TWI584133B - Static bending deflection and free vibration analytical system of symmetric laminated plates - Google Patents

Description

Static bending deflection and free vibration analysis system and method for symmetric composite laminate

The invention relates to a static bending deflection and free vibration analysis system and method for a symmetric composite laminate, in particular to a method for providing a fast and accurate method by a nonlinear optimization programming module to effectively realize the dynamics of the composite laminate. Behavior analysis and simulation, and calculate the minimum rate of change of transverse shear stress.

According to the composite material, the composite material is composed of two or more kinds of fiber reinforced materials, which are better than the traditional materials, low in specific gravity, high in rigidity, high in strength and long in service life, because the characteristics of the composite material are tough and not easy to corrode. Therefore, its application range is quite wide, and it can be applied in the fields of industry, construction, people's livelihood, medical care, and military. The fuselage of the aircraft, the missile shell, the reinforcement of the house, and the sporting goods can all use composite materials, and the progress of human civilization. Greatly helpful, and the most widely used composite material is laminate, which is a composite sheet of reinforced fibers composed of high-density parallel layers, such as graphite and epoxy or graphite and glass fiber stacks. The composite material, whereby the geometry of the composite laminate is easier to achieve mechanically optimized structural properties than conventional materials, and can obtain more excellent mechanical properties.

Since composite laminates are composed of different materials, the distribution of structure and force is much more complicated than that of traditional materials. In the past, some analytical theories of laminates have been proposed, such as classical laminated plate theory. It is based on the displacement theory and the Kirchhoff hypothesis to analyze the deformation of composite laminates. However, classical composite laminate theory can only analyze thin isotropic laminates, and cannot be effectively analyzed for thicker laminates. After that, more laminate analysis theories have been proposed, such as 3D elasticity equilibrium equations, layerwise plate theories, zigzag theories, first-order shear deformation theory. (first-order impedance deformation theory) and higher-order training deformation plate theories; the first-order shear deformation theory can simplify the calculation of shear stress between layers compared with other analytical theories. It only considers a transverse shear stress constant with respect to thickness, and introduces a shear correction factor in the transverse shear stress to derive the stress change between the layers; however, the first-order shear deformation theory Although there is a high computational efficiency, the accuracy cannot be better than the lack of high-order shear deformation theory, and the high-order shear deformation theory requires a large amount of computer computing resources, so the analysis method is too complicated and the calculation efficiency is low. The lack of cost is too high.

Nowadays, the numerical approximation method combined with the first-order shear deformation theory is used to analyze the composite laminate and solve its dynamic response. The main technical systems include: finite difference method, finite element method, differential value product method, Rayleigh-Ritz method. For example, the Republic of China Bulletin No. TW I494783 B "Composite Plate Vibration Analysis System with Resilient Foundation" is a prior patent of the inventor, which is provided by a multi-dimensional digital digital filtering network model system with parallel processing architecture. Accurate partial differential equation solution; however, compared with other current numerical approximation methods, although the partial differential equation solution can be obtained quickly and accurately, it does not have the function of calculating the static bending deflection of the composite laminate, and its network Structural, stability, convergence and accuracy still need to be improved.

Nowadays, the inventor is in view of the above-mentioned existing laminate analysis method (such as numerical approximation), which still has many defects in actual implementation, so it is a tireless spirit, and with its rich professional knowledge and many years of practice The experience was assisted and improved, and the present invention was developed based on this.

The main object of the present invention is to provide a static bending deflection and free vibration analysis system and method for a symmetric composite laminate, which provides a fast and accurate method for realizing the dynamic behavior of the composite laminate by a nonlinear optimization programming module. Analysis and simulation, and calculate the minimum rate of change of transverse shear stress

In order to achieve the above-mentioned object, the present invention provides a static bending deflection and free vibration analysis system for a symmetric composite laminate, which comprises an input module for converting a composite laminate system into three interactive subsystems; The multi-dimensional digital signal processing unit connected to the input module comprises a series of junction electron lumping circuits, a multi-dimensional linear inductor, a multi-dimensional Kirchhoff circuit, a discrete module, a nonlinear optimization programming module and a Digital filter, in which the tandem electronic lumping circuit is used to represent the physical behavior of the composite laminate system in continuous time, the multi-dimensional linear inductance system generates system variables to maintain operation, and the multi-dimensional Kirchhoff circuit system converts the string-connected electrons The lumped circuit is a plurality of passive digital circuit components formed a main loop subcircuit of three system variables, and connected by two coupled gyrators , a total of five system variables of the main loop subcircuit and The main loop sub-circuit of five system variables, the discrete module system makes the passive digital circuit component achieve the discrete approximation by the trapezoidal rule in the displacement domain, and the nonlinear optimization programming module provides an optimization method. To obtain the best convergence, the digital filter system ensures that the discrete multi-dimensional Kirchhoff circuit has passiveness and algorithm robustness at each turn; and an output module electrically connects the multi-dimensional digital signal processing unit Each of the multi-dimensional Kirchhoff circuits is converted and synthesized into a complete multi-dimensional digital-digital filtering network.

Another object of the present invention is to provide a static bending deflection and free vibration analysis method for a symmetric composite laminate comprising: step one: taking a composite laminate having a long thickness ratio of ≤ 20; and step 2: utilizing the composite laminate The partial differential equation is converted into three interactive subsystems; Step 3: is the output of the receiving subsystem, and the network operation is realized by a multi-dimensional Kirchhoff circuit by a series of junction lumped circuits and a multi-dimensional linear inductor. Step 4: The multi-dimensional Kirchhoff circuit is operated and filtered by a discrete module, a nonlinear optimization programming module and a digital filter; and step 5: an output module receives the discrete module, nonlinear optimization The operation results of the programming module and the digital filter enable each 埠 of the multi-dimensional Kirchhoff circuit to be converted and synthesized into a complete multi-dimensional digital digital filtering network.

In one embodiment of the invention, the input module is a time-varying partial differential equation such that the static bending deflection and free vibration analysis system of the symmetric composite laminate is represented by the following three interacting subsystems: , and Where q(x, y) is the external lateral load q _x (x, y), q _y (x, y) is the distributed force moving along the x-axis and the y-axis direction, Is the lateral speed, versus It is the transverse bending rotational speed of the x-axis and the y-axis, I is the mass inertia, and Z _Q , Z _M and Z _N are respectively defined as ;among them, Matrix Determinant, versus a transverse shear stress stiffness matrix, Then the bending stiffness matrix, and Department representative A cofactor for a triple conjugate matrix.

In an embodiment of the invention, the lateral velocity as well as versus As a graphical voltage, the proportional variable variables i _k , k = 1, 2, ..., 13 are combined and the constants r _j = 1, 2, ..., 5 are defined by the following relationship: The static bending deflection and free vibration analysis system of the symmetric composite laminate realizes the network operation by the multi-dimensional Kirchhoff circuit, which is defined by the following relationship: Combining auxiliary constants by the above relationship , j=1, 2,...,10 with =1, 2,...,5 to define multi-dimensional partial derivatives : Where D _x , D _y and D _t are expressed as the partial derivative derivatives of the space coordinates x, y and the time coordinate t.

In one embodiment of the invention, the multi-dimensional linear inductance system comprises, for example, a resistor, a linear inductor, an ideal transformer, and a gyrator, and is a single turn or a plurality of turns.

In an embodiment of the present invention, the multi-dimensional wave digital filtering network is constrained by CFL (Courant-Friedrichs-Lewy) condition, so that the network has good stability and provides an initial value. To make the minimum of the time-to-space ratio limited Between the nonlinear optimization programming module adopts an optimization method of active set strategy, and the iterative operation is performed to calculate the optimal solution of the initial value of a guess. The nonlinear constraint is determined by the following Jiahua module obtained:

In an embodiment of the invention, the C _g is defined as: Theoretical derivation The values are as follows: Theoretical derivation The values are as follows:

In an embodiment of the invention, the optimal solution of the shear correction factor (SCF) is found when the plane stress state changes minimally, versus Define the maximum lateral deflection of nondimensionalized And the best transverse shear stress , , are: Among them, the symbol The system is Euclidean norm, and arg opt represents the optimum argument. The next step is to observe the minimum change of transverse shear energy through some Euclidean distance calculations. , , meaning is defined as follows:

;among them, Represents the discrete region of the SCF to evaluate the minimum rate of change of transverse shear energy, and the function Calculate and compare current changes With previous changes difference between.

The object of the present invention and its structural and functional advantages will be explained in conjunction with the specific embodiments according to the structure shown in the following drawings, so that the reviewing committee can have a more in-depth and specific understanding of the present invention.

For the continuation, please refer to the first to fourth figures, the static bending deflection and free vibration analysis system of the symmetric composite laminate of the present invention, which comprises an input module (1) and an electrical connection input module (1) for multi-dimensional digital signal processing. a unit (2), and an output module (3) electrically connected to the multi-dimensional digital signal processing unit (2); wherein the multi-dimensional digital signal processing unit (2) comprises a series of junction electronic lumping circuits (21), a multi-dimensional linear inductor (22), a multi-dimensional Kirchhoff circuit (23), a discrete module (24), a nonlinear optimization programming module (25), and a digital filter (26), The tandem electronic lumped circuit (21) is used to indicate the physical behavior of the composite laminate system in continuous time, and the multi-dimensional linear inductor (22) generates system variables to maintain operation, multi-dimensional Kirchhoff circuit (23) The conversion string junction electron lumping circuit (21) is a plurality of passive digital circuit components, and the plurality of passive digital circuit components and the corresponding digital filtering network are defined as shown in the second figure to form a third figure. Shown Main loop subcircuit of three system variables, connected by two coupled gyrators The main loop subcircuit of five system variables The main loop sub-circuit of five system variables, the discrete module (24) makes the passive digital circuit components achieve the discrete approximation by the trapezoidal rule in the displacement domain, and the nonlinear optimization programming module (25) provides the best. The method is used to obtain the best convergence. The digital filter (26) ensures that the discrete multi-dimensional Kirchhoff circuit (23) has passiveness and algorithm robustness at each turn, as shown in the fourth figure. Each of the multi-dimensional Kirchhoff circuits (23) is converted and synthesized into a complete multi-dimensional digital-digital filtering network.

Furthermore, referring to the fifth figure, the present invention also provides a static bending deflection and free vibration analysis method for a symmetric composite laminate, comprising: Step 1 (S1): taking a composite laminate having a long thickness ratio ≤ 20; Second (S2): Converting the composite laminate into three interactive subsystems using partial differential equations; Step 3 (S3): receiving the output of the subsystem, and using a series of junction lumped circuits (21) and more The dimensional linear inductance (22) implements network operation by a multi-dimensional Kirchhoff circuit (23); step 4 (S4): the multi-dimensional Kirchhoff circuit (23) is a discrete module (24), a The nonlinear optimization programming module (25) and a digital filter (26) perform operations and filtering; and the fifth step (S5): an output module (3) receives the discrete module (24), and the nonlinear optimization programming module ( 25) and the result of the digital filter (26), each of the multi-dimensional Kirchhoff circuit (23) is converted and synthesized into a complete multi-dimensional digital-digit filtering network; wherein, multi-dimensional linear inductance (22 ) can be, for example, a single turn or a plurality of turns, and includes a resistor, a linear inductor, and an ideal And a rotary pressure filter, if a single-port-based linear dimension as much inductance, the inductance value based And obtain the impedance value of a single 藉 by the trapezoidal rule :

If the system is a multi-dimensional linear inductor (22), the impedance value of 埠 is defined as: .

The research method adopted by the present invention is to add the classical laminate theory to the first-order shear deformation theory, and the transverse shear and tangential strain forces are neglected in the laminate of the classical laminate theory, so it is insufficient to predict the medium-thickness laminated composite board. Or the overall response characteristics of a highly heterogeneous composite laminate; the Hamilton principle combined with the linear strain-displacement relationship produces the Euler-Lagrange equation, which is the dynamic form of the virtual work. And contains virtual displacement This governing equation can be used to integrate the displacement gradient and set a coefficient to the virtual displacement, that is, to derive the displacement component and the partial differential equation of the combined stress, as shown in the following equation: ;

Converting a composite laminate system into three interactive subsystems according to the above equation, expressed as: , and Where q(x, y) is the external lateral load q _x (x, y), q _y (x, y) is the distributed force moving along the x-axis and the y-axis direction, 𝑣 speed, versus It is the transverse bending rotational speed of the x-axis and the y-axis, I is the mass inertia, and Z _Q , Z _M and Z _N are respectively defined as ;among them, Matrix Determinant, versus a transverse shear stress stiffness matrix, Then the bending stiffness matrix, and Department representative A cofactor for a triple conjugate matrix.

Also, the lateral speed as well as versus Considering the graphical voltage, the proportional variable variable i _k , k = 1, 2, ..., 13 is introduced and combined with the constant r _j = 1, 2, ..., 5, defined by the following relationship:

And define the partial derivative operators D _x , D _y and D _t , relative to the space coordinates x, y and the time coordinate t, to convert the three interactive subsystems into the Kirchhoff circuit equation of the network operation, such as the following equation Shown as follows:

Among them, the auxiliary constant is combined by the above relationship , j=1, 2,...,10 with =1, 2,...,5, know for:

In addition, the scope of the invention may be further exemplified by the following specific examples, which are not intended to limit the scope of the invention.

Firstly, a static bending deflection and free vibration analysis system of a symmetric composite laminate is constructed, which comprises an input module (1), a multi-dimensional digital signal processing unit (2) electrically connected to the input module, comprising a string of junction electron sets a total circuit (21), a multi-dimensional linear inductor (22), a multi-dimensional Kirchhoff circuit (23), a discrete module (24), a nonlinear optimization programming module (25), and a digital filter (25); and an output module (3) electrically connected to the multi-dimensional digital signal processing unit (2).

The input module (1) is a time-varying partial differential equation to represent the static bending deflection and free vibration analysis system of the symmetric composite laminate as the following three interactive subsystems: , and Where q(x,y) , q _x (x,y), q _y (x,y) are external lateral loads, Is the lateral speed, versus The transverse bending rotational speed is the x-axis and the y-axis, I is the mass inertia, and Z _Q , Z _M , Z _N are defined as .

Factor and Multiplied by Z _Q , Z _M , and Z _N described above to obtain versus The equations, in order to achieve conversion to the network, enable the components to be symmetrically arranged.

Matrix Determinant, versus a transverse shear stress stiffness matrix, Then the bending stiffness matrix, and Department representative The cofactors of the three conjugate matrix can be listed as:

Also, the lateral speed as well as versus As a graphical voltage, the proportional variable variables i _k , k = 1, 2, ..., 13 are combined and the constants r _j = 1, 2, ..., 5 are defined by the following relationship:

In order to make the static bending deflection and free vibration analysis system of the symmetric composite laminate realize the network operation by the multi-dimensional Kirchhoff circuit (23), the following relationship is defined: Combining auxiliary constants by the above relationship , j=1, 2,...,10 with =1, 2,...,5 to define multi-dimensional partial derivatives : Where D _x , D _y and D _t are expressed as the partial derivative derivatives of the space coordinates x, y and the time coordinate t.

Furthermore, the CFL (Courant-Friedrichs-Levy) condition is used to constrain the multi-dimensional wave digital filtering network to make the network have better stability. Since the digital filter (26) is capable of ensuring that the discrete multi-dimensional Kirchhoff circuit (23) is passive at each turn, the multi-dimensional digital filtering network ensures a certain stability. In addition, the minimum time-to-space ratio is limited by the necessary conditions of the CFL. , where C _g is defined as:

Theoretical derivation The values are as follows:

Theoretical derivation The values are as follows:

After numerical simulation, the minimum time-space ratio is limited by the CFL value range. The stability of the multi-dimensional digital filtering network can be maintained in this numerical interval, and the overall calculation speed can be improved; therefore, the optimal solution of the multi-dimensional digital filtering network can be optimized by a simple nonlinear optimization. The nonlinear constraints of the programming module (25) are obtained, for example:

Provide an initial value based on the above CFL The nonlinearly constrained nonlinear optimization programming module (25) can perform an iterative operation by a golden section search method to calculate an initial value of the guess. Further, the nonlinear optimization programming mode Group (25) adopts an optimization method of active set strategy, which is a quadratic programming method based on the iterative calculation process. The module (25) ensures the information needed for accurate prediction of the multi-dimensional digital filtering network and the stability of the network.

In the continuation, the best solution of the shear correction factor (SCF) must be found. Based on the first-order shear deformation theory, the optimal solution or correct solution of SCF should be found. The key system provides compensation for the first-order shear. The difference between the transverse shear stress value predicted by the shear deformation theory and the value calculated by the 3D elastic equilibrium equation is obtained to obtain a more accurate optimal solution.

Since different SCF systems mispredict the stress state, it is necessary to find the SCF when the plane stress state changes minimally, that is, when the transverse shear energy has the smallest change ratio, thereby reducing the predicted transverse shear stress value. The error, the best solution obtained is more accurate.

First versus Define the maximum lateral deflection of nondimensionalized And the best transverse shear stress , , are:

Which symbol The system is Euclidean norm, and arg opt represents the optimum argument. The next step is to observe the minimum change of transverse shear energy through some Euclidean distance calculations. ;

, meaning is defined as follows:

among them, Represents the discrete region of the SCF to evaluate the minimum rate of change of transverse shear energy, and the function Calculate and compare current changes With previous changes The difference between the two, obviously, from the above equation, can be found after the comparison calculation, the SCF corresponding to the minimum error can be found, which is the optimal solution required.

In actual implementation, first of all, take two different materials, the mechanical properties of materials one and two are: ;among them For the Yang modular, Shear modulus of the main coordinate, It is Baisongbi.

In order to make the prediction more accurate, for the measurement of materials with different stiffness ratios, the reference value as well as Define error percentage for: .

The following is a description of the free vibration analysis and static bending analysis of different types of composite laminates by the static bending deflection and free vibration analysis system of the above symmetric composite laminate, and the optimization method of the nonlinear optimization programming module can be realized. Fast calculation process.

Free vibration analysis:

In order to properly perform free vibration analysis, first define the periodic solution as:

,

Spatial variation of lateral displacement And the radial frequency of natural plate vibration Can be written as:

.

(i) A three-layer square composite laminate and a rectangular composite laminate are alternately stacked with the material one, and the stacking angle is [0°/90°/0°], and the Young's modulus ratio E ₁ /E ₂ =40 The long thickness ratio a/h is 5, 10, 20, respectively, by comparing the five approximation methods with the multidimensional wave digital filtering network (MDWDF) method of the present invention. The error, as shown in Table 1 and Table 2, the five approximation methods are IRBF (Integrated Radial Basis Function), RBF-PS (radial basis function pseudospectral), NS-DSG3, NS-DSG3*, and p-Ritz, where NS -DSG3 and NS-DSG3* are high-order shear deformation theory, while Tables 1 and 2 are based on p-Ritz.

Table I:

Table II:

It can be seen from Tables 1 and 2 that the results of the present invention are quite close to other approximation methodologies, and the resulting errors are quite small.

(ii) A four-layer, simply-supported square composite laminate with a stack of materials at a stacking angle of [0°/90°/0°/90°] and a poplar modulus ratio E ₁ /E ₂ of 3, 10, 20, 30, and 40, the length-to-thickness ratio a/h is 5, 10, by comparing the four approximation methods with the multi-dimensional digital-digital filtering network method of the present invention. The error is shown in Table 3. The four approximation methods are 3D ELS (3D elasticity solution), HSDT-FEM (higher-order salad forming plate theories and finite element method), FSDT-FEM (first-order The theory finite element method) and CLPT (classical laminated plate theory), while the third table uses 3D ELS as a reference value.

Table 3:

According to the results of Table 3, the present invention is obviously superior to CLPT. Compared with HSDT-FEM and FSDT-FEM, the values are quite close and the error is quite small.

Best SCFs Perform static bending analysis:

The invention further finds the optimal shear correction factor (SCF) as an important target, so the test is analyzed by the composite laminate made of material 2, and the best solution is found from many different SCFs. First, define the maximum bending deflection of the composite laminate as:

, among them According to the numerical value of the SDL (Sinusoidally Distributed Load) of the uppermost layer of the composite laminate, that is, And the distribution force of the composite laminate can correspond And expressed as .

See Table 4 for the following examples.

(i) A two-layer square composite laminate with a stack of three layers is made of material 2, the stacking angle is [0°/90°/0°], and the long-thickness ratio a/h is 4, 10 respectively. First, one will be Group SCFs to range to The average is divided into 13 SCFs and compared with each other. It can be clearly seen from the table that the composite laminate of a/h = 4 Spatial variation error of lateral displacement Is the minimum value, Also the minimum rate of change; a/h = 4 composite laminate Spatial variation error of lateral displacement Is the minimum value, Also the minimum rate of change. Therefore, it can be known that the optimum SCF of composite laminates with a/h = 4 and 10 is 1 and 8/12, respectively.

(ii) Two-layer square composite laminates are alternately stacked with material 2, the stacking angle is [0°/90°/90°/0°], and the long-thickness ratio a/h is 4, 10, respectively. , will set a range of SCFs to The average is divided into 13 SCFs and compared with each other. It can be clearly seen from the table that the composite laminate of a/h = 4 Spatial variation error of lateral displacement Is the minimum value, Also the minimum rate of change; a/h = 4 composite laminate Spatial variation error of lateral displacement Is the minimum value, Also the minimum rate of change. Therefore, it can be known that the optimum SCF of composite laminates with a/h = 4 and 10 is 1 and 8/12, respectively.

(iii) Prepare a five-layer square composite laminate with material stacking at a material angle of [0°/90°/0°/90°/0°] and a length-to-thickness ratio of a/h of 4, respectively. 10, first, set a range of SCFs to The average is divided into 13 SCFs and compared with each other. It can be clearly seen from the table that the composite laminate of a/h = 4 Spatial variation error of lateral displacement Is the minimum value, Also the minimum rate of change; a/h = 4 composite laminate Spatial variation error of lateral displacement Is the minimum value, Also the minimum rate of change. Therefore, it can be known that the optimum SCF of composite laminates with a/h = 4 and 10 is 14/12 and 11/12, respectively.

Table 4: Among them Is the amount of change in transverse shear stress energy, For the minimum rate of change.

The invention continuously tests different SCFs to smoothly predict the maximum bending deflection of the three-to-five-layer square composite laminates, and can also be found Time, The smaller the change, the system The larger the lateral deflection w, the smaller.

See Table 5 for the following examples.

(iv) Prepare a three-layer square composite laminate with two layers of material, with a stacking angle of [0°/90°/0°] and a length-to-thickness ratio of a/h of 20. First, a set of SCFs is range to Averaged into 25 SCFs and compared with each other, as can be clearly seen from the table, composite laminates Spatial variation error of lateral displacement Is the minimum value, Also the minimum rate of change, and And This data was obtained by HSDT-zigzag, so it can be proved that the calculation method of this case is smaller and more accurate than HSDT-zigzag.

(v) Two-layer square composite laminates are alternately stacked with material 2, the stacking angle is [0°/90°/90°/0°], and the long thickness ratio a/h is 20. First, one will be Group SCFs to range to Averaged into 25 SCFs and compared with each other, as can be clearly seen from the table, composite laminates Spatial variation error of lateral displacement Is the minimum value, and And This data is obtained by 3D-ELS, so it can be proved that the calculation method of this case is smaller and more accurate than the 3D-ELS error.

Table 5:

It can be seen from the above description that the present invention has the following advantages compared with the prior art:

1. The free vibration analysis of the present invention, which analyzes a stack of three-layer and four-layer composite laminates, and the calculation error is quite close to the current approximation method, and is significantly superior to the classical laminate theory.

2. Compared with the current theory of laminate analysis (first-order shear deformation theory (FSDT), high-order deformation theory (HSDT), etc.), the predicted maximum bending deflection is more accurate and the error is smaller.

3. The present invention solves the problem that the best SCF of the current first-order shear deformation theory is difficult to obtain, and is a breakthrough in the field of laminate analysis.

In summary, the static bending deflection and free vibration analysis system and method of the symmetric composite laminate of the present invention can achieve the intended use efficiency by the above disclosed embodiments, and the present invention has not been disclosed in the application. Before, Cheng has fully complied with the requirements and requirements of the Patent Law.爰Issuing an application for a patent for invention in accordance with the law, and asking for a review, and granting a patent, is truly sensible.

The illustrations and descriptions of the present invention are merely preferred embodiments of the present invention, and are not intended to limit the scope of the present invention; those skilled in the art, which are characterized by the scope of the present invention, Equivalent variations or modifications are considered to be within the scope of the design of the invention.

(1)‧‧‧ Input Module

(2) ‧‧‧Multi-dimensional digital signal processing unit

(21)‧‧‧Segmented electron lumping circuit

(22)‧‧‧Multidimensional linear inductance

(23) ‧‧‧Multidimensional Kirchhoff circuit

(24)‧‧‧Discrete modules

(25)‧‧‧Nonlinear optimization programming module

(26)‧‧‧Digital Filter

(3)‧‧‧Output module

(S1)‧‧‧Step one

(S2)‧‧‧Step 2

(S3) ‧ ‧ Step 3

(S4)‧‧‧Step four

(S5) ‧ ‧ step five

First Figure: Schematic diagram of the modeling process of the preferred embodiment of the present invention.

Second: A schematic diagram of the definition of a passive digital circuit component and a corresponding digital filtering network in accordance with a preferred embodiment of the present invention.

Third Figure: Kirchhoff circuit diagram of a preferred embodiment of the invention.

Fourth Figure: A multi-dimensional digital filtering network diagram of a preferred embodiment of the present invention.

Figure 5 is a flow chart showing the steps of a preferred embodiment of the present invention.

(1)‧‧‧ Input Module

(2) ‧‧‧Multi-dimensional digital signal processing unit

(21)‧‧‧Segmented electron lumping circuit

(22)‧‧‧Multidimensional linear inductance

(23) ‧‧‧Multidimensional Kirchhoff circuit

(24)‧‧‧Discrete modules

(25)‧‧‧Nonlinear optimization programming module

(26)‧‧‧Digital Filter

(3)‧‧‧Output module

Claims (10)

A static bending deflection and free vibration analysis system for a symmetric composite laminate includes at least: an input module for converting a composite laminate system into three interactive subsystems; and electrically connecting the multiple dimensions of the input module The digital signal processing unit comprises a series of junction electron lumping circuits, a multi-dimensional linear inductor, a multi-dimensional Kirchhoff circuit, a discrete module, a nonlinear optimization programming module and a digital filter, wherein The tandem junction lumped circuit is used to represent the physical behavior of the composite laminate system in a continuous time, the multi-dimensional linear inductance is to generate a system variable to maintain operation, and the multi-dimensional Kirchhoff circuit converts the string junction electron set The total circuit is a plurality of passive digital circuit components, forming The main loop subcircuit of the three system variables is connected to the two coupled gyrators , a total of five system variables of the main loop subcircuit and a main loop sub-circuit of five system variables, the discrete module is such that the passive digital circuit components achieve discrete approximation by the trapezoidal rule in the displacement domain, and the nonlinear optimization programming module provides optimization The method is used to obtain the best convergence. The digital filter ensures that the discrete multi-dimensional Kirchhoff circuit has passiveness and algorithm robustness at each turn; and an output module is electrically connected. The dimensional digital signal processing unit converts each of the multi-dimensional Kirchhoff circuits into a complete multi-dimensional digital-digit filtering network. The static bending deflection and free vibration analysis system of the symmetric composite laminate according to claim 1, wherein the input module is a time-varying partial differential equation to make the static bending deflection of the symmetric composite laminate And the free vibration analysis system is represented as the following three interactive subsystems: , and Where q(x, y) is the external lateral load q _x (x, y), q _y (x, y) is the distributed force moving along the x-axis and the y-axis direction, Is the lateral speed, versus It is the transverse bending rotational speed of the x-axis and the y-axis, I is the mass inertia, and Z _Q , Z _M and Z _N are respectively defined as ;among them, Matrix Determinant, versus a transverse shear stress stiffness matrix, Then the bending stiffness matrix, and Department representative A cofactor for a triple conjugate matrix. The static bending deflection and free vibration analysis system of the symmetric composite laminate according to claim 2, wherein the lateral velocity as well as versus As a graphical voltage, the proportional variable variables i _k , k = 1, 2, ..., 13 are combined and the constants r _j = 1, 2, ..., 5 are defined by the following relationship: The static bending deflection and free vibration analysis system of a symmetric composite laminate is implemented in a multi-dimensional Kirchhoff circuit for network operation, which is defined by the following relationship: Combining auxiliary constants by the above relationship , j=1, 2,...,10 with =1, 2,...,5 to define multi-dimensional partial derivatives : Where D _x , D _y and D _t are expressed as the partial derivative derivatives of the space coordinates x, y and the time coordinate t. The static bending deflection and free vibration analysis system of the symmetric composite laminate according to claim 1, wherein the multi-dimensional linear inductance comprises a resistor, a linear inductor, an ideal transformer, and a gyrator, and is a single crucible or A number of 埠. A static bending deflection and free vibration analysis method for a symmetric composite laminate, comprising: Step 1: taking a medium thickness composite laminate; Step 2: converting the composite laminate into three interactive subsystems using a partial differential equation; Step 3: Receive the output of the subsystem, and implement network operation by a multi-dimensional Kirchhoff circuit by a series of junction lumped circuits and a multi-dimensional linear inductor; Step 4: The multi-dimensional Kilhoo The circuit is operated and filtered by a discrete module, a nonlinear optimization programming module and a digital filter; and step 5: receiving the discrete module, the nonlinear optimization programming module and the digital filter by using an output module As a result of the operation, each of the multi-dimensional Kirchhoff circuits is converted and synthesized into a complete multi-dimensional digital-digit filtering network. The method of static bending deflection and free vibration analysis of a symmetric composite laminate according to claim 5, wherein the subsystem is expressed as: , and Where q(x, y) is the external lateral load q _x (x, y), q _y (x, y) is the distributed force moving along the x-axis and the y-axis direction, Is the lateral speed, versus It is the transverse bending rotational speed of the x-axis and the y-axis, I is the mass inertia, and Z _Q , Z _M and Z _N are respectively defined as ;among them, Matrix Determinant, versus a transverse shear stress stiffness matrix, Then the bending stiffness matrix, and Department representative A cofactor for a triple conjugate matrix. The method of static bending deflection and free vibration analysis of a symmetric composite laminate according to claim 6 of the patent application, wherein the lateral velocity is , the versus Considering the graphical voltage, the proportional variable variables ik, k = 1, 2, ..., 13 are combined and the constants r _j = 1, 2, ..., 5 are defined by the following relationship: The static bending deflection and free vibration analysis system of a symmetric composite laminate is implemented in a multi-dimensional Kirchhoff circuit for network operation, which is defined by the following relationship: Combining auxiliary constants by the above relationship , j=1, 2,...,10 with =1, 2,...,5 to define multi-dimensional partial derivatives : Where D _x , D _y and D _t are expressed as the partial derivative derivatives of the space coordinates x, y and the time coordinate t. The static bending deflection and free vibration analysis method of the symmetric composite laminate according to claim 5, wherein the static bending deflection and free vibration analysis method of the composite laminate further provides an initial value To make the minimum of the time-to-space ratio limited Between the nonlinear optimization programming module adopts an optimization method of active set strategy, and the iterative operation is performed to calculate the optimal solution of the initial value of a guess. The nonlinear constraint is determined by the following Jiahua module obtained: The static bending deflection and free vibration analysis method of the symmetric composite laminate according to claim 8 of the patent application, wherein the C _g system is defined as: Theoretical derivation The values are as follows: Theoretical derivation The values are as follows: . The method of static bending deflection and free vibration analysis of a symmetric composite laminate according to claim 8 wherein the effective set strategy is to find an optimal solution of a shear correction factor (SCF). versus Define the maximum lateral deflection of nondimensionalized And the best transverse shear stress , , are: Among them, the symbol The system is Euclidean norm, and arg opt represents the optimum argument. The following is the calculation of the Euclidean distance to observe the minimum change of transverse shear energy. , meaning is defined as follows: ;among them, Represents the discrete region of the SCF to evaluate the minimum rate of change of transverse shear energy, and the function Calculate and compare current changes With previous changes difference between.