PRODUCTION VERIFICATION METHOD USING FIXED-CORD CURVES

Abstract

The invention is a manufacturing verification method developed to produce technical outputs that can be directly used in industrial design and engineering applications, and for the computer-aided generation and / or analysis of geometric structures defined in polar coordinates.

Description

1 TARIFF PRODUCTION VERIFICATION USING FIXED-CORD CURVES METHOD TECHNICAL FIELD The invention is for use in computer-aided design (CAD) and manufacturing (CAM) environments. In this way, the physical and / or mathematically defined curve parameters 10 based on the structural correlation with mechanical function and performance criteria This relates to the field of profile production and / or analysis methods. The invention specifically relates to the design magnitudes of the mathematical parameters in question. technical profiles that can be produced through transformation and that enable this transformation It deals with a structural production method that makes it possible to obtain it. 15 PREVIOUS TECHNIQUE In the current state of the art, geometric design processes are mostly pre-planned. defined closed mathematical expressions or limited parametric models 20 This approach is carried out through a specific geometric form. The technical benefit is obtained incidentally, but under the same functional conditions systematic generation or verification of alternative geometries that provide It is not possible. Mathematically defined curves, except for some special and limited cases, structural features that are valuable from an industrial perspective — especially fixed cords mechanically significant geometric characteristics — in a clear and systematic form It does not reveal. Furthermore, these mathematical expressions are related to mechanical function and performance criteria. Techniques that can be produced, verified, and repeated by systematically relating them. A method for converting profiles is not described in the literature. 2 In conclusion, all the problems mentioned above are functional in the relevant technical field. a new structural production based on conditions and applicable in a computer environment This has made it necessary to develop a new method. BRIEF DESCRIPTION OF THE INVENTION 5 The present invention aims to eliminate the aforementioned disadvantages and the related technical to bring new advantages to the field, specific functional and physical conditions algorithmic generation and / or analysis of profiles providing It is related to a method. 10 The main purpose of the invention is to industrially apply the profiles obtained by the aforementioned method. technical outputs that can be directly used in design and engineering applications The goal is to enable it to produce. Another objective of the invention is to address predefined geometric conditions, such as those associated with a fixed cord. and the curves satisfying the physical conditions, and the profiles derived from these curves, in a single system. The goal is to enable systematic production without being bound by mathematical expression. Another objective of the invention is to enable any given profile to perform the specified functional 20 It allows for the algorithmic determination of whether the conditions are met or not. It is to recognize. All the purposes mentioned above and those that will emerge from the detailed explanation below. to realize the existing invention, industrial design and engineering 25 developed to produce technical outputs that can be used directly in applications for the computer-aided generation and / or analysis of profiles It is a production verification method. Accordingly, the characteristic of the invention in question is that the method these sub-steps a) a constant radial parameter defined relative to the reference center of the profile Determining R, 30 b) defining the parameter C, which determines the fixed cord length, 3 c) creating the external profile according to the general functional template r(θ)=R+f(θ), d) the function f(θ) f(0)=0, f(π)=C>0 and It should be defined such that the conditions f(θ)+f(θ+π)=C are satisfied, 5 e) The formula K = 2R + C proves that the generated profile satisfies the constant cord condition. numerical verification through its use, f) area and / or volume calculation for profiles that provide verification to be done, g) evaluation of profiles according to calculated technical metrics and 10 the selection It is characterized by its inclusion. Another preferred construction of the invention is that the function f(θ);  trigonometric functions,  Fourier series or harmonic expansions, 15  linear or nonlinear combinations of these It is characterized by being selected from among many others. Another preferred configuration of the invention is the fixed cord condition, external profile. This will apply to all pairs of points located opposite each other along the line. It is characterized by its provision. BRIEF DESCRIPTION OF THE FIGURE Figure 1 shows a general view of the curves obtained using the method described in the invention. 25 It has been given. Figure 2 shows an alternative view of the curves obtained using the method described in the invention. It has been given. 4 Figure 3 shows a general 3D representation of the curves obtained using the method described in the invention. The appearance is given. Figure 4a shows the standard profile of a rotary valve pump. Figure 4b shows the improved profile of the rotary valve pump system, which is the subject of this invention. r(θ) = a – b cos (θ) is given. Figure 4c shows the comparison of the two casings in the rotary valve pump example. A representative view is given. 10 DETAILED DESCRIPTION OF THE INVENTION This detailed explanation describes the invention's algorithm-based generation and verification. The method is solely aimed at a better understanding of the subject, with no limiting effects. 15 This is explained with examples that will not create a conflict. The invention requires at least one processor and at least one It relates to a computer-based method with memory. The invention involves at least one processor, and at most... a computer-based system consisting of a small amount of memory and software instructions stored in that memory It is a system. The terms used in the invention are explained as follows. Profile → the physical structure obtained by the method described in the invention. r(θ) → defines the entire external profile, external profile function. 25 R → constant radial parameter defined relative to the reference center of the profile. f(θ) → a functional that determines the volumetric behavior and angular distribution of the profile. It is a component. K→ is the constant cord length between points at opposite angles on the external profile. 30 This magnitude defines the fundamental symmetry requirement of the profile and the upper limit of the static internal volume. The parameter K forms the basis of the invention's volumetric optimization capability. The cord is a linear connection that links pairs of points in opposite angular positions on the profile. It is defined as a segment. Reference center → fixed coordinate where the profile is defined in the polar coordinate system. This is the starting point and the point from which all radial measurements are based. 5 C→ is the functional design parameter that defines the fixed cord. In other words, C is a constant numerical value under the fixed cord condition. The inverse of the function f(θ) is... It represents the sum of the angles between two opposite points on a curve. Geometrically, it represents two opposite points on a curve. It is the magnitude that ensures the stability of the cord between them. Therefore, C is the design constraint, 10 For example, it determines outcomes such as symmetry, balance, and volume limits. ∀θ → constant cord condition for all values ​​of θ ([-π,+π]), i.e. the curve It indicates that it is valid everywhere. The L→ parameter enables the conversion of two-dimensional area calculations into three-dimensional volume. It is the geometric dimension that provides. All of these parameters are inputs to the algorithmic generation and validation method. They form and each carries a physical design meaning. 20 The invention is not in a specific formula itself, but in the curve defined in polar coordinates. Producing profiles for technical purposes according to functional conditions using family types, This relates to the method of verification and evaluation. By using these curves, for example, the center of gravity of the building and the wall thickness can be determined. or the largest or most efficient volume of a reservoir located on a device Structural features such as its geometry can be determined. The invention describes a method for designing architectural load-bearing surfaces with a surface area and / or volume of 30%. It allows for balance control. 6 The invention also covers the application of pressure vessels, flow channels or special volumes. It also allows for the determination of the structural characteristics of the structures that require it. It provides. The invention describes a three-dimensional model obtained using a specific method, involving volume, contact continuity, and 5. It enables the calculation of surface area or center of mass. The invention describes a method of extruding a two-dimensional profile into a third dimension, or It is characterized by its transformation into a three-dimensional solid model with rotational symmetry. In this text, functions like f(θ) are only samplers, and different functions If the same conditions are provided as with their families, the method is applied in the same way. f(θ) A single mathematical expression is not required for the function. Trigonometric functions (sin(θ), cos(θ)); Fourier series and harmonic expansions; this Function 15, which refers to linear or nonlinear combinations of functions. Families can be used as examples. The common characteristics of these functions are as follows:  They are closed and continuous,  They can be defined in polar coordinates,  They ensure a fixed cord condition at opposite angles, 20 The manufacturing and verification method used in this invention is based on certain fundamental principles. They are defined through parameters. These parameters are directly physical and It has a mechanical meaning. 1. Basic Geometric Definitions and Parameters The invention involves a curve with a fixed cord property, in polar coordinates. This can be expressed with the following general formula: r(θ) = R + f(θ), θ = [-π,+π] R > 0 7 The distinguishing feature of these curves is the points at opposite angular positions on the curve. The condition is that the cord length between them is constant. This condition applies to the function f(θ). This is provided under the following conditions:  f(0) = 0 5  f(π) = C > 0  f(θ) + f(θ+π) = C, ∀θ If these conditions are met, the curve will have a fixed cord characteristic, and the cord Length: 10  K = 2R + C It is obtained as follows. Here, it is possible for the function f(θ) to be a constant function. It is not and should not be. Because one of the conditions is f(0)=0, and the other is f(π)=C>0. If 15 If f(θ) were constant, all its values ​​would be the same; but these two different values ​​(0 and C) make it different. This makes it impossible. Thus, the method uses not an ordinary circle, but a new one with a fixed cord. It generates original curve families. If the condition f(θ) ≥ 0 is satisfied for all θ values, then the profile reference point is 20. It includes an interior circle whose center is tangent to radius R at least at one point. 2. Profile Generation – Algorithmic Generation Flow Multiple candidate curves are generated at this stage. Example steps of the production flow are as follows: 25 Parameter selection: R > 0 and C > 0 values ​​are selected. Function definition: The function f(θ) is defined with the conditions f(0)=0, f(π)=C and f(θ)+f(θ+π)=C It is defined in such a way as to be provided. 30 Profile generation: The curve is formed as r(θ) = R + f(θ). 8 Generating candidate profiles: different function families / parameters for f(θ) A set of candidate curves is created by testing them. This selection is limited to a single mathematical expression. It is not; trigonometric / harmonic / Fourier type expansions or any of these Combinations can be used. Example definitions of f(θ) (not exhaustive):  f(θ) = (C / 2).(1 − cosθ)  f(θ) = b·sin(θ) + (C / 2)(1 − cosθ) This example illustrates function definitions that satisfy the condition; methods use these. It is not limited. Verification and elimination: Generated candidates are numerically evaluated according to the fixed cord condition. Candidates are tested, and those who do not meet the requirements are eliminated. Candidates who meet the requirements are selected from among 15. They are evaluated, ranked, and deemed appropriate according to technical metrics (e.g., volume / area / balance, etc.). Those who are eligible are selected. Depending on the requirements of the technical application, candidate profiles may have additional geometric features. It can also be filtered according to the conditions. In particular, the reference point is 20 within the profile. In cases where it is desired that the expression r(θ)=R+f(θ) remains positive for all θ values, parameter selection or such that the condition f(θ) > (-R) remains unchanged (e.g., the condition f(θ) > (-R)) The elimination process takes place. In some applications, it may be preferable for the profile to be a closed curve that does not intersect itself; 25 In this case, appropriate geometric control and filtering steps are applied. 3. Curve Analysis (Detection) of Potential Profiles – Given a Determining the Fixed Cord Property of the Curve The other key function of the invention is to have a fixed cord property on an existing curve. The goal is to algorithmically verify that this is not the case and to extract the relevant parameters. Example Verification procedure: 9 a. The value of R = r(0) is determined for the given curve. b. f(θ) = r(θ) − R is defined. c. f(0)=0 is checked and it is confirmed that the expression f(θ)+f(θ+π)=C gives a constant C. d. If the conditions are met, the curve has a constant cord and K = 2R + C is calculated. With this procedure, the method not only “generates a new curve”; it also removes curves that were previously different. used for defined purposes or as part of an existing design It also allows for the analysis of geometric structures. Within the scope of the method, Whether a given curve satisfies the constant cord condition is determined algorithmically. It is being verified; 10 corresponding geometric structures satisfying the condition in question. Functional parameters are determined. Using these parameters, the design is carried out. geometric quantities such as area and volume, as well as the characteristics of equilibrium and mass distribution. can be evaluated in architectural, industrial or mechanical designs. obtaining the most suitable geometric structures for the targeted technical requirements opportunities are provided. In this respect, the method is not only a means of production, but also 15 among the existing designs at the time, one based on objective and measurable technical criteria. It offers an evaluation and selection mechanism. 4. Equivalence Under Transformations In a preferred configuration of the invention, a curve with a fixed cord feature; The same applies under distance-preserving geometric transformations (isometric transformations). These transformations are evaluated within the family. Examples of these transformations include translation, rotation, and It is a reflection. Thus, the method does not depend on only a single position of the curve, It also includes variants obtained through transformations. 25 Specifically, curves can be written in parametric form.  x(θ)=r(θ)cosθ,  y(θ)=r(θ)sinθ 30 If a curve is in a different position than its original position, a simple translation can return it to the origin. It can be tested by taking it. x'(θ) = x(θ) – x0 , y'(θ) = y(θ) – y0 This process does not break the fixed cord feature. Therefore, algorithmic validation only It works for the cases defined at the origin; however, thanks to parametric translation, any An existing curve can first be plotted back to the origin and then tested. 5 5. Technical Impact Assessment: Area / Volume / Balance Metrics The method produces measurable technical outputs. The field for the generated r(θ) profile is numerical. It can be calculated and converted to the selected design metric. This allows for 10 The method ceases to be an “abstract mathematical definition”; it becomes testable in CAD / CAE. It transforms into a technical tool that can generate design decisions through numerical verification. These curves also exhibit a particular characteristic of mass distribution: mass density, 15 within the framework of an appropriate functional relationship with the distance from the reference center This can be defined. In this case, the center of reference and the center of mass coincide. This feature allows curves to be viewed not only geometrically, but also... also in physical and engineering applications (e.g., equilibrium, resonance, and flow) (control) enables its use. Constant cord curves have a wide range of applications in various disciplines. In mechanics... It provides volume increase and flow balance in pump and compressor designs; in architecture. and in civil engineering, it forms load-bearing surfaces and stable structures; optical and It offers focus control and resonance properties in acoustics; and in art and design, it offers new... It can be used to produce symmetrical and aesthetic forms. 25 7. Software Integration and Outputs The algorithmic generation and verification method described in this invention uses computer-aided technology. It is suitable for implementation as a software module. The method can be integrated into a software environment. integration, repeatability in both design and production processes and 30 It provides automation. As part of the software integration, the system operates as follows: 11 1. System with fixed radial parameter (R), fixed cord length (K), system The length (L) and the preferred function family are defined. 2. Candidate r(θ) = R + f(θ) profiles are automatically generated according to the selected function family. It is produced. 3. The constant cord condition is numerically tested for each candidate profile. 5 4. Depending on technical application requirements, candidate profiles may have additional geometric features. can be filtered according to conditions (e.g., the expression r(θ) must be positive for all θ values). (either remaining or the profile being a closed curve that does not intersect itself). For profiles that meet condition 5, the static internal volume is calculated and the volumetric efficiency is determined. The metric is created. 10 6. Profiles are sorted and presented to the user based on volumetric efficiency. 7. The selected profile can be exported in CAD / CAM compatible data formats. Thanks to this structure, the method is not only a theoretical design tool; it is also an industrial tool. It becomes a technical system that can be directly integrated into production. CONCRETE APPLICATION – ROTARY VANE PUMP (EXAMPLE APPLICATION) In this section, the algorithmic generation and validation method described is of the rotary vane type. its application to a rotary displacement pump, through quantitative calculations is being concretized. 20 Reference System Definition For comparison purposes, two different external rotors with the same rotor radius and cord length were used. Profile geometry has been considered: In this example application, the function f(θ) satisfies the condition f(θ) ≥ 0 for all values ​​of θ. selected in such a way as to provide, in this case, R centered at the profile reference point. It includes an inscribed circle with radius. • Reference profile: Classic circular stator profile 30 • Recommended profile: An algorithmically generated outer casing that ensures a fixed cord condition. profile In both cases: 12 • rotor radius, • eccentricity conditions, • fin structure, • pump length (L) They were kept the same. Thus, the difference obtained is only from the external profile geometry. It originates from. Parameters Used In the example application, the following parameters were selected: 10 • Rotor radius: R = 1 • Target diameter for the outer stator: Ds = 2.8 • Profile width b = 0.4 • System dimensions: L = 2 15 These values ​​are consistent with the ratios used in classic rotary vane systems. and clearly demonstrates the volumetric effect of the method. Area and Static Volume Calculations 20 For a circular stator profile: The annular area between the outer field and the rotor field: In contrast, static internal volume: For the recommended profile with a fixed cord: r(θ) = a – b cos(θ) and a = 1.4 13 Area of ​​the algorithmically generated external profile: Static internal volume when rotor area is removed: 5 Quantitative Comparison • Circular profile: 6.0319 10 • Fixed cord profile: 6.5359 • Relative increase: approximately 8.3% An example of quantitatively demonstrating the technical impact of the method is fixed cord. When comparing a profile with a classic circular profile, only the profile geometry differs by 15 An increase in static interior volume can be achieved by algorithmic optimization. This is shown in the example calculation. For a circular profile, the static volume is approximately 6.0319. For the fixed cord profile, approximately 6.5359 was obtained, representing a relative increase of approximately 8.3%. It is given as follows. This difference is achieved solely through the external profile geometry, without altering the mechanical structure. This was stated in the technical assessment. This result was achieved without changing the rotor-stator relationship, mechanical structure, and eccentricity; 25 obtained solely by algorithmic optimization of the external profile geometry This increase in static internal volume is directly reflected in volumetric efficiency, and It provides a significant improvement in system performance. This example demonstrates that the described method is industrially applicable and measurable. This clearly demonstrates that it has created a technical impact. 30 Figure 4a shows the circular stator profile used in classic rotary vane systems. This is shown. In this structure 14 stator diameter Ø2.8 (R = 1.4) and rotor diameter Ø2.0 (R = 1.0) They are positioned in such a way as to ensure the rotor-stator operating condition. 5 The stator center is eccentrically shifted to the point (-0.4, 0). Continuity of flow... to provide input and output ports within an angular range of approximately ±(35°–40°). It is positioned in this geometry, which serves as a reference for classic rotary vane pump systems. It is used as a building structure. Figure 4b shows an example of a stator profile that satisfies the fixed cord condition. The configuration is shown. In this profile, the stator geometry is shown in polar coordinates. r(θ) = 1.4 − 0.4 cosθ It is defined by the expression. The fundamental characteristic of the defined profile is the reciprocal relationship between θ and θ+π. the linear distance between points located at angular positions is constant This is the case. This constant cord length is obtained as K = 2.8. Profile, Ø2.0 It is shown with a rotor of a certain diameter, and the input and output ports are again ±(35°–40°) angular. It is located within the range. The straight line shown in the figure is 20, which remains constant along the profile. It represents a fixed-length cord. Figure 4c shows the circular stator profile given in Figure 4a and the fixed profile given in Figure 4b. The stator profile that satisfies the cord condition is on the same axis system. A comparative (overlay) representation is given. The comparison includes 25 profile segments. under correct geometric intersection conditions within the stator boundaries has been evaluated. As a result of quantitative calculations, the circular configuration was found to be suitable. The static internal volume is calculated to be approximately 6.0319, while for the fixed cord profile... This value was obtained as approximately 6.5359. The corresponding relative volume... The increase is approximately 8.3%, and this improvement is solely due to the 30% external profile geometry. It appears to be due to the modification. Curves that satisfy the constant cord condition control the surface area and volume relationships. It can be used in architectural and industrial design applications where necessary. In structures where mass distribution is geometrically oriented, symmetry and Balance characteristics can be made into design parameters. The method described is not limited to two-dimensional profiles. The same principle applies to three-dimensional profiles. This can be generalized to geometries. For example, under fixed cord-like conditions, three 5 Three-dimensional surfaces can be produced. This approach is suitable for rotating bodies, pressure vessels, or custom applications. It is applicable for structures that require a certain amount of space. A suitable functional relationship between the mass distribution along the profile and the function r(θ) If defined within this framework, the profile's reference center and center of mass must be 10 This overlap can be achieved. This provides additional benefits in terms of mechanical balance and vibration reduction. It offers advantages and increases the potential for interdisciplinary use of the method. The scope of protection of the invention is set out in the attached claims and is strictly adhered to. The detailed explanation cannot be limited to the examples given. Because in technique, 15 A specialist, without deviating from the main theme of the invention, will do what is described above. It is clear that similar structures can emerge in light of this.

Claims

16 REQUESTS 1. Can be used directly in industrial design and engineering applications. 5 developed to produce technical outputs and defined in polar coordinates computer-aided generation and / or analysis of geometric structures It is a production verification method aimed at ensuring that the characteristic feature is... the following sub-steps of the method a) a constant radial parameter defined relative to the reference center of the profile Determining the value of R, 10 b) defining the parameter C, which determines the fixed cord length, c) creating the external profile according to the general functional template r(θ)=R+f(θ), d) the function f(θ) f(0)=0, f(π)=C>0 and 15 It should be defined such that the conditions f(θ)+f(θ+π)=C are satisfied, e) numerically demonstrating that the generated profile satisfies the fixed cord condition. verification, f) area and / or volume calculation for profiles that provide verification to be done, 20 g) evaluation of profiles according to calculated technical metrics and the selection It is characterized by its inclusion.

2. It is a production verification method according to claim 1, and its characteristic is that the function f(θ) is;  Trigonometric functions, 25  Fourier series or harmonic expansions,  linear or nonlinear combinations of these It is characterized by being selected from among many others. 17 3. A production verification method according to Claim 1, characterized by its fixed cord. the condition for all pairs of points located opposite each other along the outer profile It is characterized by being provided in a way that will be valid.

4. The method according to Claim 1 is characterized by the polarity of the closed curve profile in question. through Cartesian, parametric, or implicit mathematical representations If defined, ensuring the geometric invariant of the fixed cord. It is characterized by the preservation of structural conditions aimed at achieving this.

5. Providing a fixed cord invariant, manufactured or verified according to Claim 1. It is a closed curve profile; its characteristic feature is that the profile in question is a technical system. volumetric capacity, contact continuity or structural balance within a structural characterized by its applicability in a way that will create It is an element. 15