CN113434972B - Method for calculating axial static stiffness of planetary roller screw - Google Patents
Method for calculating axial static stiffness of planetary roller screw Download PDFInfo
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Abstract
The embodiment of the invention discloses a method for calculating axial static stiffness of a planetary roller screw, which comprises load decomposition, stiffness decomposition and stiffness combination, wherein the load decomposition comprises the steps of decomposing the load borne by the whole planetary roller screw into all threads to obtain the load value borne by each thread; the rigidity decomposition comprises modeling processes such as mathematical modeling of a thread surface of the planetary roller screw, modeling of contact rigidity of the thread of the planetary roller screw, modeling of deformation rigidity of the thread of the planetary roller screw, modeling of rigidity of a cylindrical entity of the planetary roller screw and the like; the rigidity combination comprises the steps of connecting the rigidity of each thread of the planetary roller screw in parallel to obtain an integral rigidity model of the planetary roller screw, and simplifying the model from complexity to complexity so as to calculate the rigidity. The invention can realize the calculation and analysis of the axial static rigidity of the planetary roller screw and can conveniently carry out parametric design on the planetary roller screw. And a dynamic analysis theoretical basis is provided for improving the precision, reducing the transmission error and reducing the vibration.
Description
Technical Field
The invention relates to the field of manufacturing of planetary roller screws, in particular to a method for calculating axial static stiffness of a planetary roller screw.
Background
The planetary roller screw is a high-precision transmission device which mutually converts rotation and linear motion, and is widely applied to the fields of aviation, aerospace, medical treatment and the like. Since the precision requirements of the fields on the used equipment are very high, and the planetary roller screw is used as the main transmission component, the accuracy of the transmission precision is very important. When the planetary roller screw bears the axial force, the planetary roller screw can generate axial deformation to influence the transmission precision of the planetary roller screw, and the research on the deformation rule when the planetary roller screw is subjected to different tensile forces can generate important influence on the precision of the improved device.
At present, the axial rigidity of the planetary roller screw is researched at home and abroad mainly by simplifying the actual design size of the planetary roller screw according to a ball screw design method, so that the axial rigidity of the planetary roller screw is obtained through calculation. They do not give a realistic profile model of the contact thread of the planetary roller screw, but rather the model of the contact position is regarded as the contact of the ball with the curved surface, rather than the point contact of two threads during actual operation, the contact curvature radii of the two contact surfaces being the radius of the ball and the radius of the curved surface, respectively. The difference between the calculation result obtained by using the traditional method and the actual axial static rigidity of the planetary roller screw pair is large, and the actual rigidity characteristic of the planetary roller screw cannot be reflected. It mainly shows that: 1. the axial static stiffness of the planetary roller screw is calculated inaccurately; 2. some parameters of the planetary roller screw cannot be reflected in the stiffness characteristics; 3. when the planetary roller screw is subjected to parametric design, the rigidity change of the planetary roller screw cannot be truly and accurately reflected. This will seriously affect the transmission accuracy of the device.
Disclosure of Invention
The technical problem to be solved by the embodiment of the invention is to provide a method for calculating the axial static stiffness of a planetary roller screw. The method can be used for analyzing the axial static stiffness of the planetary roller screw.
In order to solve the technical problem, the embodiment of the invention provides a method for calculating axial static stiffness of a planetary roller screw, which comprises a load decomposition method, a stiffness decomposition method and a stiffness combination method;
the load decomposition method comprises the steps of decomposing the load borne by the whole planetary roller screw into the load borne by each thread in the planetary roller screw, wherein the load decomposition formula is as follows:
in the formula, fave is the load borne by a single thread, F is the load borne by the whole screw rod, n is the number of threads of a single roller, and n is the number of threads of a single roller r The number of the rollers;
the stiffness decomposition method includes:
simplifying and decomposing the rigidity of the planetary roller screw, changing the rigidity model of the planetary roller screw into a specific expression relational expression, and converting the rigidity into deformation, wherein the integral deformation delta 1 of a single thread is as follows:
δ 1 ＝δ BS +δ TS +δ HS +δ TRS +δ BR +δ TRN
+δ HN +δ TN +δ BN
in the formula, delta T ，δ B ，δ H Respectively representing thread deformation, solid deformation and contact deformation, T representing threads, H representing contacts, B representing solids, N representing a nut, R representing a roller and S representing a lead screw;
the rigidity combination method comprises the following steps of combining the obtained single thread rigidity to obtain an integral axial static rigidity model of the planetary roller screw, wherein the integral axial rigidity K of the planetary roller screw is as follows:
wherein the stiffness decomposition method comprises:
modeling the outline mathematical model of the planetary roller screw thread with an equivalent curvature radius R Ei Comprises the following steps:
in the formula, the theta i and the Xi are the main curvatures of the contact curved surfaces;
establishing a planetary roller screw thread contact rigidity model:
in the above formula, P is the pressure, F 2 Correction factor for displacement in Hertz contact, E 1 And E 2 Is the modulus of elasticity of two contacting objects, E * Is the equivalent elastic modulus of the two contacting objects.
Wherein the stiffness combination method comprises: and connecting the single thread contact rigidity models of the planetary roller screw in parallel for establishing an integral rigidity model of the planetary roller screw.
The embodiment of the invention has the following beneficial effects:
1. the mathematical model of the thread profile of the planetary roller screw is accurately established, and the modeling is more accurate compared with the simplified modeling of the profile of the other planetary roller screws;
2. the axial static stiffness model of the planetary roller screw is accurately established, and is more accurate compared with the regional static stiffness model of the planetary roller screw;
3. the established axial static stiffness model of the planetary roller screw can consider various factors influencing axial static stiffness, and then the influence of the factors on the axial static stiffness is analyzed, so that the axial static stiffness model has important significance on the parametric design of the planetary roller screw.
Drawings
FIG. 1 is a schematic view of a planetary roller screw contact connection according to the present invention;
FIG. 2 is a flow chart of the present invention for establishing a mathematical model of the thread profile of a planetary roller screw;
FIG. 3 is a flow chart of the present invention for establishing a planetary roller screw overall axial static stiffness model;
FIG. 4 is a graph of stiffness as a function of load modeled by the present invention.
Detailed Description
To make the objects, technical solutions and advantages of the present invention more apparent, the present invention will be described in further detail with reference to the accompanying drawings.
FIG. 1 is a schematic view of a planetary roller screw contact connection. As shown in the figure, the nut, the roller and the screw rod are arranged from outside to inside in sequence. In actual operation, the nut is connected with the outside, and the roller IThe side is contacted with the nut, and one side is connected with the lead screw which is connected with the motor. In the diagram delta T ，δ B ，δ H Respectively representing thread deformation, solid deformation and contact deformation.
Wherein T represents a thread (thread), H represents a contact (Hertz), B represents a solid (body), N represents a nut (nut), R represents a roller (roller), and S represents a screw (screw), so that δ in the figure represents TN It represents the deformation between the thread and the nut. The others are combined as defined above.
The steps required by each stage in the modeling and calculating process of the method are as follows:
A. and (3) a load decomposition stage:
the load borne by the planetary roller screw as a whole needs to be divided into the loads borne by the respective threads. The actual planetary roller screw contains different numbers of rollers according to different models, and the different models lead to different lengths of the rollers and different thread pitches, so that the load needs to be uniformly distributed according to the actual model of the planetary roller screw through initial conditions, and the load borne by each thread is determined through an iterative calculation mode according to the obtained rigidity model, namely the whole load/thread number is the uniformly distributed load borne by the thread. The planetary roller screw can be regarded as being formed by connecting the actually contained rollers in parallel, and each roller can be regarded as being formed by connecting all threads contained in parallel.
The load decomposition formula is:
in the formula F ave Load for a single thread, F load for a whole screw, n number of threads for a single roller, n r The number of the rollers.
B. A rigidity decomposition stage:
the rigidity of the planetary roller screw is decomposed into the rigidity of each thread, and through analysis of actual conditions, abstract simplification deduces that the rigidity of each thread is decomposed into contact rigidity, thread rigidity and cylinder solid rigidity. On one hand, the decomposition method can reasonably consider factors influencing the axial static rigidity of the planetary roller screw and introduce more comprehensive influencing factors; on the other hand, the influence degree of the components of the axial rigidity of the planet roller screw and various rigidities can be clearly reflected, and the part with the largest influence on the axial static rigidity of the planet roller screw is analyzed, so that the part with weak rigidity can be strengthened.
The rigidity decomposition method comprises the following steps:
simplifying and decomposing the rigidity of the planetary roller screw, changing the rigidity model of the planetary roller screw into a specific expression relational expression, and converting the rigidity into deformation, wherein the integral deformation delta 1 of a single thread is as follows:
δ 1 ＝δ BS +δ TS +δ HS +δ TRS +δ BR +δ TRN
+δ HN +δ TN +δ BN
in the formula, delta T ，δ B ，δ H Respectively represent thread deformation, solid deformation and contact deformation, T represents threads, H represents contacts, B represents solids, N is a nut, R is a roller and S is a lead screw.
1. Contact stiffness modeling for a single thread of a planetary roller screw
Contact stiffness modeling first requires a mathematical model of the planetary roller screw thread profile. The contact threads of the planetary roller screw are all nonstandard threads. The threads of the roller in the planetary roller screw are formed by an upper small semicircle and a lower small semicircle, wherein the actual parameters of the two semicircles are obtained by the type of the planetary roller screw; and the threads of the nut and the screw in the planetary roller screw are right-angled triangular threads.
The thread section is an inferior circle, the section needs to be raised along the spiral line direction in a Cartesian coordinate system, due to the complex structure, the modeling of the thread surface needs to use the related knowledge of space differential geometry, through the first and second expression forms of the space lead screw thread curved surface, if the rigidity is required, the curvature is required to be known, and the curvature is required to be solved through a differential geometry method, at the moment, the following coefficients are required:
E＝1+tan 2 β S ，
L＝0，
M＝-cosβ S sinα S ，N＝-u sin β s
in the formula, E, F, G, L, M and N are all coefficients of 1 and 2 basic form equations in differential geometry.
And (3) solving the main curvature and the main curvature radius of the two contacted curved surfaces at the contact point position through the expression of the contact position:
the principal curvature formula of a curved surface is:
the two principal radii of curvature of the curved surface are:
from the hertzian contact equation:
θ i and x i Are all the coefficients of the expression of the contact surface,
and R 1i Is the length of the major axis and the minor axis of the 1 plane,
and R 2i The length of the long axis and the short axis of the 2 planes.
Wherein gamma is the included angle of two main curvature axes.
Equivalent radius of curvature:
because the respective material properties of the two curved contact surfaces are known, the contact deformation generated by the two contact surfaces under the rated load condition during contact can be obtained through the classical hertzian contact theory.
In which P is the pressure, F 2 As a displacement correction factor in Hertz contact, E 1 And E 2 Is the elastic modulus of the two contact objects, and E is the equivalent elastic modulus of the two contact objects.
The modeling of the thread contact rigidity of the planetary roller screw is used for establishing the rigidity generated when all threads in the integral rigidity model of the planetary roller screw are contacted and is an important component of the integral rigidity of the planetary roller screw,
the thread deformation rigidity of the planetary roller screw is used for establishing rigidity generated when all threads in the overall rigidity model of the planetary roller screw deform and is an important component of the overall rigidity of the planetary roller screw;
the rigidity of the cylindrical entity of the planetary roller screw is used for establishing the rigidity of all the cylindrical entities except the threads in the overall rigidity model of the planetary roller screw and is an important component of the overall rigidity of the planetary roller screw.
Deformation = load/stiffness
The contact stiffness of the contact surfaces of the two threads can be obtained by dividing the load by the generated contact deformation, and therefore the above deformations are solved for the final stiffness solution.
2. The calculation of the stiffness of the threads may be considered to be a minor cantilever beam for a single thread, and some well-established empirical formula may be used for the calculation of the flexural stiffness of the cantilever beam. The deformation and the rigidity of each special thread of the planetary roller screw can be obtained by fully considering the outline of the thread and using an empirical formula and bringing in thread related parameters.
The screw thread deformation calculation formula:
in the formula:
v is Poisson's ratio, ω is applied load, r 0 The initial radial position, r is the radius of the position where the mirror image position applies force, a is the outer diameter of the circle, b is the inner diameter of the disc, C 8 ，C 9 ，L 9 ，G 3 ，F 2 ，F 3 Are all coefficients.
3. And calculating the rigidity of the cylindrical solid part of each component of the planetary roller screw by using knowledge of material mechanics. The cylindrical solid part refers to each component of the planetary roller screw and is actually formed by combining a part of cylinder and threads. The rigidity of the cylindrical solid part is obtained by not considering the complex thread part and only considering the rigidity of the remaining solid part, and the rigidity of the part is the tensile rigidity of the cylinder or the ring by using simple knowledge of material mechanics.
The formula for the bulk stiffness of the screw, nut and roller:
n R representing the number of rollers in the model, E S Is the modulus of elasticity of the screw, A S Is the cross-sectional area of the screw, P is the length of the thread, E N Is the modulus of elasticity of the nut, A N Is the cross-sectional area of the nut, E R Is the modulus of elasticity of the roller, A R Is the cross-sectional area of the roller.
C. Combined stages of stiffness
1. Modeling of a planetary roller screw stiffness model for single thread contact
All the component rigidity models of the axial static rigidity of the planetary roller screw obtained in the above process are connected in series, so that the total rigidity of the model with only one screw thread, one roller thread, one nut thread and cylindrical entities thereof is obtained, and all the contact models are required to be connected in series.
Single thread contact stiffness model:
2. modeling of whole planetary roller screw stiffness model
And combining the rigidity models of the planetary roller screws contacted by the single threads, and considering that the rigidity model of the whole planetary roller screw is formed by connecting all contacted threads contained in the planetary roller screw pair in parallel, wherein the contact rigidity model of the single thread is connected with the number of threads contained in the parallel. Meanwhile, the load borne by the whole planetary roller screw needs to be converted into the load borne by a single thread, the load is brought into parallel calculation of the model, and the total load borne by the whole planetary roller screw needs to be considered during the rigidity calculation of the whole model.
The rigidity model of the whole planetary roller screw is as follows:
the above description is only exemplary of the present invention and should not be taken as limiting the invention, as any modification, equivalent replacement, or improvement made within the spirit and scope of the present invention is also included in the present invention.
The above calculation process will be implemented using Matlab programming. To more particularly illustrate the effectiveness of the method, the present invention provides a calculation example result.
Parameters of the planetary roller screw:
the axial static rigidity of the planetary roller screw obtained by calculation is compared with a calculation result of an empirical formula provided by a working manual of the planetary roller screw.
The axial rigidity of the planetary roller screw obtained by the invention is larger than the screw rigidity of an empirical formula. Compared with an empirical formula, the model can consider more factors influencing the rigidity of the planetary roller screw, such as design factors like contact angle and the like, and is convenient for carrying out parametric design on the planetary roller screw.
While the invention has been described in connection with what is presently considered to be the most practical and preferred embodiment, it is to be understood that the invention is not to be limited to the disclosed embodiment, but on the contrary, is intended to cover various modifications and equivalent arrangements included within the spirit and scope of the appended claims.
Claims (2)
1. A method for calculating axial static stiffness of a planetary roller screw is characterized by comprising a load decomposition method, a stiffness decomposition method and a stiffness combination method;
the load decomposition method comprises the steps of decomposing the load borne by the whole planetary roller screw into the load borne by each thread in the planetary roller screw, wherein the load decomposition formula is as follows:
in the formula F ave Load for a single thread, F load for a whole screw, n number of threads for a single roller, n r The number of the rollers;
the stiffness decomposition method includes:
the rigidity of the planet roller screw is simplified and decomposed, and the method is used for changing a rigidity model of the planet roller screw into a specific expression relational expression, converting the rigidity into deformation and integrally deforming a single thread
Comprises the following steps:
in the formula, delta T ，δ B ，δ H Respectively representing thread deformation, solid deformation and contact deformation, T representing threads, H representing contacts, B representing solids, N representing a nut, R representing a roller and S representing a lead screw;
the rigidity combination method comprises the following steps of combining the obtained single thread rigidity to obtain an integral axial static rigidity model of the planetary roller screw, wherein the integral axial rigidity K of the planetary roller screw is as follows:
and connecting the single thread contact rigidity models of the planetary roller screw in parallel for establishing an integral rigidity model of the planetary roller screw.
2. The method for calculating axial static stiffness of a planetary roller screw according to claim 1, wherein the stiffness decomposition method comprises:
modeling the outline mathematical model of the planetary roller screw thread with an equivalent curvature radius R Ei Comprises the following steps:
wherein Θ i and β i are the principal curvatures of the curved contact surfaces;
establishing a planetary roller screw thread contact rigidity model:
in the above formula, P is the pressure, F 2 As a displacement correction factor in Hertz contact, E 1 And E 2 Is the modulus of elasticity of two contacting bodies, E * Is the equivalent elastic modulus of the two contacting objects.
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| CN110188446A (en) * | 2019-05-24 | 2019-08-30 | 大连理工大学 | A kind of composite panel drilling layering axis critical force calculation method considering deformation |
| CN110795800A (en) * | 2019-10-22 | 2020-02-14 | 广州广电计量检测股份有限公司 | Screw rigidity determination method |
| CN112016196A (en) * | 2020-08-11 | 2020-12-01 | 西北工业大学 | Double-nut planetary roller screw dynamics research method based on elastic deformation |
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| Publication number | Priority date | Publication date | Assignee | Title |
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| CN110188446A (en) * | 2019-05-24 | 2019-08-30 | 大连理工大学 | A kind of composite panel drilling layering axis critical force calculation method considering deformation |
| CN110795800A (en) * | 2019-10-22 | 2020-02-14 | 广州广电计量检测股份有限公司 | Screw rigidity determination method |
| CN112016196A (en) * | 2020-08-11 | 2020-12-01 | 西北工业大学 | Double-nut planetary roller screw dynamics research method based on elastic deformation |
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