CN113434972B - Method for calculating axial static stiffness of planetary roller screw - Google Patents

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

A Calculation Method of Axial Static Stiffness of Planetary Roller Screw

technical field

The invention relates to the field of planetary roller screw manufacturing, in particular to a method for calculating the axial static stiffness of a planetary roller screw.

Background technique

Planetary roller screw is a high-precision transmission device that converts rotation and linear motion into each other. Planetary roller screw is widely used in aviation, aerospace, medical and other fields. Because these fields have very high precision requirements for the equipment used, and the planetary roller screw is its main transmission component, the accuracy of its transmission accuracy is very important. When the planetary roller screw is subjected to axial force, it will produce axial deformation, which will affect its transmission accuracy. Studying its deformation law when it is subjected to different tensile forces will have an important impact on improving the accuracy of the device.

At present, the research on the axial stiffness of the planetary roller screw at home and abroad is mainly based on the design method of the ball screw to simplify the actual design size of the planetary roller screw, so as to calculate the axial stiffness of the planetary roller screw. They cannot obtain the actual shape model of the contact thread of the planetary roller screw, but regard the model of the contact position as the contact between the ball and the curved surface, rather than the point contact of the two threads in the actual working process, and the two contact surfaces The contact curvature radii of are the radii of the ball and the radii of the curved surface, respectively. There is a large gap between the calculation results obtained by using the traditional method and the actual axial static stiffness of the planetary roller screw pair, which cannot reflect the stiffness characteristics of the actual planetary roller screw. The main manifestations are: 1. The calculation of the axial static stiffness of the planetary roller screw is inaccurate; 2. Some parameters of the planetary roller screw cannot be reflected in the stiffness characteristics; 3. When parameterizing the planetary roller screw It cannot be truly and accurately reflected in the rigidity change of the planetary roller screw during design. This will seriously affect the transmission accuracy of the device.

Contents of the invention

The technical problem to be solved by the embodiments of the present invention is to provide a method for calculating the axial static stiffness of a planetary roller screw. It can be used to analyze the axial static stiffness of planetary roller screws.

In order to solve the above technical problems, an embodiment of the present invention provides a method for calculating the axial static stiffness of a planetary roller screw, including load decomposition, stiffness decomposition and stiffness combination methods;

The load decomposition method includes decomposing the load borne by the planetary roller screw as a whole into the load borne by each thread in the planetary roller screw, and the load decomposition formula is:

In the formula, Fave is the load borne by a single thread, F is the load borne by the overall screw, n is the number of threads of a single roller, and n _r is the number of rollers;

The stiffness decomposition method includes:

The stiffness of the planetary roller screw is simplified and decomposed, which is used to turn the stiffness model of the planetary roller screw into a specific expression relation, and convert the stiffness into deformation. The overall deformation δ1 of a single thread is:

δ ₁ = δ _BS + δ _TS + δ _HS + δ _TRS + δ _BR + δ _TRN

+δ _HN +δ _TN +δ _BN

In the formula, δ _T , δ _B , δ _H represent thread deformation, solid deformation and contact deformation respectively, T represents thread, H represents contact, B represents entity, N represents nut, R represents roller, S for the lead screw;

The stiffness combination method includes combining the obtained single thread stiffness to obtain the overall static axial stiffness model of the planetary roller screw, and the overall axial stiffness K of the planetary roller screw is:

Wherein, the stiffness decomposition method includes:

Modeling the shape mathematical model of the planetary roller screw thread, the equivalent radius of curvature R _Ei is:

where Θi and Xi are the principal curvatures of the contact surface;

Establish a planetary roller screw thread contact stiffness model:

In the above formula, P is the applied pressure, F ₂ is the displacement correction factor in Hertz contact, E ₁ and E ₂ are the elastic modulus of the two contact objects, and E ^* is the equivalent elastic modulus of the two contact objects.

Wherein, the stiffness combination method includes: parallel connection of a single thread contact stiffness model of the planetary roller screw to establish an overall stiffness model of the planetary roller screw.

Implementing the embodiment of the present invention has the following beneficial effects:

1. Accurately established the mathematical model of the thread shape of the planetary roller screw, which is more accurate than the simplified modeling of the other planetary roller screw shapes;

2. Accurately established the axial static stiffness model of the planetary roller screw, which is more accurate than the regional planetary roller screw static stiffness model;

3. The established axial static stiffness model of the planetary roller screw can consider a variety of factors affecting the axial static stiffness, and then analyze the influence of these factors on the axial static stiffness, which is very important for the parameterization of the planetary roller screw Design matters.

Description of drawings

Fig. 1 is a schematic diagram of the contact connection of the planetary roller screw involved in the present invention;

Fig. 2 is the flow chart that the present invention establishes the thread shape mathematical model of planetary roller screw;

Fig. 3 is the flowchart of establishing the overall axial static stiffness model of the planetary roller screw in the present invention;

Fig. 4 is a graph showing the variation of stiffness with load of the model established by the present invention.

detailed description

In order to make the object, technical solution and advantages of the present invention clearer, the present invention will be further described in detail below in conjunction with the accompanying drawings.

Figure 1 is a schematic diagram of the contact connection of the planetary roller screw. As shown in the figure, from outside to inside are nuts, rollers, and leadscrews. In actual work, the nut is connected to the outside, one side of the roller is in contact with the nut, and one side is connected to the lead screw, and the lead screw is connected to the motor. In the figure, δ _T , δ _B , and δ _H represent thread deformation, solid deformation and contact deformation, respectively.

Where T stands for thread (thread), H stands for contact (Hertz), B stands for body (body), N stands for nut (nut), R stands for roller (roller), S stands for screw (screw) , so δ _TN in the figure represents the deformation between the thread and the nut. The rest are combined as defined above.

The steps required for each stage of the method of the present invention in the modeling and calculation process are as follows:

A. Load decomposition stage:

For the load borne by the planetary roller screw as a whole, it is necessary to decompose it into the load borne by each thread. The actual planetary roller screw contains different numbers of rollers according to different models, and different models lead to different lengths of rollers and different thread pitches. Therefore, it is necessary to pass the initial conditions according to the actual model of the planetary roller screw, and then Uniform load, according to the obtained stiffness model, determine the load borne by each thread through iterative calculation, that is, the overall load/number of threads is the uniform load it receives. The planetary roller screw can be regarded as the parallel connection of the actual rollers, and each roller can be regarded as the parallel connection of all the threads.

The load decomposition formula is:

In the formula, F _ave is the load borne by a single thread, F is the load borne by the overall screw, n is the number of threads of a single roller, and n _r is the number of rollers.

B. Stiffness decomposition stage:

The stiffness of the planetary roller screw is decomposed into the stiffness of each thread. Through the analysis of the actual situation, the abstract simplification deduces that the stiffness of each thread should be decomposed into contact stiffness, thread stiffness and cylindrical solid stiffness. On the one hand, this decomposition method can reasonably consider the factors affecting the axial static stiffness of the planetary roller screw, and introduce more comprehensive influencing factors; on the other hand, it can also clearly reflect the components of the axial stiffness of the planetary roller screw and The degree of influence of various stiffnesses is analyzed to analyze the part that has the greatest impact on the axial static stiffness of the planetary roller screw, so that the weaker parts can be strengthened.

Stiffness decomposition methods include:

The stiffness of the planetary roller screw is simplified and decomposed, which is used to turn the stiffness model of the planetary roller screw into a specific expression relation, and convert the stiffness into deformation. The overall deformation δ1 of a single thread is:

δ ₁ = δ _BS + δ _TS + δ _HS + δ _TRS + δ _BR + δ _TRN

+δ _HN +δ _TN +δ _BN

In the formula, δ _T , δ _B , δ _H represent thread deformation, solid deformation and contact deformation respectively, T represents thread, H represents contact, B represents entity, N represents nut, R represents roller, S For the lead screw.

1. Contact stiffness modeling for a single thread of a planetary roller screw

The contact stiffness modeling needs to obtain the mathematical model of the thread shape of the planetary roller screw at first. Since the contact threads of the planetary roller screw are non-standard threads. The thread of the roller in the planetary roller screw is composed of two upper and lower semicircles, and the actual parameters of these two semicircles also need to be obtained from the model of the planetary roller screw; The thread of the bar is a right angle triangular thread.

The thread section itself is an inferior circle, and in the Cartesian coordinate system, the section needs to be raised along the helical direction. Due to the complex structure, the modeling of the thread surface needs to use the relevant knowledge of space differential geometry. Through the spatial screw thread surface The first and second types of expressions, if you want to find the stiffness, you must know the curvature. To solve the curvature, you need to solve it by differential geometry. At this time, the following coefficients are required:

L＝0，E=1+tan ² β _S ,

L=0,

M＝ _{-cosβS sinαS} , N＝-u _sinβs

E, F, G, L, M, and N in the formula are all the coefficients of the 1st and 2nd types of basic formal equations in differential geometry.

The principal curvature and the principal curvature radius of two contacting surfaces at the contact point are obtained by the expression of the contact position:

The principal curvature formula of a surface is:

The two principal radii of curvature of the surface are:

From the Hertz contact formula can be obtained:

和R_1i为1面的长轴与短轴长度，

和R_2i为2面的长轴与短轴长度。 _{θi and xi are} both coefficients of the contact surface expression,

and R _1i is the major axis and minor axis length of 1 plane,

and R _2i are the lengths of the major and minor axes of the two planes.

where γ is the angle between the two principal axes of curvature.

Equivalent radius of curvature:

Because the respective material properties of the two contact surfaces are known, the contact deformation produced by the two contact surfaces under the rated load condition can be obtained through the classical Hertzian contact theory.

In the formula, P is the applied pressure, F ₂ is the displacement correction factor in Hertz contact, E ₁ and E ₂ are the elastic modulus of the two contact objects, and E* is the equivalent elastic modulus of the two contact objects.

Planetary roller screw thread contact stiffness modeling is used to establish the stiffness generated when all threads in the overall stiffness model of the planetary roller screw are in contact, and is an important part of the overall stiffness of the planetary roller screw.

The thread deformation stiffness of the planetary roller screw is used to establish the stiffness generated when all threads in the overall stiffness model of the planetary roller screw are deformed, and is an important part of the overall stiffness of the planetary roller screw;

The rigidity of the cylindrical body of the planetary roller screw is used to establish the rigidity of all the remaining cylindrical bodies except the threads in the overall rigidity modulus of the planetary roller screw, and is an important part of the overall stiffness of the planetary roller screw.

Deformation = Load/Stiffness

The contact stiffness of the two thread contact surfaces can be obtained by dividing the load by the resulting contact deformation, so the above deformation is solved for the final stiffness solution.

2. For the calculation of thread stiffness, a single thread can be regarded as a tiny cantilever beam. For the calculation of the bending stiffness of the cantilever beam, some mature empirical formulas can be used. Fully considering the profile of the thread, using the empirical formula and bringing in the relevant parameters of the thread, the deformation and stiffness of each special thread of the planetary roller screw can be obtained.

Calculation formula for thread deformation:

In the formula:

v is Poisson's ratio, ω is the applied load, r ₀ is the initial radial position, r is the radius of the position where the force is applied at the mirror position, 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 , and F ₃ are all coefficients.

3. For the calculation of the rigidity of the cylindrical solid part of each component of the planetary roller screw, it is obtained by using the knowledge of material mechanics. The solid part of the cylinder refers to the various components of the planetary roller screw, which is actually composed of a part of the cylinder and the thread. The rigidity of the solid part of the cylinder means that the complex thread part is not considered, and only the stiffness of the remaining solid part is considered. The stiffness of this part is the tensile stiffness of the cylinder or ring, which can be obtained by using simple knowledge of material mechanics.

Body stiffness formulas for screws, nuts and rollers:

n _R represents the number of rollers in the model, E _S is the elastic modulus of the screw, A _S is the cross-sectional area of the screw, P is the thread length, E _N is the elastic modulus of the nut, A _N is the nut Cross-sectional area, E _R is the elastic modulus of the roller, and _AR is the cross-sectional area of the roller.

C. Stiffness combination stage

1. Modeling the stiffness model of a planetary roller screw with a single thread contact

Connecting the stiffness models of the components of the axial static stiffness of the planetary roller screw obtained from the above process in series, the total stiffness of the model with only one screw thread, one roller thread, one nut thread and their cylindrical entities is obtained. Here we need Concatenate the contact models.

Single thread contact stiffness model:

2. Modeling the stiffness model of the entire planetary roller screw

Combining the stiffness models of the planetary roller screw with single thread contact, it is considered that the stiffness model of the entire planetary roller screw is the parallel connection of all the contact threads contained in the planetary roller screw pair, and the contact stiffness model of a single thread is connected in parallel. Contains the number of threads. At the same time, it is necessary to convert the load borne by the entire planetary roller screw into the load borne by a single thread, which is brought into the parallel calculation of the model, and the total load borne by the entire planetary roller screw needs to be considered when calculating the stiffness of the overall model .

The entire planetary roller screw stiffness model:

The above descriptions are only preferred embodiments of the present invention, and are not intended to limit the present invention. Any modifications, equivalent replacements and improvements made within the principles and spirit of the present invention are all included in the protection of the present invention. within range.

The above calculation process will be implemented using Matlab programming. To illustrate the effectiveness of the method more specifically, the present invention provides a calculation example result.

Planetary roller screw parameters:

The comparison between the static axial stiffness of the planetary roller screw calculated by the present invention and the calculation result of an empirical formula provided by the working manual of the planetary roller screw.

The axial stiffness of the planetary roller screw obtained by using the invention is greater than the stiffness of the screw according to the empirical formula. Compared with the empirical formula, the model of the present invention can consider more factors affecting the rigidity of the planetary roller screw, such as design factors such as contact angle, so as to facilitate the parametric design of the planetary roller screw.

The above disclosure is only a preferred embodiment of the present invention, which certainly cannot limit the scope of rights of the present invention. Therefore, equivalent changes made according to the claims of the present invention still fall within the scope of the present invention.

Claims (2)

1. A calculation method for static axial stiffness of a planetary roller screw, characterized in that, comprising load decomposition, stiffness decomposition and stiffness combination methods; The load decomposition method includes decomposing the load borne by the planetary roller screw as a whole into the load borne by each thread in the planetary roller screw, and the load decomposition formula is:

In the formula, F _ave is the load borne by a single thread, F is the load borne by the overall screw, n is the number of threads of a single roller, and n _r is the number of rollers; The stiffness decomposition method includes: The rigidity of the planetary roller screw is simplified and decomposed, which is used to turn the rigidity model of the planetary roller screw into a specific expression relation, and convert the stiffness into deformation, and the deformation of a single thread as a whole

for:

In the formula, δ _T , δ _B , δ _H represent thread deformation, solid deformation and contact deformation respectively, T represents thread, H represents contact, B represents entity, N represents nut, R represents roller, S for the lead screw; The stiffness combination method includes combining the obtained single thread stiffness to obtain the overall static axial stiffness model of the planetary roller screw, and the overall axial stiffness K of the planetary roller screw is:

; The single thread contact stiffness model of the planetary roller screw is connected in parallel to establish the overall stiffness model of the planetary roller screw. 2. The method for calculating the axial static stiffness of the planetary roller screw according to claim 1, wherein the stiffness decomposition method comprises: Modeling the shape mathematical model of the planetary roller screw thread, the equivalent radius of curvature R _Ei is:

Where Θi and Χi are the principal curvatures of the contact surface; Establish a planetary roller screw thread contact stiffness model:

In the above formula, P is the applied pressure, F ₂ is the displacement correction factor in Hertz contact, E ₁ and E ₂ are the elastic modulus of the two contact objects, and E ^* is the equivalent elastic modulus of the two contact objects.

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