CN110414056A - A kind of Cycloid tooth profile correction method compensating flexible deformation - Google Patents
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Abstract
The invention discloses a kind of Cycloid tooth profile correction methods for compensating flexible deformation, by design based on equidistant plus modification of moved distance flank profil, and cycloidal gear teeth profile is sought using optimization algorithm, this method considers influence of the flexible deformation to Cycloid tooth profile, can improve Cycloidal Wheel performance；Analog simulation is carried out convenient for exploitation mathematics software, facilitates Cycloid tooth profile correction of the flank shape, there is very high use value；And method can be embedded into RV Gear Reducer Optimal Design software, have good social value and economic value.
Description
Technical Field
The invention belongs to the technical field of mechanical design and manufacture, and relates to an optimal design method for modifying the tooth profile of a cycloidal gear, which is suitable for modifying the tooth profile of the cycloidal gear of an RV reducer.
Background
The cycloidal gear is used as a core part of the RV reducer and directly influences the important performances of the RV reducer, such as transmission precision, efficiency, bearing capacity, service life, reliability and the like. The modification of the tooth profile of the cycloidal gear is an important part in the design and manufacturing process of the cycloidal gear, and the research and determination of the optimal modification amount of the cycloidal gear have important significance and practical value for improving various performances of the cycloidal gear. Researchers at home and abroad make a great deal of research on the modification of the tooth profile of the cycloidal gear. The He Weidong and the like adopt a combined optimization shape-modifying method of negative equidistance and negative displacement to obtain a multi-tooth conjugate meshed tooth shape and reduce side gaps; the Nie-Shao-Wen and the like provide a modification method based on the equal distance, the displacement and the tooth height, and compared with other modification methods, the modification method has higher fitting degree with a corner modification tooth profile; zhao Bo et al have proposed a parabolic profile modification method, in the cycloidal gear work tooth profile part, the tooth profile after modifying is closer to the conjugate tooth profile; chmurawa et al achieved better transmission performance through optimization of the cycloid wheel tooth profile, and studied the influence of the profile modification parameters on load distribution, stress, etc.
Cycloidal gear tooth profile modification theory
FIG. 1 is a standard theoretical tooth profile of a cycloidal gear, which has the equation:
in the formula, rpIs the central radius of the pin gear rrpIs the radius of the pin teeth, zpWith number of teeth, i being cycloid and pin gearThe relative transmission ratio of the gear to the gear,the meshing phase angle of the cycloid wheel is shown, a is the eccentricity of the cycloid wheel, and K is the short-amplitude coefficient of the cycloid wheel.
The standard cycloidal gear and the needle teeth are in gapless meshing, manufacturing errors cannot be compensated, and lubricating conditions cannot be met, so that the standard cycloidal gear cannot be directly used in actual use, and the tooth profile of the cycloidal gear must be modified. Common modification methods include equidistant modification, distance modification, corner modification and combination modification. The corner profile modification is the most ideal profile modification mode, the tooth profile after the profile modification is a conjugate meshing tooth profile, the side gap is uniform, but no meshing gap exists between the tooth top and the tooth bottom. Because the machining process of modifying the corner with the equal distance or the corner with the shifting distance is complex and the modification time is long, the modification mode of the equal distance and the shifting distance is usually adopted to fit the tooth profile of the corner modification so as to obtain the conjugate meshing tooth profile with reasonable meshing gaps at the tooth top and the tooth bottom.
The modification tooth profile equation with the modification quantity of delta corner is as follows:
the equation of the tooth profile with equal distance and displacement modification is as follows:
in the formula,. DELTA.rp、ΔrrpRespectively a displacement modification quantity and an equidistant modification quantity,K'＝azp/(rp+Δrp)。
during no-load, the cycloidal gear is meshed with the single teeth of the needle teeth, after torque is applied, the meshed cycloidal gear and the needle teeth generate elastic deformation, the tooth profile of the cycloidal gear is concave in consideration of the influence of the elastic deformation on the tooth profile, the tooth profile is not a theoretical tooth profile any more, the conjugate meshing relation between the cycloidal gear and the needle teeth is damaged, the transmission characteristic of the RV reducer is influenced, and the transmission precision is reduced. The heavy-load RV reducer has large transmission torque, is larger than the elastic deformation of a common RV reducer, and seriously influences the performance of the RV reducer, so that the elastic deformation is very necessary to be considered in the process of the shape modification design of the RV reducer.
Disclosure of Invention
Aiming at the defects of the prior art, the invention provides an optimal design method for modifying the tooth profile of the cycloidal gear, and the performance of the cycloidal gear is improved.
In order to achieve the aim, the invention provides a method for modifying the tooth profile of a cycloidal gear for compensating elastic deformation, which is characterized in that:
s1, designing a cycloidal gear tooth profile based on the equidistant and displacement modified tooth profile:
rotating the meshing point of the cycloidal gear to the same contour, converting the meshing point of the cycloidal gear to a first gear tooth by utilizing a coordinate transformation principle, and fitting the meshing point to obtain a cycloidal gear tooth profile after contact deformation, wherein the coordinates of the cycloidal gear tooth profile are as follows:
wherein the coordinates (x) of the meshing pointj,yj) Δ x, Δ y are components of contact deformation in the x, y directions, zcThe number of teeth of the cycloid gear is set as the jth needle tooth, and the rotation angle of the center C of the needle tooth relative to the rotating arm is
S2, calculating the meshing characteristic of the cycloidal gear by using the deviation degree of the actual modified tooth profile with the equal distance and the displacement and the corner modified tooth profile, wherein the calculation formula is as follows:
in the formula,. DELTA.rp、ΔrrpRespectively a displacement modification amount and an equidistant modification amount, wherein m is the number of simultaneous meshing teeth, and i is 1,2 and 3 … m;
s3, designing an optimization model and seeking an optimal solution by using the model:
wherein an objective function of the optimization model is: using the degree of deviation of the equidistant-addendum-modification profile from the angular-modification profile as a target function, i.e.
z＝min f(Δrrp,Δrp)
Design variables of the optimization model are as follows:
the objective function being with respect to Δ rpΔrrpTo ensure that a reasonable radial clearance delta r is left between the tooth root and the tooth crest of the cycloidal gear, the function of (a) is that delta r is equal to delta rrp-ΔrpThen Δ rp＝Δrrp- Δ r, with equidistant modification Δ rrpAnd the radial clearance delta r is a design variable;
the constraint conditions of the optimization model are as follows:
further, in the step s3, a genetic algorithm is used to calculate the optimal solution, and the algorithm is set as follows:
the population size N is 50, the number of iterations N is 100, the crossover rate is 0.9, and the variation rate is 0.1.
The invention has the advantages that:
the contact deformation of the cycloidal gear and the pin teeth is considered, so that the tooth profile of the cycloidal gear is closer to the corner modification tooth profile in actual work, the meshing of the cycloidal gear and the pin teeth is closer to the conjugate meshing, the transmission is more stable, the equidistant and displacement modification mode is adopted, the processing is convenient, the genetic algorithm is applied to carry out optimization solution, and the optimization efficiency is improved.
Drawings
Fig. 1 is a schematic diagram of a standard theoretical tooth profile structure of a cycloidal gear.
Fig. 2 is a schematic view of the meshing of the cycloid gears.
Fig. 3 is the initial backlash of the cycloid gear.
FIG. 4 is a flow chart of the genetic algorithm of the present invention.
FIG. 5 is an iteration chart of the genetic algorithm of the present invention.
Fig. 6 is a graph of the theoretical tooth profile of the cycloidal gear.
FIG. 7 is a graph of initial meshing clearance and elastic deformation of a modified cycloid gear.
FIG. 8 is a graph of a contact force profile of the face of a cycloidal gear.
Fig. 9 is a comparison graph of theoretical tooth profile considering elastic deformation and corner modification.
Fig. 10 is a comparison diagram of an actual tooth profile in consideration of elastic deformation and corner modification.
Fig. 11 is a partially enlarged view of the actual working tooth profile of the cycloid gear.
Detailed Description
The invention is described in further detail below with reference to the following figures and specific examples:
in order to reduce the influence of elastic deformation generated by meshing of the cycloidal gear and the pin gear on the performance of the RV reducer, during the modification design of the tooth profile of the cycloidal gear, the elastic deformation is compensated, and the equal distance and the displacement modification amount are optimized, so that the modified tooth profile of the cycloidal gear approaches to the theoretical conjugate meshing tooth profile after the elastic deformation is generated.
S1, designing a cycloidal gear tooth profile based on the equidistant and displacement modified tooth profile:
because the cycloidal gear and the needle teeth are meshed in a multi-tooth mode, contact deformation exists on the teeth of the cycloidal gear, in order to visually reflect the influence of the contact deformation on the profile of the cycloidal gear, the meshing point of the cycloidal gear is rotated to the same profile, and the tooth profile of the cycloidal gear subjected to the contact deformation is obtained through the fitting of the meshing point. Firstly, the contact point position of the cycloidal gear and the pin gear is determined, the meshing point of the cycloidal gear or the point to be meshed intersects with the central connecting line of the pin gear at a node P, as shown in figure 2, the intersection point D is the cycloidal gearThe mesh point of the wire wheel and the needle tooth is set as the jth needle tooth, and the corner of the center C of the needle tooth relative to the rotating arm is DO can be determined by the cosine theoremcAnd OcThe included angle gamma of P, the meshing phase angle alpha of the D point on the cycloidal gear is zcGamma, and taking alpha into the formulas (6) and (7) to obtain the coordinate (x) of the meshing pointj,yj) (ii) a Considering the influence of contact deformation, after the cycloidal gear is meshed with the needle teeth, contact deformation along a common normal direction (a connecting line between a node P and the center of the needle teeth) is generated, and the coordinate of the point of the cycloidal gear after the contact deformation is generated is (x)j-Δx,yj- Δ y), Δ x, Δ y are the components of the contact deformation in the x, y direction, as known from the geometry
Δx＝εsin(π-γ) (9)
Δy＝εcos(π-γ) (10)
Wherein epsilon is the contact deformation along the common normal direction generated by the meshing of the cycloid wheel and the needle teeth; finally, the contact point of the cycloidal gear is converted into the first gear tooth by utilizing the coordinate transformation principle, and the coordinate of the contact point is
Wherein, the parameters related to the contact deformation in the formulas (9) and (10) are calculated as follows:
after the tooth profile of the cycloidal gear is modified by adding displacement at equal intervals, a certain gap exists between the cycloidal gear and the needle teeth, and the cycloidal gear rotates through a corner beta relative to the needle teethtCan be engaged with the needle teeth, and only has to be engaged with the needle teeth when the needle is unloadedThe needle teeth are contacted with the cycloid wheel, and gaps with different sizes are formed on the other needle teeth and the cycloid wheel, as shown in figure 3, the initial meshing gap is usedThe clearance between the cycloid wheel and the needle teeth in the direction of the theoretical common normal is shown.
The first contact point of the cycloid wheel and the needle teeth in no load isThe relative rotation angle at that position is
In the formula, zcThe number of teeth of the cycloid gears is,
at this time, the initial meshing clearance of the cycloid wheel
After loading, considering the compensation effect of the elastic deformation of the cycloidal gear and the needle teeth, when the contact deformation of the cycloidal gear and the needle teeth is larger than the initial meshing clearance, the needle teeth are meshed with the cycloidal gear, otherwise, the needle teeth are not meshed. The contact force of the needle teeth with the first contact point is set as FTDeformation by contact of epsilonTAt an arbitrary positionThe contact force of the needle teeth is FiDeformation by contact of epsiloniThe contact force of the pin-teeth being linear with the difference between deformation and initial backlash, i.e.
Wherein,
is balanced by moment
Wherein T is an output torque, m and n are a starting mesh tooth number and an ending mesh tooth number, respectively, and LiFrom the common normal line of the ith needle tooth meshing point to the center O of the cycloidal gearcThe distance of (c).
The deformation comprises two parts, namely contact deformation of the cycloid wheel and the needle teeth, and contact deformation of the needle teeth and the needle teeth shell, wherein the deformation can be calculated by a Hertz formula
Wherein u and E are respectively Poisson's ratio and elastic modulus of the material of the cycloidal gear (E is 2.06 x 10^5MPa, u is 0.3), b is the width of the cycloidal gear, R1 and R2 are respectively the curvature radius of two contact cylinders, and when the cycloidal gear is contacted with the needle teeth, the equivalent curvature radius rho isD＝2|ρ|rrp/(|ρ|+rrp) When the pin gear contacts with the pin gear shell, the equivalent curvature radius rhoD＝2|ρ|rrp/(|ρ|-rrp) Rho is a cycloid wheelThe radius of curvature of (d).
Calculating epsilonTWhen needed to know FTCalculating FTWhen necessary to know epsilonTTherefore, the maximum contact force cannot be obtained directly, usually an iterative method is adopted for calculation, and an initial value F is given firstT0Calculating epsilonT0From epsilonT0Find FT1And compare FT0And FT1If the absolute value of the difference between the two is greater than 0.1% FT1The iterative calculation continues until | FTk-FTk-1|＜0.1％FTkTake FT＝(FTk+FTk-1) (ii)/2, finding FTThen, the distribution of the contact force of the tooth surface of the cycloidal gear and the contact deformation can be obtained.
S2, calculating the meshing characteristic of the cycloidal gear by using the deviation degree of the actual modified tooth profile with the equal distance and the displacement and the corner modified tooth profile, wherein the calculation formula is as follows:
obtaining the actual working tooth profile of the cycloidal gear by fitting the contact point coordinates of the cycloidal gear, obtaining the meshing intervals (A and B) of the cycloidal gear from the meshing start point and the meshing stop point to measure the fitting degree of the actual working tooth profile with different modification quantities and the corner modification tooth profile, equally dividing the points (A and B), wherein m is the number of simultaneous meshing teeth, namelySubstituting the formula (4) and (5) to obtain the coordinates (x) of the corner modification tooth profilei,yi) 1,2,3 … m, let yi＝yi', the degree of deviation of the equidistant-addition-displacement actual modified tooth profile from the angular modified tooth profile is calculated by the formula (12)
The smaller the deviation degree from the corner modification curve is, namely the higher the fitting degree of the equidistant modified distance actual tooth profile and the corner modification tooth profile is, the closer the equidistant modified distance actual tooth profile curve approaches to the corner modification tooth profile curve and the closer the equidistant modified distance actual tooth profile curve approaches to the conjugate meshing tooth profile, the more stable the transmission is, and the better the meshing characteristic of the cycloidal gear is.
S3, designing an optimization model and seeking an optimal solution by using the model:
3.1 objective function of the optimization model:
adopting modification mode of equal distance and displacement to make the deviation degree of the modified tooth profile curve of equal distance and displacement after elastic deformation and modified tooth profile curve of corner be used as target function, i.e. adopting modification mode of equal distance and displacement
z＝min f(Δrrp,Δrp) (21)
3.2 design variables of the optimization model:
the objective function being with respect to Δ rpΔrrpTo ensure that a reasonable radial clearance delta r is left between the tooth root and the tooth crest of the cycloidal gear, the function of (a) is that delta r is equal to delta rrp-ΔrpThen Δ rp＝Δrrp- Δ r, with equidistant modification Δ rrpAnd the radial clearance deltar are design variables.
3.3 the constraint conditions of the optimization model:
on one hand, in order to fully exert the characteristic of wide optimization range of the genetic algorithm, the value range of the equidistant modification quantity is expanded as much as possible, and on the other hand, the excessive modification quantity can increase the side clearance of the cycloid wheel and reduce the transmission precision of the cycloid wheel, so the modification quantity is not suitable to be too large. Constraining the amount of pitch modification
|Δrrp|＜0.2mm (22)
Besides constraining the distance modification amount, the radial clearance is also constrained:
0.01≤Δr≤0.1mm (23)
therefore, the constraint here is
3.4 the optimization algorithm adopted by the optimization model is as follows:
the genetic algorithm is adopted for optimizing and solving, and the algorithm parameters are set as follows: the population size N is 50, the number of iterations N is 100, the crossover rate is 0.9, the mutation rate is 0.1, and the algorithm flow is shown in fig. 4.
The specific embodiment is as follows:
the RV-550E type reducer is a typical precision heavy-load RV reducer, has the characteristics of large bearing capacity, high transmission precision, long service life and the like, is widely applied to the fields of robots, machine tools and the like, and has the basic parameters of a cycloid wheel of the RV-550E type reducer as shown in Table 1. The method adopts a genetic algorithm to carry out shape modification optimization design on the RV-550E type reducer cycloid wheel, utilizes MATLAB to write an optimization algorithm for solving, and carries out comparative analysis with an optimization shape modification method without considering elastic deformation.
TABLE 1 cycloid wheel parameter table
Theoretical tooth profile: the working tooth section has no tooth profile with elastic deformation;
actual tooth profile: the working tooth section has an elastically deformed tooth profile;
the theoretical modification method comprises the following steps: the deviation degree of the equidistant modified distance theoretical tooth profile and the corner modified tooth profile is minimum without considering elastic deformation;
the elastic deformation compensation modification method comprises the following steps: and considering elastic deformation, the deviation degree of the equidistant modified distance actual tooth profile and the corner modified tooth profile is minimum.
Fig. 5 is an algorithm iteration graph, with the increase of iteration times, the deviation degree of the actual tooth profile of the cycloidal gear and the corner modification curve is smaller and smaller, the actual tooth profile of the cycloidal gear tends to converge after the iteration is carried out for 25 times, and the optimal solution is that the equidistant modification amount is 0.0886mm, and the radial gap is 0.0105 mm. FIG. 6 is a theoretical profile of a modified cycloid gear, it can be seen that a certain radial clearance exists between the tooth top and the tooth root for compensating manufacturing errors and satisfying lubrication conditions, the working tooth profile portion is outside a corner modification curve, is concave after being deformed by stress, and the actual tooth profile is closer to the corner modification curve and is close to a conjugate meshing relationship; fig. 7 is a distribution diagram of an initial meshing gap and elastic deformation of a cycloidal gear, when the elastic deformation is larger than the initial meshing gap, the cycloidal gear is meshed with a pin gear, otherwise, the cycloidal gear is not meshed, the intersection point of two curves is a starting meshing point and a stopping meshing point of the cycloidal gear, the number of meshing teeth is 26, and a standard cycloidal gear is meshed with half the number of teeth, namely, the number of meshing teeth is 30, which is more than that of a trimmed cycloidal gear, so that the maximum tooth surface contact force of the cycloidal gear is increased after trimming, as shown in fig. 8, but the tooth profile of the trimmed cycloidal gear is a 'reverse bow' tooth profile, the tooth surface contact force is distributed uniformly, and compared with a common trimming method, the.
FIG. 9 is a theoretical tooth profile curve for two modification methods, i.e. no contact deformation, from which it can be seen that, without considering the elastic deformation of the cycloid wheel, the tooth profile curve of the theoretical modification method more approximates the tooth profile curve of the corner modification than the tooth profile curve of the compensation elastic deformation optimization method; fig. 10 is a curve of actual tooth profile of two modification methods, and fig. 11 is a partially enlarged view of actual working tooth profile of the cycloidal gear, so that it can be seen that the optimization method for compensating elastic deformation is closer to the curve of corner modification. The compensation elastic deformation optimization method is characterized in that a theoretical tooth profile curve is arranged on the outer side of a corner modification curve, when a cycloidal gear is meshed with a needle tooth, contact deformation along the normal direction is generated, the tooth profile is concave inwards, elastic deformation is compensated by a part exceeding the corner modification tooth profile curve, and the actual working tooth profile is closer to the corner modification tooth profile; and the tooth profile curve of the theoretical modification method is positioned at the inner side of the corner modification curve, when the cycloid wheel is meshed with the needle teeth, the elastic deformation cannot be compensated, but the cycloid wheel is recessed because of the elastic deformation and is far away from the corner modification curve.
Table 2 is a parameter table of the cycloid wheel tooth profile, and from the numerical point of view, compared with the theoretical modification method, the deviation degree of the actual tooth profile and the corner modification tooth profile of the compensation elastic deformation modification method is reduced by 34.38%. Meanwhile, the maximum contact force of the tooth surface of the cycloidal gear of the compensation elastic modification method is reduced by 8.27 percent compared with that of a theoretical modification method, and the maximum contact force of the tooth surface of the cycloidal gear is increased mainly because the modification amount and the radial clearance of the theoretical modification method are larger, so that the number of meshing teeth is smaller, the number of bearing teeth is reduced; the modification method for compensating elastic deformation has relatively small modification amount and radial clearance, more meshing teeth, increased bearing teeth and reduced maximum contact force of the tooth surface of the cycloidal gear.
TABLE 2 cycloidal gear tooth profile parameter table
The invention has the beneficial effects that:
1) the invention provides an optimal design method for modifying the tooth profile of a cycloidal gear for compensating elastic deformation, which considers the influence of the elastic deformation on the tooth profile of the cycloidal gear and can improve the performance of the cycloidal gear;
2) the method provided by the invention is convenient for developing computer mathematical software for analog simulation, is beneficial to the modification of the tooth profile of the cycloidal gear, and has high use value;
3) the method provided by the invention can be embedded into RV reducer optimization design software, and has good social value and economic value.
The above embodiments are only used for illustrating the design idea and features of the present invention, and the purpose of the present invention is to enable those skilled in the art to understand the content of the present invention and implement the present invention accordingly, and the protection scope of the present invention is not limited to the above embodiments. Therefore, all equivalent changes and modifications made in accordance with the principles and concepts disclosed herein are intended to be included within the scope of the present invention.
Claims (2)
1. A method for modifying the tooth profile of cycloidal gear for compensating elastic deformation is characterized in that:
s1, designing a cycloidal gear tooth profile based on the equidistant and displacement modified tooth profile:
rotating the meshing point of the cycloidal gear to the same contour, converting the contact point of the cycloidal gear to a first gear tooth by utilizing a coordinate transformation principle, and fitting the meshing point to obtain the cycloidal gear tooth profile after contact deformation, wherein the coordinates of the cycloidal gear tooth profile are as follows:
wherein the coordinates (x) of the meshing pointj,yj) The delta x and the delta y are components of contact deformation along the x and y directions, the meshing point is the jth pin tooth, and the rotation angle of the pin tooth center C relative to the rotating arm iszcThe number of teeth of the cycloid gear is shown;
s2, calculating the meshing characteristic of the cycloidal gear by using the deviation degree of the actual modified tooth profile with the equal distance and the displacement and the corner modified tooth profile, wherein the calculation formula is as follows:
in the formula,. DELTA.rp、ΔrrpAre respectively provided withThe displacement modification amount and the equidistant modification amount are shown, m is the number of simultaneous meshing teeth, and i is 1,2,3 … m;
s3, designing an optimization model and seeking an optimal solution by using the model:
wherein an objective function of the optimization model is: using the degree of deviation of the equidistant-addendum-modification profile from the angular-modification profile as a target function, i.e.
z＝min f(Δrrp,Δrp)
Design variables of the optimization model are as follows:
the objective function being with respect to Δ rpΔrrpTo ensure that a reasonable radial clearance delta r is left between the tooth root and the tooth crest of the cycloidal gear, the function of (a) is that delta r is equal to delta rrp-ΔrpThen Δ rp＝Δrrp- Δ r, with equidistant modification Δ rrpAnd the radial clearance delta r is a design variable;
the constraint conditions of the optimization model are as follows:
2. the method of modifying a cycloid gear tooth profile that compensates for elastic deformation of claim 1 wherein: in step s3, a genetic algorithm is used to calculate the optimal solution, and the algorithm is set as follows:
the population size N is 50, the number of iterations N is 100, the crossover rate is 0.9, and the variation rate is 0.1.
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| CN107742054A (en) * | 2017-12-02 | 2018-02-27 | 北京工业大学 | A kind of RV decelerators flank profil optimization correction method based on genetic algorithm |
| CN108662115A (en) * | 2018-05-23 | 2018-10-16 | 湖北工业大学 | Cycloid tooth profile correction of the flank shape multi-objective optimization design of power method |
| US20190186487A1 (en) * | 2017-12-19 | 2019-06-20 | Jiangnan University | Method for Modifying Performance of Rotor Profile by Adjusting Meshing Line Segments |
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| Publication number | Priority date | Publication date | Assignee | Title |
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| RU2202694C1 (en) * | 2002-06-13 | 2003-04-20 | Общество с ограниченной ответственностью фирма "Радиус-Сервис" | Screw hydraulic machine helical gear rotation mechanism |
| CN107742054A (en) * | 2017-12-02 | 2018-02-27 | 北京工业大学 | A kind of RV decelerators flank profil optimization correction method based on genetic algorithm |
| US20190186487A1 (en) * | 2017-12-19 | 2019-06-20 | Jiangnan University | Method for Modifying Performance of Rotor Profile by Adjusting Meshing Line Segments |
| CN108662115A (en) * | 2018-05-23 | 2018-10-16 | 湖北工业大学 | Cycloid tooth profile correction of the flank shape multi-objective optimization design of power method |
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