Rocket sled impact speed dynamic envelope line construction method for preventing chipping

Abstract

The invention provides a rocket sled impact speed dynamic envelope line construction method capable of preventing chipping, and mainly solves the problem of boundary condition control deficiency caused by chipping of a rocket sled running at a high speed. According to the scheme adopted by the invention, a limit impact speed dynamic envelope line is obtained by establishing a rocket sled coupling material point model, and n typical course speeds are selected from a speed section where chipping may occur as calculation points; and solving the minimum lateral speed # imgabs0 # and the minimum vertical speed # imgabs1 # which are corresponding to each speed point vi under various adverse factors and cause chipping, and then solving a limit impact speed function by applying a cubic spline interpolation method so as to obtain an envelope curve of an impact speed limit value.

Description

Technical Field

[0001] The invention belongs to the field of basic experimental testing technology, and in particular relates to a method for constructing a dynamic envelope of a rocket sled's ultimate impact velocity. Background Art

[0002] To achieve the transition of weapons and equipment from supersonic speed to hypersonic speed, it is necessary to carry out functional assessment and performance testing of systems / key components. Rocket sled test is a ground high dynamic test method between laboratory test and flight test. It not only overcomes the limitations of laboratory test that cannot fully reflect the working mechanical environment of aircraft and cannot obtain the complete dynamic performance parameters of weapons and equipment, but also makes up for the objective shortcomings of high cost, difficult process control and difficult reproduction of typical dynamic working conditions in flight test, and becomes an important test method to support the development of hypersonic weapons and equipment. However, the current rocket sled test is limited by the increase in speed. This is mainly because when the running speed of the sled car is close to hypersonic speed, the sled body is more likely to run unstably under external excitation. When the skid hits the rail, it is possible to produce "raindrop-shaped" or "spindle-shaped" chiseling damage on the rail. When chiseling occurs, it will cause a sharp deterioration of the mechanical environment of the rocket sled, and even cause test failure in severe cases. Only by solving the chiseling problem in the hypersonic rocket sled test can the requirements of the rocket sled test speed for hypersonic weapon equipment testing be met. The impact speed of the rocket sled shoe on the track is one of the most important factors affecting the occurrence of chiseling. Different heading speeds cause different impact speeds to produce chiseling on the track. Therefore, there must be a limit impact speed envelope. As long as the impact speed of the rocket sled shoe on the track is controlled within the envelope, chiseling will not occur. Therefore, the limit impact speed envelope of the rocket sled shoe on the track is used as the boundary condition of the scheme design to prevent the occurrence of chiseling. However, in the prior art, the current research on the high-speed chiseling problem of the rocket sled is mostly the analysis of the occurrence mechanism, phenomenon reproduction or influencing factors, or the use of theoretical formulas to calculate the threshold speed of chiseling, such as the method mentioned in CN112580144A. The prior art has not yet proposed boundary conditions for controlling the occurrence of chiseling. Therefore, when designing the hypersonic rocket sled system scheme, it is impossible to perform structural design analysis or system adjustment for the chiseling problem. Summary of the invention

[0003] In order to overcome the shortcomings of the prior art and address the problem of lack of boundary condition control caused by gouging of a high-speed rocket sled, the present invention provides a method for constructing a dynamic envelope of the impact velocity of a rocket sled to prevent gouging.

[0004] A method for constructing a dynamic envelope of a rocket sled impact velocity to prevent gouging, comprising the following steps:

[0005] Step 1): Obtain basic information of the rocket sled test system;

[0006] Step 2): Determine the calculation point;

[0007] Step 3): Calculate the minimum lateral velocity of the impact velocity limit corresponding to each calculation point and minimum vertical speed

[0008] Step 4): Construct the minimum lateral speed and minimum vertical speed The envelope of .

[0009] Furthermore, the basic information includes the material used for the rocket sled slide, the material used for the slide rail, the matching method of the slide shoe and the slide rail, and the maximum heading speed v of the sled body. max , the hardness of the rocket sled slide shoe material and the hardness of the slide rail material.

[0010] Furthermore, the step of determining the calculation point is:

[0011] Threshold speed v cv Defined as the minimum heading speed that causes chiseling to occur, the threshold speed v cv Calculated by Stefain-Parker formula;

[0012] Compare the hardness of the rocket sled shoe material and the hardness of the rail material, select the material with the greatest hardness, and substitute the hardness value of the material with the greatest hardness into the relationship between pressure amplitude and material Vickers hardness:

[0013] p=0.073HV max +9.343,

[0014] Among them, HV max =max(hardness of the sliding shoe material, hardness of the sliding rail material), p is the pressure amplitude of the high-pressure core when the chiseling occurs; max() is the maximum value;

[0015] The Gruneisen equation of state for the shoe rail material is:

[0016] U=c+s*D

[0017] Among them, U is the shock wave velocity, parameter c is the sound velocity of the material, s is the material constant, and D is the particle velocity;

[0018] The threshold velocity v is solved by the relationship between velocity and collision interface pressure. cv :

[0019]

[0020] Among them, C A is the speed of sound of the sliding shoe material, c B is the speed of sound of the track material, ρA is the density of the sliding shoe material, ρ B is the density of the track material, s A is the material constant of the sliding shoe material, s B is the material constant of the track material;

[0021] The maximum heading speed v of the rocket sled max Obtained in step 1);

[0022] The threshold speed v calculated by the theoretical formula cv The maximum heading speed v of the rocket sled max Select n heading speeds v in the interval i As the calculation point, where i = 1, 2, ..., n.

[0023] Furthermore, the impact velocity limit value and the minimum lateral velocity are obtained. and minimum vertical speed The specific steps are:

[0024] ① Establish a simplified coupled three-dimensional model of the rocket sled system;

[0025] ②Calculate the velocity point v i The corresponding minimum lateral speed and minimum vertical speed

[0026] Furthermore, the steps of establishing a simplified coupled three-dimensional model of the rocket sled system are:

[0027] According to the matching method of the sliding shoe and the sliding rail in step 1), the shoe-rail coupling structure is simplified and modeled by 3D design software, and is placed according to the actual position relationship to obtain a simplified rocket sled coupling 3D model;

[0028] The rocket sled coupled three-dimensional model is imported into the finite element software, and meshing is performed to obtain the node information and unit information of the finite element model; the model is discretized according to the node information and unit information of the finite element model to obtain the material point model.

[0029] Furthermore, the calculation speed point v i The corresponding minimum lateral speed and minimum vertical speed The specific steps are:

[0030] Traverse all heading speeds v i The steps to calculate whether a chiseling pit with a depth greater than 1.5 mm appears at different lateral speeds are:

[0031] Step a.1: Heading speed is v i ,

[0032] Step a.2: Starting from 0 m / s, the lateral speed is gradually increased in steps of 0.1 m / s;

[0033] Step a.3: Determine whether a chiseling pit with a depth greater than 1.5 mm appears in the material point model at this time. If a chiseling pit with a depth greater than 1.5 mm appears, record the lateral velocity at this time and set Jump to step a.1 and traverse the next heading speed; if there is no chisel pit with a depth greater than 1.5 mm, and the maximum value of the lateral speed does not exceed 3 m / s; then jump to step a.2 and continue to traverse the next lateral speed; if there is no chisel pit with a depth greater than 1.5 mm, and the value of the lateral speed is greater than 3 m / s, then jump to step a.1 and traverse the next heading speed;

[0034] Traverse all heading speeds v i The steps to calculate whether a chiseling pit with a depth greater than 1.5 mm appears at different vertical speeds are:

[0035] Step b.1: For heading velocity v i ,

[0036] Step b.2: Starting from 0 m / s, the vertical speed is gradually increased with a step length of 0.1 m / s;

[0037] Step b.3: Determine whether a chiseling pit with a depth greater than 1.5 mm appears in the material point model at this time. If a chiseling pit with a depth greater than 1.5 mm appears, record the vertical velocity at this time and set Jump to step b.1 and traverse the next heading speed; if there is no chiseling pit with a depth greater than 1.5mm and the maximum vertical speed does not exceed 3m / s; then jump to step b.2 and continue to traverse the next vertical speed; if there is no chiseling pit with a depth greater than 1.5mm and the value of the vertical speed is greater than 3m / s, then jump to step b.1 and traverse the next heading speed.

[0038] Furthermore, the minimum lateral speed is established The envelope method is:

[0039] At n speed points v i The corresponding impact velocity limit As the basic data, i = 1, 2, ..., n, the impact velocity limit value is calculated by cubic spline interpolation method. Solve and get the high-speed running section of the rocket sled [v cv ,v max ] is the limiting impact velocity cubic spline function f(x);

[0040] The cubic spline function f(x) of the limiting impact velocity satisfies the following in each heading speed interval [v i ,v i+1 ] is a cubic polynomial with second-order continuous derivatives at each interior node;

[0041] According to the definition of cubic spline function, the cubic spline function of the limit impact velocity f(x) is in each heading speed interval [v i ,v i+1 The expression on ] is f(x)=a i0 +a i1 x+a i2 x 2 +a i3 x 3 ;

[0042] Solve f(x)=a by interpolation conditions and connection conditions i0 +a i1 x+a i2 x 2 +a i3 x 3 The four unknown parameters a in i0 、a i1 、a i2 and a i3 ;

[0043] The interpolation conditions are:

[0044]

[0045] The connection conditions are:

[0046]

[0047] Solve f(x) = a i0 +a i1 x+a i2 x 2 +a i3 x 3 The four unknown parameters a in i0 、a i1 、a i2 and a i3 ;

[0048] By a i0 、a i1 、a i2 and a i3 The obtained f(x) is the envelope curve of the required minimum lateral velocity;

[0049] In formula (1), Replace with That is, the envelope curve of the minimum vertical velocity is obtained.

[0050] The beneficial effects of the present invention are as follows: the present invention proposes boundary conditions for controlling the occurrence of chiseling, and structural design analysis or system adjustment can be performed on the chiseling problem when designing a hypersonic rocket sled system, thereby improving the accuracy of the design of the hypersonic rocket sled test system. BRIEF DESCRIPTION OF THE DRAWINGS

[0051] Figure 1 It is a flow chart for constructing the dynamic envelope of the rocket sled's ultimate impact velocity;

[0052] Figure 2 It is a simplified coupled three-dimensional model schematic diagram of the rocket sled system;

[0053] Figure 3 It is the dynamic envelope of the limiting impact velocity. DETAILED DESCRIPTION

[0054] The scheme adopted by the present invention is as follows: the method obtains the dynamic envelope of the limit impact velocity by establishing a rocket sled coupled material point model, and selects n typical heading velocities v in the velocity segment where chiseling may occur. i As calculation points, where i = 1, 2, ..., n, solve for each speed point v under various unfavorable factors i The corresponding minimum lateral speed that causes chiseling to occur and minimum vertical speed Then the cubic spline interpolation method is used to solve the limiting impact velocity function f(x).

[0055] A method for constructing a dynamic envelope of a rocket sled impact velocity to prevent gouging, comprising the following steps:

[0056] Step 1): Obtain basic information of the rocket sled test system;

[0057] Obtain basic information according to the overall plan of the rocket sled test system;

[0058] The basic information includes the materials used for the rocket sled's slides, the materials used for the rails, the matching method of the slides and rails, and the maximum heading speed v of the sled. max , the hardness of the rocket sled shoe material and the hardness of the slide rail material;

[0059] Step 2): Determine the calculation point;

[0060] Threshold speed v cv Defined as the minimum heading speed that causes chiseling to occur, the threshold speed v cv Calculated by Stefain-Parker formula;

[0061] Compare the hardness of the rocket sled shoe material and the hardness of the rail material, select the material with the greatest hardness, and substitute the hardness value of the material with the greatest hardness into the relationship between pressure amplitude and material Vickers hardness:

[0062] p=0.073HV max +9.343,

[0063] Among them, HV max =max(hardness of the sliding shoe material, hardness of the sliding rail material), p is the pressure amplitude of the high-pressure core when the chiseling occurs; max() is the maximum value;

[0064] The Gruneisen equation of state for the shoe rail material is:

[0065] U=c+s*D

[0066] Among them, U is the shock wave velocity, parameter c is the sound velocity of the material, s is the material constant, and D is the particle velocity;

[0067] The threshold velocity v is solved by the relationship between velocity and collision interface pressure. cv :

[0068]

[0069] Among them, c A is the speed of sound of the sliding shoe material, c B is the speed of sound of the track material, ρ A is the density of the sliding shoe material, ρ B is the density of the track material, s A is the material constant of the sliding shoe material, s B is the material constant of the track material;

[0070] The maximum heading speed v of the rocket sled max Obtained in step 1);

[0071] The threshold speed v calculated by the theoretical formula cv The maximum heading speed v of the rocket sled max Select n heading speeds v in the interval i As calculation points, where i = 1, 2, ..., n;

[0072] Step 3): Calculate the minimum lateral velocity of the impact velocity limit corresponding to each calculation point and minimum vertical speed

[0073] The impact speed limit is the minimum lateral speed at which the shoe impacts the rail and causes chiseling at a certain heading speed. and minimum vertical speed When the heading speed is constant, the limit value of the lateral impact speed is equal to the limit value of the vertical impact speed. These two values ​​are collectively referred to as the impact speed limit value.

[0074] Get the impact velocity limit value minimum lateral velocity and minimum vertical speed The specific steps are:

[0075] ③Establish a simplified coupled three-dimensional model of the rocket sled system

[0076] According to the matching method of the sliding shoe and the sliding rail in step 1), the shoe-rail coupling structure is simplified and modeled by 3D design software, and is placed according to the actual position relationship to obtain a simplified rocket sled coupling 3D model;

[0077] Import the rocket sled coupled 3D model into the finite element software and perform meshing to obtain the node information and unit information of the finite element model; discretize the model according to the node information and unit information of the finite element model to obtain the material point model;

[0078] ④Calculate velocity point v i The corresponding minimum lateral speed and minimum vertical speed

[0079] Traverse all heading speeds v i The steps to calculate whether a chiseling pit with a depth greater than 1.5 mm appears at different lateral speeds are:

[0080] Step a.1: Heading speed is v i ,

[0081] Step a.2: Starting from 0 m / s, the lateral speed is gradually increased in steps of 0.1 m / s;

[0082] Step a.3: Determine whether a chiseling pit with a depth greater than 1.5 mm appears in the material point model at this time. If a chiseling pit with a depth greater than 1.5 mm appears, record the lateral velocity at this time and set Jump to step a.1 and traverse the next heading speed; if there is no chisel pit with a depth greater than 1.5 mm, and the maximum value of the lateral speed does not exceed 3 m / s; then jump to step a.2 and continue to traverse the next lateral speed; if there is no chisel pit with a depth greater than 1.5 mm, and the value of the lateral speed is greater than 3 m / s, then jump to step a.1 and traverse the next heading speed;

[0083] Traverse all heading speeds v i The steps to calculate whether a chiseling pit with a depth greater than 1.5 mm appears at different vertical speeds are:

[0084] Step b.1: For heading velocity v i ,

[0085] Step b.2: Starting from 0 m / s, the vertical speed is gradually increased with a step length of 0.1 m / s;

[0086] Step b.3: Determine whether a chiseling pit with a depth greater than 1.5 mm appears in the material point model at this time. If a chiseling pit with a depth greater than 1.5 mm appears, record the vertical velocity at this time and set Jump to step b.1 and traverse the next heading speed; if there is no chiseling pit with a depth greater than 1.5 mm, and the maximum vertical speed does not exceed 3 m / s; then jump to step b.2 and continue to traverse the next vertical speed; if there is no chiseling pit with a depth greater than 1.5 mm, and the value of the vertical speed is greater than 3 m / s, then jump to step b.1 and traverse the next heading speed;

[0087] Step 4): Construct the minimum lateral speed and minimum vertical speed The envelope of

[0088] Build minimum lateral speed The envelope method is:

[0089] At n speed points v i The corresponding impact velocity limit As the basic data, i = 1, 2, ..., n, the impact velocity limit value is calculated by cubic spline interpolation method. Solve and get the high-speed running section of the rocket sled [v cv ,v max ] is the limiting impact velocity cubic spline function f(x);

[0090] The cubic spline function f(x) of the limiting impact velocity satisfies the following in each heading speed interval [v i ,v i+1 ] is a cubic polynomial with second-order continuous derivatives at each interior node;

[0091] According to the definition of cubic spline function, the cubic spline function of the limit impact velocity f(x) is in each heading speed interval [v i ,v i+1 The expression on ] is f(x)=a i0 +a i1 x+a i2 x 2 +a i3 x 3 ;

[0092] Solve f(x)=a by interpolation conditions and connection conditionsi0 +a i1 x+a i2 x 2 +a i3 x 3 The four unknown parameters a in i0 、a i1 、a i2 and a i3 ;

[0093] The interpolation conditions are:

[0094]

[0095] The connection conditions are:

[0096]

[0097] Solve f(x) = a i0 +a i1 x+a i2 x 2 +a i3 x 3 The four unknown parameters a in i0 、a i1 、a i2 and a i3 ;

[0098] By a i0 、a i1 、a i2 and a i3 The obtained f(x) is the envelope curve of the required minimum lateral velocity;

[0099] In formula (1), Replace with That is, the envelope curve of the minimum vertical velocity is obtained.

[0100] The present invention is further described below in conjunction with the accompanying drawings and embodiments.

[0101] Taking a hypersonic monorail rocket sled test as the object, the main steps of implementing the method of constructing the dynamic envelope of the rocket sled's limit impact velocity to prevent chiseling are as follows:

[0102] Step 1: Determine basic information based on the overall plan. The material used for the rocket sled's slide shoe is 30CrMnSiNi2A steel, the material of the slide rail is U71Mn steel, the five surfaces of the slide shoe match the five surfaces of the slide rail, and the maximum heading speed of the sled body is 7Ma.

[0103] Step 2: Calculate the threshold speed v using the Stefain-Parker formula cvAbout 5Ma. Six points are taken as calculation points in the velocity range of 5Ma to 7Ma, namely: 5Ma, 5.5Ma, 6Ma, 6.5Ma, 6.75Ma, and 7Ma.

[0104] Step 3: Establish a simplified coupled 3D model of the rocket sled system, such as Figure 2 The impact speed limit values ​​corresponding to the six speed points are calculated. When the heading speed is 5 Ma, the impact speed limit value is 2.4 m / s; when the heading speed is 5.5 Ma, the impact speed limit value is 2.4 m / s; when the heading speed is 6 Ma, the impact speed limit value is 2.1 m / s; when the heading speed is 6.5 Ma, the impact speed limit value is 1.8 m / s; when the heading speed is 6.75 Ma, the impact speed limit value is 1.1 m / s; when the heading speed is 7 Ma, the impact speed limit value is 0.5 m / s.

[0105] Step 4: Solve the impact velocity limit value by cubic spline interpolation method and construct the dynamic envelope of the limit impact velocity, such as Figure 3 The cubic spline function of the limiting impact velocity f(v) is: f(v) = 62.5-33.5v+6.2v 2 -0.4v 3 .

Claims

1. A method for constructing a dynamic envelope of a rocket sled impact velocity to prevent chiseling, characterized in that: The following steps are involved: Step 1): Obtain basic information of the rocket sled test system; Step 2): Determine the calculation point; Step 3): Calculate the minimum lateral velocity of the impact velocity limit value corresponding to each calculation point and minimum vertical speed Step 4): Construct the minimum lateral speed and minimum vertical speed The envelope of .

2. A method for constructing a dynamic envelope of a rocket sled impact velocity for preventing chiseling according to claim 1, characterized in that: The basic information includes the material used for the rocket sled's slide, the material used for the slide rail, the matching method of the slide and the slide rail, and the maximum heading speed v of the sled body. max , the hardness of the rocket sled slide shoe material and the hardness of the slide rail material.

3. The method for constructing a dynamic envelope of a rocket sled impact velocity for preventing chiseling according to claim 1, characterized in that: The steps of determining the calculation point are: Threshold speed v cv Defined as the minimum heading speed that causes chiseling to occur, the threshold speed v cv Calculated by Stefain-Parker formula; Compare the hardness of the rocket sled shoe material and the hardness of the rail material, select the material with the greatest hardness, and substitute the hardness value of the material with the greatest hardness into the relationship between pressure amplitude and material Vickers hardness: p=0.073HV max +9.343, Among them, HV max =max(hardness of the sliding shoe material, hardness of the sliding rail material), p is the pressure amplitude of the high-pressure core when the chiseling occurs; max() is the maximum value; The Gruneisen equation of state for the shoe rail material is: U=c+s*D Among them, U is the shock wave velocity, parameter c is the sound velocity of the material, s is the material constant, and D is the particle velocity; The threshold velocity v is solved by the relationship between velocity and collision interface pressure. cv : Among them, C A is the sound velocity of the sliding shoe material, C B is the speed of sound of the track material, ρ A is the density of the sliding shoe material, ρ B is the density of the track material, s A is the material constant of the sliding shoe material, S B is the material constant of the track material; The maximum heading speed v of the rocket sled max Obtained in step 1); The threshold speed v calculated by the theoretical formula cv The maximum heading speed v of the rocket sled max Select n heading speeds v in the interval i As the calculation point, where i = 1, 2, ..., n.

4. The method for constructing a dynamic envelope of a rocket sled impact velocity for preventing chiseling according to claim 1, characterized in that: Minimum lateral velocity to obtain the impact velocity limit and minimum vertical speed The specific steps are: ① Establish a simplified coupled three-dimensional model of the rocket sled system; ②Calculate the velocity point v i The corresponding minimum lateral speed and minimum vertical speed 5. A method for constructing a dynamic envelope of a rocket sled impact velocity for preventing chiseling according to claim 4, characterized in that: The steps of establishing a simplified coupled three-dimensional model of the rocket sled system are: According to the matching method of the sliding shoe and the sliding rail in step 1), the shoe-rail coupling structure is simplified and modeled by 3D design software, and is placed according to the actual position relationship to obtain a simplified rocket sled coupling 3D model; The rocket sled coupled three-dimensional model is imported into the finite element software, and meshing is performed to obtain the node information and unit information of the finite element model; the model is discretized according to the node information and unit information of the finite element model to obtain the material point model.

6. A method for constructing a dynamic envelope of a rocket sled impact velocity for preventing chiseling according to claim 4, characterized in that: The calculated speed point v i The corresponding minimum lateral speed and minimum vertical speed The specific steps are: Traverse all heading speeds v i The steps to calculate whether a chiseling pit with a depth greater than 1.5 mm appears at different lateral speeds are: Step a.1: Heading speed is v i , Step a.2: Starting from 0 m / s, the lateral speed is gradually increased in steps of 0.1 m / s; Step a.3: Determine whether a chiseling pit with a depth greater than 1.5 mm appears in the material point model at this time. If a chiseling pit with a depth greater than 1.5 mm appears, record the lateral velocity at this time and set Jump to step a.1 and traverse the next heading speed; if there is no chisel pit with a depth greater than 1.5 mm, and the maximum value of the lateral speed does not exceed 3 m / s; then jump to step a.2 and continue to traverse the next lateral speed; if there is no chisel pit with a depth greater than 1.5 mm, and the value of the lateral speed is greater than 3 m / s, then jump to step a.1 and traverse the next heading speed; Traverse all heading speeds v i The steps to calculate whether a chiseling pit with a depth greater than 1.5 mm appears at different vertical speeds are: Step b.1: For a heading velocity v i , Step b.2: Starting from 0 m / s, the vertical speed is gradually increased with a step length of 0.1 m / s; Step b.3: Determine whether a chiseling pit with a depth greater than 1.5 mm appears in the material point model at this time. If a chiseling pit with a depth greater than 1.5 mm appears, record the vertical velocity at this time and set Jump to step b.1 and traverse the next heading speed; if there is no chiseling pit with a depth greater than 1.5mm and the maximum vertical speed does not exceed 3m / s; then jump to step b.2 and continue to traverse the next vertical speed; if there is no chiseling pit with a depth greater than 1.5mm and the value of the vertical speed is greater than 3m / s, then jump to step b.1 and traverse the next heading speed.

7. The method for constructing a dynamic envelope of a rocket sled impact velocity for preventing chiseling according to claim 1, characterized in that: Build minimum lateral speed The envelope method is: At n speed points v i The corresponding impact velocity limit As the basic data, i = 1, 2, ..., n, the impact velocity limit value is calculated by cubic spline interpolation method. Solve and get the high-speed running section of the rocket sled [v cv ,v max ] is the limiting impact velocity cubic spline function f(x); The cubic spline function f(x) of the limiting impact velocity satisfies the following in each heading speed interval [v i ,v i+1 ] is a cubic polynomial with second-order continuous derivatives at each interior node; According to the definition of cubic spline function, the cubic spline function of the limit impact velocity f(x) is in each heading speed interval [v i ,v i+1 The expression on ] is f(x)=a i0 +a i1 x+a i2 x 2 +a i3 x 3 , where a i0 、a i1 、a i2 and a i3 is an unknown parameter; Solve f(x)=a by interpolation conditions and connection conditions i0 +a i1 x+a i2 x 2 +a i3 x 3 The four unknown parameters a in i0 、a i1 、a i2 and a i3 ; The interpolation conditions are: The connection conditions are: Solve f(x) = a i0 +a i1 x+a i2 x 2 +a i3 x 3 The four unknown parameters a in i0 、a i1 、a i2 and a i3 ; By a i0 、a i1 、a i2 and a i3 The obtained f(x) is the envelope curve of the required minimum lateral velocity; In formula (1), Replace with That is, the envelope curve of the minimum vertical velocity is obtained.

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