 3: as shown in Fig. 2, two end axles bearing bore successively installs end cap, mandrel, expansion sleeve 4, sleeve 5, housing screw 6,
Every side axle bearing bore uses one each 4 two pairs of expansion sleeve, mandrel one, front and rear cover, 5 one, sleeve, housing screw 6 several.Core
Axle suspension stretches out bearing hole, controls 6 screw-down torque of housing screw rationally to avoid compression deformation is generated, it is secured to be installed rear mandrel
It does not loosen.
Step 4: reading, take as shown in figure 3, beating periphery highest point on the mandrel 2 of table measurement left side along mandrel generatrix direction
If doing, δ is rememberedzs1、δzs2、δzs3、δzs4..., it is minimized δzs-min。
Step 5: reading, take as shown in figure 3, beating table measurement left side 2 side periphery highest point of mandrel along mandrel generatrix direction
If doing, δ is rememberedzc1、δzc2、δzc3、δzc4..., it is minimized δzc-min。
Step 6: turntable rotates 180 degree, right side mandrel 7 is transferred to measured zone, 9 running accuracy of turntable can pass through machinery
Compensation guarantees.
Step 7: referring to step 4 and step 5, beating periphery and side periphery reading on the mandrel 7 of table measurement right side, remember respectively
δys2、δys2、δys3、δys4... and δyc1、δyc2、δyc3、δyc4..., it is maximized δys-maxAnd δyc-max。
Step 8: the comparison of left and right sides mandrel measurement data resolves, and compensates diameter difference axle diameter difference Δ, two axial lines height side
To maximum deviation δs=| δys-max-δzs-min|-Δ, lateral maximum deviation δc=| δyc-max-δzc-min|-Δ resolves to obtain concentricity t.
In the description of the present application, it is to be understood that term " on ", "front", "rear", "left", "right", " is erected at "lower"
Directly ", the orientation or positional relationship of the instructions such as "horizontal", "top", "bottom", "inner", "outside" is orientation based on the figure or position
Relationship is set, description the application is merely for convenience of and simplifies description, rather than the device or element of indication or suggestion meaning are necessary
It with specific orientation, is constructed and operated in a specific orientation, therefore should not be understood as the limitation to the application.
Specific embodiments of the present invention are described above.It is to be appreciated that the invention is not limited to above-mentioned
Particular implementation, those skilled in the art can make a variety of changes or modify within the scope of the claims, this not shadow
Ring substantive content of the invention.In the absence of conflict, the feature in embodiments herein and embodiment can any phase
Mutually combination.
Claims (10)
1. a kind of measuring device of the U-shaped frame bearing hole concentricity of two axis, the bearing hole for measuring the tested U-shaped frame of two axis (1) are same
Axis degree, which is characterized in that including left side mandrel (2), rear end cap (3), expansion sleeve (4), sleeve (5), housing screw (6), right side
Mandrel (7), drive end bearing bracket (8) and turntable (9);
The tested U-shaped frame of two axis (1) is mounted on turntable (9);The left side mandrel (2), right side mandrel (7) are separately mounted to
In the left side of the tested U-shaped frame of two axis (1), right side bearing hole;The sleeve (5) is mounted at first position, and two sides are also set up
There are expansion sleeve (4);The expansion sleeve (4) and sleeve (5) pass through rear end cap (3) and drive end bearing bracket (8) is fixed on first position;Institute
Rear end cap (3) and drive end bearing bracket (8) is stated to be fixed by housing screw (6);
The first position be left side mandrel (2) outer diameter be tested the U-shaped frame of two axis (1) left side bearing hole internal diameter between position and
Position between right side mandrel (7) outer diameter and the right side bearing hole internal diameter of the tested U-shaped frame of two axis (1).
2. the measuring device of the U-shaped frame bearing hole concentricity of two axis according to claim 1, which is characterized in that the left side
Mandrel (2), right side mandrel (7) pass through expansion sleeve (4) and the left side of the tested U-shaped frame of two axis (1), right side bearing hole gapless respectively
Cooperation, and the axial length of left side mandrel (2), right side mandrel (7) is respectively greater than left side, the right side of the tested U-shaped frame of two axis (1)
The axial length of bearing hole.
3. the measuring device of the U-shaped frame bearing hole concentricity of two axis according to claim 1, which is characterized in that the left side
Mandrel (2), right side mandrel (7) are supported compression using two pairs of expansion sleeves (4).
4. the measuring device of the U-shaped frame bearing hole concentricity of two axis according to claim 1, which is characterized in that the left side
The bearing hole tight fit of mandrel (2), right side mandrel (7), expansion sleeve (4) and the U-shaped frame of two axis (1), cooperation precision satisfaction are set
Fixed requirement.
5. the measuring device of the U-shaped frame bearing hole concentricity of two axis according to claim 1, which is characterized in that the sleeve
(5), rear end cap (3) and drive end bearing bracket (8) and other parts clearance fit, do not influence measurement accuracy.
6. a kind of measurement method of the U-shaped frame bearing hole concentricity of two axis, utilizes two axis U described in any one of any one of claims 1 to 55
The measuring device of type frame bearing hole concentricity characterized by comprising
Measuring device installation steps: the tested U-shaped frame of two axis (1) is installed on turntable (9), and adjusts datum level and turntable (9) work
Surface is parallel, and azimuth axis and turntable (9) shaft are coaxial, by left side mandrel (2), rear end cap (3), expansion sleeve (4), sleeve (5),
Housing screw (6), right side mandrel (7) and drive end bearing bracket (8) are installed to the two end axles bearing bore of the tested U-shaped frame of two axis (1);
Measuring process: periphery and side periphery on table measurement left side mandrel (2) are beaten along left side mandrel (2) axis direction and is read simultaneously
Record, turntable (9) rotate 180 degree, right side mandrel (7) are transferred to measured zone;Table is played along right side mandrel (7) axis direction to survey
Periphery and side periphery read and record on amount right side mandrel (7);
Calculate analytical procedure: two sides measurement data compares, and resolves concentricity.
7. the measurement method of the U-shaped frame bearing hole concentricity of two axis according to claim 6, which is characterized in that the measurement
In device installation steps, the left end bearing hole diameter of phi D for measuring and recording the tested U-shaped frame of two axis (1) is also needed1, right end bearing hole
Diameter of phi D2, left side mandrel (2) diameter of phi d1And the diameter of phi d of right side mandrel (7)2。
8. the measurement method of the U-shaped frame bearing hole concentricity of two axis according to claim 7, which is characterized in that the measurement
In step:
Along left side mandrel (2) axis direction, reading n times in periphery highest point on table measurement left side mandrel (2) are beaten, and be denoted as δzsi,
Along left side mandrel (2) axis direction, table measurement left side mandrel (2) side periphery highest point reading n times are repeatedly beaten, and are denoted as δzci,
Wherein i is the positive integer no more than n;
After turntable (9) rotation, along right side mandrel (7) axis direction, periphery highest point reading on table measurement right side mandrel (7) is beaten
M times, and it is denoted as δysj, along right side mandrel (7) axis direction, beats table measurement right side mandrel (7) side periphery highest point and reads m times,
And it is denoted as δycj, wherein j is the positive integer no more than m.
9. the measurement method of the U-shaped frame bearing hole concentricity of two axis according to claim 8, which is characterized in that take δ respectivelyzsi
Minimum value δzs-min、δzciMinimum value δzc-min、δysjMaximum value δys-maxAnd δycjMaximum value δyc-maxAs measurement number
According to for calculating analytical procedure resolving.
10. the measurement method of the U-shaped frame bearing hole concentricity of two axis according to claim 9, which is characterized in that the calculating
In analytical procedure, concentricity t is calculated by the first formula:
In formula, δsFor two axial lines short transverse maximum deviation, can be calculated by the second formula, δcLaterally most for two axial lines
Large deviation can be calculated by third formula;
Second formula are as follows:
δs=| δys-max-δzs-min|-Δ
Third formula are as follows:
δc=| δyc-max-δzc-min|-Δ
Wherein, Δ is that axle diameter is poor, can be calculated by the 4th formula:
Δ=| Φ d1-Φd2|。
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