CN112815900A - Coordinate system establishing method and rigid body centroid and inertia parameter testing method - Google Patents

Coordinate system establishing method and rigid body centroid and inertia parameter testing method Download PDF

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CN112815900A
CN112815900A CN202110146097.7A CN202110146097A CN112815900A CN 112815900 A CN112815900 A CN 112815900A CN 202110146097 A CN202110146097 A CN 202110146097A CN 112815900 A CN112815900 A CN 112815900A
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coordinate system
axis
wobble plate
inertia
measured object
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CN112815900B (en
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陈雪艳
龚志豪
上官文斌
袁焱龙
南毅
张宁
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Shiyan Fengsheng Automotive Rubber & Plastic Products Co ltd
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Shiyan Fengsheng Automotive Rubber & Plastic Products Co ltd
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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
    • G01B21/00—Measuring arrangements or details thereof, where the measuring technique is not covered by the other groups of this subclass, unspecified or not relevant
    • G01B21/02—Measuring arrangements or details thereof, where the measuring technique is not covered by the other groups of this subclass, unspecified or not relevant for measuring length, width, or thickness
    • G01B21/04—Measuring arrangements or details thereof, where the measuring technique is not covered by the other groups of this subclass, unspecified or not relevant for measuring length, width, or thickness by measuring coordinates of points
    • G—PHYSICS
    • G01—MEASURING; TESTING
    • G01M—TESTING STATIC OR DYNAMIC BALANCE OF MACHINES OR STRUCTURES; TESTING OF STRUCTURES OR APPARATUS, NOT OTHERWISE PROVIDED FOR
    • G01M1/00—Testing static or dynamic balance of machines or structures
    • G01M1/10—Determining the moment of inertia
    • G—PHYSICS
    • G01—MEASURING; TESTING
    • G01M—TESTING STATIC OR DYNAMIC BALANCE OF MACHINES OR STRUCTURES; TESTING OF STRUCTURES OR APPARATUS, NOT OTHERWISE PROVIDED FOR
    • G01M1/00—Testing static or dynamic balance of machines or structures
    • G01M1/12—Static balancing; Determining position of centre of gravity
    • G01M1/122—Determining position of centre of gravity

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  • Engineering & Computer Science (AREA)
  • Aviation & Aerospace Engineering (AREA)
  • Length Measuring Devices With Unspecified Measuring Means (AREA)

Abstract

The invention discloses a coordinate system establishing method and a rigid body mass center and inertia parameter testing method, which are characterized in that a geodetic coordinate system, a reference coordinate system, a connected coordinate system and a wobble plate coordinate system are established, and the position relation among the coordinate systems is solved; measuring the positions of the axes of the pendulums in different placing postures in a reference coordinate system; calculating the coordinates of the centroid of the measured object in the reference coordinate system; measuring the torsional pendulum period and the direction cosine of the axis of the pendulum plate in the reference coordinate system under different placing postures; and calculating inertia parameters of the measured object relative to the centroid coordinate system. The method can measure the mass center and the inertia parameters of the irregular-shaped rigid body, and decouple the measurement of the mass center and the measurement of the inertia parameters; the mass center of the object to be tested is ensured to fall on the axis of the wobble plate, and the test requirement is met; the three-coordinate measuring instrument and the inclinometer are adopted for measurement, and data processing methods such as a least square method and the like are combined, so that the device has the characteristic of high testing precision.

Description

Coordinate system establishing method and rigid body centroid and inertia parameter testing method
Technical Field
The invention relates to the technical field of rigid body mass center and inertia parameter testing, in particular to a coordinate system establishing method and a rigid body mass center and inertia parameter testing method.
Background
The centroid position of the rigid body and the inertial parameters (including the moment of inertia and the product of inertia) with respect to the centroid coordinate system are important dynamic parameters of the rigid body. For the dynamic analysis of a mechanical system, an important premise is to obtain the mass center position and inertia parameters of each component in the mechanical system. In addition, some vibration isolation mechanisms, such as suspension mechanisms of automobile power assemblies, require input of the center of mass and inertial parameters of the controlled object. Therefore, the method has important significance for the dynamic analysis, design and the like of the system by accurately measuring the mass center and the inertia parameters of the rigid body.
Common methods for obtaining the centroid and inertial parameters can be divided into two categories: a calculation method based on a CAD model and an experimental test method. Because the shape, material density and other factors of the real object have certain differences with the CAD model, the centroid and the inertial parameters calculated by the calculation method based on the CAD model have larger errors. Experimental test methods can be further divided into two categories: static methods and dynamic methods. When the static method is used for testing the inertia parameters of an object, only two parameters of mass and mass center position can be obtained. Because all the inertia parameters of the object cannot be acquired at one time, the static method has low testing efficiency and is rarely used. All inertial parameters of the object can be obtained simultaneously by using a dynamic method for testing, and the most common dynamic methods comprise a torsion pendulum method and a modal parameter identification method. When the torsional pendulum method is used for measuring the inertial parameters of an object, the object to be measured needs to perform torsional pendulum motion of a small angle around the axis of the wobble plate, the rotational inertia of the object to be measured about the axis of the wobble plate is obtained through testing the torsional pendulum period, and the rest inertial parameters are obtained through a certain calculation method. When the modal parameter identification method is used for measuring the inertia parameters of an object, the frequency response function of the measured object under the free boundary condition needs to be measured, and the inertia parameters of the measured object are identified through the frequency response function. The modal parameter identification method has high requirements on test equipment, and test results are influenced by boundary conditions and frequency response functions.
For a rigid body with a complex shape, the centroid coordinate and the inertia parameter of the rigid body are difficult to obtain by a calculation method, so a three-line pendulum method is often adopted to measure the inertia parameter. The measured object is placed on the three-line pendulum in different postures, and the position of the measured object is adjusted to enable the readings of the three force sensors connected with the cycloid to be the same, so that the mass center of the measured object is ensured to fall on the axis of the wobble plate. The mass center position of the measured object can be determined by solving the intersection point of the axes of the lower pendulums in different placing postures. And the inertia parameters of the object to be measured can be obtained by measuring the torsional pendulum period of the object about the axis of the wobble plate and utilizing the inertia ellipsoid principle and the least square method.
Disclosure of Invention
Aiming at the defects in the prior art, the invention aims to provide a coordinate system establishing method and a rigid body centroid and inertia parameter testing method.
In order to achieve the above purposes, the technical scheme adopted by the invention is as follows: a coordinate system establishing method, comprising the steps of:
s101, fixing a three-coordinate measuring instrument on a horizontal ground, establishing a geodetic coordinate system, and defining coordinates obtained by measuring with the three-coordinate measuring instrument in the geodetic coordinate system;
s102, fixing a measured object on the ground, measuring the geometric characteristics of the measured object by using a three-coordinate measuring instrument, and establishing a reference coordinate system on the measured object; in the testing process, a mass center coordinate system is defined, the origin of the mass center coordinate system is positioned at the mass center of the object to be tested, and the directions of all coordinate axes are the same as the reference coordinate system;
s103, fixing the object to be measured at the unchanged position in the step S102, and mounting a tool with three mutually vertical planes on the object to be measured; measuring the positions of three mutually adjacent planes on the tool in a geodetic coordinate system by using a three-coordinate measuring instrument to establish a connected coordinate system;
s104, obtaining a coordinate conversion relation between the connected coordinate system and the reference coordinate system according to the positions of the reference coordinate system and the connected coordinate system in the geodetic coordinate system;
s105, placing a measured object on the three-line pendulum, adjusting the position of the measured object to enable the readings of the three force sensors connected with the cycloid to be the same, and ensuring that the mass center of the measured object falls on the axis of the wobble plate; and fixing the measured object, and measuring the positions of at least three characteristic points on the wobble plate by using a three-coordinate measuring instrument to establish a wobble plate coordinate system.
On the basis of the technical scheme, the reference coordinate system origin O is obtained by the following formularIn a reference coordinate system OrXrYrZrCoordinate of (x)rob,yrob,zrob)：
Figure RE-GDA0002996098940000031
Then, the reference coordinate system O is obtained by the following formularXrYrZrEach coordinate axis is in a connected coordinate system ObXbYbZbDirectional cosine of (1):
Figure RE-GDA0002996098940000032
for any point W, assuming the point W is in the connected coordinate system ObXbYbZbHas the coordinate of (x)wb,ywb,zwb) Then, thenPoint W in reference coordinate system OrXrYrZrThe coordinates in (c) can be calculated by:
Figure RE-GDA0002996098940000033
the invention provides a rigid body centroid parameter testing method based on the coordinate system, which comprises the following steps:
s201, placing a measured object on a balance according to a specific posture, measuring three adjacent planes on a tool by using a three-coordinate measuring instrument, and establishing a connected coordinate system according to the mode of the step 103;
s202, solving the position of the axis of the wobble plate in the connected coordinate system by using the position relation between the connected coordinate system and the wobble plate coordinate system; then, according to the coordinate conversion relation between the reference coordinate system and the connected coordinate system established in the step 104, the position of the axis of the wobble plate in the reference coordinate system is obtained;
s203, changing the placing posture of the measured object on the swinging plate, and repeating the steps S201 to S202; until the positions of the axes of the wobble plate of the object to be measured in at least two placing postures in the reference coordinate system are obtained;
s204, if the axes of the wobble plates have an intersection point, the intersection point is the mass center of the object to be measured; if no intersection point exists between any two wobble plate axes, the midpoint of a common perpendicular line segment of the two wobble plate axes is taken as a feasible centroid; the average position of the feasible centroids can be taken, and a calculated centroid can be determined.
Based on the above technical solution, in step S202, it is assumed that the origin O of the coordinate system of the connected body is located at the ith placement positionbThe coordinates in the geodetic coordinate system are: (x)bogi,ybogi,zbogi) Coordinate axis ObXb，ObYb，ObZbThe direction cosines in the geodetic coordinate system are respectively: (cos alpha.)rxg，cosβrxg，cosγrxg)，(cosαbygi,cosβbygi,cosγbygi)， (cosαbzgi,cosβbzgi,cosγbzgi)；
Origin OpThe coordinates in the global coordinate system are:
Figure RE-GDA0002996098940000041
OpZpthe direction cosine of the axis in the connected coordinate system is:
Figure RE-GDA0002996098940000051
origin OpThe coordinates in the reference coordinate system are:
Figure RE-GDA0002996098940000052
OpZpthe directional cosine of the axes in the reference coordinate system is:
Figure RE-GDA0002996098940000053
based on the above technical solution, in step S204, if there is no intersection point between any two wobble plate axes, O is obtainedp1And Op2The coordinates in the reference coordinate system are respectively (x)por1,ypor1,zpor1) And (x)por2,ypor2,zpor2)，Op1Zp1Shaft and Op2Zp2The direction cosine of the axes in the reference coordinate system is (cos)pzr1,cospzr1,cospzr1) And (cos)pzr2,cospzr2,cospzr2) The coordinates of the feasible centroid in the reference coordinate system are:
Figure RE-GDA0002996098940000054
wherein Q, R and S are respectively expressed as
Figure RE-GDA0002996098940000055
A feasible center of mass can be found from any two wobble plate axes, and then a calculated center of mass can be determined from the feasible center of masses according to the following formula:
Figure RE-GDA0002996098940000061
wherein N is the number of the placing postures.
The invention provides a rigid body inertia parameter testing method based on the coordinate system, which comprises the following steps:
s301, placing a measured object on a swing disc according to a specific posture, adjusting the position of the measured object to enable the readings of the three force sensors to be the same, then enabling the measured object and the swing disc to do small-angle torsional pendulum motion around the axis of the swing disc, and measuring the torsional pendulum period of the measured object;
s302, measuring included angles between the three planes on the tool selected in the step 103 and a horizontal plane by using an inclinometer; the direction cosine of the axis of the wobble plate in the connected coordinate system is obtained according to the angles of the three included angles, and then the direction cosine of the axis of the wobble plate in the reference coordinate system is obtained through the coordinate conversion relation between the reference coordinate system and the connected coordinate system established in the step 104;
s303, changing the placing posture of the object to be measured on the swinging plate, and repeating the steps S301 to S302; until the direction cosine of the axis of the wobble plate of the measured object in at least six placing postures in the reference coordinate system is obtained;
s304, obtaining the rotational inertia of the object to be tested about the axis of the wobble plate according to the torsion period obtained by the test; and then establishing equations of the rotational inertia of the measured object about the axis of the wobble plate and the inertial parameters of the measured object about the mass center coordinate system according to the principle of the inertia ellipsoid, and solving the equations by using a least square method to obtain the inertial parameters of the measured object about the mass center coordinate system.
On the basis of the above technical solution, in step S302, a plane S on the tool is measured by using an inclinometer1，S2，S3The angles with the horizontal plane are respectively marked as theta1，θ2，θ3(0≤θ<Pi); thus, OpZpThe directional cosine of the axis in the global coordinate system can be calculated by:
Figure RE-GDA0002996098940000062
then O ispZpThe directional cosine of the axis in the centroid coordinate system can be calculated by:
Figure RE-GDA0002996098940000071
on the basis of the above technical solution, in step S304, the moment of inertia of the torsional pendulum system about the axis of the wobble plate is:
Figure RE-GDA0002996098940000072
in the formula, m is the mass of the torsional pendulum system, including the mass of the pendulum plate, the cycloid and the measured object; r is the torsional pendulum radius, l is the pendulum length, and g is the gravity acceleration;
moment of inertia J of the pendulum mass about the axis of the pendulum when no object is placed0The moment of inertia of the measured object about the axis of the wobble plate can be obtained through a calibration experiment:
JPi＝Ji-J0 (13)；
in the formula, JPi、JiThe moment of inertia of the measured object and the torsion pendulum system about the axis of the wobble plate under the ith placing posture is respectively.
Based on the above technical solution, in step S304, according to the principle of inertia ellipsoid, the moment of inertia of an object under the coordinate system of the center of mass thereof is associated withProduct of inertia (J)x、Jy、Jz、Jxy、Jyz、 Jzx) Moment of inertia J with an axis passing through the center of massPi(i ═ 1,2, … …, N) has the following relationship:
Figure RE-GDA0002996098940000073
li＝cosαpzci，mi＝cosβpzci，ni＝cosγpzci (14b)；
in the formula, N is the total times of the test;
therefore, the inertia parameters of the measured object about the centroid coordinate system can be obtained by using the least square method as follows:
J＝(ATA)-1ATJP (15a)；
J＝[Jx Jy Jz Jxy Jyz Jzx]T (15b)；
JP＝[JP1 JP2 … JPN]T (15c)；
Figure RE-GDA0002996098940000081
the invention has the beneficial effects that:
1) the centroid and inertial parameters of the irregularly-shaped rigid body can be measured, and the measurement of the centroid and the measurement of the inertial parameters are decoupled.
2) The readings of the three force sensors connected with the cycloid are the same by adjusting the position of the measured object, so that the center of mass of the measured object is ensured to fall on the axis of the wobble plate, and the test requirement is met. In addition, the three-coordinate measuring instrument and the inclinometer are adopted for measurement, and data processing methods such as a least square method and the like are combined, so that the device has the characteristic of high testing precision.
3) The mass center and inertia parameters of the object to be tested can be obtained through one complete test process, and the method has high test accuracyAnd testing efficiency. The 9 parameters specifically refer to coordinates of the measured centroid in a reference coordinate system, and comprise three parameters of an x coordinate, a y coordinate and a z coordinate; and inertial parameters of the measured object about a centroid coordinate system, including three moments of inertia Jx，Jy，JzAnd three products of inertia Jxy，Jyz，Jzx。
4) When the inertia parameters are tested, the included angle between the upper plane of the tool and the horizontal plane is measured by the inclinometer, so that the use times and time of the articulated arm are reduced, and the test efficiency is improved.
Drawings
FIG. 1 is a flow chart of the present invention.
Fig. 2 is a schematic diagram of a method for establishing a global coordinate system according to the present invention.
Fig. 3 is a schematic diagram of the method for establishing the wobble plate coordinate system in the present invention.
FIG. 4 is a schematic diagram of the positional relationship of the coordinate systems in the present invention.
FIG. 5 is a schematic diagram of a method for selecting a feasible centroid in the present invention.
Detailed Description
Reference will now be made in detail to the embodiments of the present invention, examples of which are illustrated in the accompanying drawings, wherein like reference numerals refer to the same or similar elements or elements having the same or similar functions throughout.
In the description of the present invention, it should be noted that, for the terms of orientation, such as "central", "lateral (X)", "longitudinal (Y)", "vertical (Z)", "length", "width", "thickness", "upper", "lower", "front", "rear", "left", "right", "vertical", "horizontal", "top", "bottom", "inner", "outer", "clockwise", "counterclockwise", etc., indicate that the orientation and positional relationship are based on the orientation or positional relationship shown in the drawings, and are only for the convenience of describing the present invention and simplifying the description, but do not indicate or imply that the referred device or element must have a specific orientation, be constructed and operated in a specific orientation, and should not be construed as limiting the specific scope of the present invention.
The technical scheme and the beneficial effects of the invention are clearer and clearer by further describing the specific embodiment of the invention with the accompanying drawings of the specification. The embodiments described below are exemplary and are intended to be illustrative of the invention, but are not to be construed as limiting the invention.
Referring to fig. 1, an embodiment of the present invention provides a coordinate system establishing method, including the following steps:
the method comprises the following steps: four coordinate systems are established, and the position relation between the coordinate systems is solved.
(1) The three-coordinate measuring instrument is fixed on a horizontal ground, and a geodetic coordinate system O can be automatically established in the three-coordinate measuring instrumentgXgYgZgThe positive Z-axis direction of the geodetic coordinate system is by default directed downward.
(2) The method comprises the steps of fixing a measured object on the ground, measuring geometrical characteristics of a plurality of points, lines, surfaces and the like on the measured object by using a three-coordinate measuring instrument, and establishing a space rectangular coordinate system on the measured object by using the characteristics obtained by the measurement to serve as a reference coordinate system. The function of the reference coordinate system is to define the coordinates of the centroid of the measured object. In the test process, a mass center coordinate system is defined, the origin of the mass center coordinate system is located at the mass center of the measured object, and the directions of all coordinate axes are the same as those of the reference coordinate system. Assuming that the coordinates of the origin of the reference coordinate system measured by the coordinate measuring machine in the geodetic coordinate system are: (x)rog,yrog,zrog) Coordinate axis OrXr，OrYr，OrZrThe direction cosines in the geodetic coordinate system are respectively: (cos alpha.)rxg,cosβrxg,cosγrxg)，(cosαryg,cosβryg,cosγryg)， (cosαrzg,cosβrzg,cosγrzg)。
(3) And (3) fixing the object to be measured in the step (2) at a constant position, and mounting a high-precision tool with three mutually vertical planes on the object to be measured, wherein a cuboid tool is taken as an example. Selecting three mutually adjacent planes on the tool, and respectively naming the three planes as S1，S2，S3. Defining a connected coordinate system O on the toolrXrYrZrAs shown in fig. 2. Measuring the positions of three planes in the geodetic coordinate system by using a three-coordinate measuring instrument, determining the origin position of the connected coordinate system by calculating the intersection point of the three planes, and determining the position of one coordinate axis of the connected coordinate system by calculating the intersection line of any two planes, such as the plane S2And S3The intersection line of is OrXrAxis, plane S1And S3The intersection line of is OrYrAxis, plane S1And S2The intersection line of is OrZrA shaft. The intersection point and the intersection line can be calculated on the operating software of the three-coordinate measuring machine, and the coordinates of the origin of the connected coordinate system in the geodetic coordinate system are assumed as follows: (x)bog,ybog,zbog) Coordinate axis ObXb，ObYb， ObZbThe direction cosines in the geodetic coordinate system are respectively: (cos alpha.)bxg,cosβbxg,cosγbxg)， (cosαbyg,cosβbyg,cosγbyg)，(cosαbzg,cosβbzg,cosγbzg)。
(4) Establishing a reference coordinate system OrXrYrZrAnd a connected coordinate system ObXbYbZbCoordinate conversion formula between. First, the origin Or of the reference coordinate system in the reference coordinate system O is determined by the following formulabXbYbZbCoordinate of (x)rob,yrob,zrob)：
Figure RE-GDA0002996098940000111
Then, the reference coordinate system O is obtained by the following formularXrYrZrEach coordinate axis is in a connected coordinate system ObXbYbZbDirectional cosine of (1):
Figure RE-GDA0002996098940000112
for any point W, assuming the point W is in the connected coordinate system ObXbYbZbHas the coordinate of (x)wb,ywb,zwb) Then the point W is in the reference coordinate system OrXrYrZrThe coordinates in (c) can be calculated by:
Figure RE-GDA0002996098940000113
(5) establishing a wobble plate coordinate system OpXpYpZp. The measured object is placed on the three-line pendulum, the position of the measured object is adjusted to enable the readings of the three force sensors connected with the cycloid to be the same, and the fact that the center of mass of the measured object falls on the axis of the wobble plate is guaranteed. And fixing the measured object, and measuring the positions of the three characteristic points on the wobble plate by using a three-coordinate measuring instrument. Using the central positions of the three characteristic points as the origin O of the wobble plate coordinate systempTaking the connecting line direction of the characteristic point 1 and the characteristic point 2 as a wobble plate coordinate system OpXpDirection of axis, with origin OpThe direction of the connecting line with the characteristic point 3 is taken as a wobble plate coordinate system OpYpThe direction of the shaft is determined by the right hand rule to determine the coordinate system O of the wobble platepZpThe direction of the axis, as shown in fig. 3. Assumed measured origin O of wobble plate coordinate systempThe coordinate in the geodetic coordinate system is (x)pog,ypog,zpog)，OpZpThe direction cosine of the axis in the geodetic coordinate system is (0,0, 1). The relationship between the various coordinate systems is shown in figure 4.
The embodiment of the invention also provides a rigid body centroid parameter testing method based on the coordinate system, which comprises the following steps:
step two: and measuring the positions of the axes of the pendulums in the reference coordinate system under different placing postures.
(1) Measuring S on the tool using a three-coordinate measuring machine1，S2，S3Plane surfaceAnd establishing a connected coordinate system according to the mode of the step one (3). Suppose that the origin O of the connected coordinate system is at the ith placement attitudebThe coordinates in the geodetic coordinate system are: (x)bogi,ybogi,zbogi) Coordinate axis ObXb， ObYb，ObZbThe direction cosines in the geodetic coordinate system are respectively: (cos alpha.)bxgi,cosβbxgi,cosγbxgi)， (cosαbygi,cosβbygi,cosγbygi)，(cosαbzgi,cosβbzgi,cosγbzgi)。
(2) And solving the position of the axis of the wobble plate in the connected coordinate system by using the position relation between the connected coordinate system and the wobble plate coordinate system. Thus, the origin OpThe coordinates in the global coordinate system are:
Figure RE-GDA0002996098940000121
OpZpthe direction cosine of the axis in the connected coordinate system is:
Figure RE-GDA0002996098940000122
and then, according to the coordinate conversion relation between the reference coordinate system and the connected coordinate system established in the step one (4), the position of the axis of the wobble plate in the reference coordinate system is obtained. Origin OpThe coordinates in the reference coordinate system are:
Figure RE-GDA0002996098940000123
OpZpthe directional cosine of the axes in the reference coordinate system is:
Figure RE-GDA0002996098940000131
(3) changing the placing posture of the measured object on the balance, adjusting the position of the measured object to enable the readings of the three force sensors to be the same, and then repeating the second step. At least 2 placing postures are needed to measure the centroid position, and in order to reduce the test error, 4 placing postures can be generally selected for testing.
Step three: calculating the mass center of the measured object in the reference coordinate system OrXrYrZrCoordinates of (2).
(1) In a reference coordinate system OrXrYrZrIn theory, any two wobble plate axes (i.e., O)pZpAxes) is the centroid of the object being measured. Due to measurement errors, there may be no intersection between any two wobble plate axes, and therefore the midpoint of the common perpendicular line segment of the two wobble plate axes is taken as a feasible centroid, as shown in fig. 5. Taking the axis 1 and 2 of the wobble plate as an example, O can be obtained by the second stepp1And Op2The coordinates in the reference coordinate system are respectively (x)por1,ypor1,zpor1) And (x)por2,ypor2,zpor2)，Op1Zp1Shaft and Op2Zp2The direction cosine of the axes in the reference coordinate system is (cos)pzr1,cospzr1,cospzr1) And (cos)pzr2,cospzr2,cospzr2). Thus, the coordinates of a feasible centroid in the reference coordinate system are:
Figure RE-GDA0002996098940000132
wherein Q, R and S are respectively expressed as
Figure RE-GDA0002996098940000133
(2) A feasible center of mass can be found from any two wobble plate axes, and then a calculated center of mass can be determined from the feasible center of masses according to the following formula:
Figure RE-GDA0002996098940000141
wherein N is the number of the placing postures.
The embodiment of the invention also provides a rigid body inertia parameter testing method based on the coordinate system, which comprises the following steps:
step four: and measuring the torsional pendulum period and the direction cosine of the torsional pendulum axis in the reference coordinate system under different placing postures.
(1) The measured object is placed on the swing disc, the position of the measured object is adjusted to enable the readings of the three force sensors to be the same, then the measured object and the swing disc do small-angle torsional pendulum motion around the axis of the swing disc, and the torsional pendulum amplitude is required to be less than 5 degrees. Measuring the torsional pendulum period T of a measured object by means of a photoelectric sensori。
(2) Fixing the object to be measured, and measuring the plane S on the tool by using the inclinometer1，S2，S3The angles with the horizontal plane are respectively marked as theta1，θ2，θ3(0≤θ<Pi). Thus, OpZpThe directional cosine of the axis in the global coordinate system can be calculated by:
Figure RE-GDA0002996098940000142
then O ispZpThe directional cosine of the axis in the centroid coordinate system can be calculated by:
Figure RE-GDA0002996098940000143
(3) changing the placing posture of the measured object on the balance and repeating the step four. At least 6 sets of placement poses are required, and 8 sets are generally selected in order to reduce the number of test errors.
Step five: and calculating inertia parameters of the measured object relative to the centroid coordinate system.
(1) Firstly, the moment of inertia of a measured object about the axis of the wobble plate is calculated, and the moment of inertia of the torsional pendulum system about the axis of the wobble plate is as follows:
Figure RE-GDA0002996098940000151
in the formula, m is the mass of the torsional pendulum system, including the mass of the pendulum plate, the cycloid and the measured object; r is the torsional pendulum radius, l is the pendulum length, and g is the gravitational acceleration.
Moment of inertia J of the pendulum mass about the axis of the pendulum when no object is placed0The moment of inertia of the measured object about the axis of the wobble plate can be obtained through a calibration experiment:
JPi＝Ji-J0 (13)
in the formula, JPi、JiThe moment of inertia of the measured object and the torsion pendulum system about the axis of the wobble plate under the ith placing posture is respectively.
(2) When the number of the placing postures is more than 6, the inertia parameters of the measured object relative to the centroid coordinate system need to be calculated by using a least square method. Firstly, according to the principle of inertia ellipsoid, the rotational inertia and the inertia product (J) of an object under the coordinate system of the centroidx、Jy、Jz、Jxy、Jyz、Jzx) Moment of inertia J with an axis passing through the center of massPi(i ═ 1,2, … …, N) has the following relationship:
Figure RE-GDA0002996098940000152
li＝cosαpzci，mi＝cosβpzci，ni＝cosγpzci (14b)
in the formula, N is the total number of tests.
Therefore, the inertia parameters of the measured object about the centroid coordinate system can be obtained by using the least square method as follows:
J＝(ATA)-1ATJP (15a)
J＝[Jx Jy Jz Jxy Jyz Jzx]T (15b)
JP＝[JP1 JP2 … JPN]T (15c)
Figure RE-GDA0002996098940000161
the following describes the data processing and calculation process in detail by taking a certain test as an example. The data measured in the test are as follows (in the test data, the unit of coordinates is mm, and the unit of inertia parameter is kg · m2)：
In the step one (2), the coordinate of the origin of the reference coordinate system in the geodetic coordinate system is measured as
[xrog,yrog,zrog]＝[-514.152,835.713,-440.124]
Measured coordinate axis OrXr，OrYr，OrZrThe direction cosines in the geodetic coordinate system are respectively:
[cosαrxg,cosβrxg,cosγrxg]＝[-0.4603,-0.8877,-0.0074]
[cosαryg,cosβryg,cosγryg]＝[0.8877,-0.4603,0.0024]
[cosαrzg,cosβrzg,cosγrzg]＝[-0.0049,-0.0043,0.9999]
in the step one (3), the coordinate of the origin of the connected coordinate system in the geodetic coordinate system is measured by using a three-coordinate measuring instrument as follows:
[xbog,ybog,zbog]＝[-632.582,875.251,-412.407]
measured coordinate axis ObXb，ObYb，ObZbThe direction cosines in the geodetic coordinate system are respectively:
[cosαbxg,cosβbxg,cosγbxg]＝[0.7585,-0.4035,0.5117]
[cosαbyg,cosβbyg,cosγbyg]＝[0.4020,0.9078,0.1200]
[cosαbzg,cosβbzg,cosγbzg]＝[-0.5129,0.1147,0.8508]
then, according to the equations (1-3) in the step one (4), the reference coordinate system O can be establishedrXrYrZrAnd a connected coordinate system ObXbYbZbCoordinate conversion formula between:
Figure RE-GDA0002996098940000171
in the step one (5), a three-coordinate measuring instrument is used for measuring the origin O of the coordinate system of the wobble platepThe coordinates in the geodetic coordinate system are:
[xpog,ypog,zpog]＝[587.744,855.308,1221.158]
in the second step (1), the origin O of the coordinate system of the connected body in the first placing posture is measured by using the three-coordinate measuring instrumentbThe coordinates in the geodetic coordinate system are:
[xbog1,ybog1,zbog1]＝[486.689,958.409,837.988]
measured coordinate axis ObXb，ObYb，ObZbThe direction cosines in the geodetic coordinate system are respectively:
[cosαbxg1,cosβbxg1,cosγbxg1]＝[0.4726,-0.7936,0.3836]
[cosαbyg1,cosβbyg1,cosγbyg1]＝[0.8710,0.4869,-0.0652]
[cosαbzg1,cosβbzg1,cosγbzg1]＝[-0.1350,0.3649,0.9212]
then, according to the formulas (4,5) in the step two (2), the origin O of the wobble plate coordinate system can be obtainedp1The coordinates in the global coordinate system are:
[xpob1,ypob1,zpob1]＝[276.5384,12.8242,301.7024]
derived pendulum coordinate system Op1Zp1The direction cosine of the axis in the connected coordinate system is:
[cosαpzb1,cosβpzb1,cosγpzb1]＝[0.3836,-0.0652,0.9212]
then, according to the coordinate transformation relation (3) between the reference coordinate system and the connected coordinate system obtained in the step one (4), a wobble plate coordinate system O can be obtainedp1The coordinates in the reference coordinate system are:
[xpor1,ypor1,zpor1]＝[46.5591,-38.8015,427.8176]
derived pendulum coordinate system Op1Zp1The directional cosine of the axes in the reference coordinate system is:
[cosαpzr1,cosβpzr1,cosγpzr1]＝[0.1846,-0.1322,0.9736]
in the same way, under the second placing posture, the measured origin O of the wobble plate coordinate systemp2The coordinates in the reference coordinate system are:
[xpor2,ypor2,zpor2]＝[214.8637,-130.6814,294.2495]
balance coordinate system Op2Zp2The directional cosine of the axes in the reference coordinate system is:
[cosαpzr2,cosβpzr2,cosγpzr2]＝[0.5848,-0.3399,0.7358]
then, the coordinates of one of the feasible centroids in the reference coordinate system can be obtained according to equation (8) in step three (1):
[xcr1,ycr1,zcr1]＝[-40.2284,20.4271,-27.1063]
in the test, 4 placing postures are provided, and any two placing postures can determine a feasible centroid, so that 6 feasible centroids are provided, and the coordinates in the reference coordinate system are respectively as follows:
[xcr1,ycr1,zcr1]＝[-40.2284,20.4271,-27.1063]
[xcr2,ycr2,zcr2]＝[-39.9803,20.7854,-27.3153]
[xcr3,ycr3,zcr3]＝[-40.4935,21.4031,-28.0308]
[xcr4,ycr4,zcr4]＝[-39.4581,21.2537,-26.5139]
[xcr5,ycr5,zcr5]＝[-41.0310,19.9509,-27.4356]
[xcr6,ycr6,zcr6]＝[-39.8689,19.7002,-26.6638]
then, the coordinates of the calculated centroid in the reference coordinate system can be obtained according to equation (9) in step three (2):
[xcr,ycr,zcr]＝[-40.1767,20.5868,-27.1776]
in the step four (1), the torsional pendulum period measured in the first placing posture is T1＝1.4151s。
In step four (2), in the first placement posture, the plane S measured by the inclinometer1，S2，S3The included angles with the horizontal plane are respectively theta1＝113.17°，θ2＝85.27°，θ3When the angle is 156.33 °, the wobble plate coordinate system O can be determined from equations (10) and (11)pZpThe direction cosine of the axis in the coordinate system of the center of mass is
[cosαpzc1,cosβpzc1,cosγpzc1]＝[-0.2010,0.1199,-0.9722]
In step five (1), the mass m of the torsional pendulum system is 147.26kg, the torsional pendulum radius R is 310mm, the pendulum length l is 3266.49mm, and the moment of inertia J of the masses of the wobble plate and the cycloid about the wobble plate axis0＝0.9107kg·m2. Then, according to the equations (12) and (13), the moment of inertia of the object to be measured about the axis of the wobble plate is calculated as Jp1＝1.2429kg·m2。
In step five (2), 8 sets of placement postures are totally obtained, and the A matrix and the J matrix obtained by measurement are obtainedpThe matrices are respectively:
Figure RE-GDA0002996098940000191
JP＝[1.2429 1.1663 0.9722 1.0163 1.1612 1.3199 1.2587 1.0689]T
then, the inertial parameters of the measured object with respect to the centroid coordinate system can be obtained according to equation (14) as follows:
J＝[0.9956 1.3076 1.1808 0.1475 -0.0462 -0.1791]T
in the description of the specification, reference to the description of "one embodiment", "preferably", "an example", "a specific example" or "some examples", etc., means that a particular feature, structure, material, or characteristic described in connection with the embodiment or example is included in at least one embodiment or example of the invention, and schematic representations of the terms in this specification do not necessarily refer to the same embodiment or example. Furthermore, the particular features, structures, materials, or characteristics described may be combined in any suitable manner in any one or more embodiments or examples.
The present invention is not limited to the above-described embodiments, and it will be apparent to those skilled in the art that various modifications and improvements can be made without departing from the principle of the present invention, and such modifications and improvements are also considered to be within the scope of the present invention. Those not described in detail in this specification are within the skill of the art.

Claims (9)

1. A method for establishing a coordinate system, comprising the steps of:
s101, fixing a three-coordinate measuring instrument on a horizontal ground, establishing a geodetic coordinate system, and defining coordinates obtained by measuring with the three-coordinate measuring instrument in the geodetic coordinate system;
s102, fixing a measured object on the ground, measuring the geometric characteristics of the measured object by using a three-coordinate measuring instrument, and establishing a reference coordinate system on the measured object; in the testing process, a mass center coordinate system is defined, the origin of the mass center coordinate system is positioned at the mass center of the object to be tested, and the directions of all coordinate axes are the same as the reference coordinate system;
s103, fixing the object to be measured at the unchanged position in the step S102, and mounting a tool with three mutually vertical planes on the object to be measured; measuring the positions of three mutually adjacent planes on the tool in a geodetic coordinate system by using a three-coordinate measuring instrument to establish a connected coordinate system;
s104, obtaining a coordinate conversion relation between the connected coordinate system and the reference coordinate system according to the positions of the reference coordinate system and the connected coordinate system in the geodetic coordinate system;
s105, placing a measured object on the three-line pendulum, adjusting the position of the measured object to enable the readings of the three force sensors connected with the cycloid to be the same, and ensuring that the mass center of the measured object falls on the axis of the wobble plate; and fixing the measured object, and measuring the positions of at least three characteristic points on the wobble plate by using a three-coordinate measuring instrument to establish a wobble plate coordinate system.
2. The coordinate system establishing method according to claim 1, wherein: in step S104, the origin O of the reference coordinate system is obtained by the following formularIn a reference coordinate system OrXrYrZrCoordinate of (x)rob,yrob,zrob)：
Figure FDA0002930448170000011
Then, the reference coordinate system O is obtained by the following formularXrYrZrEach coordinate axis is in a connected coordinate system ObXbYbZbDirectional cosine of (1):
Figure FDA0002930448170000021
for any point W, assuming the point W is in the connected coordinate system ObXbYbZbHas the coordinate of (x)wb,ywb,zwb) Then the point W is in the reference coordinate system OrXrYrZrThe coordinates in (c) can be calculated by:
Figure FDA0002930448170000022
3. a rigid body centroid parameter testing method based on the coordinate system of claim 1, characterized by comprising the following steps:
s201, placing a measured object on a balance according to a specific posture, measuring three adjacent planes on a tool by using a three-coordinate measuring instrument, and establishing a connected coordinate system according to the mode of the step 103;
s202, solving the position of the axis of the wobble plate in the connected coordinate system by using the position relation between the connected coordinate system and the wobble plate coordinate system; then, according to the coordinate conversion relation between the reference coordinate system and the connected coordinate system established in the step 104, the position of the axis of the wobble plate in the reference coordinate system is obtained;
s203, changing the placing posture of the measured object on the swinging plate, and repeating the steps S201 to S202; until the positions of the axes of the wobble plate of the object to be measured in at least two placing postures in the reference coordinate system are obtained;
s204, if the axes of the wobble plates have an intersection point, the intersection point is the mass center of the object to be measured; if no intersection point exists between any two wobble plate axes, the midpoint of a common perpendicular line segment of the two wobble plate axes is taken as a feasible centroid; the average position of the feasible centroids can be taken, and a calculated centroid can be determined.
4. The rigid body centroid parameter testing method of claim 3, wherein: in step S202, it is assumed that the origin O of the coordinate system of the connected body is set in the ith placement posturebThe coordinates in the geodetic coordinate system are: (x)bogi,ybogi,zbogi) Coordinate axis ObXb，ObYb，ObZbAt the earth groundThe direction cosines in the coordinate system are respectively: (cos alpha.)bxgi,cosβbxgi,cosγbxgi)，(cosαbygi,cosβbygi,cosγbygi)，(cosαbzgi,cosβbzgi,cosγbzgi)；
Origin OpThe coordinates in the global coordinate system are:
Figure FDA0002930448170000031
OpZpthe direction cosine of the axis in the connected coordinate system is:
Figure FDA0002930448170000032
origin OpThe coordinates in the reference coordinate system are:
Figure FDA0002930448170000033
OpZpthe directional cosine of the axes in the reference coordinate system is:
Figure FDA0002930448170000034
5. the rigid body centroid parameter testing method of claim 4, wherein: in step S204, if there is no intersection point between any two wobble plate axes, O is obtainedp1And Op2The coordinates in the reference coordinate system are respectively (x)por1,ypor1,zpor1) And (x)por2,ypor2,zpor2)，Op1Zp1Shaft and Op2Zp2The direction cosine of the axes in the reference coordinate system is (cos)pzr1,cospzr1,cospzr1) And (cos)pzr2,cospzr2,cospzr2) The coordinates of the feasible centroid in the reference coordinate system are:
Figure FDA0002930448170000041
wherein Q, R and S are respectively expressed as
Figure FDA0002930448170000042
A feasible center of mass can be found from any two wobble plate axes, and then a calculated center of mass can be determined from the feasible center of masses according to the following formula:
Figure FDA0002930448170000043
wherein N is the number of the placing postures.
6. A rigid body inertia parameter testing method based on the coordinate system of claim 1, comprising the steps of:
s301, placing a measured object on a swing disc according to a specific posture, adjusting the position of the measured object to enable the readings of the three force sensors to be the same, then enabling the measured object and the swing disc to do small-angle torsional pendulum motion around the axis of the swing disc, and measuring the torsional pendulum period of the measured object;
s302, measuring included angles between the three planes on the tool selected in the step 103 and a horizontal plane by using an inclinometer; the direction cosine of the axis of the wobble plate in the connected coordinate system is obtained according to the angles of the three included angles, and then the direction cosine of the axis of the wobble plate in the reference coordinate system is obtained through the coordinate conversion relation between the reference coordinate system and the connected coordinate system established in the step 104;
s303, changing the placing posture of the object to be measured on the swinging plate, and repeating the steps S301 to S302; until the direction cosine of the axis of the wobble plate of the measured object in at least six placing postures in the reference coordinate system is obtained;
s304, obtaining the rotational inertia of the object to be tested about the axis of the wobble plate according to the torsion period obtained by the test; and then establishing equations of the rotational inertia of the measured object about the axis of the wobble plate and the inertial parameters of the measured object about the mass center coordinate system according to the principle of the inertia ellipsoid, and solving the equations by using a least square method to obtain the inertial parameters of the measured object about the mass center coordinate system.
7. The rigid body inertia parameter testing method of claim 6, wherein: in step S302, a plane S on the tool is measured by using an inclinometer1，S2，S3The angles with the horizontal plane are respectively marked as theta1，θ2，θ3(0≤θ<Pi); thus, OpZpThe directional cosine of the axis in the global coordinate system can be calculated by:
Figure FDA0002930448170000051
then O ispZpThe directional cosine of the axis in the centroid coordinate system can be calculated by:
Figure FDA0002930448170000052
8. a rigid body inertia parameter testing method according to claim 7, wherein in step S304, the moment of inertia of the torsional pendulum system about the axis of the wobble plate is:
Figure FDA0002930448170000053
in the formula, m is the mass of the torsional pendulum system, including the mass of the pendulum plate, the cycloid and the measured object; r is the torsional pendulum radius, l is the pendulum length, and g is the gravity acceleration;
moment of inertia J of the pendulum mass about the axis of the pendulum when no object is placed0The moment of inertia of the measured object about the axis of the wobble plate can be obtained through a calibration experiment:
JPi＝Ji-J0 (13)；
in the formula, JPi、JiThe moment of inertia of the measured object and the torsion pendulum system about the axis of the wobble plate under the ith placing posture is respectively.
9. The rigid body inertia parameter testing method of claim 8, wherein in step S304, the product of inertia (J) and the moment of inertia of an object in the centroid coordinate system thereof is determined according to the principle of inertia ellipsoidx、Jy、Jz、Jxy、Jyz、Jzx) Moment of inertia J with an axis passing through the center of massPi(i ═ 1,2, … …, N) has the following relationship:
Figure FDA0002930448170000061
li＝cosαpzci，mi＝cosβpzci，ni＝cosγpzci (14b)；
in the formula, N is the total times of the test;
therefore, the inertia parameters of the measured object about the centroid coordinate system can be obtained by using the least square method as follows:
J＝(ATA)-1ATJP (15a)；
J＝[Jx Jy Jz Jxy Jyz Jzx]T (15b)；
JP＝[JP1 JP2 L JPN]T (15c)；
Figure FDA0002930448170000062
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