Aviation Engine Rotor Assembly Phase Optimization Method and Matching Selection Method

Abstract

The present invention discloses an optimization method and a selection method for the assembly phase of an aero-engine rotor. The optimization method for the assembly phase of the aero-engine rotor includes: obtaining the axial length of the compressor rotor, the front-end eccentricity and the front-end eccentricity phase of the compressor rotor, the axial length of the turbine rotor, the rear-end eccentricity and the rear-end eccentricity phase of the turbine rotor; obtaining the relationship between the eccentricity of the high-pressure combined rotor joint surface and the assembly phase angle according to a mathematical model; adjusting the assembly phase angle according to the requirement of the eccentricity of the high-pressure combined rotor joint surface. A mathematical model is established according to the dimensional characteristics of the compressor rotor and the turbine rotor, so as to establish the relationship between the high-pressure combined rotor joint surface and the assembly phase angle. According to this relationship, the range of the eccentricity or the runout value at the joint surface of the combined rotor formed by the compressor rotor and the turbine rotor can be quickly evaluated, and the optimal assembly phase angle can be given, effectively guiding the actual assembly work and improving the assembly efficiency.

Description

Technical Field

[0001] This application belongs to the field of aero-engines, and particularly relates to a method for optimizing the assembly phase of an aero-engine rotor and a method for selecting and matching an aero-engine rotor. Background Art

[0002] There are many factors that cause the vibration of the whole engine. The magnitude of the rotor unbalance and the distribution state of the internal unbalance of the rotor are one of the key factors. And the most direct and important factor affecting the rotor unbalance is the assembly quality of the rotor. Generally speaking, the straighter a rotor is installed, the smaller its axial runout is, which means that the degree of deviation of each stage of disk / disk-drum, etc. inside the rotor assembly from the rotation axis is smaller, and thus the initial unbalance of the rotor assembly is also smaller. Controlling the rotor runout is one of the important means to control the initial unbalance of the rotor. The high-pressure combined rotor of a civil turbofan dual-rotor aero-engine is composed of a high-pressure compressor maintenance unit rotor (referred to as the compressor rotor for short) and a high-pressure turbine maintenance unit rotor (referred to as the turbine rotor for short). With the introduction of more new models of aero-engine products, the high-pressure combined rotor shows a trend of becoming more and more slender in structure. The change of the runout of the maintenance unit rotor itself, especially the runout value at the middle joint of the combined rotor, will have a more sensitive impact on the straightness of the combined rotor. On the one hand, this will require higher requirements for the runout accuracy of the maintenance unit rotor, which will inevitably lead to the further improvement of the manufacturing accuracy of the rotor components and the assembly accuracy of the rotor assembly, but this will reduce the qualified rate of part manufacturing and the assembly efficiency of the rotor assembly. On the other hand, in the prior art, it is difficult to quickly estimate the runout at the middle joint of the high-pressure combined rotor, which has hindered the progress of the research and development of aero-engine rotors and the improvement of the batch production qualified rate of aero-engine rotors in models. Summary of the Invention

[0003] The technical problem to be solved by the present invention is to overcome the defects that the existing assembly methods are difficult to meet the requirements of rotor runout accuracy and it is difficult to quickly estimate the runout at the middle joint of the high-pressure combined rotor in the prior art, and to provide a method for optimizing the assembly phase of an aero-engine rotor and a method for selection and matching.

[0004] The present invention solves the above technical problems through the following technical solutions:

[0005] A method for optimizing the assembly phase of an aero-engine rotor, the method for optimizing the assembly phase of an aero-engine rotor includes:

[0006] Obtaining the axial length of the compressor rotor, the front-end eccentricity and the front-end eccentricity phase of the compressor rotor, the axial length of the turbine rotor, the rear-end eccentricity and the rear-end eccentricity phase of the turbine rotor;

[0007] According to the mathematical model, obtaining the relationship between the eccentricity of the joint surface of the high-pressure combined rotor and the assembly phase angle;

[0008] Adjust the assembly phase angle according to the requirement of the eccentricity of the joint surface of the high-pressure combined rotor.

[0009] In this solution, a mathematical model is established based on the dimensional characteristics of the compressor rotor and the turbine rotor. Through the mathematical model, the relationship between the joint surface of the high-pressure combined rotor and the assembly phase angle is established. According to this relationship, the eccentricity or the range of the runout value at the joint surface of the combined rotor formed by the compressor rotor and the turbine rotor can be quickly evaluated, and the optimal assembly phase angle can be given, effectively guiding the actual assembly work and improving the assembly efficiency.

[0010] Preferably, obtaining the axial length of the compressor rotor, the front-end eccentricity and the front-end eccentricity phase of the compressor rotor, the axial length of the turbine rotor, the rear-end eccentricity and the rear-end eccentricity phase of the turbine rotor includes:

[0011] Measure the runout value at the front-end bearing journal of the compressor rotor with the mating spigot at the rear-end joint surface as the reference.

[0012] Fit the front-end eccentricity and the front-end eccentricity phase of the compressor rotor by the least squares method.

[0013] Preferably, obtaining the axial length of the compressor rotor, the front-end eccentricity and the front-end eccentricity phase of the compressor rotor, the axial length of the turbine rotor, the rear-end eccentricity and the rear-end eccentricity phase of the turbine rotor includes:

[0014] Measure the runout value at the rear-end bearing journal of the turbine rotor with the mating spigot at the front-end joint surface as the reference.

[0015] Fit the rear-end eccentricity and the rear-end eccentricity phase of the turbine rotor by the least squares method.

[0016] Preferably, the formula of the mathematical model is:

[0017]

[0018] Wherein:

[0019] S is the eccentricity of the joint surface of the high-pressure combined rotor;

[0020] L1 is the axial length of the compressor rotor;

[0021] P1 is the front-end eccentricity of the compressor rotor;

[0022] L2 is the axial length of the turbine rotor;

[0023] P2 is the eccentricity at the rear end of the turbine rotor;

[0024] θ is the assembly phase angle.

[0025] Preferably, according to the formula, a relationship curve between the eccentricity of the high-pressure combined rotor joint surface and the assembly phase angle is obtained.

[0026] A method for matching and selecting aero-engine rotors, the method for matching and selecting aero-engine rotors comprising:

[0027] Providing m compressor rotors and n turbine rotors, where both m and n are integers greater than or equal to one;

[0028] Measuring the axial length of each compressor rotor, the front-end eccentricity and front-end eccentricity phase of each compressor rotor, the axial length of each turbine rotor, the rear-end eccentricity and rear-end eccentricity phase of each turbine rotor;

[0029] Arbitrarily combining the compressor rotors and the turbine rotors in pairs to obtain m×n groups of rotor combinations;

[0030] Using the aero-engine rotor assembly phase optimization method as described above, obtaining the relationship between the eccentricity of the high-pressure combined rotor joint surface and the assembly phase angle in each group of rotor combinations;

[0031] Screening out all rotor combinations that can meet the eccentricity requirement of the high-pressure combined rotor joint surface;

[0032] Selecting a rotor combination according to the optimization goal to complete the rotor matching and selection.

[0033] In this solution, all combinations of the compressor rotors and the turbine rotors are substituted into the aforementioned mathematical model to obtain the relationship between the eccentricity of the high-pressure combined rotor joint surface and the assembly phase angle in all rotor combinations, and corresponding rotor combinations are selected according to the optimization goal.

[0034] Preferably, the selecting a rotor combination according to the optimization goal to complete the rotor matching and selection includes:

[0035] If the optimization goal is to obtain the matching and selection method with the largest number of rotor combinations that can meet the eccentricity requirement of the high-pressure combined rotor joint surface among m compressor rotors and n turbine rotors, then list all the matching and selection methods, obtain the number of rotor combinations that can meet the eccentricity requirement of the high-pressure combined rotor joint surface in each matching and selection method, and select the matching and selection method with the largest number of rotor combinations that can meet the eccentricity requirement of the high-pressure combined rotor joint surface.

[0036] Preferably, the selecting a rotor combination according to the optimization goal to complete the rotor matching and selection includes:

[0037] If the optimization objective is to obtain a rotor combination with the minimum eccentricity at the joint surface of the high-pressure combined rotor, calculate the minimum eccentricity that can be achieved at the joint surface of the high-pressure combined rotor in each rotor combination, and select the rotor combination that can achieve the minimum eccentricity.

[0038] The positive and progressive effects of the present invention are as follows: The method for optimizing the assembly phase of the aero-engine rotor of the present invention can quickly evaluate the range of eccentricity or runout value at the joint surface of the combined rotor formed by the compressor rotor and the turbine rotor, and can give the optimal assembly phase angle, effectively guiding the actual assembly work and improving the assembly efficiency. The method for selecting aero-engine rotors of the present invention can be used to pair and screen two high-pressure maintenance unit rotors during the mass production of engines, improving the qualified rate of assembly. The optimization method of the present invention is more reasonable and closer to the actual requirements of the high-pressure combined rotor of the engine. Description of the Drawings

[0039] Figure 1A With Figure 1B It is a schematic diagram of the docking assembly state of the high-pressure combined rotor.

[0040] Figure 2A With Figure 2B It is a schematic diagram of the assembly state when the compressor rotor and the turbine rotor are in different assembly phases.

[0041] Figure 3 It is a schematic diagram of the spatial geometric relationship between the axis of the compressor rotor and the axis of the turbine rotor in the high-pressure combined rotor.

[0042] Figure 4 It is a flowchart of the method for optimizing the assembly phase of the aero-engine rotor according to an embodiment of the present invention.

[0043] Figure 5 It is the runout evaluation method of the compressor rotor in the method for optimizing the assembly phase of the aero-engine rotor according to an embodiment of the present invention.

[0044] Figure 6 It is the runout evaluation method of the turbine rotor in the method for optimizing the assembly phase of the aero-engine rotor according to an embodiment of the present invention.

[0045] Figure 7 It is a relationship diagram between the eccentricity of the joint surface of the high-pressure combined rotor and the assembly phase angle drawn according to the method for optimizing the assembly phase of the aero-engine rotor according to an embodiment of the present invention.

[0046] Figure 8 It is a flowchart of the method for selecting aero-engine rotors according to an embodiment of the present invention.

[0047] Description of the Reference Numerals

[0048] Compressor rotor 1

[0049] Compressor rotor axis 11

[0050] Inner stop 12

[0051] Turbine rotor 2

[0052] Turbine rotor rotation axis 21

[0053] Outer stop 22

[0054] High-pressure combined rotor 3

[0055] Actual rotation axis 31 Specific implementation manner

[0056] The present invention will be further described below by way of embodiments, but the present invention is not limited to the scope of the described embodiments.

[0057] To ensure the assembly quality of the rotor, different runout control measures are taken at different engine assembly stages. During the sub-assembly stage, runout control is separately performed on the rotors of the two maintenance unit bodies. The runout requirement for the compressor rotor is as follows: taking two cylindrical surfaces or one cylindrical surface plus one shaft shoulder end face on the front journal of the compressor rotor as the reference, it is required that the end face runout and the radial runout of the cylindrical surface at the rear stop of the labyrinth disc are both less than or equal to a certain value R1. The runout requirement for the turbine rotor is as follows: taking the end face and the cylindrical surface at the front drum shaft stop of the turbine rotor as the reference, it is required that the runout of the cylindrical surface and the shaft shoulder end face at the rear support journal are both less than or equal to a certain value R2. During the general assembly stage, the runout requirement for the high-pressure combined rotor is as follows: taking the cylindrical surfaces at the two support journals of the combined rotor as the combined reference, it is required that the runout of the labyrinth disc at the intermediate joint part or the runout of the outer cylindrical surface at the front end of the drum shaft is less than or equal to a certain value R3.

[0058] There are two problems with the above method: First, it is impossible to quickly estimate the possible range of runout values of the combined rotor, and it is impossible to directly identify the optimal docking phase of the two maintenance unit body rotors that can meet the runout requirements of the combined rotor; Second, the rotor runout value cannot reflect the true state of the rotor, which may mislead the assembly process. For example, when there is a perpendicularity error or a concentricity error between the measurement references, during the process of finding the runout measurement alignment reference, due to the influence of the rotor's own structural characteristics, non-critical measurement extension errors are generated, resulting in a false appearance of unqualified rotor runout.

[0059] This embodiment provides a method for optimizing the rotor assembly phase, which can effectively solve the runout extension error caused by the error between the measurement references, and can also achieve a quick estimate of the runout range of the combined rotor, realizing the efficient selection and matching of the maintenance unit body rotors.

[0060] The compressor rotor 1 and the turbine rotor 2 of the high-pressure combined rotor 3 are as Figure 1AAs shown, the compressor rotor 1 and the turbine rotor 2 are combined to form a new actual rotation axis 31 (as shown in FIG. Figure 1B As shown), that is, when the high-pressure combined rotor 3 rotates, it will rotate around the actual rotation axis 31. Due to the influence of the rotor runout, the eccentricity of the joint surface of the high-pressure combined rotor 3 is recorded as S, which is defined as the distance that the center of the middle joint of the high-pressure combined rotor deviates from the actual rotation axis. When the roundness error of the parts is not considered, the runout value is twice the eccentricity S.

[0061] Generally speaking, the selectable assembly phase of two rotors connected by bolts is equal to the number of bolt holes, and the runout size of the combined rotor joint surface is different when they are connected at different phases. The bolt holes are generally evenly distributed in the circumferential direction. Therefore, when the number of bolt holes is i, the step value of adjusting the assembly phase angle is 360° / i, where i is a positive integer greater than 1.

[0062] In order to quickly estimate the runout at the joint surface of the high-pressure combined rotor and find the optimal runout combination, we can change an angle to analyze the runout characteristics of the combined rotor. Figure 2A As shown in FIG. 1 , when the compressor rotor and the turbine rotor are in a certain assembly phase, the eccentricity S of the high-pressure combined rotor joint surface is large. When the turbine rotor 2 is rotated to be in another assembly phase, as shown in FIG. Figure 2B As shown, the eccentricity S of the high-pressure combined rotor joint surface is reduced.

[0063] like Figure 3 As shown in the figure, X, Y, and Z are mutually perpendicular coordinate axes, and point O is the intersection of the three coordinate axes. First, adjust the joint surface of the high-pressure combined rotor 3 to Figure 4 On the YOZ plane in the high-pressure combined rotor 3, the compressor rotor axis 11 is adjusted to the XOZ plane, and the turbine rotor axis 21 may be oriented to any angle in space. This expression can also be equivalently understood as: the high-pressure combined rotor 3 is based on the joint surface of the two maintenance unit rotors (i.e., the compressor rotor 1 and the turbine rotor 2) and the eccentricity of the support journal at the other end of each rotor is staggered by a certain phase angle and then butt-jointed and assembled. The relationship expression between the combined rotor eccentricity S and the relevant parameters of the compressor rotor and the turbine rotor can be quickly obtained from the spatial geometric relationship. The expression is as follows:

[0064]

[0065] Wherein: S is the eccentricity of the high-pressure combined rotor joint surface; L1 is the axial length of the compressor rotor; P1 is the front end eccentricity of the compressor rotor; L2 is the axial length of the turbine rotor; P2 is the eccentricity at the rear end of the turbine rotor; θ is the assembly phase angle, and its value range is 0 to 360°.

[0066] When making a quick assessment, the radial runout value of the combined high-pressure rotor joint surface can be taken as twice the S value.

[0067] As Figure 4 shown, this embodiment provides a method for optimizing the assembly phase of an aero-engine rotor. The method for optimizing the assembly phase of the aero-engine rotor includes:

[0068] Obtain the axial length L1 of the compressor rotor, the front-end eccentricity P1 and the front-end eccentricity phase of the compressor rotor, the axial length L2 of the turbine rotor, the rear-end eccentricity P2 and the rear-end eccentricity phase of the turbine rotor;

[0069] According to the mathematical model, obtain the relationship between the eccentricity S of the combined high-pressure rotor joint surface and the assembly phase angle θ;

[0070] According to the requirement of the eccentricity S of the combined high-pressure rotor joint surface, adjust the assembly phase angle θ.

[0071] In this embodiment, a mathematical model is established based on the dimensional characteristics of the compressor rotor 1 and the turbine rotor 2. Through the mathematical model, the relationship between the combined high-pressure rotor joint surface and the assembly phase angle θ is established. According to this relationship, the eccentricity or the range of the runout value at the joint surface of the combined rotor formed by the compressor rotor 1 and the turbine rotor 2 can be quickly evaluated, and the optimal assembly phase angle can be given, effectively guiding the actual assembly work and improving the assembly efficiency. If the actual bolt hole distribution cannot reach the optimal assembly phase, the assembly phase closest to the optimal assembly phase is selected.

[0072] The measurement principle of the front-end eccentricity and the front-end eccentricity phase of the compressor rotor 1 is as Figure 5 shown. Specifically, the runout value at the front-end support journal of the compressor rotor 1 is measured with the mating spigot at the rear-end joint surface as the reference, and the front-end eccentricity P1 and the front-end eccentricity phase of the compressor rotor 1 are fitted by the least squares method. This front-end eccentricity can also be approximately considered as 0.5 times the runout value.

[0073] Similarly, the measurement principle of the rear-end eccentricity and the rear-end eccentricity phase of the turbine rotor 2 is as Figure 6 shown. The runout value at the rear-end support journal of the turbine rotor 2 is measured with the mating spigot at the front-end joint surface as the reference, and the rear-end eccentricity P2 and the rear-end eccentricity phase of the turbine rotor 2 are fitted by the least squares method. This rear-end eccentricity can also be approximately considered as 0.5 times the runout value.

[0074] The optimization method of this embodiment optimizes the evaluation method of the runout of the compressor rotor in the sub-assembly stage to the requirement of the eccentricity at the front-end support journal with the outer spigot 12 at the rear-end joint surface as the reference; at the same time, it optimizes the evaluation method of the runout of the turbine rotor to the requirement of the eccentricity at the rear-end support journal with the inner spigot 22 at the front-end joint surface as the reference.

[0075] According to the above formula (1), the relationship curve between the eccentricity of the high-pressure combined rotor joint surface and the assembly phase angle is obtained (such as Figure 7 As shown), according to the curve, the range of the eccentricity S of the high-pressure combined rotor joint surface can be quickly and intuitively obtained, and the optimal assembly phase angle θ can be obtained.

[0076] This embodiment also provides a method for selecting and assembling rotors for an aircraft engine, when there are a number of compressor rotors (for example, m) and a number of turbine rotors (for example, n) to be selected and assembled, where m and n are both integers greater than or equal to 1. If m ≥ n, at most n high-pressure combined rotors that meet the requirements can be matched.

[0077] like Figure 8 As shown, the aircraft engine rotor selection method of this embodiment includes:

[0078] Providing m compressor rotors and n turbine rotors, wherein m and n are both integers greater than or equal to one;

[0079] Combining compressor rotor 1 and turbine rotor 2 in pairs arbitrarily, we can obtain m×n rotor combinations;

[0080] By using the above-mentioned aircraft engine rotor assembly phase optimization method, the relationship between the eccentricity of the high-pressure combined rotor joint surface and the assembly phase angle in each rotor combination is obtained;

[0081] Screen out all rotor combinations that can meet the eccentricity requirements of the high-pressure combined rotor joint surface;

[0082] According to the optimization target, select the rotor combination and complete the rotor matching.

[0083] In this scheme, all combinations of compressor rotors and turbine rotors are substituted into the aforementioned mathematical model, i.e., formula (1), to obtain the relationship between the eccentricity of the high-pressure combined rotor joint surface and the assembly phase angle in all rotor combinations, and the corresponding rotor combination is selected according to the optimization target.

[0084] If the optimization goal is to obtain a matching method with the largest number of rotor combinations that can meet the eccentricity requirements of the high-pressure combined rotor joint surface among m compressor rotors and n turbine rotors, then all matching methods are listed, the number of combinations that can be successfully paired for each compressor rotor is found, and the compressor rotors are sorted in order from small to large in the number of combinations (when m≤n, they are sorted according to the situation of the turbine rotor, and at most m combined rotors can be successfully paired), and then the compressor rotor and the turbine rotor are paired in turn, wherein the docking phase of the compressor rotor and the turbine rotor is executed according to the θ angle corresponding to the eccentricity S (the runout value is 2 times the S value) that the pair of combined rotors meet the requirements, and the rotor matching is completed.

[0085] Therefore, when the eccentricity or the radial runout value at the middle joint surface of the given high-pressure combined rotor is required, the matching method of this embodiment can quickly match the batch-maintained unit body rotors.

[0086] If the optimization goal is to obtain the rotor combination with the minimum eccentricity at the joint surface of the high-pressure combined rotor, calculate the minimum eccentricity that can be achieved at the joint surface of the high-pressure combined rotor in each rotor combination, and select the rotor combination that can achieve the minimum eccentricity.

[0087] Although the specific embodiments of the present invention have been described above, those skilled in the art should understand that this is only an example. The protection scope of the present invention is defined by the appended claims. Without departing from the principle and essence of the present invention, those skilled in the art can make various changes or modifications to these embodiments, but these changes and modifications all fall within the protection scope of the present invention.

Claims

1. An optimization method for the assembly phase of an aero-engine rotor, characterized in that, The method for optimizing the assembly phase of an aero-engine rotor includes: Obtaining the axial length of the compressor rotor, the front-end eccentricity and front-end eccentricity phase of the compressor rotor, the axial length of the turbine rotor, the rear-end eccentricity and rear-end eccentricity phase of the turbine rotor; Obtaining the relationship between the eccentricity of the high-pressure combined rotor joint surface and the assembly phase angle according to a mathematical model; Adjusting the assembly phase angle according to the requirement of the eccentricity of the high-pressure combined rotor joint surface; The formula of the mathematical model is: Where: S is the eccentricity of the high-pressure combined rotor joint surface; L1 is the axial length of the compressor rotor; P1 is the front-end eccentricity of the compressor rotor; L2 is the axial length of the turbine rotor; P2 is the rear-end eccentricity of the turbine rotor; θ is the assembly phase angle.

2. The method for optimizing the assembly phase of an aeroengine rotor according to claim 1, characterized in that The obtaining of the axial length of the compressor rotor, the front-end eccentricity and front-end eccentricity phase of the compressor rotor, the axial length of the turbine rotor, the rear-end eccentricity and rear-end eccentricity phase of the turbine rotor includes: Measuring the runout value at the front-end support journal of the compressor rotor with the mating spigot at the rear-end joint surface as the reference; Fitting the front-end eccentricity and rear-end eccentricity phase of the compressor rotor by the least squares method.

3. The method for optimizing the assembly phase of an aero-engine rotor as claimed in claim 1, wherein The obtaining of the axial length of the compressor rotor, the front-end eccentricity and front-end eccentricity phase of the compressor rotor, the axial length of the turbine rotor, the rear-end eccentricity and rear-end eccentricity phase of the turbine rotor includes: Measuring the runout value at the rear-end support journal of the turbine rotor with the mating spigot at the front-end joint surface as the reference; Fitting the rear-end eccentricity and front-end eccentricity phase of the turbine rotor by the least squares method.

4. The method for optimizing the assembly phase of an aero-engine rotor according to claim 1, characterized in that, Obtaining the relationship curve between the eccentricity of the high-pressure combined rotor joint surface and the assembly phase angle according to the formula.

5. A method for matching and selecting an aero-engine rotor, characterized in that The method for selecting an aero-engine rotor includes: Providing m compressor rotors and n turbine rotors, where both m and n are integers greater than or equal to one; Arbitrarily combining the compressor rotors and turbine rotors in pairs to obtain m×n groups of rotor combinations; Using the method for optimizing the assembly phase of an aero-engine rotor according to any one of claims 1 to 4 to obtain the relationship between the eccentricity of the high-pressure combined rotor joint surface and the assembly phase angle in each group of rotor combinations; Screening out all rotor combinations that can meet the requirement of the eccentricity of the high-pressure combined rotor joint surface; Selecting a rotor combination according to the optimization objective to complete rotor selection.

6. The method for matching an aero-engine rotor according to claim 5, wherein The selecting a rotor combination according to the optimization objective to complete rotor selection includes: If the optimization objective is to obtain the selection method with the largest number of rotor combinations that can meet the requirement of the eccentricity of the high-pressure combined rotor joint surface among m compressor rotors and n turbine rotors, listing all selection methods, obtaining the number of rotor combinations that can meet the requirement of the eccentricity of the high-pressure combined rotor joint surface in each selection method, and selecting the selection method with the largest number of rotor combinations that can meet the requirement of the eccentricity of the high-pressure combined rotor joint surface.

7. The method for matching an aero-engine rotor according to claim 5, characterized in that, The selecting a rotor combination according to the optimization objective to complete rotor selection includes: If the optimization objective is to obtain a rotor combination with the minimum eccentricity of the high-pressure combined rotor mating surface, calculate the minimum eccentricity of the high-pressure combined rotor mating surface that can be achieved in each rotor combination, and select the rotor combination that can achieve the minimum eccentricity.

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