Mathematical modeling method of FPC contact deformation based on segment analysis and solution of nonlinear equations

Abstract

The invention relates to a mathematical modeling method for FPC contact deformation based on segment analysis and solution of nonlinear equation groups, which is used for analyzing the deformation of a flexible printed circuit (FPC) when it contacts with other components. The core steps include: dividing the FPC into different segments based on contact points; linearizing the relationship between the curvature of the FPC and the length of a path along the surface of the FPC in different FPC segments, and obtaining a series of unknowns, such as the slope and intercept in the linear relationship; in addition, the specific coordinates of the contact points and the length of each FPC in different segments are also used as unknown parameters to be solved; the known relationships include the coordinate difference and angle difference at the two ends of different segments, and the curvature at the connecting points of the head and the tail of different segments must be equal; and at the same time, using the Jacobian matrix iteration method and the size iteration method to solve all unknowns, and finally obtaining the deformation curves of the FPC under different contact conditions with other components that need to be solved.

Description

Technical field:

[0001] The present invention belongs to the field of electronic engineering and focuses on developing a mathematical model for predicting the contact bending morphology of a flexible printed circuit (FPC). Specifically, the present invention proposes a mathematical modeling method for the contact bending morphology of FPC based on segmented analysis and solving a nonlinear equation system, aiming to improve the accuracy and efficiency of FPC design. Background technology:

[0002] With the rapid development of electronic engineering, especially in the application of flexible printed circuits (FPC), there is a growing demand for efficient and accurate bending shape prediction technology. FPC plays a vital role in the design of modern electronic products with its unique flexibility and thinness. It is widely used in various high-end electronic devices, including but not limited to smart wearable devices, smart phones, medical devices and aerospace equipment, where the performance requirements of FPC are particularly stringent. Accurately predicting the bending behavior of FPC in actual use is crucial to ensure the reliability and durability of these products.

[0003] In the prior art, the finite element method (FEM) is a common means of analyzing the bending performance of FPC. However, this method faces several significant limitations. First, the operation process of FEM is complex and time-consuming, and needs to be performed by engineers with high expertise, which to some extent limits its application in rapid product development cycles. Second, FEM has low computational efficiency when dealing with large-scale or complex simulations, which not only increases design costs but also prolongs the time from concept to market for products. In addition, when dealing with particularly complex FPC deformation situations, another challenge of the finite element method (FEM) is that its analysis process may be difficult to converge. Complex FPC deformations often involve variable geometric shapes and complex stress distributions, which may lead to numerical instability in the calculation process, making it difficult for FEM analysis to reach an accurate solution within a reasonable time, or even errors. This computational uncertainty not only increases the complexity of the analysis, but also may lead to unreliable analysis results, thereby affecting the design decision-making process. Therefore, when dealing with complex FPC contact deformation problems, it becomes particularly important to invent more stable and effectively convergent prediction methods.

[0004] In view of these challenges, new mathematical models are urgently needed for predicting the contact bending morphology of FPCs. Summary of the invention:

[0005] The present invention relates to a mathematical modeling method for FPC contact deformation by segmentation analysis and solution of nonlinear equation groups. The method divides the FPC into segments by taking the contact points of the FPC and other components as dividing points. In each segment, a linear relationship between the bending curvature of the FPC and its length is proposed. A nonlinear equation group is established by combining multiple known and unknown conditions, and is solved based on a Jacobian matrix iteration method. Finally, the deformation curve of the FPC after contact is accurately predicted, thereby providing a powerful and efficient tool for the design and manufacture of the FPC. The method solves the technical problems that the finite element method used in the prior art has a complex and time-consuming operation process, needs to be performed by engineers with high professional knowledge, has low calculation efficiency when processing large-scale or complex simulations, and the analysis process may be difficult to converge.

[0006] To achieve the above object, the present invention adopts the following technical solutions:

[0007] The FPC contact deformation mathematical modeling method based on segment analysis and solution of nonlinear equations is characterized by comprising the following steps:

[0008] S1. Based on the finite element tool, the corresponding FPC contact bending morphology is modeled and simulated, and the relationship between its curvature change and the distance along the length direction of the FPC is obtained;

[0009] S2, dividing the curvature change trend curve into segments based on the contact point position;

[0010] S3, linearizing the curvature of the FPC and the changing trend along the length of the FPC in different sections;

[0011] S4, combining the FPC parameters in different sections into a nonlinear equation system;

[0012] S5, solving the nonlinear equation group to obtain a series of parameters required to determine the contact deformation of the FPC;

[0013] S6. Calculate the deformed shape of the FPC after contact based on the solution of the nonlinear equations.

[0014] The step S2 specifically includes the following steps:

[0015] S2.1. Divide the curvature change trend curve into segments according to the contact points;

[0016] S2.2. Observe whether the curvature change of FPC is monotonically changing in the same section. If not, continue to divide it further to ensure that in each small section, the curvature is monotonically increasing or monotonically decreasing along the length of FPC.

[0017] The step S3 specifically includes the following steps:

[0018] S3.1. Linearizing the relationship between the FPC curvature and the length along the FPC in each small segment will produce unknowns such as the slope and intercept.

[0019] S3.2. Within the large segment defined by the contact point, it is determined that the absolute value of the slope of the relationship between the FPC curvature and the length of the FPC in all small segments is equal.

[0020] The step S4 specifically includes the following steps:

[0021] S4.1. In a large section with the contact point as the dividing point, the X, Y coordinate difference and tangent angle difference of its two end points and the X, Y coordinate difference and tangent angle difference of the two end points of the entire FPC curve are taken as known conditions;

[0022] S4.2, the slope and intercept of the FPC curvature change curve in each section, the specific coordinates of the contact point, and the length of the FPC in each section with the contact point as the decomposition point are taken as unknowns and combined into a nonlinear equation system;

[0023] The step S5 specifically includes the following steps:

[0024] S5.1. The solution of the corresponding complex nonlinear equations is mainly based on the Jacobi matrix iteration method. The solution process needs to be segmented iterative to ensure that the corresponding iterative process can search for the correct solution nearby.

[0025] S5.2. If the contact deformation is too large, the position of the contacted parts is analyzed in sections, gradually reaching the specified position to complete the analysis of the final FPC contact shape.

[0026] Compared with the prior art, the present invention has the following beneficial effects:

[0027] This invention proposes an innovative method to accurately predict the deformation characteristics of flexible printed circuits (FPCs) under various contact conditions through mathematical models. This technology is of great significance for improving the design and production process of FPCs, and can significantly improve their performance and reliability. It provides engineers and designers in multiple industries with a powerful tool, enabling them to more accurately grasp and control the bending characteristics of FPCs, thereby achieving significant results in improving the overall performance and reliability of electronic products.

[0028] The mathematical model of the present invention is easy to operate, has fast calculation speed and high accuracy, and is suitable for rapid design iteration and prototype development. It reduces the professional skill requirements for operators, allowing a wider range of engineers and designers to directly apply it to the design and analysis of FPC. This not only helps to reduce the time and cost of product development, but also improves the reliability and innovation of product design. Description of the drawings:

[0029] Figure 1 Comparative analysis of FPC before and after deformation; 1(a) is the shape of FPC before bending; 1(b) is the shape of FPC after bending;

[0030] Figure 2 The finite element model of FPC bending deformation and its size diagram; 2(a) is the FPC unbending shape model; 2(b) is the FPC bending shape model; 2(c) is the size diagram of the FPC after bending;

[0031] Figure 3 is the path tracked along the FPC model surface; 3(a) is the deformed FPC and the tracking path created along its surface; 3(b) is a partial enlarged view of the tracking path along the FPC surface;

[0032] Figure 4 The stress changes in the tracking path and their adjustment results; 4(a) is the Mises stress distribution extracted along the tracking path; 4(b) is the bending stress change trend after adjustment according to the stress properties;

[0033] Figure 5 is the variation trend of FPC bending curvature along the path distance;

[0034] Figure 6 It is the FPC deformation analysis based on differential principle;

[0035] Figure 7 This is the flowchart of Jacobi matrix iteration method;

[0036] Figure 8 This is the flow chart of the size iteration method;

[0037] Fig. 9 It is a comparison between the iterative process of mathematical analytical analysis of FPC free bending and the finite element results; 9(a) is a series of FPC curves obtained by the dimensional iteration process; 9(b) is a comparison between the finite element simulation and mathematical modeling analysis results;

[0038] Fig.10 Figure 10 is the Mises stress variation trend of the FPC single-point contact bending surface and its stress adjustment result; 10(a) is the FPC single-point contact deformation finite element simulation model; 10(b) is the Mises stress variation trend extracted based on the tracking path; 10(c) is the bending stress variation trend after adjustment according to the stress properties;

[0039] Fig.11 The bending curvature of the FPC single point contact deformation varies with its length;

[0040] Fig.12Comparison of the results of mathematical analysis and finite element analysis of the FPC single-point contact bending deformation curve;

[0041] Fig.13 are the finite element models of FPC free bending and double-point contact bending; 13(a) is the finite element model of FPC free bending; 13(b) is the finite element model of FPC double-point contact bending;

[0042] Fig.14 The bending stress variation trend and stress adjustment results of FPC double-point contact; 14(a) is the Mises stress variation trend extracted based on the tracking path; 14(b) is the bending stress variation trend after adjustment according to the stress properties;

[0043] Fig.15 It is the changing trend of the FPC double-point contact bending curvature along the tracking path;

[0044] Fig.16 FPC bending curve analysis: mathematical modeling iteration process and comparison of finite element and mathematical modeling analysis results; 16(a) is a series of curves obtained from the FPC double-point contact deformation iterative calculation and analysis process; 16(b) is a comparison of the FPC double-point contact deformation curve finite element simulation analysis and mathematical modeling analysis results. Specific implementation method:

[0045] 1. Exploration of the relationship between FPC curvature and surface bending stress

[0046] Understanding the deformation behavior of flexible printed circuits (FPCs) is crucial, especially the curvature variation trend under different bending conditions. Therefore, analyzing the relationship between the curvature of the FPC and the resulting bending stress is particularly important.

[0047] exist Figure 1 (a) shows the unbent state of the FPC, where NN is the neutral layer. At this time, the FPC has not deformed. Figure 1 (b) shows the FPC after bending, at which time the arc E'F' in the neutral layer (corresponding to the straight line segment EF before bending) maintains its length δx unchanged. In the bent state, the length of E'F' can be calculated as R·θ.

[0048] Figure 1 The line segment GH in (a) is Figure 1 The arc G'H' in (b) corresponds to the arc G'H', which extends from δx to (R+y)·θ after bending. This length change Δl can be expressed as:

[0049]

[0050] Assuming that no plastic deformation occurs during bending, the elastic strain ε in the material layer at a distance 'y' from the neutral layer can be calculated:

[0051]

[0052] The corresponding bending stress σ can be expressed as:

[0053]

[0054] Where E is Young's modulus, an intrinsic property of the material. This expression expresses the relationship between bending stress σ, Young's modulus E, strain ε, radial distance y, and radius of curvature R.

[0055] In the finite element method (FEM) analysis, three key assumptions are used to simplify the analysis of FPC:

[0056] ① Uniform thickness (h): The thickness of FPC is consistent, which is 'h', that is, the material thickness is uniform.

[0057] ② Neutral layer distance (h / 2): The distance from any surface point to the neutral layer is half of the thickness 'h'.

[0058] ③ Constant elastic modulus (E): The elastic modulus of FPC is considered to be constant.

[0059] As shown in equation (4), the product of the elastic modulus ('E') of the material and the distance ('y') from a point on the surface to the neutral layer remains constant as shown in equation (3), which leads to mathematical simplification and establishes a corresponding relationship between bending stress (σ) and curvature (1 / R):

[0060]

[0061] This expression highlights the important relationship between bending stress ('σ') and curvature ('1 / R') in a flexible printed circuit (FPC).

[0062] 2. Finite element simulation and stress analysis

[0063] The finite element model FEM was established based on Abaqus 2020. The model dimensions include length 60mm, width 2mm and thickness 0.1mm. The global element size was set to 0.2, and the model was divided into 10 equal parts along the thickness direction. The C3D8R element type was selected. The analysis step used static and general steps, and nonlinear geometry (Nlgeom) was enabled. The final model configuration contained 30,000 elements.

[0064] Figure 2(a) shows the initial unbent state of the FPC, with a displacement of (25mm, 0) and a rotation angle of 90° applied to end A. At the same time, a displacement of (25mm, -5mm) and a rotation angle of -90° were applied to end B.

[0065] The resulting bent FPC state is similar to a 'U' shape, such as Figure 2 (b) The coordinate difference (Δx and Δy) between point A' and point B' at both ends of the FPC is as follows: Figure 2 (c) as shown.

[0066] In order to observe the trend of bending stress on the FPC surface, a path is drawn along the curved surface of the FPC in the model. The path consists of a series of points, such as Figure 3 (a) shown. Figure 3 (b) Yes Figure 3 A magnified image of the area within the blue circle in (a). It can be seen that the path is composed of a series of discrete points.

[0067] Figure 4 (a) is the Von Mises stress variation curve along the corresponding path extracted based on the corresponding finite element model. The calculation formula of Von Mises stress is:

[0068]

[0069] Among them, σ1, σ2 and σ3 represent the principal stresses in three-dimensional stress analysis, and τ1, τ2, τ3 are the corresponding shear stresses.

[0070] The Von Mises stress value is non-negative, and the present invention adjusts it accordingly. Specifically, the compressive stress is regarded as a negative value, and the tensile stress is regarded as a positive value. Figure 4 As shown in (b), the corresponding modification can more reasonably describe the changing trends of stress and curvature.

[0071] 3. Mathematical modeling and analysis of FPC free bending

[0072] 3.1 Mathematical modeling

[0073] right Figure 4 The analysis of the data shown in (b) shows that the FPC curvature variation trend curve can be divided into two parts, namely S1 and S2. Figure 5 shown.

[0074] In each segment (S1 and S2), there is a relatively obvious linear relationship between the curvature of the FPC and the corresponding path length. For the S1 segment, the relationship between curvature and length is:

[0075] c1=m·l+b1 (6)

[0076] Where c1 is the curvature of segment S1, m represents the slope of the linear correlation, l is the length along the FPC, and b1 is the intercept of the expression for c1.

[0077] In segment S2, a similar relationship was observed, as shown in

[0078] c2=-m·l+b2 (7) Where c2 is the curvature of segment S2 and b2 is the intercept of the expression for c2. The coordinates of the intersection point T of c1 and c2 are

[0079] To predict the deformation curve of FPC based on the coordinates of points A and B and the total length of FPC, three key parameters need to be known: slope 'm' and intercepts 'b1' and 'b2'. To determine these parameters, it is necessary to analyze the curvature change of FPC and the positional relationship between its two end points, especially focusing on the different sections S1 and S2 determined earlier.

[0080] The differential increments of FPC along the X and Y directions can be expressed as:

[0081] dx=dl·cosθ (8)

[0082] dy=dl·sinθ (9)

[0083] Where, 'dl' represents the micro increment along the FPC. 'θ' is the cut angle at any point on the FPC curve, such as Figure 6 shown.

[0084] The cut angle 'θ' can be expressed as a function of the distance 'l' from the starting point to a given point along the FPC. Therefore, the distance between point A' and point B' along the X and Y axes can be calculated by integration:

[0085]

[0086]

[0087] according to Figure 5 The relationship between the FPC curvature and the corresponding length is shown. It is necessary to distinguish the changes in θ(l) in different sections.

[0088] For segment S1, the tangent angle dθ S1 The micro-increment is:

[0089] dθ S1 =-dl·c1 (12)

[0090] The tangent angle of any point on segment S1 can be expressed as:

[0091]

[0092] In the interval S1, the size differentials in the X and Y directions can be expressed as:

[0093] dx1=cos(θ S1 (l))·dl (14)

[0094] dy1=sin(θ S1 (l))·dl (15)

[0095] Based on this, the change of the coordinates of the two end points of segment S1 can be expressed as:

[0096]

[0097]

[0098] like Figure 5 As shown, point T marks the end point of segment S1, and the corresponding tangent angle θ T It can be expressed as:

[0099]

[0100] Similarly, in the S2 segment, the tangent angle differential at any point can be expressed as:

[0101] dθ S2 =-dl·c2=-dl·(-m·l+m·L / 2) (19)

[0102] The tangent angle at any point in segment S2 can be expressed as:

[0103]

[0104] The size differentials in the X and Y directions within the S2 segment can be expressed as:

[0105] d x 2=cos(θ S2 (l))·dl (21)

[0106] dy2=sin(θ S2 (l))·dl (22)

[0107] Based on this, the coordinate changes at both ends of the FPC in the S2 segment can be expressed as:

[0108]

[0109]

[0110] The dimensions of the entire FPC curve in the X and Y directions can be expressed as:

[0111]

[0112]

[0113] Among them, θ S1 and θ S2 As shown in equations (13) and (20), respectively. Δx and Δy represent the dimensional differentials of the FPC terminal along the X and Y axes, respectively. 'L' represents the total length of the FPC, while 'b1', 'b2' and 'm' correspond to the intercept and slope of the curvature expression, as described above.

[0114] like Figure 2 As shown, the tangent angle at the starting point A' of the curve after the FPC is bent can be expressed as:

[0115] θ A' =π / 2 (27)

[0116] The corresponding tangent angle at endpoint B' is:

[0117]

[0118] The difference between the tangent angles at the starting point A' and the end point B' can be expressed as:

[0119]

[0120] The system of equations (25), (26) and (29) can determine three variables, namely 'm', 'b1' and 'b2'. Based on equations (13), (18) and (20), equations (25), (26) and (29) can form the following system of equations:

[0121]

[0122] At this time, F(u) is a nonlinear equation system, where u = [m, b1, b2], which consists of three uncertain variables: 'm', 'b1' and 'b2'. Accurately solving the values ​​of these variables is crucial to predicting the curve of FPC.

[0123] 3.2 Solving nonlinear equations

[0124] To solve the above nonlinear equations, the Jacobi matrix iteration method is used here. The specific process is as follows:

[0125] (1) Initialization:

[0126] First, determine the initial value of the unknown variable, that is,

[0127] (2) Iteration process:

[0128] a. Calculate the Jacobian matrix value J(u (k) ):

[0129] At each iteration stage k, the corresponding Jacobian matrix J(u (k) In this matrix, each component element (denoted as J ij (u (k) ) represents the partial derivative of the i-th equation with respect to the j-th unknown, and its expression is:

[0130]

[0131] b. Calculate the equation value F(u (k) ):

[0132] Calculate and current estimate u (k) The corresponding function value vector F(u (k) ). This process is crucial in the iterative solution process because it can gradually improve the accuracy of the unknown variable estimation and gradually improve the unknowns.

[0133] c. Solve the linear equations:

[0134] Solve J(u (k) )Δu=-F(u (k) ) to determine the update vector Δu. In this equation, J(u (k) ) is the Jacobian matrix, and Δu represents the correction vector. Solving this linear system is an important process in the iterative method. (k) Continuous updating. Through repeated updates, the estimates of unknown variables in the nonlinear equations become more accurate.

[0135] d. Update the estimate u(k+1):

[0136] Based on the formula u (k+1) =u (k) +Δu improves the estimate. In this framework, u (k+1) represents the estimate after the (k+1)th iteration, and u (k) represents the previous estimate at the kth iteration. Incorporating the correction vector Δu into the current estimate is a key step in the iterative process. This step gradually improves the accuracy of the iterative solution.

[0137] e. Convergence check:

[0138] Iteration termination criteria is an indispensable part of the iterative solution process. It is divided into the following three criteria. As long as one of the criteria is met, the iteration process will be terminated:

[0139] ① Standard 1: Function value F(u (k+1) ) is lower than the preset tolerance level, indicating that the function value is sufficiently close to zero.

[0140] ② Criterion 2: The update vector Δu is less than the specified critical value, indicating that the convergence of the solution is satisfactory.

[0141] ③ Standard 3: If the pre-set maximum number of iterations has been reached, the iteration is forcibly terminated to prevent entering an infinite loop.

[0142] f. Continue iteration or terminate:

[0143] If any of the specified iteration termination criteria are not met, the iteration sequence must be restarted by returning to step 2a. This iterative process ensures that the solution is constantly improved to ensure that the specified accuracy criteria are met. Conversely, when the predetermined criteria are met, the iteration process ends. This strict termination method is essential to ensure the accuracy and reliability of the numerical solution.

[0144] (3) Output:

[0145] After the iteration, the estimated value u (k+1) will be the final solution to the nonlinear system of equations. This result is obtained through rigorous iterative optimization. Figure 7 A schematic diagram of the process of the Jacobi matrix iteration method is shown.

[0146] When applying the Jacobi matrix iteration method to solve a nonlinear system of equations, it is important to determine a suitable initial guess for the unknown variables. The initial guess should be close to the actual value so that the method can converge to the exact solution effectively. Without a sufficiently accurate initial guess, it will be difficult or even impractical to obtain the correct solution to the nonlinear system of equations.

[0147] In order to solve this problem, the present invention introduces an iterative process called "size iteration method". Initially, the fixed size of the FPC is determined, the total length L0 = 50mm, the coordinate difference between the two ends of the FPC is Δx0 = 1.9mm, Δy0 = -15mm, and the initial value m (0) =0.01003, In practical applications, the actual total length L and the coordinate differences Δx and Δy at both ends of the FPC often deviate from these preset values ​​L0, Δx0, and Δy0. (0) , and The value may not effectively represent the true value, or even be far from the true value.

[0148] Here, a positive integer N is selected to divide the changes of parameters L, Δx0, and Δy0 into N equal parts. The corresponding expression is:

[0149] dL=(L-L0) / N (32)

[0150] dΔx=(Δx-Δx0) / N (33)

[0151] dΔy=(Δy-Δy0) / N (34)

[0152] During the dimensional iteration process, the initial FPC curve is characterized by the following parameters: L1 = L0 + dL, Δx1 = Δx0 + dΔx, Δy1 = Δy0 + dΔy. (0) , and As the initial value and combined with the Jacobian matrix iteration method, the parameter m after the first size iteration is determined (1) , and Then, according to m (1) , and Methodically derive the parameters of subsequent FPC curves (such as m of the second curve) (2) , This iterative process continues until the Nth iteration ends, and the m corresponding to the final FPC curve is obtained. (N) , and The value of .

[0153] Based on the derived values ​​of slope 'm' and intercepts 'b1' and 'b2', the coordinates of any point on the FPC can be calculated according to equations (35) and (36):

[0154]

[0155]

[0156] Where (x0, y0) represents the coordinates of the starting point of the FPC, (x, y) represents the coordinates of any point on the FPC, and θ represents the tangent angle of the corresponding point on the FPC. If the corresponding point is located in the S1 segment, the mathematical expression of its tangent angle is as shown in formula (13), and if the point is located in the S2 segment, the mathematical expression of its tangent angle is as shown in formula (20).

[0157] The final curve is called the Nth curve, which is the actual FPC curve. Figure 8 Flowchart of the iterative process for the corresponding dimensions.

[0158] If the values ​​of 'm', 'b1' and 'b2' are still difficult to determine during the corresponding iteration (e.g., there is a significant deviation between their estimated values ​​and actual values ​​in the previous iteration), an effective strategy is to increase the value of N. Generally, N = 50 is considered sufficient to meet the conditions in most cases.

[0159] 3.3 Iterative analysis and its calculation results

[0160] Here, N is assigned a value of 10, indicating that the calculation process will go through 10 iterations, such as Fig. 9 (a). Curve 0 represents the initial shape of the FPC defined in the program and is the initial state of the iterative process. After each iteration, a unique FPC curve appears. Curve 10 is the result of the tenth iteration, which is the actual curve of the FPC to be solved. The entire iterative process went smoothly, highlighting the robustness and reliability of the corresponding method.

[0161] Fig. 9 (b) is a comparison between the finite element method (FEM) simulation result curve and the mathematical modeling analysis result proposed in the present invention.

[0162] 4. FPC contact deformation analysis modeling

[0163] 4.1 Single point contact model

[0164] The mathematical model described above is specifically used to model the free bending FPC. However, in practical applications, there may be situations where the FPC comes into contact with other components. Fig.10 As shown in (a), this interaction will make the mathematical model mentioned above no longer applicable.

[0165] To solve the problem, the present invention adopts a similar method to analyze the curvature variation trend of the FPC in contact with other components. Similarly, it relies on the analysis of the surface bending stress (such as Fig.10 (b)) and the adjustment of the corresponding Von Mises stress (as shown in Fig.10 (c)).

[0166] Fig.10 The bending stress variation trend shown in (c) is related to the curvature variation trend and is segmented according to different slope values. Fig.11 Different curvature segments are clearly divided.

[0167] like Fig.11 As shown, the contact point C on the FPC corresponds to the point T on the curvature curve. Based on the contact point C, the FPC curve can be divided into two sections, S1 and S2, which are located on the left and right sides of the contact point, respectively.

[0168] The S1 segment can be divided into and They are all located on the left side of the contact point, with their absolute slope values ​​being m1 and their intercepts being b1 and b2. The S2 segment can be further divided into and Two segments, both located on the right, have absolute slope values ​​of m2 and intercepts of b3 and b4.

[0169] about Fig.10The differences of the X, Y coordinates and tangent angles of the two end points of the S1 segment shown in (a) are Δx1=x C -x A , Δy1=y C -y A and Δθ1=-π / 2, where (x A ,y A ) and (x C ,y C ) represent the coordinates of points A and C respectively.

[0170] Using the method described above, the values ​​of Δx1, Δy1, and Δθ1 can be expressed based on the curvature value change function:

[0171]

[0172]

[0173]

[0174] Where L1 is the length of the FPC in segment S1. Similarly, for segment S2, the differences between the X, Y coordinates and the tangent angle of its endpoints are Δx2=x B -x C , Δy2=y B -y C and Δθ2 = -π / 2. Here, (x B ,y B ) represents the coordinates of point B, such as Fig.10 (a). These differences can be expressed as:

[0175]

[0176]

[0177]

[0178] Where L1 is the length of the FPC in the S2 segment. It is worth noting that the curvature at the end of the S1 segment is equal to the curvature at the beginning of the S2 segment. Fig.11 This relationship is intuitively demonstrated. This relationship can be expressed mathematically as:

[0179] -m1·L1+b2=m2·L1+b3 (43)

[0180] According to equations (37)-(43), the vector function F(u) can be expressed as:

[0181]

[0182] Where 'u' represents the unknown vector, defined as [m1,m2,b1,b2,b3,b4,x C ,L1]. The solution process of this function can also be realized by using the Jacobian matrix iteration method. The initial value of the 'u' vector is determined based on the following assumption: the highest point of the FPC curve is considered as the contact point when there is no contact with any other parts.

[0183] After solving based on the Jacobian matrix iteration method, the coordinates of any point on the S1 segment can be expressed as:

[0184]

[0185]

[0186] Here, the range of 'l' is [0,L1], (x A ,y A ) is the coordinate of point A. Similarly, for any point on segment S2, its coordinates can be expressed as:

[0187]

[0188]

[0189] In this case, 'l' is in the range [L1,L]. Based on these expressions, the final curves obtained are compared with the finite element simulation results. Fig.12 .

[0190] As can be seen, the two curves agree very well, but there are some minor differences. The main reason for these differences is the assumption that the FPC curvature is linear with the length values ​​in different segments and along the FPC. However, the results obtained based on this assumption are accurate enough for most practical needs.

[0191] 4.2 Two-point contact model

[0192] In actual situations, there may even be multiple contact points between the FPC and adjacent components, such as Fig.13 shown.

[0193] Fig.13 (a) is the FPC in a free bending state, and Fig.13 (b) is the case where the FPC has two contact points (C1, C2) with the adjacent component.

[0194] In this scenario, the complexity is greatly increased, and more unknown variables must be solved to accurately predict the deformation curve of FPC. Despite the increase in complexity, the corresponding solution ideas are still similar to the previously mentioned model.

[0195] Similarly, the trend of the bending stress (Von Mises stress) and the subsequent adjustment results based on the stress characteristics can be seen in Fig.14 .

[0196] According to the relationship between bending stress and FPC curvature derived earlier (Formula (4)), a similar change trend of FPC curvature can be obtained. Similarly, the curvature curve can be segmented according to different slope values, such as Fig.15 shown.

[0197] Fig.13 (b) The contact points C1 and C2 on the FPC correspond to Fig.15 Points T1 and T2 on the middle curvature curve. Due to the existence of these contact points, the FPC curve can be divided into three sections: S1, S2 and S3, which are located at the top, middle and bottom of the curve respectively.

[0198] The S1 segment can be divided into and The absolute value of the slope of each is m1, and the intercepts are b1 and b2 respectively; the absolute value of the slope of the S2 segment is m2, and the intercept is b3; the S3 segment can be divided into and The absolute values ​​of the slopes are all m3, and the intercepts are b4 and b5 respectively.

[0199] Regarding the S1 segment, Fig.13 (b) shows that the differences between the X and Y coordinates and the tangent angle of the two endpoints are Δx1=x C1 -x A , Δy1=y C1 -y A and Δθ1 = -π / 2. By applying the method described above, the values ​​of Δx1, Δy1, and Δθ1 can be determined based on the change in curvature value, respectively:

[0200]

[0201]

[0202]

[0203] For segment S2, the differences between the X, Y coordinates and tangent angles of its two endpoints are: Δx2 = x C2 -x C1 , Δy2=y C2 -y C1 , Δθ2=0, which can be expressed as:

[0204]

[0205]

[0206]

[0207] For the S3 segment, the differences between the X, Y coordinates and tangent angles of its two endpoints are: Δx3 = x B -x C2 , Δy3=y B -y C2 and Δθ3=π / 2, which can be expressed as:

[0208]

[0209]

[0210]

[0211] Similarly, the curvature of the end point of S1 is equal to the curvature of the start point of S2, and the curvature of the end point of S2 is equal to the curvature of the start point of S3. Fig.15 This continuity is intuitively demonstrated, and the corresponding mathematical expression is:

[0212] -m1·L1+b2=m2·L1+b3 (58)

[0213] m2·(L1+L2)+b3=m3·(L1+L2)+b4 (59)

[0214] Based on equations (49)-(59), the vector function F(u) can be expressed as:

[0215]

[0216] Where 'u' represents the unknown parameter vector, which is [m1,m2,m3,b1,b2,b3,b4,b5,y C1 ,y C2 ,L1,L2], where L1, L2 and L3 are the lengths of the FPC in the S1 segment, S2 segment and S3 segment respectively. Similarly, the nonlinear equations can be solved based on the Jacobian matrix iteration method. Fig.13 As shown in (a), the initial value of vector 'u' is obtained based on the following assumptions: contact point C1 is the rightmost point of the upper segment of the FPC free bending curve, and contact point C2 is the leftmost point of its lower segment.

[0217] If the initial value of the vector 'u' deviates significantly from its true value, making it difficult to solve for its true value using the Jacobian matrix iteration method, the iteration process can be divided into N discrete steps. In this method, the position of the contacting parts is gradually adjusted in N incremental steps, from its initial position (as indicated by the initial value of the vector 'u') to the specified position.

[0218] Fig.16(a) Describes the changing FPC curves during the iterative adjustment of the contact component positions. Here, the number of iterations N is set to 5, generating six different FPC curves, including the initial curve without any contact and the final curve, ‘Curve 5’, which represents the target deformation FPC curve to be determined. Fig.16 (b) Comparative analysis between the final curves obtained by finite element simulation and mathematical analytical model.

[0219] Fig.16 (b) shows that there is very little difference between the finite element simulation results and the results predicted by the mathematical analytical model. The consistency between the two strongly proves the reliability of the corresponding mathematical model, especially its ability to accurately capture the deformation behavior of FPC under various bending conditions. This consistency confirms the practicality of the corresponding mathematical model in engineering applications and its importance as an accurate and reliable FPC deformation analysis tool.

[0220] The present invention is an algorithm problem for calculating the bending deformation of an FPC under the condition of contact with other components. The mathematical model adopted by the present invention divides the FPC into segments based on the contact points, and the curvature change slope of each segment after segmentation is inconsistent.

[0221] This paper introduces an innovative mathematical model to efficiently and accurately predict the bending deformation of flexible printed circuit boards (FPCs) under different contact conditions. The main advantages of this model include:

[0222] ① Simplified mathematical relationship construction: The FPC is divided into different sections based on the contact points, and the relationship between curvature and length is linearized in each section. This model simplifies the traditional complex calculation process.

[0223] ② Improved computational efficiency: Compared with the finite element method (FEM), this model has a significant advantage in computational speed and is suitable for rapid design iteration and prototyping.

[0224] ③Easy to apply: The model is designed to be intuitive and simple, easy to understand and implement, and does not require deep expertise or complex software support.

[0225] ④Wide applicability: Suitable for handling complex FPC contact bending situations and can meet different design requirements and application scenarios.

[0226] This mathematical model has great significance in the field of electronic product design, which can improve the reliability and performance of flexible electronic products while reducing design and production costs. It provides designers with an efficient and flexible tool, enabling them to realize more innovative and complex product designs in a shorter time and at a lower cost.

[0227] It should be emphasized that although the basic principles and main features of the invention are shown and described in detail herein, these only represent preferred examples of the present invention. It should be understood by those skilled in the art that the present invention is not limited to these embodiments, and that there may be various changes and modifications without departing from the spirit and scope of the present invention. These improvements and changes should be regarded as the scope of the present invention, which is defined by the attached claims and their equivalents.

Claims

1. A mathematical modeling method for FPC contact deformation based on segment analysis and solution of nonlinear equations, characterized in that: The following steps are involved: S1. Based on the finite element tool, the corresponding FPC contact bending morphology is modeled and simulated, and the relationship between its curvature change and the distance of the FPC along the length direction is obtained; S2, dividing the curvature change trend curve into segments based on the contact point position; S3, linearizing the curvature of the FPC and the changing trend along the length of the FPC in different sections; S4, combining the FPC parameters in different sections into a nonlinear equation system; S5, solving the nonlinear equation group to obtain a series of parameters required to determine the contact deformation of the FPC; S6. Calculate the deformed shape of the FPC after contact according to the solution of the nonlinear equation group; The step S2 specifically includes the following steps: S2.

1. Divide the curvature change trend curve into segments according to the contact points; S2.

2. Observe whether the curvature change of FPC is monotonically changing in the same section. If not, continue to divide it further to ensure that in each small section, the curvature is monotonically increasing or monotonically decreasing along the length of FPC; The step S4 specifically includes the following steps: S4.

1. In a large section with the contact point as the dividing point, the X, Y coordinate difference and tangent angle difference of its two end points and the X, Y coordinate difference and tangent angle difference of the two end points of the entire FPC curve are taken as known conditions; S4.2, the slope and intercept of the FPC curvature change curve in each section, the specific coordinates of the contact point, and the length of the FPC in each section with the contact point as the decomposition point are taken as unknowns and combined into a nonlinear equation system; The step S5 specifically includes the following steps: S5.

1. The solution of the corresponding complex nonlinear equations is mainly based on the Jacobi matrix iteration method. The solution process needs to be segmented iterative to ensure that the corresponding iterative process can search for the correct solution nearby. S5.

2. If the contact deformation is too large, the position of the contacted parts is analyzed in sections, gradually reaching the specified position to complete the analysis of the final FPC contact shape.

2. The FPC contact deformation mathematical modeling method based on segment analysis and solution of nonlinear equations according to claim 1 is characterized in that: The step S3 specifically includes the following steps: S3.

1. Linearizing the relationship between the FPC curvature and the length along the FPC in each small segment will produce unknowns such as the slope and intercept. S3.

2. Within the large segment defined by the contact point, it is determined that the absolute value of the slope of the relationship between the FPC curvature and the length of the FPC in all small segments is equal.

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