CN108520136A - A kind of open vertical loop orthodontic force prediction model method for building up - Google Patents
A kind of open vertical loop orthodontic force prediction model method for building up Download PDFInfo
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Abstract
The invention discloses a kind of open vertical loop orthodontic force prediction model method for building up, are related to orthodontic treatment technical field, it is comprised the following steps：1) structure feature of extraction open vertical loop and load feature, conclude open vertical loop orthodontic force affecting parameters；2) upright arm deflection differential equation is established；3) top circular arc deflection differential equation is established, determines boundary condition；4) residual stress correction factor is introduced, open vertical loop orthodontic force prediction model is established.The open vertical loop correction force value that the present invention can effectively apply doctor carries out Parameter Expression, the open vertical loop correction force value that accurately prediction doctor is applied, it assists doctor to improve safety and the foresight of orthodontic treatment, improves the digitized degree of mouth cavity orthodontic diagnosis and treatment.
Description
Technical field
The present invention relates to a kind of open vertical loop orthodontic force prediction model method for building up, belong to orthodontic treatment technical field.
Background technology
Fixed orthodontics are presently the most effective orthodontic treatment method, wherein the shape of molding orthodontic wire is to influence
The determinant of orthodontic force.At present during rescuing, orthodontist is by virtue of experience fed back with the treatment of patient to determine mostly
The shape of bending of arch edgewise, orthodontic force lack quantitative criteria, treatment results place one's entire reliance upon doctor level, easily patient is damaged
And lead to the reduction of therapeutic efficiency.Therefore the prediction model for establishing orthodontic force carries out Parameter Expression, for carrying out to orthodontic force
Oral cavity digitizes diagnosis and treatment, and auxiliary doctor improves the safety of orthodontic treatment and foresight has a very important significance.
Invention content
In view of the above-mentioned problems, the technical problem to be solved in the present invention is to provide a kind of open vertical loop orthodontic force prediction models
Method for building up carries out Parameter Expression to open vertical loop orthodontic force.
Above-mentioned purpose is mainly realized by following scheme：
A kind of open vertical loop orthodontic force prediction model method for building up of the present invention, it is characterised in that：The tool of the method
Body realizes that process is：
1) structure feature of extraction open vertical loop and load feature, conclude open vertical loop orthodontic force affecting parameters；
2) upright arm deflection differential equation is established；
3) top circular arc deflection differential equation is established, determines boundary condition；
4) residual stress correction factor is introduced, open vertical loop orthodontic force prediction model is established.
Preferably, in the step 1), the characteristics of by analyzing open vertical loop, the structure of open vertical loop is extracted
Feature and load feature are respectively：Open vertical loop is made of top circular arc, upright arm and horizontal arm, whole along circular arc symmetry axis
Axisymmetricly structure, the horizontal arm of open vertical loop and the angle of upright arm are 90 °；It is horizontal by the way that pressure compression two is previously applied
The clearance distance of arm is obtained along the outward orthodontic force of horizontal arm；
It includes Toxocara species characteristic M, arch wire cross section property that concluding, which influences the major parameter of open vertical loop orthodontic force F sizes,
The characteristic parameter Q of S and open vertical loop, wherein common Toxocara species include stainless steel wire, Australia's silk and nitinol alloy wire, parameter
Cross section property for elastic modulus E, arch wire includes sectional area and cross sectional shape, and parameter is that big hang down is opened the inertia of bending axis away from I
Straight bent characteristic parameter is the clearance distance w of the height h and open vertical loop of open vertical loop, therefore obtains open vertical loop just
The citation form of abnormal power prediction model, as shown in Equation 1.
F=F (M, S, Q) (1)
Preferably, in the step 2), by open vertical loop structure feature it is found that only need to side symmetric part into
The bending radius of row analysis, open vertical loop top circular arc is that line of deflection can be obtained in the case that without advance active force w=2R in R
Approximate differential equation equation is as follows：
In formula, v is the sag of upright arm, i.e. the displacement distance of horizontal arm, M (y) be in vertical arm y apart from place by
Moment of flexure, IzIt is arch wire section to the moment of inertia of z-axis, y-axis is upright arm axis, and z-axis is and bending moment vertical with y-axis on upright arm
The axis of effect, for circle silk Iz=π DY 4/ 64, DYTo justify silk diameter, for square silk Iz=c1c2 3/ 12, c2For on rectangle silk section
With the length of z-axis parallel edges, c1For the length with z-axis vertical edges on rectangle silk section.
Formula (2) is integrated, the equations of rotating angle θ (y) and deflection equation v (y) for obtaining open vertical loop are：
In formula, C and D are integral constants, are determined by boundary condition, and the Bending Moment Equations of open vertical loop upright arm are：
M (y)=- P (h-R-y) (5)
In formula, P is that upright arm generation amount of deflection is pressure needed for the deformation of v, and formula (2) and (5) are brought into formula (3) and (4)
In integrated, can obtain：
Preferably, in the step 3), to determine the boundary condition of open vertical loop upright arm, need to upright arm and
The junction of top circular arc, i.e. deflection at y=0 are solved, and it is π/4 that the top circular arc of side, which is equivalent to radian, bent
Rate radius is the bent beam of R, and it is d α infinitesimals that one section of radian is taken to it, then the deflection differential equation of top circular arc is：
In formula, u is the displacement of top circular arc cross section in the x direction, as top circular arc and upright arm junction, the side x
To for top circular arc bending radius direction, initial deflection of the upright arm at y=0, torque M suffered by arc section0=M |Y=0=-P
(h-R), IωIt is section to the moment of inertia of ω axis, ω axis is top arc axis, due to the bending of upright arm and top circular arc
Type is consistent, there is Iω=Iz。
The ds=Rd α known to arc length formula, therefore the deflection differential equation of top circular arc can be changed to：
Solution constant coefficient non-homogeneous differential equation acquires：
Due to the symmetry of top circular arc, the boundary condition of top circular arc isIt solves：A=0,
Then the deflection differential equation of top circular arc is：
The equations of rotating angle of top circular arc is：
Therefore, by boundary conditionIt solves：
C, D, which are brought into formula (6), (7), to be obtained：
Because hard-over and maximum defluxion generate at upright arm and horizontal arm junction, i.e. y=h-R, maximum is scratched
Degree is horizontal arm displacement distance d, it is known that it is that pressure P needed for the deformation of v is that upright arm, which generates amount of deflection,：
By active force reaction force principle, orthodontic force FOAs generate the counter-force of pressure needed for the deflection：
Preferably, in the step 4), open vertical loop top arc section and upright arm and horizontal arm friendship are being bent
During meeting place, since flexural deformation will produce residual stress, certain influence can be generated to above-mentioned analytic process, therefore draw
Enter adjusted coefficient KOTo build final open vertical loop orthodontic force prediction model, wherein KOCan by experimental result with do not consider
The comparative analysis of the calculated results of correcting action obtains, and R=w/2, then has：
It defines experimental data and does not consider adjusted coefficient KOGross data difference relative to gross data ratio be reason
It is highly influenced by being fitted to the average theory adjusted rate under each parameter to obtain open vertical loop by adjusted rate
Adjusted coefficient Kh, the adjusted coefficient K that is influenced by gap widthw, the adjusted coefficient K s influenced by cross section property, and by material
The adjusted coefficient K that characteristic influencesM, since the correction effect of affecting parameters is different, the direct phase of correction factor under the conditions of inciting somebody to action respectively
The overlapping of correction effect can be caused by multiplying, therefore the height bent with 0.016 inch of most commonly seen stainless steel round wire is 7mm, wide
The correction factor of open vertical loop that degree is 2mm is that radix is converted, and calculates it in different gap width, different cross section and not
It is respectively K with the orthodontic force adjusted rate under materialw0、KS0And KM0, then adjusted coefficient KOFor：
Then the final expression formula of open vertical loop orthodontic force prediction model is：
Beneficial effects of the present invention are：
1, using the modeling method of parametrization, it can more intuitively reflect each influence factor to open vertical loop orthodontic force
Impact effect, the arch wire bent is adjusted convenient for doctor, to obtain suitable orthodontic force.
2, residual stress correction factor is introduced, and the expression formula of correction factor has been obtained by orthodontic force measurement experiment,
The error caused by residual stress during bending during model foundation can be eliminated, keeps the foundation of model more accurate.
Description of the drawings
The present invention is described in detail by following specific implementations and attached drawing for ease of explanation,.
Fig. 1 is the method for the present invention open vertical loop orthodontic force prediction model Establishing process figure；
Fig. 2 is open vertical loop upright arm mechanical analysis schematic diagram of the present invention；
Fig. 3 is circular arc mechanical analysis schematic diagram in open vertical loop top of the present invention.
Specific implementation mode
In order to make the objectives, technical solutions and advantages of the present invention clearer, below by shown in the accompanying drawings specific
Embodiment describes the present invention.However, it should be understood that these descriptions are merely illustrative, and it is not intended to limit the model of the present invention
It encloses.In addition, in the following description, descriptions of well-known structures and technologies are omitted, to avoid unnecessarily obscuring the present invention's
Concept.
As shown in Figure 1, Figure 2, Figure 3 shows, present embodiment uses following technical scheme：A kind of open vertical loop orthodontic force
Prediction model method for building up, it is characterised in that：The specific implementation process of the method is：
1) structure feature of extraction open vertical loop and load feature, conclude open vertical loop orthodontic force affecting parameters；
2) upright arm deflection differential equation is established；
3) top circular arc deflection differential equation is established, determines boundary condition；
4) residual stress correction factor is introduced, open vertical loop orthodontic force prediction model is established.
Further, in the step 1), by analyze open vertical loop the characteristics of, extract the structure of open vertical loop
Feature and load feature are respectively：Open vertical loop is made of top circular arc, upright arm and horizontal arm, whole along circular arc symmetry axis
Axisymmetricly structure, the horizontal arm of open vertical loop and the angle of upright arm are 90 °；It is horizontal by the way that pressure compression two is previously applied
The clearance distance of arm is obtained along the outward orthodontic force of horizontal arm；
It includes Toxocara species characteristic M, arch wire cross section property that concluding, which influences the major parameter of open vertical loop orthodontic force F sizes,
The characteristic parameter Q of S and open vertical loop, wherein common Toxocara species include stainless steel wire, Australia's silk and nitinol alloy wire, parameter
Cross section property for elastic modulus E, arch wire includes sectional area and cross sectional shape, and parameter is that big hang down is opened the inertia of bending axis away from I
Straight bent characteristic parameter is the clearance distance w of the height h and open vertical loop of open vertical loop, therefore obtains open vertical loop just
The citation form of abnormal power prediction model, as shown in Equation 1.
F=F (M, S, Q) (1)
Further, in the step 2), by open vertical loop structure feature it is found that only need to side symmetric part into
The bending radius of row analysis, open vertical loop top circular arc is that line of deflection can be obtained in the case that without advance active force w=2R in R
Approximate differential equation equation is as follows：
In formula, v is the sag of upright arm, i.e. the displacement distance of horizontal arm, M (y) be in vertical arm y apart from place by
Moment of flexure, IzIt is arch wire section to the moment of inertia of z-axis, y-axis is upright arm axis, and z-axis is and bending moment vertical with y-axis on upright arm
The axis of effect, for circle silk Iz=π DY 4/ 64, DYTo justify silk diameter, for square silk Iz=c1c2 3/ 12, c2For on rectangle silk section
With the length of z-axis parallel edges, c1For the length with z-axis vertical edges on rectangle silk section.
Formula (2) is integrated, the equations of rotating angle θ (y) and deflection equation v (y) for obtaining open vertical loop are：
In formula, C and D are integral constants, are determined by boundary condition, and the Bending Moment Equations of open vertical loop upright arm are：
M (y)=- P (h-R-y) (5)
In formula, P is that upright arm generation amount of deflection is pressure needed for the deformation of v, and formula (2) and (5) are brought into formula (3) and (4)
In integrated, can obtain：
Further, in the step 3), to determine the boundary condition of open vertical loop upright arm, need to upright arm and
The junction of top circular arc, i.e. deflection at y=0 are solved, and it is π/4 that the top circular arc of side, which is equivalent to radian, bent
Rate radius is the bent beam of R, and it is d α infinitesimals that one section of radian is taken to it, then the deflection differential equation of top circular arc is：
In formula, u is the displacement of top circular arc cross section in the x direction, as top circular arc and upright arm junction, the side x
To for top circular arc bending radius direction, initial deflection of the upright arm at y=0, torque M suffered by arc section0=M |Y=0=-P
(h-R), IωIt is section to the moment of inertia of ω axis, ω axis is top arc axis, due to the bending of upright arm and top circular arc
Type is consistent, there is Iω=Iz。
The ds=Rd α known to arc length formula, therefore the deflection differential equation of top circular arc can be changed to：
Solution constant coefficient non-homogeneous differential equation acquires：
Due to the symmetry of top circular arc, the boundary condition of top circular arc isIt solves：A=0
Then the deflection differential equation of top circular arc is：
The equations of rotating angle of top circular arc is：
Therefore, by boundary conditionIt solves：
C, D, which are brought into formula (6), (7), to be obtained：
Because hard-over and maximum defluxion generate at upright arm and horizontal arm junction, i.e. y=h-R, maximum is scratched
Degree is horizontal arm displacement distance d, it is known that it is that pressure P needed for the deformation of v is that upright arm, which generates amount of deflection,：
By active force reaction force principle, orthodontic force FOAs generate the counter-force of pressure needed for the deflection：
Further, in the step 4), open vertical loop top arc section and upright arm and horizontal arm friendship are being bent
During meeting place, since flexural deformation will produce residual stress, certain influence can be generated to above-mentioned analytic process, therefore draw
Enter adjusted coefficient KOTo build final open vertical loop orthodontic force prediction model, wherein KOCan by experimental result with do not consider
The comparative analysis of the calculated results of correcting action obtains, and R=w/2, then has：
It defines experimental data and does not consider adjusted coefficient KOGross data difference relative to gross data ratio be reason
It is highly influenced by being fitted to the average theory adjusted rate under each parameter to obtain open vertical loop by adjusted rate
Adjusted coefficient Kh, the adjusted coefficient K that is influenced by gap widthw, the adjusted coefficient K s influenced by cross section property, and by material
The adjusted coefficient K that characteristic influencesM, since the correction effect of affecting parameters is different, the direct phase of correction factor under the conditions of inciting somebody to action respectively
The overlapping of correction effect can be caused by multiplying, therefore the height bent with 0.016 inch of most commonly seen stainless steel round wire is 7mm, wide
The correction factor of open vertical loop that degree is 2mm is that radix is converted, and calculates it in different gap width, different cross section and not
It is respectively K with the orthodontic force adjusted rate under materialw0、KS0And KM0, then adjusted coefficient KOFor：
Then the final expression formula of open vertical loop orthodontic force prediction model is：
The above shows and describes the basic principles and main features of the present invention and the advantages of the present invention.The technology of the industry
Personnel are it should be appreciated that the present invention is not limited to the above embodiments, and the above embodiments and description only describe this
The principle of invention, without departing from the spirit and scope of the present invention, various changes and improvements may be made to the invention, these changes
Change and improvement all fall within the protetion scope of the claimed invention.The claimed scope of the invention by appended claims and its
Equivalent thereof.
Claims (5)
1. a kind of open vertical loop orthodontic force prediction model method for building up, it is characterised in that：The specific implementation process of the method
Include the following steps：
1) structure feature of extraction open vertical loop and load feature, conclude open vertical loop orthodontic force affecting parameters；
2) upright arm deflection differential equation is established；
3) top circular arc deflection differential equation is established, determines boundary condition；
4) residual stress correction factor is introduced, open vertical loop orthodontic force prediction model is established.
2. according to a kind of open vertical loop orthodontic force prediction model method for building up described in claim 1, it is characterised in that：Institute
In the step 1) stated, by analyze open vertical loop the characteristics of, extract open vertical loop structure feature and load feature difference
For：Open vertical loop is made of top circular arc, upright arm and horizontal arm, whole along circular arc symmetry axis axisymmetricly structure, is opened big
The horizontal arm of vertical loop and the angle of upright arm are 90 °；The clearance distance of two horizontal arms is compressed by the way that pressure is previously applied, and is obtained
Along the outward orthodontic force of horizontal arm；
It includes Toxocara species characteristic M that concluding, which influences the major parameter of open vertical loop orthodontic force F sizes, arch wire cross section property S and
The characteristic parameter Q of open vertical loop, wherein common Toxocara species include stainless steel wire, Australia silk and nitinol alloy wire, parameter be
The cross section property of elastic modulus E, arch wire includes sectional area and cross sectional shape, and parameter is to be opened away from I the inertia of bending axis greatly vertically
Bent characteristic parameter is the clearance distance w of the height h and open vertical loop of open vertical loop, therefore obtains open vertical loop correction
The citation form of power prediction model, as shown in Equation 1.
F=F (M, S, Q) (1).
3. according to a kind of open vertical loop orthodontic force prediction model method for building up described in claim 1, it is characterised in that：Institute
In the step 2) stated, by open vertical loop structure feature it is found that need to only analyze side symmetric part, open vertical loop top
The bending radius of end circular arc is that line of deflection approximate differential equation equation can be obtained such as in the case that without advance active force w=2R in R
Under：
In formula, v is the sag of upright arm, i.e. the displacement distance of horizontal arm, M (y) be in vertical arm y apart from place by curved
Square, IzIt is arch wire section to the moment of inertia of z-axis, y-axis is upright arm axis, and z-axis is vertical with y-axis on upright arm and bending moment work
Axis, for circle silk Iz=π DY 4/ 64, DYTo justify silk diameter, for square silk Iz=c1c2 3/ 12, c2For on rectangle silk section with z
The length of axis parallel edges, c1For the length with z-axis vertical edges on rectangle silk section；
Formula (2) is integrated, the equations of rotating angle θ (y) and deflection equation v (y) for obtaining open vertical loop are：
In formula, C and D are integral constants, are determined by boundary condition, and the Bending Moment Equations of open vertical loop upright arm are：
M (y)=- P (h-R-y) (5)
In formula, P be upright arm generate amount of deflection be pressure needed for the deformation of v, by formula (2) and (5) be brought into formula (3) and (4) into
Row integral, can obtain：
。
4. according to a kind of open vertical loop orthodontic force prediction model method for building up described in claim 1, it is characterised in that：Institute
It, need to be to the junction of upright arm and top circular arc, i.e., to determine the boundary condition of open vertical loop upright arm in the step 3) stated
Deflection at y=0 is solved, and it is π/4 that the top circular arc of side, which is equivalent to radian, and radius of curvature is the bent beam of R, right
It is d α infinitesimals that it, which takes one section of radian, then the deflection differential equation of top circular arc is：
In formula, u is the displacement of top circular arc cross section in the x direction, as top circular arc and upright arm junction, and the directions x are
Top circular arc bending radius direction, initial deflection of the upright arm at y=0, torque M suffered by arc section0=M |Y=0=-P (h-
R), IωIt is section to the moment of inertia of ω axis, ω axis is top arc axis, due to the bending types of upright arm and top circular arc
Unanimously, there is Iω=Iz。
The ds=Rd α known to arc length formula, therefore the deflection differential equation of top circular arc can be changed to：
Solution constant coefficient non-homogeneous differential equation acquires：
Due to the symmetry of top circular arc, the boundary condition of top circular arc isIt solves：
Then the deflection differential equation of top circular arc is：
The equations of rotating angle of top circular arc is：
Therefore, by boundary conditionIt solves：
C, D, which are brought into formula (6), (7), to be obtained：
Because hard-over and maximum defluxion generate at upright arm and horizontal arm junction, i.e. y=h-R, maximum defluxion is
For horizontal arm displacement distance d, it is known that it is that pressure P needed for the deformation of v is that upright arm, which generates amount of deflection,：
By active force reaction force principle, orthodontic force FOAs generate the counter-force of pressure needed for the deflection：
。
5. according to a kind of open vertical loop orthodontic force prediction model method for building up described in claim 1, it is characterised in that：Institute
In the step 4) stated, during bending open vertical loop top arc section and upright arm and horizontal arm junction, due to curved
Song deformation will produce residual stress, certain influence can be generated to above-mentioned analytic process, therefore introduce adjusted coefficient KOWith structure
Final open vertical loop orthodontic force prediction model, wherein KOExperimental result and the theoretical calculation for not considering correcting action can be passed through
As a result comparative analysis obtains, and R=w/2, then has：
It defines experimental data and does not consider adjusted coefficient KOGross data difference relative to gross data ratio be theoretical correction
Rate, by being fitted to the average theory adjusted rate under each parameter, to obtain the amendment that open vertical loop is highly influenced
COEFFICIENT Kh, the adjusted coefficient K that is influenced by gap widthw, the adjusted coefficient K s influenced by cross section property, and by material property shadow
Loud adjusted coefficient KM, since the correction effect of affecting parameters is different, will it is each under the conditions of correction factor be directly multiplied and can make
At the overlapping of correction effect, therefore the height bent with 0.016 inch of most commonly seen stainless steel round wire is 7mm, width 2mm
Open vertical loop correction factor be radix converted, calculate it in different gap width, different cross section and different materials
Under orthodontic force adjusted rate be respectively Kw0、KS0And KM0, then adjusted coefficient KOFor：
Then the final expression formula of open vertical loop orthodontic force prediction model is：
。
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| CN109359343A (en) * | 2018-09-20 | 2019-02-19 | 北京电子工程总体研究所 | A kind of high speed large span flexibility sliding rail design method |
| CN110013330A (en) * | 2019-05-20 | 2019-07-16 | 哈尔滨理工大学 | One kind is forced down with auxiliary bow tooth rotational angle prediction model method for building up |
| CN110110477A (en) * | 2019-05-20 | 2019-08-09 | 哈尔滨理工大学 | One kind is forced down with auxiliary begg appliance torque prediction model method for building up |
| CN110136830A (en) * | 2019-05-20 | 2019-08-16 | 哈尔滨理工大学 | One kind, which is forced down, rescues torque prediction model method for building up with auxiliary bow dynamic |
| CN110135087A (en) * | 2019-05-20 | 2019-08-16 | 哈尔滨理工大学 | A kind of T shape song dynamic Orthodontic force prediction model method for building up |
| CN110532708A (en) * | 2019-09-02 | 2019-12-03 | 哈尔滨工业大学（深圳） | A kind of residual stress prediction technique, device, server and storage medium |
| CN111475943A (en) * | 2020-04-04 | 2020-07-31 | 哈尔滨理工大学 | Closed-gap action box-shaped curve orthodontic force prediction model establishment method |
| CN111475946A (en) * | 2020-04-04 | 2020-07-31 | 哈尔滨理工大学 | Method for establishing positive axis action box-shaped curve orthodontic force prediction model |
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| CN109359343A (en) * | 2018-09-20 | 2019-02-19 | 北京电子工程总体研究所 | A kind of high speed large span flexibility sliding rail design method |
| CN110136830B (en) * | 2019-05-20 | 2020-01-14 | 哈尔滨理工大学 | Method for establishing dynamic correction torque prediction model of auxiliary arch for depression |
| CN110110477A (en) * | 2019-05-20 | 2019-08-09 | 哈尔滨理工大学 | One kind is forced down with auxiliary begg appliance torque prediction model method for building up |
| CN110136830A (en) * | 2019-05-20 | 2019-08-16 | 哈尔滨理工大学 | One kind, which is forced down, rescues torque prediction model method for building up with auxiliary bow dynamic |
| CN110135087A (en) * | 2019-05-20 | 2019-08-16 | 哈尔滨理工大学 | A kind of T shape song dynamic Orthodontic force prediction model method for building up |
| CN110013330A (en) * | 2019-05-20 | 2019-07-16 | 哈尔滨理工大学 | One kind is forced down with auxiliary bow tooth rotational angle prediction model method for building up |
| CN110135087B (en) * | 2019-05-20 | 2020-04-10 | 哈尔滨理工大学 | T-shaped curve dynamic correction force prediction model establishing method |
| CN110532708A (en) * | 2019-09-02 | 2019-12-03 | 哈尔滨工业大学（深圳） | A kind of residual stress prediction technique, device, server and storage medium |
| CN110532708B (en) * | 2019-09-02 | 2023-01-06 | 哈尔滨工业大学（深圳） | Residual stress prediction method, residual stress prediction device, server and storage medium |
| CN111475943A (en) * | 2020-04-04 | 2020-07-31 | 哈尔滨理工大学 | Closed-gap action box-shaped curve orthodontic force prediction model establishment method |
| CN111475946A (en) * | 2020-04-04 | 2020-07-31 | 哈尔滨理工大学 | Method for establishing positive axis action box-shaped curve orthodontic force prediction model |
| CN111475943B (en) * | 2020-04-04 | 2023-09-05 | 哈尔滨理工大学 | Method for establishing closed-gap action box-shaped Qu Zhengji force prediction model |
| CN111475946B (en) * | 2020-04-04 | 2023-12-22 | 哈尔滨理工大学 | Method for establishing positive axis acting box-shaped Qu Zhengji force prediction model |
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