CN106156409B - Two-parameter special-shaped bolt hole robust design method based on minimum contour change - Google Patents
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Abstract
The invention discloses a robust design method of two-parameter special-shaped bolt holes based on minimum contour change. In the optimization model, a special-shaped hole comprehensive optimization design model is established by reasonably weighting and summing each optimization target. Meanwhile, a dynamic weight coefficient adjusting method for balancing contribution of different design targets to the comprehensive optimization target is adopted, so that a balanced optimization design effect of the special-shaped hole under a multi-target condition is obtained, the dependence of optimization results of various design variables (namely, the size of the circular arc) of the special-shaped hole on the value range of the design variables is fundamentally eliminated, and the optimal special-shaped hole profile which meets the requirement condition of a strength design task and has the minimum profile size change can be quickly obtained through optimization. The design result has better consistency.
Description
Technical Field
The invention belongs to the structural optimization design of turbine rotor parts of an aircraft engine, and particularly relates to a multi-objective optimization design model of the outline shape of a double-shaft symmetrical special-shaped bolt hole of a turbine disc.
Background
The high-pressure turbine disc of the aero-engine has a severe long-term working environment, and the bolt round holes on the front mounting edge of the disc often generate hole edge cracks due to stress concentration, so that the disc becomes an important reason for failure of the disc.
The patent "multi-arc shaped hole" (patent No. ZL 201310287309.9) proposes that an eight-arc shaped hole structure can be adopted. The following disadvantages still remain:
1. the optimal value of the objective function in the existing optimization model has an over-optimization phenomenon (namely, the stress reduction amplitude of the hole edge exceeds the design requirement).
2. The optimization design result excessively depends on the setting of the value range of the design variable, and in the actual design, continuous trial calculation is needed to determine the reasonable value range of the design variable, so that the design efficiency is low.
3. The robustness of the optimization design result is poor, and the convergence speed in the optimization process is low.
Disclosure of Invention
The purpose of the invention is as follows: in order to overcome the defects in the prior art, the invention provides a two-parameter special-shaped bolt hole steady design method based on minimum outline change, which can avoid tedious and low-efficiency boundary trial calculation during optimization design, improve the rationality and stability of design and greatly improve the design efficiency.
The technical scheme is as follows: in order to achieve the purpose, the invention adopts the technical scheme that:
a robust design method for two-parameter special-shaped bolt holes based on minimum contour change comprises the following steps:
step 1, establishing a special-shaped hole multi-target optimization model according to a task given target value, a main arc radius and a transition arc radius and according to a stress reduction target, a special-shaped hole main arc size change target and a transition arc size change target.
And 2, determining the size constraint of the special-shaped hole multi-target optimization model in the step 1 according to the upper boundary of the radius of the main circular arc and the lower boundary of the radius of the transition circular arc.
And 3, introducing a weighting coefficient, and converting the special-shaped hole multi-target optimization model into a special-shaped hole single comprehensive target optimization model.
And 4, optimizing the single comprehensive target optimization model of the special-shaped hole according to the fixed contour change value and the allowable error to obtain each design variable of the special-shaped hole and complete the design of the special-shaped hole.
The method for establishing the special-shaped hole multi-objective optimization model in the step 1 comprises the following steps:
and 11, establishing a dimensionless stress reduction amplitude target function according to the task given target value, the radius of the main arc and the radius of the transition arc.
And step 12, establishing a contour control objective function of the dimensionless special-shaped hole with the main arc radius according to the radius of the original bolt through hole and the radius of the main arc.
And step 13, establishing a contour control target function of the dimensionless special-shaped hole with the transition arc radius according to the radius of the original bolt through hole and the radius of the transition arc.
And step 14, establishing the special-shaped hole multi-target optimization model according to the dimensionless stress reduction amplitude objective function established in the step 11, the main arc radius dimensionless special-shaped hole contour control objective function established in the step 12 and the transition arc radius dimensionless special-shaped hole contour control objective function established in the step 13.
The expression of the dimensionless stress reduction amplitude objective function in the step 11 is as follows:
f1(σmax) For a dimensionless stress-reduced amplitude objective function, σmaxMaximum principal stress at the edge of the hole, σmax＝σmax(R1,R2)，R1Radius of main circular arc, R2To transition the radius of the arc, σoIndicating a task given target value.
The expression of the control objective function of the dimensionless special-shaped hole profile of the radius of the main circular arc in the step 12 is as follows:
wherein f is2(R1) Main circular arc radius dimensionless profile control objective function, R1The radius of the main circular arc is R, and the radius of the original bolt through hole is represented by R.
The expression of the control objective function of the dimensionless profile hole profile of the transition arc radius in the step 13 is as follows:
wherein f is3(R2) Control of the target function, R, for the contour of a dimensionless profiled hole of the radius of the transition arc2And (4) representing the radius of the transition circular arc, wherein R represents the radius of the original bolt through hole.
The expression of the special-shaped hole multi-objective optimization model in the step 14 is as follows:
minf(y1＝f1(σmax),y2＝f2(R1),y3＝f3(R2))。
the size constraint established in step 2 is:
wherein R is1maxIs the upper bound of the radius of the main circular arc, R2minIs the lower boundary of the transition arc radius, R1Radius of main circular arc, R2And (4) representing the radius of the transition circular arc, wherein R represents the radius of the original bolt through hole.
The abnormal hole single comprehensive target optimization model in the step 3 is as follows:
f(f1,f2,f3)＝k1·f1(σmax)+k2·f2(R1)+k3·f3(R2)。
wherein k is1Weighting coefficients for the dimensional stress reduction amplitude objective function, f1(σmax) Is a dimensionless stress reduction amplitude objective function, k2Controlling the weighting coefficient of the objective function for the dimensionless profile of the hole having the main circular arc radius, f2(R1) Control objective function, k, of main arc radius dimensionless profile hole contour3Controlling the weighting factor of the objective function for the dimensionless profile of the hole in the radius of the transition arc, f3(R2) And controlling an objective function for the profile of the dimensionless special-shaped hole with the radius of the transition arc.
The method for determining the weighting coefficient in the step 3 is as follows:
weighting coefficient k of target function of dimensional stress reduction amplitude1Weighting coefficient k of main circular arc radius dimensionless special-shaped hole outline control objective function2Weighting coefficient k of contour control objective function of dimensionless special-shaped hole with transition circular arc radius3With R1Radius of the main arc and sigmaoThe stress target values are dynamically varied, wherein: radius of main arc R1And transition arcRadius R2When the value range is changed, the influence on the hole edge stress is different.
R1maxWhen the change is carried out:
k2＝0.008～0.012
k2and k3Satisfies the following conditions:
k3＝g1(k2,R1max)＝(1.25～2.1)·k2·R1max
k1and k is2，k3Satisfies the following conditions:
wherein σ1rThe maximum main stress value of the edge of the through hole of the original circular bolt is obtained.
When the abnormal-shaped hole single comprehensive target optimization model is optimized in the step 4, only the stress of the optimized abnormal-shaped hole edge meets the set requirement, namely, the stress of the abnormal-shaped hole edge meets the set requirementmax-σoAnd | ≦ ε, which is the allowable error. And compared with the original bolt circular through hole, the profile change of the special-shaped bolt hole is smaller than a given profile change value, and the design parameter or variable is an optimized value meeting the requirement.
Has the advantages that: compared with the prior art, the robust design method for the two-parameter special-shaped bolt hole based on the minimum profile change has the following beneficial effects:
1. the optimization target of the optimization problem of the two-variable special-shaped hole is perfected, so that the obtained optimization solution can reduce the hole edge stress according to the expected target.
2. The contour dimension of the special-shaped hole obtained by the optimized design can meet the requirement that the stress is reduced to a specified value, and the deviation from the original contour is not too large, so that the design is more stable.
3. The problem of excessive dependence of the original optimization result on the value range of the design variable is solved, and the size (R) of each circular arc of the special-shaped hole is not required to be adjusted repeatedly in the optimization design1，R2) Thereby improving the stability of the optimized design point, leading the final design scheme to be more reasonable and leading the design efficiency to be more reasonableThe improvement is obvious.
Drawings
FIG. 1 shows the weighting factor k1，k2，k3Dynamically determining a flow chart;
FIG. 2 is a profile of a shaped hole based on a multi-objective two-variable optimization model;
wherein: 1-two-variable special-shaped hole main circular arc R1(ii) a 2-two-variable special-shaped hole transition arc R2(ii) a 3-conventional bolt through hole (base circle).
Detailed Description
The present invention is further illustrated by the following description in conjunction with the accompanying drawings and the specific embodiments, it is to be understood that these examples are given solely for the purpose of illustration and are not intended as a definition of the limits of the invention, since various equivalent modifications will occur to those skilled in the art upon reading the present invention and fall within the limits of the appended claims.
A robust design method of two-parameter special-shaped bolt holes based on minimum profile change aims at the profile optimization problem of special-shaped bolt through holes (hereinafter referred to as special-shaped holes) on a high-pressure turbine disc of an aeroengine, and provides the robust design method. In the optimization model, a special-shaped hole comprehensive optimization design model is established by reasonably weighting and summing each optimization target. In the comprehensive optimization model, because the weight of each target changes along with the change of the value range of the design variable of the irregular hole, the patent inventor designs a dynamic weight coefficient adjusting method capable of keeping the contribution of different design targets to the comprehensive optimization target balanced, so that the weight coefficient (k) related to the stress control target1) Is associated with the value range of the profile design variable of the special-shaped hole, and when the latter changes, k1Can be automatically adjusted, thereby obtaining the balanced optimization design effect of the special-shaped holes under the condition of multiple targets, and fundamentally eliminating the optimization result pair of each design variable (namely the arc size) of the special-shaped holesThe dependency of the value range of the design variable can be quickly optimized to obtain the optimal special-shaped hole profile which meets the requirement condition of the strength design task and has the minimum profile size change. The invention enables the two-variable profiled hole contour design to be optimally designed according to the hole edge stress target value given by a task, meanwhile, the value range of each design variable size of the profiled hole does not need to be repeatedly tentatively adjusted, and the design result has better consistency.
The method specifically comprises the following steps:
step 1, establishing a special-shaped hole multi-target optimization model according to a task given target value, a main arc radius and a transition arc radius and according to a stress reduction target, a special-shaped hole main arc size change target and a transition arc size change target.
The method specifically comprises the following steps:
step 11, establishing a dimensionless stress reduction amplitude target function according to the task given target value, the radius of the main arc and the radius of the transition arc, f1(σmax) For the dimensionless stress reduction amplitude objective function, it is necessary to ensure that the maximum principal stress of the hole edge of the special-shaped hole sufficiently approaches to a certain target value sigma set in advance (task setting)o. The expression is as follows:
f1(σmax)＝mind
d represents f1(R) maximum principal stress σ at hole edgemaxWith a predetermined stress target value sigmaoIs measured by the dimensional difference of (a).
f1(σmax) For a dimensionless stress-reduced amplitude objective function, σmaxMaximum principal stress at the edge of the hole, σmax＝σmax(R1,R2)，R1Radius of main circular arc, R2To transition the radius of the arc, σoIndicating a task given target value. In specific implementation,. sigmaoThe stress target value is determined according to the design task target and the working condition
And step 12, establishing a main circular arc radius dimensionless special-shaped hole outline control target function according to the original bolt through hole radius and the main circular arc radius, wherein the target function enables the difference between the obtained special-shaped hole outline and the original outline to be as small as possible, and therefore a more stable design point is obtained. And taking the radius R of the original bolt through hole as a reference. The expression is as follows:
wherein f is2(R1) Main circular arc radius dimensionless profile control objective function, R1The radius of the main circular arc is R, and the radius of the original bolt through hole is represented by R.
And the objective function f2(R1)，f3(R2) Ensuring that the target f is met1(R) on the premise of obtaining the profile hole contour dimension R (R)1,R2) As close as possible to the conventional design bolt through hole profile, resulting in a more robust design point.
f2(R1)＝minδ1，f3(R2)＝minδ2；
Wherein, delta1，δ2Respectively represent main circular arcs R of the special-shaped holes1And transition arc radius R2The dimensionless difference from the base radius of the circular bolt hole is as follows.
And step 13, establishing a control objective function of the profile of the dimensionless special-shaped hole with the transition arc radius according to the radius of the original bolt through hole and the radius of the transition arc, wherein the objective function enables the difference between the profile of the special-shaped hole obtained by optimization and the original profile to be as small as possible, and therefore a more stable design point is obtained. And taking the radius R of the original bolt through hole as a reference. The expression is as follows:
wherein f is3(R2) Control of the target function, R, for the contour of a dimensionless profiled hole of the radius of the transition arc2And (4) representing the radius of the transition circular arc, wherein R represents the radius of the original bolt through hole.
In the optimization design, when the value range of the design variable is determined, R1max(the upper boundary of the radius of the main circular arc) can be 8-20R (R is the radius of the base circle) according to the actual diameter of the bolt hole; r2minThe lower boundary of the transition arc radius depends on the minimum radius under the actual processing condition, and the actual minimum diameter cutter radius can be adopted.
Step 14, establishing a special-shaped hole multi-target optimization model according to the dimensionless stress reduction amplitude target function established in the step 11, the main arc radius dimensionless special-shaped hole contour control target function established in the step 12 and the transition arc radius dimensionless special-shaped hole contour control target function established in the step 13, wherein the expression of the special-shaped hole multi-target optimization model is as follows:
minf(y1＝f1(σmax),y2＝f2(R1),y3＝f3(R2))
all design points R when solving a concrete optimization problemi(R1，R2) Corresponding to f of the objective function1，f2,f3A criterion space, denoted by F, can be formed as follows:
F＝{fi(f1,f2,f3)∈R3|f1＝f1(σmaxi),f2＝f2(R1),f3＝f3(R2)}
fi＝(f1,f2,f3) Is a vector consisting of three objective function values. The optimization process is to all vectors f in the criterion spaceiThe evaluation process of (1).
σmax＝σmax(R1,R2),R1,R2Respectively represent a main circular arc and a transition circular arc, which represent a design point R; f. of1(σmax) Is equal to the design point R (R)1,R2) A corresponding stress reduction amplitude objective function; f. of2(R1) Is a main circular arc R1Varying a control objective function; f. of3(R2) Is a transition arc R2The variation controls the objective function. R is the radius of the base circle (i.e., the conventional design bolt through hole), R1maxIs the upper bound of the radius of the main circular arc, R2minThe lower bound of the radius of the transition arc.
The special-shaped hole multi-target optimization model meets the following requirements:
(1) reducing the stress of the hole edge to the target given by the strength design task;
(2) the change of the overall size of the hole is minimal;
(3) the dependence of the optimized profile hole contour dimension on the value range of the design variable is effectively reduced, and the design robustness is improved.
And 2, determining the size constraint of the special-shaped hole multi-target optimization model in the step 1 according to the upper boundary of the radius of the main circular arc and the lower boundary of the radius of the transition circular arc.
Its dimensional constraints are:
wherein R is1maxIs the upper bound of the radius of the main circular arc, and can be arbitrarily valued in a reasonable range, R2minIs the lower boundary of the transition arc radius, R1Radius of main circular arc, R2For the transition arc radius, R denotes the base circle, i.e. the radius of the (original) bolt through hole of the conventional design.
The optimization model of the single comprehensive target of the special-shaped hole is as follows:
f(f1,f2,f3)＝k1·f1(σmax)+k2·f2(R1)+k3·f3(R2)。
wherein k is1Weighting coefficients for the dimensional stress reduction amplitude objective function, f1(σmax) Is a dimensionless stress reduction amplitude objective function, k2Controlling the objective function for the profile of a dimensionless profiled hole having a radius of the main arcWeighting coefficient, f2(R1) Control objective function, k, of main arc radius dimensionless profile hole contour3Controlling the weighting factor of the objective function for the dimensionless profile of the hole in the radius of the transition arc, f3(R2) And controlling an objective function for the profile of the dimensionless special-shaped hole with the radius of the transition arc.
The multi-objective optimization design model formed by the three new objective functions can simultaneously control the maximum main stress reduction amplitude of the hole edge and the change of the outline size of the special-shaped hole, so that the hole edge stress reduction amplitude meets the set target, the hole pattern design is more stable, and the hole pattern parameters are more stable.
By a weighting coefficient k1，k2，k3The contributions of the three single optimization objectives to the composite optimization objective may be adjusted.
As shown in fig. 1, the determination method of the weighting factor is as follows:
weighting coefficient k of target function of dimensional stress reduction amplitude1Weighting coefficient k of main circular arc radius dimensionless special-shaped hole outline control objective function2Weighting coefficient k of contour control objective function of dimensionless special-shaped hole with transition circular arc radius3With R1Radius of the main arc and sigmaoThe stress target values are dynamically varied, wherein: radius of main arc R1And transition arc radius R2When the value range is changed, the influence on the hole edge stress is different.
R1maxWhen the change is carried out:
k2＝0.008～0.012
k2and k3Satisfies the following conditions:
k3＝g1(k2,R1max)＝(1.25～2.1)·k2·R1max
k1and k is2，k3Satisfies the following conditions:
wherein σ1rIs in the shape of a circleThe maximum main stress value of the hole edge of the bolt through hole.
Generated weighting coefficient k1，k2，k3The change of each design parameter can be more balanced, and a steady design point is obtained.
And 4, optimizing the single comprehensive target optimization model of the special-shaped hole according to the fixed contour change value and the allowable error to obtain each design variable of the special-shaped hole and complete the design of the special-shaped hole.
When the optimization model of the single comprehensive target of the special-shaped hole is optimized, the stress of the special-shaped hole edge after optimization reaches the set requirement, namely, the stress of the | sigma & lt/EN & gtmax-σoAnd | ≦ ε, which is the allowable error. Compared with the original bolt circular through hole, when the profile change of the special-shaped bolt hole is smaller than a given profile change value, the design parameter or variable is an optimized value meeting the requirement, and three single targets are required to contribute to the comprehensive design result in a given multi-target optimization model in a balanced manner. Due to the weighting coefficient k1，k2，k3In relation to the value range of the design variable, a weighting factor k is given for this purpose1，k2，k3Dynamically generating a solution such that k1，k2，k3The value can follow R1maxAnd adjusting the value according to different value ranges to meet the balance among three single optimization targets, finally obtaining a design point which meets the strength design requirement and is more stable and stable, and ensuring that the three have proper influence factors, thereby ensuring the robustness of the final optimal solution.
The multi-objective optimization design method provided by the invention can obtain a hole pattern design scheme with small hole outline change and more stability on the basis of meeting a stress design target (task value); in addition, in the optimization design, when the allowable variation range of the arc size of the special-shaped hole is obviously changed, a stable optimization solution can still be kept, so that the final optimization design result is free from the dependence on the set range value of the arc design variable, and the design efficiency is improved. The structure of the two-parameter special-shaped hole obtained under the multi-objective optimization model is shown in the attached figure 2.
The invention provides a novel multi-objective optimization design model, which can be used for designing holes according to task specificationsThe edge stress reduction amplitude provides a steady and stable design scheme. The multi-objective optimization model consists of (a) a stress reduction target and (b) a main circular arc size R of a special-shaped hole1Varying the target and (c) the transition arc size R2A change target composition; in the multi-objective optimization model, the contributions of three single objectives (namely (a), (b) and (c)) to the comprehensive objective meet the equilibrium condition; the provided new model and algorithm can avoid complex and inefficient boundary trial calculation during optimization design, improve the reasonability and stability of design and greatly improve the design efficiency.
The above description is only of the preferred embodiments of the present invention, and it should be noted that: it will be apparent to those skilled in the art that various modifications and adaptations can be made without departing from the principles of the invention and these are intended to be within the scope of the invention.
Claims (5)
1. A robust design method for two-parameter special-shaped bolt holes based on minimum profile variation is characterized by comprising the following steps:
step 1, establishing a special-shaped hole multi-target optimization model according to a task given target value, a main arc radius and a transition arc radius and according to a stress reduction target, a special-shaped hole main arc size change target and a transition arc size change target;
the method for establishing the special-shaped hole multi-target optimization model comprises the following steps:
step 11, establishing a dimensionless stress reduction amplitude target function according to a task given target value, a main arc radius and a transition arc radius;
the expression of the dimensionless stress reduction amplitude objective function is as follows:
f1(σmax) For a dimensionless stress-reduced amplitude objective function, σmaxMaximum principal stress at the edge of the hole, σmax＝σmax(R1,R2)，R1Radius of main circular arc, R2Is a transition arc halfDiameter, σoRepresenting a task given target value;
step 12, establishing a main circular arc radius dimensionless special-shaped hole outline control objective function according to the original bolt through hole radius and the main circular arc radius;
the expression of the control objective function of the dimensionless special-shaped hole profile of the radius of the main circular arc in the step 12 is as follows:
wherein f is2(R1) Main circular arc radius dimensionless profile control objective function, R1The radius of the main circular arc is R, and the radius of the original bolt through hole is represented by R;
step 13, establishing a transition arc radius dimensionless special-shaped hole profile control target function according to the original bolt through hole radius and the transition arc radius;
the expression of the control objective function of the dimensionless special-shaped hole profile of the radius of the transition circular arc is as follows:
wherein f is3(R2) Control of the target function, R, for the contour of a dimensionless profiled hole of the radius of the transition arc2The radius of a transition arc is shown, and R represents the radius of an original bolt through hole;
step 14, establishing a special-shaped hole multi-target optimization model according to the dimensionless stress reduction amplitude objective function established in the step 11, the main arc radius dimensionless special-shaped hole contour control objective function established in the step 12 and the transition arc radius dimensionless special-shaped hole contour control objective function established in the step 13;
the expression of the special-shaped hole multi-objective optimization model is as follows:
minf(y1＝f1(σmax),y2＝f2(R1),y3＝f3(R2))；
step 2, determining the size constraint of the special-shaped hole multi-target optimization model in the step 1 according to the upper boundary of the radius of the main arc and the lower boundary of the radius of the transition arc;
step 3, introducing a weighting coefficient, and converting the special-shaped hole multi-target optimization model into a special-shaped hole single comprehensive target optimization model;
and 4, optimizing the single comprehensive target optimization model of the special-shaped hole according to the fixed contour change value and the allowable error to obtain each design variable of the special-shaped hole and complete the design of the special-shaped hole.
2. The two-parameter deformed bolt hole robust design method based on minimum profile variation according to claim 1, characterized in that: the size constraint established in step 2 is:
wherein R is1maxIs the upper bound of the radius of the main circular arc, R2minIs the lower boundary of the transition arc radius, R1Radius of main circular arc, R2And (4) representing the radius of the transition circular arc, wherein R represents the radius of the original bolt through hole.
3. The two-parameter deformed bolt hole robust design method based on the minimum profile variation as claimed in claim 2, characterized in that: the abnormal hole single comprehensive target optimization model in the step 3 is as follows:
f(f1,f2,f3)＝k1·f1(σmax)+k2·f2(R1)+k3·f3(R2)；
wherein k is1Weighting coefficients for the dimensional stress reduction amplitude objective function, f1(σmax) Is a dimensionless stress reduction amplitude objective function, k2Controlling the weighting coefficient of the objective function for the dimensionless profile of the hole having the main circular arc radius, f2(R1) Control objective function, k, of main arc radius dimensionless profile hole contour3Controlling the weighting factor of the objective function for the dimensionless profile of the hole in the radius of the transition arc, f3(R2) Is a transition arcThe radius dimensionless profile of the hole controls the objective function.
4. The robust design method for two-parameter special-shaped bolt holes based on minimum profile variation according to claim 3, characterized in that: the method for determining the weighting coefficient in the step 3 is as follows:
weighting coefficient k of target function of dimensional stress reduction amplitude1Weighting coefficient k of main circular arc radius dimensionless special-shaped hole outline control objective function2Weighting coefficient k of contour control objective function of dimensionless special-shaped hole with transition circular arc radius3With R1Radius of the main arc and sigmaoThe stress target values are dynamically varied, wherein: radius of main arc R1And transition arc radius R2When the value range is changed, the influence on the hole edge stress is different;
R1maxwhen the change is carried out:
k2＝0.008～0.012
k2and k3Satisfies the following conditions:
k3＝g1(k2,R1max)＝(1.25～2.1)·k2·R1max
k1and k is2，k3Satisfies the following conditions:
wherein σ1rThe maximum main stress value of the edge of the through hole of the original circular bolt is obtained.
5. The robust design method for two-parameter special-shaped bolt holes based on minimum profile variation according to claim 4, characterized in that: when the abnormal-shaped hole single comprehensive target optimization model is optimized in the step 4, only the stress of the optimized abnormal-shaped hole edge meets the set requirement, namely, the stress of the abnormal-shaped hole edge meets the set requirementmax-σoEpsilon is less than or equal to | and epsilon is an allowable error; and compared with the original bolt circular through hole, the profile change of the special-shaped bolt hole is smaller than the given profile change valueThe design parameter or variable is the optimum value that meets the requirements.
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| CN109344494B (en) * | 2018-09-28 | 2021-09-03 | 中国航发湖南动力机械研究所 | Bolt hole of bolt connection type gas turbine baffle and design method thereof |
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| CN103334798A (en) * | 2013-07-10 | 2013-10-02 | 南京航空航天大学 | Multi-arc specially-shaped hole |
| CN105046000A (en) * | 2015-07-20 | 2015-11-11 | 南京航空航天大学 | Engineering discrete variable based two-variable multi-circular arc special-shaped hole balance design method and model |
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| CN103334798A (en) * | 2013-07-10 | 2013-10-02 | 南京航空航天大学 | Multi-arc specially-shaped hole |
| CN105046000A (en) * | 2015-07-20 | 2015-11-11 | 南京航空航天大学 | Engineering discrete variable based two-variable multi-circular arc special-shaped hole balance design method and model |
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