CN112380742A - Method for calculating rotational inertia of homogeneous object in any shape around any rotating shaft - Google Patents
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Abstract
The invention discloses a method for calculating the moment of inertia of a homogeneous object in any shape around any rotating shaft, which belongs to the field of engineering mechanics and comprises the following steps: step 1, establishing a geometric model of an object and dividing grid units; step 2, extracting data such as coordinates, areas or volumes of each unit, and establishing a unit information database; step 3, determining the position information of the rotating shaft; step 4, calculating the mass of each unit, the distance between each unit and the rotating shaft and the moment of inertia of each unit around the rotating shaft; and 5, summing the rotational inertia of each unit to obtain the rotational inertia of the object. The method for calculating the rotational inertia is suitable for homogeneous objects in any shapes, particularly homogeneous objects in complex shapes, can wind any rotating shaft, and provides effective support for accurate calculation of the rotational inertia of large-scale equipment in the engineering field.
Description
Technical Field
The invention relates to the technical field of engineering mechanics, in particular to a method for calculating the moment of inertia of a homogeneous object in any shape around any rotating shaft.
Background
In the rotational dynamics, the moment of inertia is one of important parameters for researching the rotational motion of an object, is used for establishing a quantitative relation among a plurality of variables such as angular momentum, angular velocity, moment and angular acceleration, and is applied to the industrial fields such as aerospace, electric power, machinery, instruments and the like. Particularly, in the aspect of precision equipment such as the appearance design of engine blades, flywheels, gyros and spacecrafts, accurate calculation of the rotational inertia is necessary.
The current methods for obtaining the moment of inertia of an object include two methods: simple rotational inertia calculation method and experimental method. The simple rotational inertia calculation method is only suitable for homogeneous objects with regular shapes, and can only be used for measuring objects with complex shapes or uneven mass distribution through an experimental method. Aiming at a homogeneous object with a complex shape, the simple rotational inertia calculation method cannot solve the problems of centroid determination, infinitesimal division, rotational inertia solution and the like. The measurement of the rotational inertia is reviewed by the progress and the prospect of the measurement research of rotational inertia of Wang Xiao San, Liuyun, Nighingn, Zhang, etc. (J) astronavigation measurement technology, 2019,39(2): 1-5), and the measurement principle and the advantages and the disadvantages of the experimental methods of the measurement of the rotational inertia such as a three-line pendulum method, a torsion pendulum method, a falling body method, etc. are summarized and analyzed. The highest measuring accuracy of the rotational inertia measuring equipment based on the torsional pendulum method can reach 0.1%, but the experimental method needs to consider the influence of a dynamic model, air damping and the like, and the accuracy depends on the experimental equipment, and the method comprises the following steps: accuracy of the control system, accuracy of the measurement system, equipment carrying capacity, etc. Therefore, the measurement cost of the experimental method is high, and the maximum measurement mass or the maximum measurement rotational inertia limit exists in the equipment. Meanwhile, for a homogeneous object with a complex shape, the experimental method cannot realize the measurement of the rotational inertia of the object around any rotating shaft. In conclusion, the method for calculating the moment of inertia of the homogeneous object with the complex shape around any rotating shaft is established, the defects of a simple moment of inertia calculation method and an experimental method are overcome, and the method becomes a key technical problem in the rotational dynamics.
Disclosure of Invention
The invention aims to overcome the defects of the prior art and provide a method for calculating the moment of inertia of a homogeneous object in an arbitrary shape around an arbitrary rotating shaft, namely a unit superposition method. According to the method, geometric modeling is carried out on an object with any shape through CAD software, characteristics of accurate division, unit information and the like of a finite element grid are fully utilized, grid unit division is carried out on the object, a unit information database is established, the relative position relation between a unit and a rotating shaft is determined, and then the rotational inertia of the object is obtained. The core of the unit superposition method is to calculate the moment of inertia of each unit, and finally obtain the integral moment of inertia of the object through the superposition of all the units. The method is suitable for homogeneous objects with any shape, particularly homogeneous objects with complex shapes, and can wind around any rotating shaft to provide effective support for the research of rotational dynamics.
The invention is realized by at least one of the following technical schemes.
A method for calculating the moment of inertia of a homogeneous object in an arbitrary shape around an arbitrary rotation axis comprises the following steps:
step 1, establishing a geometric model of an object and dividing grid units;
step 2, extracting data such as coordinates, areas or volumes of each unit, and establishing a unit information database;
step 3, determining the position information of the rotating shaft;
step 4, calculating the mass of each unit, the distance between each unit and the rotating shaft and the moment of inertia of each unit around the rotating shaft;
and 5, summing the rotational inertia of each unit to obtain the rotational inertia of the object.
Preferably, step 1 is to apply CAD software to create the corresponding geometric model.
Preferably, the grid elements are divided in ANSYS finite element software, the geometric model of the object is introduced into the ANSYS finite element software, the element types are selected, and the grid elements are divided.
Preferably, step 2 is to extract the data information by using the Fortune language code of ANSYS software.
Preferably, the data information comprises coordinate, area or volume data for each cell.
Preferably, step 3 specifically comprises: an arbitrary rotation axis in space is expressed as an intercept equation:
wherein (x, y, z) is the coordinate of any point in accordance with the rotation axis equation (x)l，yl，zl) Is the coordinate of a certain point on the rotating shaft, (i, j, k) is the direction vector of the rotating shaft, and t is the common variable of the rotating shaft equation.
Preferably, the mass Δ m of the unit is:
Δm＝ρV (2)
Δm＝ρδS (3)
where ρ is the material density, V is the volume of the SOLID cell, S is the area of the SHELL cell, and δ is the thickness of the SHELL cell.
Preferably, the distance between each unit and the rotation axis is:
wherein (x)0，y0，z0) Is the coordinate of the cell, t is the common variable of the rotation axis equation, and Δ r is the distance between the cell and the rotation axis.
Preferably, the moment of inertia of the unit is:
ΔJ＝ΔmΔr2 (6)。
preferably, the moment of inertia of the object is:
J＝∑ΔJ (7)。
the arbitrary rotating shaft realizes the positioning of the rotating shaft in the coordinate system through an intercept equation.
The unit superposition method comprises grid unit division, information databases such as coordinates, areas or volumes of the units and the like, unit mass calculation, distance calculation between the units and a rotating shaft, rotational inertia calculation of the units and rotational inertia superposition of each unit.
The grid unit division realizes the accurate division of the micro-elements of the geometric model.
The information database of the coordinates, the areas or the volumes of the units utilizes the information characteristics of the units, and the information of the units can be directly extracted and the information database can be established through the Fortune language of finite element software.
The unit mass calculation can be realized by combining the information database such as the coordinate, the area or the volume of the unit and the like with the material density parameter.
The distance between the unit and the rotating shaft can be calculated by combining an intercept equation with an information database of the coordinate, the area or the volume of the unit and the like.
The calculation of the mass of the unit in combination with the calculation of the distance between the unit and the rotation axis enables the calculation of the moment of inertia of the unit.
The superposition of the rotational inertia of each unit is defined by using the rotational inertia, and finally the rotational inertia of the whole object is obtained.
The unit superposition method is suitable for homogeneous objects with any shape, particularly homogeneous objects with complex shapes, can wind any rotating shaft, and can verify the accuracy and the effectiveness of the homogeneous objects by a simple rotational inertia calculation method of regular homogeneous objects.
According to the invention, a corresponding geometric model is established for an object with any shape by using CAD software, accurate division of grid units is realized in finite element software, parameters such as coordinates, areas and volumes of the grid units are extracted by using information characteristics of the grid units, and further the mass of each grid unit, the distance between each grid unit and a rotating shaft and the moment of inertia of each grid unit around the rotating shaft are calculated. And finally, according to the superposable property of the rotational inertia, the rotational inertia of the homogeneous object in any shape around any rotating shaft can be obtained by superposing the rotational inertia of all the units. The core of the unit superposition method is to extract the required unit parameters by utilizing the characteristics of the finite element grids such as accurate division, unit information and the like, further calculate the rotational inertia of each unit, and finally obtain the rotational inertia of the whole object through the superposition of all the units. The method can solve the problems of determination of the mass center of the object, infinitesimal division, solution of the rotational inertia and the like in a simple rotational inertia calculation method, avoids the problems of dependence on the accuracy of experimental equipment, high measurement cost, maximum measurement limitation, incapability of realizing rotational inertia measurement around any rotating shaft and the like in an experimental method, and simplifies the rotational inertia solution problem into the object geometric model modeling problem. The method for calculating the rotational inertia is suitable for homogeneous objects in any shapes, particularly homogeneous objects in complex shapes, can wind any rotating shaft, and provides effective support for accurate calculation of the rotational inertia of large-scale equipment in the engineering field.
Compared with the prior art, the invention has the beneficial effects that:
(1) the problems that a simple rotational inertia calculation method cannot solve the problems of mass center determination, infinitesimal division, rotational inertia solving and the like of the homogeneous object with the complex shape are solved, and the application range of the rotational inertia calculation method is widened to the homogeneous object with the arbitrary shape;
(2) the problems that a three-line pendulum method, a torsion pendulum method, a falling body method and other rotational inertia measurement experiment methods depend on the accuracy of experiment equipment, the measurement cost is high, the maximum measurement quality or the maximum measurement rotational inertia is limited, the rotational inertia measurement of an object around any rotating shaft cannot be realized and the like are solved without experiments, and the rotational inertia calculation of a homogeneous object in any shape around any rotating shaft is realized;
(3) the method has the advantages that the rotational inertia calculation problem is converted into the object geometric modeling problem, the calculation process is easy to operate, and an effective method is provided for the rotational inertia calculation of increasingly large equipment such as flywheels and spacecrafts around any rotating shaft in the industrial field.
Drawings
FIG. 1 is a flow chart of an embodiment relating to a rotational inertia calculation method (unit superposition method);
FIG. 2 is a schematic view of a geometric model of an embodiment complex-shaped object;
FIG. 3 is a schematic diagram of a finite element model of an embodiment of a complex object;
FIG. 4 is a schematic view of a geometric model of a cylindrical shell according to an embodiment;
FIG. 5 is a schematic view of a geometric model of a cylinder according to an embodiment;
FIG. 6 is a schematic view of a geometric model of a disk according to an embodiment.
Detailed Description
In order to make the technical solutions and advantages of the present invention clearer, the following describes the technical solutions in the embodiments of the present invention clearly and completely with reference to the drawings in the embodiments of the present invention.
The method for calculating the moment of inertia of the homogeneous object in any shape around any rotating shaft, as shown in fig. 1, specifically comprises the following steps:
step 1, establishing a geometric model of an object and dividing grid units. In engineering practice, a corresponding geometric model is established in CAD software according to geometric parameters or design drawings of an object. And (3) introducing the geometric model of the object into ANSYS finite element software, selecting a proper unit type (the entity is an SOLID unit, and the SHELL is a SHELL unit), dividing the grid unit, and easily realizing accurate division of the object micro-elements by using the advantages of grid unit division in the finite element software. The higher the divided grid unit quality is, the more the divided grid unit quantity is, and the more accurate the finally calculated moment of inertia is. The basis for judging the quality of the grid cells is that the size of the grid cells is uniform, and the maximum or minimum grid cells are few.
In an embodiment, as shown in FIG. 2, a gear geometry model is created using the CAD software SolidWorks. The geometric model is saved in the format of x _ t, ANSYS finite element software is imported, SOLID unit types are selected, and grid units are divided, as shown in FIG. 3.
And 2, extracting data such as coordinates, areas or volumes of each unit by using a Fortune language code of ANSYS software and combining the information characteristics of the units, and establishing a unit information database. In the finite element software, the division of the grid elements simultaneously realizes the determination of information such as element positions, areas or volumes, and the like, and the realization of the functions under the normal condition is an extremely complex process. Therefore, the utilization of the information characteristics of the cell itself provides a precondition for the mass calculation of the object infinitesimal and the calculation of the distance between the cell and the rotation axis.
The Fortune language code of ANSYS software is written and comprises the following steps: GET command extraction unit total; extracting the minimum unit number by the GET command; DIM command definition unit information database table; performing unit data extraction circulation by using the DO command, wherein the circulation times are the total number of units; in the loop process, the loop is started from the minimum unit number, the GET command extracts the data such as XYZ coordinates, areas or volumes of the corresponding unit number and stores the data in the unit database table, and the ELNEXT command jumps to the next unit number. By writing codes, XYZ coordinate, area or volume and other data of each unit can be directly extracted, and a unit information database is established.
And 3, determining the position information of the rotating shaft. Any rotation axis in space can be expressed as an intercept equation:
wherein (x, y, z) is the coordinate of any point in accordance with the rotation axis equation (x)l，yl，zl) Is the coordinate of a certain point on the rotating shaft, (i, j, k) is the direction vector of the rotating shaft, and t is the common variable of the rotating shaft equation.
In an embodiment, for the gear, a central axis z-axis is selected as the rotation axis. When the coordinate of a certain point on the rotating shaft is (0, 0, 0) and the directional quantity is (0, 0, 1), the equation of the z-axis of the central axis is:
x＝y＝0 (2)。
step 4, obtaining the distance from each unit mass to the rotating shaft and the moment of inertia, specifically: and combining a unit information database, calculating unit mass through unit area or volume, and calculating the distance between the unit mass and the rotating shaft through unit coordinates. The unit mass Δ m is obtained by equation (3) and equation (4):
Δm＝ρV (3)
Δm＝ρδS (4)
where ρ is the material density, V is the volume of the SOLID cell, S is the area of the SHELL cell, and δ is the thickness of the SHELL cell.
Let the cell coordinate be (x)0，y0，z0) The distance between the cell and the rotation axis is obtained by equation (5) and equation (6):
by the formula (3) to the formula (6), the mass Δ m of each cell, the distance Δ r between each cell and the rotation axis can be obtained. On the basis, the moment of inertia of the unit is obtained by the formula (7):
ΔJ＝ΔmΔr2 (7)
the mass of the cell is obtained by using formula (3), and the distance between the cell and the z-axis of the rotating shaft is formula (8).
And 5, summing the rotational inertia of each unit to obtain the rotational inertia of the object.
J＝∑ΔJ (9)
Through the 5 steps, the moment of inertia calculation of the homogeneous object with any shape around any rotating shaft can be realized, and particularly, the homogeneous object with a complex shape can be obtained. Except that the geometric modeling of the object in the step 1 needs to be realized in CAD software, the other steps can be realized by Fortune language programming of finite element software, and the whole process is quick and efficient. Therefore, the method converts the solving problem of the rotational inertia into the object geometric modeling problem, the design drawing of the engineering structure provides shape parameters for geometric modeling, and the CAD software modeling technology is very mature.
The method will be described in detail by taking a gear, which is a homogeneous object having a complicated shape, as an example. The accuracy of the method will be verified below using as examples regular homogeneous objects-cylindrical shells (as in fig. 4), cylinders (as in fig. 5), discs (as in fig. 6). The rotational inertia of the cylindrical shell, the cylinder and the disc around the y axis can be directly obtained through a formula (9), a formula (10) and a formula (11). As shown in Table 1, the relative error between the formula solving result of the rotational inertia of the regular homogeneous object and the result of the method is extremely small, and the accuracy and the effectiveness of the method are fully illustrated. Moreover, the quality and the quantity of the grid units are improved, the relative error can be further reduced, and the accuracy is improved.
Where m represents the mass of the object.
TABLE 1 comparison of formula solving results of regular homogeneous object moment of inertia with results of the method of the present invention
The above embodiments are preferred embodiments of the present invention, but the present invention is not limited to the above embodiments, and any other changes, modifications, substitutions, combinations, and simplifications which do not depart from the spirit and principle of the present invention should be construed as equivalents thereof, and all such changes, modifications, substitutions, combinations, and simplifications are intended to be included in the scope of the present invention.
Claims (10)
1. A method for calculating the moment of inertia of a homogeneous object in any shape around any rotating shaft is characterized by comprising the following steps of:
step 1, establishing a geometric model of an object and dividing grid units;
step 2, extracting data such as coordinates, areas or volumes of each unit, and establishing a unit information database;
step 3, determining the position information of the rotating shaft;
step 4, calculating the mass of each unit, the distance between each unit and the rotating shaft and the moment of inertia of each unit around the rotating shaft;
and 5, summing the rotational inertia of each unit to obtain the rotational inertia of the object.
2. The method of calculating the moment of inertia of a homogeneous object of arbitrary shape about an arbitrary rotation axis according to claim 1, wherein: step 1 is to build a corresponding geometric model in CAD software.
3. The method of calculating the moment of inertia of a homogeneous object of arbitrary shape about an arbitrary rotation axis according to claim 2, wherein: and the grid unit division is carried out in ANSYS finite element software, a geometric model of the object is introduced into the ANSYS finite element software, the unit type is selected, and the grid unit is divided.
4. The method of calculating the moment of inertia of a homogeneous object of arbitrary shape about an arbitrary rotation axis according to claim 3, wherein: and step 2, extracting data information by using the Fortune language code of ANSYS software.
5. The method of calculating the moment of inertia of an arbitrarily-shaped homogeneous object about an arbitrary rotation axis according to claim 4, wherein: the data information includes coordinate, area or volume data for each cell.
6. The method of calculating the moment of inertia of a homogeneous object of arbitrary shape about an arbitrary rotation axis according to claim 5, wherein: the step 3 specifically comprises the following steps: an arbitrary rotation axis in space is expressed as an intercept equation:
wherein (x, y, z) is the coordinate of any point in accordance with the rotation axis equation (x)l，yl，zl) Is the coordinate of a certain point on the rotating shaft, (i, j, k) is the direction vector of the rotating shaft, and t is the common variable of the rotating shaft equation.
7. The method of calculating the moment of inertia of a homogeneous object of arbitrary shape about an arbitrary rotation axis according to claim 6, wherein: the mass Δ m of the unit is:
Δm＝ρV (2)
Δm＝ρδS (3)
where ρ is the material density, V is the volume of the SOLID cell, S is the area of the SHELL cell, and δ is the thickness of the SHELL cell.
8. The method of calculating the moment of inertia of a homogeneous object of arbitrary shape about an arbitrary rotation axis according to claim 7, wherein: the distance between each unit and the rotation axis is:
wherein (x)0，y0，z0) In the form of the coordinates of the cell,t is the common variable of the rotating axis equation and Δ r is the distance between the cell and the rotating axis.
9. The method of calculating the moment of inertia of a homogeneous object of arbitrary shape about an arbitrary rotation axis according to claim 8, wherein: the moment of inertia of the unit is:
ΔJ＝ΔmΔr2 (6)。
10. the method of calculating the moment of inertia of a homogeneous object of arbitrary shape about an arbitrary rotation axis according to claim 9, wherein: the moment of inertia of the object is:
J＝∑ΔJ (7)。
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| WO2017000396A1 (en) * | 2015-06-30 | 2017-01-05 | 中国空间技术研究院 | Truss antenna reflector deployment dynamics modelling method based on multi-body analysis test |
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